[3] | 1 | (* *********************************************************************) |
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| 2 | (* *) |
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| 3 | (* The Compcert verified compiler *) |
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| 4 | (* *) |
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| 5 | (* Xavier Leroy, INRIA Paris-Rocquencourt *) |
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| 6 | (* Sandrine Blazy, ENSIIE and INRIA Paris-Rocquencourt *) |
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| 7 | (* *) |
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| 8 | (* Copyright Institut National de Recherche en Informatique et en *) |
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| 9 | (* Automatique. All rights reserved. This file is distributed *) |
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| 10 | (* under the terms of the GNU General Public License as published by *) |
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| 11 | (* the Free Software Foundation, either version 2 of the License, or *) |
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| 12 | (* (at your option) any later version. This file is also distributed *) |
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| 13 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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| 14 | (* *) |
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| 15 | (* *********************************************************************) |
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| 16 | |
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| 17 | (* * This file develops the memory model that is used in the dynamic |
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| 18 | semantics of all the languages used in the compiler. |
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| 19 | It defines a type [mem] of memory states, the following 4 basic |
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| 20 | operations over memory states, and their properties: |
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| 21 | - [load]: read a memory chunk at a given address; |
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| 22 | - [store]: store a memory chunk at a given address; |
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| 23 | - [alloc]: allocate a fresh memory block; |
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| 24 | - [free]: invalidate a memory block. |
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| 25 | *) |
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| 26 | |
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[10] | 27 | include "arithmetics/nat.ma". |
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[474] | 28 | (*include "binary/Z.ma".*) |
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| 29 | (*include "datatypes/sums.ma".*) |
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| 30 | (*include "datatypes/list.ma".*) |
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| 31 | (*include "Plogic/equality.ma".*) |
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[3] | 32 | |
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[474] | 33 | (*include "Coqlib.ma".*) |
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[3] | 34 | include "Values.ma". |
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[474] | 35 | (*include "AST.ma".*) |
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[3] | 36 | include "extralib.ma". |
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| 37 | |
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| 38 | ndefinition update : ∀A: Type[0]. ∀x: Z. ∀v: A. ∀f: Z → A. Z → A ≝ |
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| 39 | λA,x,v,f. |
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| 40 | λy.match eqZb y x with [ true ⇒ v | false ⇒ f y ]. |
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| 41 | |
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| 42 | (* Implicit Arguments update [A].*) |
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| 43 | |
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| 44 | nlemma update_s: |
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| 45 | ∀A: Type[0]. ∀x: Z. ∀v: A. ∀f: Z -> A. |
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| 46 | update … x v f x = v. |
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| 47 | #A;#x;#v;#f;nwhd in ⊢ (??%?); |
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| 48 | nrewrite > (eqZb_z_z …);//; |
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| 49 | nqed. |
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| 50 | |
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| 51 | nlemma update_o: |
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| 52 | ∀A: Type[0]. ∀x: Z. ∀v: A. ∀f: Z -> A. ∀y: Z. |
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| 53 | x ≠ y → update … x v f y = f y. |
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| 54 | #A;#x;#v;#f;#y;#H;nwhd in ⊢ (??%?); |
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| 55 | napply eqZb_elim;//; |
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| 56 | #H2;ncases H;#H3;nelim (H3 ?);//; |
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| 57 | nqed. |
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| 58 | |
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| 59 | (* FIXME: workaround for lack of nunfold *) |
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| 60 | nlemma unfold_update : ∀A,x,v,f,y. update A x v f y = match eqZb y x with [ true ⇒ v | false ⇒ f y ]. |
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| 61 | //; nqed. |
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| 62 | |
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| 63 | |
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| 64 | (* * Structure of memory states *) |
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| 65 | |
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| 66 | (* A memory state is organized in several disjoint blocks. Each block |
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| 67 | has a low and a high bound that defines its size. Each block map |
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| 68 | byte offsets to the contents of this byte. *) |
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| 69 | |
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| 70 | (* The possible contents of a byte-sized memory cell. To give intuitions, |
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| 71 | a 4-byte value [v] stored at offset [d] will be represented by |
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| 72 | the content [Datum(4, v)] at offset [d] and [Cont] at offsets [d+1], |
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| 73 | [d+2] and [d+3]. The [Cont] contents enable detecting future writes |
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| 74 | that would partially overlap the 4-byte value. *) |
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| 75 | |
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| 76 | ninductive content : Type[0] ≝ |
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| 77 | | Undef: content (*r undefined contents *) |
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| 78 | | Datum: nat → val → content (*r first byte of a value *) |
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| 79 | | Cont: content. (*r continuation bytes for a multi-byte value *) |
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| 80 | |
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| 81 | ndefinition contentmap : Type[0] ≝ Z → content. |
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| 82 | |
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| 83 | (* A memory block comprises the dimensions of the block (low and high bounds) |
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| 84 | plus a mapping from byte offsets to contents. *) |
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| 85 | |
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| 86 | (* XXX: mkblock *) |
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| 87 | nrecord block_contents : Type[0] ≝ { |
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| 88 | low: Z; |
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| 89 | high: Z; |
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[124] | 90 | contents: contentmap; |
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[480] | 91 | space: region |
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[3] | 92 | }. |
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| 93 | |
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| 94 | (* A memory state is a mapping from block addresses (represented by [Z] |
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| 95 | integers) to blocks. We also maintain the address of the next |
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| 96 | unallocated block, and a proof that this address is positive. *) |
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| 97 | (* XXX: mkmem *) |
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| 98 | |
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| 99 | nrecord mem : Type ≝ { |
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| 100 | blocks: Z -> block_contents; |
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| 101 | nextblock: block; |
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| 102 | nextblock_pos: OZ < nextblock |
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| 103 | }. |
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| 104 | |
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| 105 | (* * Operations on memory stores *) |
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| 106 | |
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| 107 | (* Memory reads and writes are performed by quantities called memory chunks, |
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| 108 | encoding the type, size and signedness of the chunk being addressed. |
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| 109 | The following functions extract the size information from a chunk. *) |
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| 110 | |
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[483] | 111 | ndefinition size_pointer : region → Z ≝ |
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| 112 | λsp. match sp with [ Data ⇒ 1 | IData ⇒ 1 | PData ⇒ 1 | XData ⇒ 2 | Code ⇒ 2 | Any ⇒ 3 ]. |
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| 113 | |
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[3] | 114 | ndefinition size_chunk : memory_chunk → Z ≝ |
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[483] | 115 | λchunk.match chunk with |
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[3] | 116 | [ Mint8signed => 1 |
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| 117 | | Mint8unsigned => 1 |
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| 118 | | Mint16signed => 2 |
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| 119 | | Mint16unsigned => 2 |
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| 120 | | Mint32 => 4 |
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| 121 | | Mfloat32 => 4 |
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[483] | 122 | | Mfloat64 => 8 |
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| 123 | | Mpointer r ⇒ size_pointer r |
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| 124 | ]. |
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| 125 | |
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| 126 | ndefinition pred_size_pointer : region → nat ≝ |
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| 127 | λsp. match sp with [ Data ⇒ 0 | IData ⇒ 0 | PData ⇒ 0 | XData ⇒ 1 | Code ⇒ 1 | Any ⇒ 2 ]. |
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| 128 | |
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[3] | 129 | ndefinition pred_size_chunk : memory_chunk → nat ≝ |
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| 130 | λchunk.match chunk with |
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| 131 | [ Mint8signed => 0 |
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| 132 | | Mint8unsigned => 0 |
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| 133 | | Mint16signed => 1 |
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| 134 | | Mint16unsigned => 1 |
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| 135 | | Mint32 => 3 |
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| 136 | | Mfloat32 => 3 |
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[483] | 137 | | Mfloat64 => 7 |
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| 138 | | Mpointer r ⇒ pred_size_pointer r |
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| 139 | ]. |
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[3] | 140 | |
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[10] | 141 | alias symbol "plus" (instance 2) = "integer plus". |
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[3] | 142 | nlemma size_chunk_pred: |
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| 143 | ∀chunk. size_chunk chunk = 1 + pred_size_chunk chunk. |
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[483] | 144 | #chunk;ncases chunk;//; #r; ncases r; napply refl; |
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[3] | 145 | nqed. |
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| 146 | |
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| 147 | nlemma size_chunk_pos: |
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| 148 | ∀chunk. 0 < size_chunk chunk. |
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| 149 | #chunk;nrewrite > (size_chunk_pred ?);ncases (pred_size_chunk chunk); |
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| 150 | nnormalize;//; |
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| 151 | nqed. |
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| 152 | |
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| 153 | (* Memory reads and writes must respect alignment constraints: |
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| 154 | the byte offset of the location being addressed should be an exact |
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| 155 | multiple of the natural alignment for the chunk being addressed. |
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| 156 | This natural alignment is defined by the following |
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| 157 | [align_chunk] function. Some target architectures |
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| 158 | (e.g. the PowerPC) have no alignment constraints, which we could |
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| 159 | reflect by taking [align_chunk chunk = 1]. However, other architectures |
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| 160 | have stronger alignment requirements. The following definition is |
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| 161 | appropriate for PowerPC and ARM. *) |
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| 162 | |
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| 163 | ndefinition align_chunk : memory_chunk → Z ≝ |
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| 164 | λchunk.match chunk return λ_.Z with |
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| 165 | [ Mint8signed ⇒ 1 |
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| 166 | | Mint8unsigned ⇒ 1 |
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[153] | 167 | | Mint16signed ⇒ 1 |
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| 168 | | Mint16unsigned ⇒ 1 |
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| 169 | | _ ⇒ 1 ]. |
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[3] | 170 | |
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| 171 | nlemma align_chunk_pos: |
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| 172 | ∀chunk. OZ < align_chunk chunk. |
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| 173 | #chunk;ncases chunk;nnormalize;//; |
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| 174 | nqed. |
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| 175 | |
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| 176 | nlemma align_size_chunk_divides: |
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| 177 | ∀chunk. (align_chunk chunk ∣ size_chunk chunk). |
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[483] | 178 | #chunk;ncases chunk;##[##8:#r; ncases r ##]nnormalize;/3/; |
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[3] | 179 | nqed. |
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| 180 | |
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| 181 | nlemma align_chunk_compat: |
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| 182 | ∀chunk1,chunk2. |
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| 183 | size_chunk chunk1 = size_chunk chunk2 → |
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| 184 | align_chunk chunk1 = align_chunk chunk2. |
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[483] | 185 | #chunk1;#chunk2; |
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| 186 | ncases chunk1; ntry ( #r1 ncases r1 ); |
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| 187 | ncases chunk2; ntry ( #r2 ncases r2 ); |
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| 188 | nnormalize;//; |
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[3] | 189 | nqed. |
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| 190 | |
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| 191 | (* The initial store. *) |
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| 192 | |
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[10] | 193 | ndefinition oneZ ≝ pos one. |
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[3] | 194 | |
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| 195 | nremark one_pos: OZ < oneZ. |
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| 196 | //; |
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| 197 | nqed. |
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| 198 | |
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[480] | 199 | ndefinition empty_block : Z → Z → region → block_contents ≝ |
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[124] | 200 | λlo,hi,bty.mk_block_contents lo hi (λy. Undef) bty. |
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[3] | 201 | |
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| 202 | ndefinition empty: mem ≝ |
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[125] | 203 | mk_mem (λx.empty_block OZ OZ Any) (pos one) one_pos. |
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[3] | 204 | |
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| 205 | ndefinition nullptr: block ≝ OZ. |
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| 206 | |
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| 207 | (* Allocation of a fresh block with the given bounds. Return an updated |
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| 208 | memory state and the address of the fresh block, which initially contains |
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| 209 | undefined cells. Note that allocation never fails: we model an |
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| 210 | infinite memory. *) |
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| 211 | |
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| 212 | nremark succ_nextblock_pos: |
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| 213 | ∀m. OZ < Zsucc (nextblock m). (* XXX *) |
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| 214 | #m;nlapply (nextblock_pos m);nnormalize; |
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| 215 | ncases (nextblock m);//; |
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| 216 | #n;ncases n;//; |
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| 217 | nqed. |
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| 218 | |
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[480] | 219 | ndefinition alloc : mem → Z → Z → region → mem × block ≝ |
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[124] | 220 | λm,lo,hi,bty.〈mk_mem |
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[3] | 221 | (update … (nextblock m) |
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[124] | 222 | (empty_block lo hi bty) |
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[3] | 223 | (blocks m)) |
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| 224 | (Zsucc (nextblock m)) |
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| 225 | (succ_nextblock_pos m), |
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| 226 | nextblock m〉. |
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| 227 | |
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| 228 | (* Freeing a block. Return the updated memory state where the given |
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| 229 | block address has been invalidated: future reads and writes to this |
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| 230 | address will fail. Note that we make no attempt to return the block |
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| 231 | to an allocation pool: the given block address will never be allocated |
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| 232 | later. *) |
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| 233 | |
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| 234 | ndefinition free ≝ |
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| 235 | λm,b.mk_mem (update … b |
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[125] | 236 | (empty_block OZ OZ Any) |
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[3] | 237 | (blocks m)) |
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| 238 | (nextblock m) |
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| 239 | (nextblock_pos m). |
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| 240 | |
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| 241 | (* Freeing of a list of blocks. *) |
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| 242 | |
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| 243 | ndefinition free_list ≝ |
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| 244 | λm,l.foldr ?? (λb,m.free m b) m l. |
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| 245 | (* XXX hack for lack of reduction with restricted unfolding *) |
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| 246 | nlemma unfold_free_list : ∀m,h,t. free_list m (h::t) = free (free_list m t) h. |
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| 247 | nnormalize; //; nqed. |
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| 248 | |
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| 249 | (* Return the low and high bounds for the given block address. |
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| 250 | Those bounds are 0 for freed or not yet allocated address. *) |
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| 251 | |
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| 252 | ndefinition low_bound : mem → block → Z ≝ |
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| 253 | λm,b.low (blocks m b). |
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| 254 | ndefinition high_bound : mem → block → Z ≝ |
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| 255 | λm,b.high (blocks m b). |
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[480] | 256 | ndefinition block_space: mem → block → region ≝ |
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[125] | 257 | λm,b.space (blocks m b). |
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[3] | 258 | |
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| 259 | (* A block address is valid if it was previously allocated. It remains valid |
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| 260 | even after being freed. *) |
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| 261 | |
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| 262 | ndefinition valid_block : mem → block → Prop ≝ |
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| 263 | λm,b.b < nextblock m. |
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| 264 | |
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| 265 | (* FIXME: hacks to get around lack of nunfold *) |
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| 266 | nlemma unfold_low_bound : ∀m,b. low_bound m b = low (blocks m b). |
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| 267 | //; nqed. |
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| 268 | nlemma unfold_high_bound : ∀m,b. high_bound m b = high (blocks m b). |
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| 269 | //; nqed. |
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| 270 | nlemma unfold_valid_block : ∀m,b. (valid_block m b) = (b < nextblock m). |
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| 271 | //; nqed. |
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| 272 | |
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| 273 | (* Reading and writing [N] adjacent locations in a [contentmap]. |
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| 274 | |
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| 275 | We define two functions and prove some of their properties: |
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| 276 | - [getN n ofs m] returns the datum at offset [ofs] in block contents [m] |
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| 277 | after checking that the contents of offsets [ofs+1] to [ofs+n+1] |
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| 278 | are [Cont]. |
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| 279 | - [setN n ofs v m] updates the block contents [m], storing the content [v] |
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| 280 | at offset [ofs] and the content [Cont] at offsets [ofs+1] to [ofs+n+1]. |
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| 281 | *) |
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| 282 | |
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| 283 | nlet rec check_cont (n: nat) (p: Z) (m: contentmap) on n : bool ≝ |
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| 284 | match n return λ_.bool with |
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| 285 | [ O ⇒ true |
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| 286 | | S n1 ⇒ |
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| 287 | match m p with |
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| 288 | [ Cont ⇒ check_cont n1 (Zplus p 1) m (* FIXME: cannot disambiguate + properly *) |
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| 289 | | _ ⇒ false ] ]. |
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| 290 | |
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| 291 | (* XXX : was +, not ∨ logical or |
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| 292 | is used when eqb is expected, coq idiom, is it necessary?? *) |
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| 293 | ndefinition eq_nat: ∀p,q: nat. p=q ∨ p≠q. |
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| 294 | napply decidable_eq_nat; (* // not working, why *) |
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| 295 | nqed. |
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| 296 | |
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| 297 | ndefinition getN : nat → Z → contentmap → val ≝ |
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| 298 | λn,p,m.match m p with |
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| 299 | [ Datum n' v ⇒ |
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| 300 | match andb (eqb n n') (check_cont n (p + oneZ) m) with |
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| 301 | [ true ⇒ v |
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| 302 | | false ⇒ Vundef ] |
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| 303 | | _ ⇒ |
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| 304 | Vundef ]. |
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| 305 | |
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| 306 | nlet rec set_cont (n: nat) (p: Z) (m: contentmap) on n : contentmap ≝ |
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| 307 | match n with |
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| 308 | [ O ⇒ m |
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| 309 | | S n1 ⇒ update ? p Cont (set_cont n1 (p + oneZ) m) ]. |
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| 310 | |
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| 311 | ndefinition setN : nat → Z → val → contentmap → contentmap ≝ |
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| 312 | λn,p,v,m.update ? p (Datum n v) (set_cont n (p + oneZ) m). |
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| 313 | |
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[391] | 314 | (* Nonessential properties that may require arithmetic |
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[3] | 315 | (* XXX: the daemons *) |
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| 316 | naxiom daemon : ∀A:Prop.A. |
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| 317 | |
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| 318 | nlemma check_cont_spec: |
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| 319 | ∀n,m,p. |
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| 320 | match (check_cont n p m) with |
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| 321 | [ true ⇒ ∀q.p ≤ q → q < p + n → m q = Cont |
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| 322 | | false ⇒ ∃q. p ≤ q ∧ q < (p + n) ∧ m q ≠ Cont ]. |
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| 323 | #n;nelim n; |
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| 324 | ##[#m;#p;#q;#Hl;#Hr; |
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| 325 | (* derive contradiction from Hl, Hr; needs: |
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| 326 | - p + O = p |
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| 327 | - p ≤ q → q < p → False *) |
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| 328 | napply daemon |
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| 329 | ##|#n1;#IH;#m;#p; |
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| 330 | (* nwhd : doesn't work either *) |
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| 331 | ncut (check_cont (S n1) p m = match m p with [ Undef ⇒ false | Datum _ _ ⇒ false | Cont ⇒ check_cont n1 (Zplus p oneZ) m ]) |
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| 332 | ##[@ |
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| 333 | ##|#Heq;nrewrite > Heq;nlapply (refl ? (m p)); |
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| 334 | ncases (m p) in ⊢ (???% → %); |
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| 335 | ##[#Heq1;@; |
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| 336 | ##[napply p |
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| 337 | ##|@; ##[napply daemon |
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| 338 | ##|napply nmk;#Hfalse;nrewrite > Hfalse in Heq1;#Heq1; |
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| 339 | ndestruct ] |
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| 340 | ##] |
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| 341 | ##|#n2;#v;#Heq1; @; |
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| 342 | ##[napply p |
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| 343 | ##| @; ##[ (* refl≤ and p < p + S n1 *)napply daemon |
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| 344 | ##|napply nmk;#Hfalse;nrewrite > Hfalse in Heq1;#Heq1; |
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| 345 | ndestruct ] |
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| 346 | ##] |
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| 347 | ##|#Heq1;nlapply (IH m (p + 1)); |
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| 348 | nlapply (refl ? (check_cont n1 (p + 1) m)); |
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| 349 | (* napply daemon *) |
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| 350 | ncases (check_cont n1 (p + 1) m) in ⊢ (???% → %); |
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| 351 | nwhd in ⊢ (? → % → %); |
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| 352 | ##[#H;#H1;#q;#Hl;#Hr; |
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| 353 | ncut (p = q ∨ p + 1 ≤ q) |
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| 354 | ##[(* Hl *) napply daemon |
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| 355 | ##|*; |
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| 356 | ##[// |
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| 357 | ##|#Hl2;napply H1;//;(*Hr*)napply daemon ##] ##] |
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| 358 | ##|#H;#H1;ncases H1;#x;*;*;#Hl;#Hr;#Hx; |
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| 359 | @ x;@ |
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| 360 | ##[@ |
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| 361 | ##[(*Hl*) napply daemon |
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| 362 | ##|(*Hr*) napply daemon ##] |
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| 363 | ##|//##]##]##]##] |
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| 364 | nqed. |
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| 365 | |
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| 366 | nlemma check_cont_true: |
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| 367 | ∀n:nat.∀m,p. |
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| 368 | (∀q. p ≤ q → q < p + n → m q = Cont) → |
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| 369 | check_cont n p m = true. |
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| 370 | #n;#m;#p;#H;nlapply (check_cont_spec n m p); |
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| 371 | ncases (check_cont n p m);//; |
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| 372 | nwhd in ⊢ (%→?);*; |
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| 373 | #q;*;*;#Hl;#Hr;#Hfalse;ncases Hfalse;#H1;nelim (H1 ?);napply H;//; |
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| 374 | nqed. |
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| 375 | |
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| 376 | nlemma check_cont_false: |
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| 377 | ∀n:nat.∀m,p,q. |
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| 378 | p ≤ q → q < p + n → m q ≠ Cont → |
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| 379 | check_cont n p m = false. |
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| 380 | #n;#m;#p;#q;nlapply (check_cont_spec n m p); |
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| 381 | ncases (check_cont n p m);//; |
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| 382 | nwhd in ⊢ (%→?);#H; |
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| 383 | #Hl;#Hr;#Hfalse;ncases Hfalse;#H1;nelim (H1 ?);napply H;//; |
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| 384 | nqed. |
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| 385 | |
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| 386 | nlemma set_cont_inside: |
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| 387 | ∀n:nat.∀p:Z.∀m.∀q:Z. |
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| 388 | p ≤ q → q < p + n → |
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| 389 | (set_cont n p m) q = Cont. |
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| 390 | #n;nelim n; |
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| 391 | ##[#p;#m;#q;#Hl;#Hr;(* by Hl, Hr → False *)napply daemon |
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| 392 | ##|#n1;#IH;#p;#m;#q;#Hl;#Hr; |
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| 393 | ncut (p = q ∨ p+1 ≤ q) |
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| 394 | ##[napply daemon |
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| 395 | ##|*; |
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| 396 | ##[#Heq;nrewrite > Heq;napply update_s; |
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| 397 | ##|#Hl2;nwhd in ⊢ (??%?);nrewrite > (? : eqZb q p = false) |
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| 398 | ##[nwhd in ⊢ (??%?);napply IH |
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[15] | 399 | ##[napply Hl2 |
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[3] | 400 | ##|(* Hr *) napply daemon ##] |
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| 401 | ##|(*Hl2*)napply daemon ##] |
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| 402 | ##] |
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| 403 | ##] |
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| 404 | ##] |
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| 405 | nqed. |
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| 406 | |
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| 407 | nlemma set_cont_outside: |
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| 408 | ∀n:nat.∀p:Z.∀m.∀q:Z. |
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| 409 | q < p ∨ p + n ≤ q → |
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| 410 | (set_cont n p m) q = m q. |
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| 411 | #n;nelim n |
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| 412 | ##[#p;#m;#q;#_;@ |
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| 413 | ##|#n1;#IH;#p;#m;#q; |
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| 414 | #H;nwhd in ⊢ (??%?);nrewrite > (? : eqZb q p = false); |
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| 415 | ##[nwhd in ⊢ (??%?);napply IH;ncases H; |
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| 416 | ##[#Hl;@;napply daemon |
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| 417 | ##|#Hr;@2;napply daemon##] |
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| 418 | ##|(*H*)napply daemon ##] |
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| 419 | ##] |
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| 420 | nqed. |
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| 421 | |
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| 422 | nlemma getN_setN_same: |
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| 423 | ∀n,p,v,m. |
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| 424 | getN n p (setN n p v m) = v. |
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| 425 | #n;#p;#v;#m;nchange in ⊢ (??(???%)?) with (update ????); |
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| 426 | nwhd in ⊢ (??%?); |
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| 427 | nrewrite > (update_s content p ??);nwhd in ⊢ (??%?); |
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| 428 | nrewrite > (eqb_n_n n); |
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| 429 | nrewrite > (check_cont_true ????) |
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| 430 | ##[@ |
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| 431 | ##|#q;#Hl;#Hr;nrewrite > (update_o content …); |
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| 432 | ##[/2/; |
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| 433 | ##|(*Hl*)napply daemon ##] |
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| 434 | ##] |
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| 435 | nqed. |
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| 436 | |
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| 437 | nlemma getN_setN_other: |
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| 438 | ∀n1,n2:nat.∀p1,p2:Z.∀v,m. |
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| 439 | p1 + n1 < p2 ∨ p2 + n2 < p1 → |
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| 440 | getN n2 p2 (setN n1 p1 v m) = getN n2 p2 m. |
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| 441 | #n1;#n2;#p1;#p2;#v;#m;#H; |
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| 442 | ngeneralize in match (check_cont_spec n2 m (p2 + oneZ)); |
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| 443 | nlapply (refl ? (check_cont n2 (p2+oneZ) m)); |
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| 444 | ncases (check_cont n2 (p2+oneZ) m) in ⊢ (???% → %); |
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| 445 | #H1;nwhd in ⊢ (% →?); nwhd in ⊢ (?→(???%)); nrewrite > H1; |
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| 446 | ##[#H2; |
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| 447 | nchange in ⊢ (??(???%)?) with (update ????); |
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| 448 | nwhd in ⊢(??%%);nrewrite > (check_cont_true …); |
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| 449 | ##[ nrewrite > (check_cont_true … H2); |
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| 450 | nrewrite > (update_o content ?????); |
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| 451 | ##[ nrewrite > (set_cont_outside ?????); //; (* arith *) napply daemon |
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| 452 | ##| (* arith *) napply daemon |
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| 453 | ##] |
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| 454 | ##| #q;#Hl;#Hh; nrewrite > (update_o content ?????); |
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| 455 | ##[ nrewrite > (set_cont_outside ?????); /2/; (* arith *) napply daemon |
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| 456 | ##| (* arith *) napply daemon |
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| 457 | ##] |
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| 458 | ##] |
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| 459 | ##| *; #q;*;#A;#B; |
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| 460 | nchange in ⊢ (??(???%)?) with (update ????); |
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| 461 | nwhd in ⊢(??%%); |
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| 462 | nrewrite > (check_cont_false n2 (update ? p1 (Datum n1 v) (set_cont n1 (p1 + 1) m)) (p2 + 1) q …); |
---|
| 463 | ##[ nrewrite > (update_o content ?????); |
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| 464 | ##[ nrewrite > (set_cont_outside ?????); //; (* arith *) napply daemon |
---|
| 465 | ##| napply daemon |
---|
| 466 | ##] |
---|
| 467 | ##| nrewrite > (update_o content ?????); |
---|
| 468 | ##[ nrewrite > (set_cont_outside ?????); //; (* arith *) napply daemon |
---|
| 469 | ##| napply daemon |
---|
| 470 | ##] |
---|
| 471 | ##| napply daemon |
---|
| 472 | ##| napply daemon |
---|
| 473 | ##] |
---|
| 474 | ##] nqed. |
---|
| 475 | |
---|
| 476 | nlemma getN_setN_overlap: |
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| 477 | ∀n1,n2,p1,p2,v,m. |
---|
| 478 | p1 ≠ p2 → |
---|
| 479 | p2 ≤ p1 + Z_of_nat n1 → p1 ≤ p2 + Z_of_nat n2 → |
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| 480 | getN n2 p2 (setN n1 p1 v m) = Vundef. |
---|
| 481 | #n1;#n2;#p1;#p2;#v;#m; |
---|
| 482 | #H;#H1;#H2; |
---|
| 483 | nchange in ⊢ (??(???%)?) with (update ????); |
---|
| 484 | nwhd in ⊢(??%?);nrewrite > (update_o content ?????); |
---|
| 485 | ##[nlapply (Z_compare_to_Prop p2 p1); |
---|
| 486 | nlapply (refl ? (Z_compare p2 p1)); |
---|
| 487 | ncases (Z_compare p2 p1) in ⊢ (???% → %);#H3; |
---|
| 488 | ##[nchange in ⊢ (% → ?) with (p2 < p1);#H4; |
---|
| 489 | (* [p1] belongs to [[p2, p2 + n2 - 1]], |
---|
| 490 | therefore [check_cont n2 (p2 + 1) ...] is false. *) |
---|
| 491 | nrewrite > (check_cont_false …); |
---|
| 492 | ##[ncases (set_cont n1 (p1+oneZ) m p2) |
---|
| 493 | ##[##1,3:@ |
---|
| 494 | ##|#n3;#v1;nwhd in ⊢ (??%?); |
---|
| 495 | ncases (eqb n2 n3);@ ##] |
---|
| 496 | ##|nrewrite > (update_s content …);napply nmk; |
---|
| 497 | #Hfalse;ndestruct |
---|
| 498 | ##|(*H2*) napply daemon |
---|
| 499 | ##|(*H4*) napply daemon |
---|
| 500 | ##|##skip ##] |
---|
| 501 | ##|nwhd in ⊢ (% → ?);#H4;nelim H;#H5;nelim (H5 ?);//; |
---|
| 502 | ##|nchange in ⊢ (% → ?) with (p1 < p2);#H4; |
---|
| 503 | (* [p2] belongs to [[p1 + 1, p1 + n1 - 1]], |
---|
| 504 | therefore [ |
---|
| 505 | set_cont n1 (p1 + 1) m p2] is [Cont]. *) |
---|
| 506 | nrewrite > (set_cont_inside …); |
---|
| 507 | ##[@ |
---|
| 508 | ##|(*H1*)napply daemon |
---|
| 509 | ##|(*H4*)napply daemon##] |
---|
| 510 | ##] |
---|
| 511 | ##|//##] |
---|
| 512 | nqed. |
---|
| 513 | |
---|
| 514 | nlemma getN_setN_mismatch: |
---|
| 515 | ∀n1,n2,p,v,m. |
---|
| 516 | n1 ≠ n2 → |
---|
| 517 | getN n2 p (setN n1 p v m) = Vundef. |
---|
| 518 | #n1;#n2;#p;#v;#m;#H; |
---|
| 519 | nchange in ⊢ (??(???%)?) with (update ????); |
---|
| 520 | nwhd in ⊢(??%?);nrewrite > (update_s content …); |
---|
[10] | 521 | nwhd in ⊢(??%?);nrewrite > (not_eq_to_eqb_false … (sym_neq … H));//; |
---|
[3] | 522 | nqed. |
---|
| 523 | |
---|
| 524 | nlemma getN_setN_characterization: |
---|
| 525 | ∀m,v,n1,p1,n2,p2. |
---|
| 526 | getN n2 p2 (setN n1 p1 v m) = v |
---|
| 527 | ∨ getN n2 p2 (setN n1 p1 v m) = getN n2 p2 m |
---|
| 528 | ∨ getN n2 p2 (setN n1 p1 v m) = Vundef. |
---|
| 529 | #m;#v;#n1;#p1;#n2;#p2; |
---|
| 530 | nlapply (eqZb_to_Prop p1 p2); ncases (eqZb p1 p2); #Hp; |
---|
| 531 | ##[nrewrite > Hp; |
---|
[10] | 532 | napply (eqb_elim n1 n2); #Hn; |
---|
[3] | 533 | ##[nrewrite > Hn;@; @; //; |
---|
| 534 | ##|@2;/2/] |
---|
| 535 | ##|nlapply (Z_compare_to_Prop (p1 + n1) p2); |
---|
| 536 | ncases (Z_compare (p1 + n1) p2);#Hcmp; |
---|
| 537 | ##[@;@2;napply getN_setN_other; /2/ |
---|
| 538 | ##|nlapply (Z_compare_to_Prop (p2 + n2) p1); |
---|
| 539 | ncases (Z_compare (p2 + n2) p1);#Hcmp2; |
---|
| 540 | ##[@;@2;napply getN_setN_other;/2/ |
---|
| 541 | ##|@2;napply getN_setN_overlap; |
---|
| 542 | ##[// |
---|
| 543 | ##|##*:(* arith *) napply daemon] |
---|
| 544 | ##|@2;napply getN_setN_overlap; |
---|
| 545 | ##[// |
---|
| 546 | ##|##*:(* arith *) napply daemon] |
---|
| 547 | ##] |
---|
| 548 | ##|nlapply (Z_compare_to_Prop (p2 + n2) p1); |
---|
| 549 | ncases (Z_compare (p2 + n2) p1);#Hcmp2; |
---|
| 550 | ##[@;@2;napply getN_setN_other;/2/ |
---|
| 551 | ##|@2;napply getN_setN_overlap; |
---|
| 552 | ##[// |
---|
| 553 | ##|##*:(* arith *) napply daemon] |
---|
| 554 | ##|@2;napply getN_setN_overlap; |
---|
| 555 | ##[// |
---|
| 556 | ##|##*:(* arith *) napply daemon] |
---|
| 557 | ##] |
---|
| 558 | ##] |
---|
| 559 | ##] |
---|
| 560 | nqed. |
---|
| 561 | |
---|
| 562 | nlemma getN_init: |
---|
| 563 | ∀n,p. |
---|
| 564 | getN n p (λ_.Undef) = Vundef. |
---|
| 565 | #n;#p;//; |
---|
| 566 | nqed. |
---|
[391] | 567 | *) |
---|
[125] | 568 | (* pointer_compat block_space pointer_space *) |
---|
[3] | 569 | |
---|
[480] | 570 | ninductive pointer_compat : region → region → Prop ≝ |
---|
[156] | 571 | | same_compat : ∀s. pointer_compat s s |
---|
| 572 | | pxdata_compat : pointer_compat PData XData |
---|
[125] | 573 | | unspecified_compat : ∀p. pointer_compat Any p |
---|
| 574 | | universal_compat : ∀b. pointer_compat b Any. |
---|
| 575 | |
---|
| 576 | nlemma pointer_compat_dec : ∀b,p. pointer_compat b p + ¬pointer_compat b p. |
---|
[124] | 577 | #b p; ncases b; |
---|
| 578 | ##[ ##1: @1; //; |
---|
| 579 | ##| ##*: ncases p; /2/; @2; @; #H; ninversion H; #e1 e2; ndestruct; #e3; ndestruct; |
---|
| 580 | ##] nqed. |
---|
| 581 | |
---|
[480] | 582 | ndefinition is_pointer_compat : region → region → bool ≝ |
---|
[125] | 583 | λb,p. match pointer_compat_dec b p with [ inl _ ⇒ true | inr _ ⇒ false ]. |
---|
[124] | 584 | |
---|
[125] | 585 | (* [valid_access m chunk psp b ofs] holds if a memory access (load or store) |
---|
[3] | 586 | of the given chunk is possible in [m] at address [b, ofs]. |
---|
| 587 | This means: |
---|
| 588 | - The block [b] is valid. |
---|
| 589 | - The range of bytes accessed is within the bounds of [b]. |
---|
| 590 | - The offset [ofs] is aligned. |
---|
[125] | 591 | - The pointer classs [psp] is compatible with the class of [b]. |
---|
[3] | 592 | *) |
---|
| 593 | |
---|
[480] | 594 | ninductive valid_access (m: mem) (chunk: memory_chunk) (psp: region) (b: block) (ofs: Z) |
---|
[3] | 595 | : Prop ≝ |
---|
| 596 | | valid_access_intro: |
---|
| 597 | valid_block m b → |
---|
| 598 | low_bound m b ≤ ofs → |
---|
| 599 | ofs + size_chunk chunk ≤ high_bound m b → |
---|
| 600 | (align_chunk chunk ∣ ofs) → |
---|
[125] | 601 | pointer_compat (block_space m b) psp → |
---|
| 602 | valid_access m chunk psp b ofs. |
---|
[3] | 603 | |
---|
| 604 | (* The following function checks whether accessing the given memory chunk |
---|
| 605 | at the given offset in the given block respects the bounds of the block. *) |
---|
| 606 | |
---|
[15] | 607 | (* XXX: Using + and ¬ instead of Sum and Not causes trouble *) |
---|
| 608 | nlet rec in_bounds |
---|
[480] | 609 | (m:mem) (chunk:memory_chunk) (psp:region) (b:block) (ofs:Z) on b : |
---|
[125] | 610 | Sum (valid_access m chunk psp b ofs) (Not (valid_access m chunk psp b ofs)) ≝ ?. |
---|
[3] | 611 | napply (Zltb_elim_Type0 b (nextblock m)); #Hnext; |
---|
| 612 | ##[ napply (Zleb_elim_Type0 (low_bound m b) ofs); #Hlo; |
---|
| 613 | ##[ napply (Zleb_elim_Type0 (ofs + size_chunk chunk) (high_bound m b)); #Hhi; |
---|
| 614 | ##[ nelim (dec_dividesZ (align_chunk chunk) ofs); #Hal; |
---|
[125] | 615 | ##[ ncases (pointer_compat_dec (block_space m b) psp); #Hcl; |
---|
[124] | 616 | ##[ @1; @; // ##] |
---|
| 617 | ##] |
---|
[3] | 618 | ##] |
---|
| 619 | ##] |
---|
| 620 | ##] |
---|
[124] | 621 | @2; napply nmk; *; #Hval; #Hlo'; #Hhi'; #Hal'; #Hcl'; napply (absurd ???); //; |
---|
[3] | 622 | nqed. |
---|
| 623 | |
---|
| 624 | nlemma in_bounds_true: |
---|
[125] | 625 | ∀m,chunk,psp,b,ofs. ∀A: Type[0]. ∀a1,a2: A. |
---|
| 626 | valid_access m chunk psp b ofs -> |
---|
| 627 | (match in_bounds m chunk psp b ofs with |
---|
[3] | 628 | [ inl _ ⇒ a1 | inr _ ⇒ a2 ]) = a1. |
---|
[125] | 629 | #m chunk psp b ofs A a1 a2 H; |
---|
| 630 | ncases (in_bounds m chunk psp b ofs);nnormalize;#H1; |
---|
[3] | 631 | ##[// |
---|
| 632 | ##|nelim (?:False);napply (absurd ? H H1)] |
---|
| 633 | nqed. |
---|
| 634 | |
---|
| 635 | (* [valid_pointer] holds if the given block address is valid and the |
---|
| 636 | given offset falls within the bounds of the corresponding block. *) |
---|
| 637 | |
---|
[480] | 638 | ndefinition valid_pointer : mem → region → block → Z → bool ≝ |
---|
[125] | 639 | λm,psp,b,ofs. Zltb b (nextblock m) ∧ |
---|
[3] | 640 | Zleb (low_bound m b) ofs ∧ |
---|
[124] | 641 | Zltb ofs (high_bound m b) ∧ |
---|
[125] | 642 | is_pointer_compat (block_space m b) psp. |
---|
[3] | 643 | |
---|
| 644 | (* [load chunk m b ofs] perform a read in memory state [m], at address |
---|
| 645 | [b] and offset [ofs]. [None] is returned if the address is invalid |
---|
| 646 | or the memory access is out of bounds. *) |
---|
| 647 | |
---|
[480] | 648 | ndefinition load : memory_chunk → mem → region → block → Z → option val ≝ |
---|
[125] | 649 | λchunk,m,psp,b,ofs. |
---|
| 650 | match in_bounds m chunk psp b ofs with |
---|
[3] | 651 | [ inl _ ⇒ Some ? (load_result chunk |
---|
| 652 | (getN (pred_size_chunk chunk) ofs (contents (blocks m b)))) |
---|
| 653 | | inr _ ⇒ None ? ]. |
---|
| 654 | |
---|
| 655 | nlemma load_inv: |
---|
[125] | 656 | ∀chunk,m,psp,b,ofs,v. |
---|
| 657 | load chunk m psp b ofs = Some ? v → |
---|
| 658 | valid_access m chunk psp b ofs ∧ |
---|
[3] | 659 | v = load_result chunk |
---|
| 660 | (getN (pred_size_chunk chunk) ofs (contents (blocks m b))). |
---|
[125] | 661 | #chunk m psp b ofs v; nwhd in ⊢ (??%? → ?); |
---|
| 662 | ncases (in_bounds m chunk psp b ofs); #Haccess; nwhd in ⊢ ((??%?) → ?); #H; |
---|
[3] | 663 | ##[ @;//; ndestruct; //; |
---|
| 664 | ##| ndestruct |
---|
| 665 | ##] |
---|
| 666 | nqed. |
---|
| 667 | |
---|
| 668 | (* [loadv chunk m addr] is similar, but the address and offset are given |
---|
| 669 | as a single value [addr], which must be a pointer value. *) |
---|
| 670 | |
---|
[15] | 671 | nlet rec loadv (chunk:memory_chunk) (m:mem) (addr:val) on addr : option val ≝ |
---|
[3] | 672 | match addr with |
---|
[125] | 673 | [ Vptr psp b ofs ⇒ load chunk m psp b (signed ofs) |
---|
[3] | 674 | | _ ⇒ None ? ]. |
---|
| 675 | |
---|
| 676 | (* The memory state [m] after a store of value [v] at offset [ofs] |
---|
| 677 | in block [b]. *) |
---|
| 678 | |
---|
| 679 | ndefinition unchecked_store : memory_chunk → mem → block → Z → val → mem ≝ |
---|
| 680 | λchunk,m,b,ofs,v. |
---|
| 681 | let c ≝ (blocks m b) in |
---|
| 682 | mk_mem |
---|
| 683 | (update ? b |
---|
| 684 | (mk_block_contents (low c) (high c) |
---|
[125] | 685 | (setN (pred_size_chunk chunk) ofs v (contents c)) (space c)) |
---|
[3] | 686 | (blocks m)) |
---|
| 687 | (nextblock m) |
---|
| 688 | (nextblock_pos m). |
---|
| 689 | |
---|
| 690 | (* [store chunk m b ofs v] perform a write in memory state [m]. |
---|
| 691 | Value [v] is stored at address [b] and offset [ofs]. |
---|
| 692 | Return the updated memory store, or [None] if the address is invalid |
---|
| 693 | or the memory access is out of bounds. *) |
---|
| 694 | |
---|
[480] | 695 | ndefinition store : memory_chunk → mem → region → block → Z → val → option mem ≝ |
---|
[125] | 696 | λchunk,m,psp,b,ofs,v. |
---|
| 697 | match in_bounds m chunk psp b ofs with |
---|
[3] | 698 | [ inl _ ⇒ Some ? (unchecked_store chunk m b ofs v) |
---|
| 699 | | inr _ ⇒ None ? ]. |
---|
| 700 | |
---|
| 701 | nlemma store_inv: |
---|
[125] | 702 | ∀chunk,m,psp,b,ofs,v,m'. |
---|
| 703 | store chunk m psp b ofs v = Some ? m' → |
---|
| 704 | valid_access m chunk psp b ofs ∧ |
---|
[3] | 705 | m' = unchecked_store chunk m b ofs v. |
---|
[125] | 706 | #chunk m psp b ofs v m'; nwhd in ⊢ (??%? → ?); |
---|
[3] | 707 | (*9*) |
---|
[125] | 708 | ncases (in_bounds m chunk psp b ofs);#Hv;nwhd in ⊢(??%? → ?);#Heq; |
---|
[3] | 709 | ##[@; ##[//|ndestruct;//] |
---|
| 710 | ##|ndestruct] |
---|
| 711 | nqed. |
---|
| 712 | |
---|
| 713 | (* [storev chunk m addr v] is similar, but the address and offset are given |
---|
| 714 | as a single value [addr], which must be a pointer value. *) |
---|
| 715 | |
---|
| 716 | ndefinition storev : memory_chunk → mem → val → val → option mem ≝ |
---|
| 717 | λchunk,m,addr,v. |
---|
| 718 | match addr with |
---|
[125] | 719 | [ Vptr psp b ofs ⇒ store chunk m psp b (signed ofs) v |
---|
[3] | 720 | | _ ⇒ None ? ]. |
---|
| 721 | |
---|
| 722 | |
---|
| 723 | (* Build a block filled with the given initialization data. *) |
---|
| 724 | ndefinition contents_init_data_step : |
---|
| 725 | Z → init_data → (Z → contentmap) → contentmap ≝ |
---|
| 726 | λpos,data,reccall. |
---|
| 727 | match data with |
---|
| 728 | [ Init_int8 n ⇒ |
---|
| 729 | setN 0 pos (Vint n) (reccall (pos + oneZ)) |
---|
| 730 | | Init_int16 n ⇒ |
---|
| 731 | setN 1 pos (Vint n) (reccall (pos + oneZ)) |
---|
| 732 | | Init_int32 n ⇒ |
---|
| 733 | setN 3 pos (Vint n) (reccall (pos + oneZ)) |
---|
| 734 | | Init_float32 f ⇒ |
---|
| 735 | setN 3 pos (Vfloat f) (reccall (pos + oneZ)) |
---|
| 736 | | Init_float64 f ⇒ |
---|
| 737 | setN 7 pos (Vfloat f) (reccall (pos + oneZ)) |
---|
| 738 | | Init_space n ⇒ reccall (pos + Zmax n 0) (*??*) |
---|
[484] | 739 | | Init_null r ⇒ setN (pred_size_pointer r) pos (Vnull r) (reccall (pos + oneZ)) |
---|
[3] | 740 | | Init_addrof s n ⇒ |
---|
| 741 | (* Not handled properly yet *) |
---|
| 742 | reccall (pos + 4) |
---|
| 743 | | Init_pointer x ⇒ |
---|
| 744 | (* Not handled properly yet *) |
---|
| 745 | reccall (pos + 4)]. |
---|
| 746 | |
---|
| 747 | nlet rec contents_init_data (pos: Z) (i_data: list init_data) on i_data |
---|
| 748 | : contentmap ≝ |
---|
| 749 | match i_data with |
---|
| 750 | [ nil ⇒ (λ_.Undef) |
---|
| 751 | | cons data i_data' ⇒ |
---|
| 752 | contents_init_data_step pos data (λn.contents_init_data n i_data') ]. |
---|
| 753 | |
---|
| 754 | ndefinition size_init_data : init_data → Z ≝ |
---|
| 755 | λi_data.match i_data with |
---|
| 756 | [ Init_int8 _ ⇒ 1 |
---|
| 757 | | Init_int16 _ ⇒ 2 |
---|
| 758 | | Init_int32 _ ⇒ 4 |
---|
| 759 | | Init_float32 _ ⇒ 4 |
---|
| 760 | | Init_float64 _ ⇒ 8 |
---|
| 761 | | Init_space n ⇒ Zmax n 0 |
---|
[483] | 762 | | Init_null r ⇒ size_pointer r |
---|
[3] | 763 | | Init_addrof _ _ ⇒ 4 |
---|
| 764 | | Init_pointer _ ⇒ Z_of_nat (*** can't use implicit coercion??? *)4]. |
---|
| 765 | |
---|
| 766 | ndefinition size_init_data_list : list init_data → Z ≝ |
---|
| 767 | λi_data.foldr ?? (λi_data,sz. size_init_data i_data + sz) OZ i_data. |
---|
| 768 | |
---|
[391] | 769 | (* Nonessential properties that may require arithmetic |
---|
[3] | 770 | nremark size_init_data_list_pos: |
---|
| 771 | ∀i_data. OZ ≤ size_init_data_list i_data. |
---|
| 772 | #i_data;nelim i_data;//; |
---|
| 773 | #a;#tl;ncut (OZ ≤ size_init_data a) |
---|
| 774 | ##[ncases a;nnormalize;//; |
---|
| 775 | #x;ncases x;nnormalize;//; |
---|
| 776 | ##|#Hsize;#IH;nchange in ⊢ (??%) with (size_init_data a + (foldr ??? OZ tl)); |
---|
| 777 | ncut (size_init_data a = OZ ∨ ∃m.size_init_data a = pos m) |
---|
| 778 | ##[ncases (size_init_data a) in Hsize IH; |
---|
| 779 | ##[##1,2:/3/ |
---|
| 780 | ##|nnormalize;#n;#Hfalse;nelim Hfalse] |
---|
| 781 | ##|#Hdisc;ncases Hdisc |
---|
| 782 | ##[#Heq;nrewrite > Heq;//; |
---|
| 783 | ##|#Heq;ncases Heq;#x;#Heq2;nrewrite > Heq2; |
---|
| 784 | (* TODO: arithmetics *) napply daemon] |
---|
| 785 | ##] |
---|
| 786 | ##] |
---|
| 787 | nqed. |
---|
[391] | 788 | *) |
---|
[3] | 789 | |
---|
[480] | 790 | ndefinition block_init_data : list init_data → region → block_contents ≝ |
---|
[124] | 791 | λi_data,bcl.mk_block_contents |
---|
| 792 | OZ (size_init_data_list i_data) (contents_init_data OZ i_data) bcl. |
---|
[3] | 793 | |
---|
[480] | 794 | ndefinition alloc_init_data : mem → list init_data → region → mem × block ≝ |
---|
[124] | 795 | λm,i_data,bcl. |
---|
[3] | 796 | 〈mk_mem (update ? (nextblock m) |
---|
[124] | 797 | (block_init_data i_data bcl) |
---|
[3] | 798 | (blocks m)) |
---|
| 799 | (Zsucc (nextblock m)) |
---|
| 800 | (succ_nextblock_pos m), |
---|
| 801 | (nextblock m)〉. |
---|
| 802 | |
---|
| 803 | nremark block_init_data_empty: |
---|
[124] | 804 | ∀bcl. block_init_data [ ] bcl = empty_block OZ OZ bcl. |
---|
[3] | 805 | //; |
---|
| 806 | nqed. |
---|
| 807 | |
---|
| 808 | (* * Properties of the memory operations *) |
---|
| 809 | |
---|
| 810 | (* ** Properties related to block validity *) |
---|
| 811 | |
---|
| 812 | nlemma valid_not_valid_diff: |
---|
| 813 | ∀m,b,b'. valid_block m b → ¬(valid_block m b') → b ≠ b'. |
---|
| 814 | #m;#b;#b';#H;#H';napply nmk;#e;nrewrite > e in H;#H; |
---|
| 815 | napply (absurd ? H H'); |
---|
| 816 | nqed. |
---|
| 817 | |
---|
| 818 | nlemma valid_access_valid_block: |
---|
[125] | 819 | ∀m,chunk,psp,b,ofs. valid_access m chunk psp b ofs → valid_block m b. |
---|
| 820 | #m;#chunk;#psp;#b;#ofs;#H; |
---|
[3] | 821 | nelim H;//; |
---|
| 822 | nqed. |
---|
| 823 | |
---|
| 824 | nlemma valid_access_aligned: |
---|
[125] | 825 | ∀m,chunk,psp,b,ofs. |
---|
| 826 | valid_access m chunk psp b ofs → (align_chunk chunk ∣ ofs). |
---|
| 827 | #m;#chunk;#psp;#b;#ofs;#H; |
---|
[3] | 828 | nelim H;//; |
---|
| 829 | nqed. |
---|
| 830 | |
---|
| 831 | nlemma valid_access_compat: |
---|
[125] | 832 | ∀m,chunk1,chunk2,psp,b,ofs. |
---|
[3] | 833 | size_chunk chunk1 = size_chunk chunk2 → |
---|
[125] | 834 | valid_access m chunk1 psp b ofs → |
---|
| 835 | valid_access m chunk2 psp b ofs. |
---|
| 836 | #m;#chunk;#chunk2;#psp;#b;#ofs;#H1;#H2; |
---|
[124] | 837 | nelim H2;#H3;#H4;#H5;#H6;#H7; |
---|
[3] | 838 | nrewrite > H1 in H5;#H5; |
---|
| 839 | @;//; |
---|
| 840 | nrewrite < (align_chunk_compat … H1);//; |
---|
| 841 | nqed. |
---|
| 842 | |
---|
| 843 | (* Hint Resolve valid_not_valid_diff valid_access_valid_block valid_access_aligned: mem.*) |
---|
| 844 | |
---|
| 845 | (* ** Properties related to [load] *) |
---|
| 846 | |
---|
| 847 | ntheorem valid_access_load: |
---|
[125] | 848 | ∀m,chunk,psp,b,ofs. |
---|
| 849 | valid_access m chunk psp b ofs → |
---|
| 850 | ∃v. load chunk m psp b ofs = Some ? v. |
---|
| 851 | #m;#chunk;#psp;#b;#ofs;#H;@; |
---|
[3] | 852 | ##[##2:nwhd in ⊢ (??%?);napply in_bounds_true;//; |
---|
| 853 | ##|##skip] |
---|
| 854 | nqed. |
---|
| 855 | |
---|
| 856 | ntheorem load_valid_access: |
---|
[125] | 857 | ∀m,chunk,psp,b,ofs,v. |
---|
| 858 | load chunk m psp b ofs = Some ? v → |
---|
| 859 | valid_access m chunk psp b ofs. |
---|
| 860 | #m;#chunk;#psp;#b;#ofs;#v;#H; |
---|
[3] | 861 | ncases (load_inv … H);//; |
---|
| 862 | nqed. |
---|
| 863 | |
---|
| 864 | (* Hint Resolve load_valid_access valid_access_load.*) |
---|
| 865 | |
---|
| 866 | (* ** Properties related to [store] *) |
---|
| 867 | |
---|
| 868 | nlemma valid_access_store: |
---|
[125] | 869 | ∀m1,chunk,psp,b,ofs,v. |
---|
| 870 | valid_access m1 chunk psp b ofs → |
---|
| 871 | ∃m2. store chunk m1 psp b ofs v = Some ? m2. |
---|
| 872 | #m1;#chunk;#psp;#b;#ofs;#v;#H; |
---|
[3] | 873 | @; |
---|
| 874 | ##[##2:napply in_bounds_true;// |
---|
| 875 | ##|##skip] |
---|
| 876 | nqed. |
---|
| 877 | |
---|
| 878 | (* section STORE *) |
---|
| 879 | |
---|
| 880 | nlemma low_bound_store: |
---|
[125] | 881 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 882 | ∀b'.low_bound m2 b' = low_bound m1 b'. |
---|
[125] | 883 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 884 | #b';ncases (store_inv … STORE); |
---|
| 885 | #H1;#H2;nrewrite > H2; |
---|
| 886 | nwhd in ⊢ (??(?%?)?);nwhd in ⊢ (??%?); |
---|
| 887 | nwhd in ⊢ (??(?%)?);nlapply (eqZb_to_Prop b' b); |
---|
| 888 | ncases (eqZb b' b);nnormalize;//; |
---|
| 889 | nqed. |
---|
| 890 | |
---|
| 891 | nlemma nextblock_store : |
---|
[125] | 892 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 893 | nextblock m2 = nextblock m1. |
---|
[125] | 894 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 895 | ncases (store_inv … STORE); |
---|
| 896 | #Hvalid;#Heq; |
---|
| 897 | nrewrite > Heq;@; |
---|
| 898 | nqed. |
---|
| 899 | |
---|
| 900 | nlemma high_bound_store: |
---|
[125] | 901 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 902 | ∀b'. high_bound m2 b' = high_bound m1 b'. |
---|
[125] | 903 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 904 | #b';ncases (store_inv … STORE); |
---|
| 905 | #Hvalid;#H; |
---|
| 906 | nrewrite > H; |
---|
| 907 | nwhd in ⊢ (??(?%?)?);nwhd in ⊢ (??%?); |
---|
| 908 | nwhd in ⊢ (??(?%)?);nlapply (eqZb_to_Prop b' b); |
---|
| 909 | ncases (eqZb b' b);nnormalize;//; |
---|
| 910 | nqed. |
---|
| 911 | |
---|
[480] | 912 | nlemma region_store: |
---|
[125] | 913 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 914 | ∀b'. block_space m2 b' = block_space m1 b'. |
---|
| 915 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[124] | 916 | #b';ncases (store_inv … STORE); |
---|
| 917 | #Hvalid;#H; |
---|
| 918 | nrewrite > H; |
---|
| 919 | nwhd in ⊢ (??(?%?)?);nwhd in ⊢ (??%?); |
---|
| 920 | nwhd in ⊢ (??(?%)?);nlapply (eqZb_to_Prop b' b); |
---|
| 921 | ncases (eqZb b' b);nnormalize;//; |
---|
| 922 | nqed. |
---|
| 923 | |
---|
[3] | 924 | nlemma store_valid_block_1: |
---|
[125] | 925 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 926 | ∀b'. valid_block m1 b' → valid_block m2 b'. |
---|
[125] | 927 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 928 | #b';nwhd in ⊢ (% → %);#Hv; |
---|
| 929 | nrewrite > (nextblock_store … STORE);//; |
---|
| 930 | nqed. |
---|
| 931 | |
---|
| 932 | nlemma store_valid_block_2: |
---|
[125] | 933 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 934 | ∀b'. valid_block m2 b' → valid_block m1 b'. |
---|
[125] | 935 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 936 | #b';nwhd in ⊢ (%→%); |
---|
| 937 | nrewrite > (nextblock_store … STORE);//; |
---|
| 938 | nqed. |
---|
| 939 | |
---|
| 940 | (*Hint Resolve store_valid_block_1 store_valid_block_2: mem.*) |
---|
| 941 | |
---|
| 942 | nlemma store_valid_access_1: |
---|
[125] | 943 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 944 | ∀chunk',psp',b',ofs'. |
---|
| 945 | valid_access m1 chunk' psp' b' ofs' → valid_access m2 chunk' psp' b' ofs'. |
---|
| 946 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
| 947 | #chunk';#psp';#b';#ofs'; |
---|
[3] | 948 | * Hv; |
---|
[124] | 949 | #Hvb;#Hl;#Hr;#Halign;#Hptr; |
---|
[3] | 950 | @;//; |
---|
| 951 | ##[napply (store_valid_block_1 … STORE);// |
---|
| 952 | ##|nrewrite > (low_bound_store … STORE …);// |
---|
[124] | 953 | ##|nrewrite > (high_bound_store … STORE …);// |
---|
[480] | 954 | ##|nrewrite > (region_store … STORE …);//] |
---|
[3] | 955 | nqed. |
---|
| 956 | |
---|
| 957 | nlemma store_valid_access_2: |
---|
[125] | 958 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 959 | ∀chunk',psp',b',ofs'. |
---|
| 960 | valid_access m2 chunk' psp' b' ofs' → valid_access m1 chunk' psp' b' ofs'. |
---|
| 961 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
| 962 | #chunk';#psp';#b';#ofs'; |
---|
[3] | 963 | * Hv; |
---|
[124] | 964 | #Hvb;#Hl;#Hr;#Halign;#Hcompat; |
---|
[3] | 965 | @;//; |
---|
| 966 | ##[napply (store_valid_block_2 … STORE);// |
---|
| 967 | ##|nrewrite < (low_bound_store … STORE …);// |
---|
[124] | 968 | ##|nrewrite < (high_bound_store … STORE …);// |
---|
[480] | 969 | ##|nrewrite < (region_store … STORE …);//] |
---|
[3] | 970 | nqed. |
---|
| 971 | |
---|
| 972 | nlemma store_valid_access_3: |
---|
[125] | 973 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 974 | valid_access m1 chunk psp b ofs. |
---|
| 975 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 976 | ncases (store_inv … STORE);//; |
---|
| 977 | nqed. |
---|
| 978 | |
---|
| 979 | (*Hint Resolve store_valid_access_1 store_valid_access_2 |
---|
| 980 | store_valid_access_3: mem.*) |
---|
| 981 | |
---|
[124] | 982 | nlemma load_compat_pointer: |
---|
[125] | 983 | ∀chunk,m,psp,psp',b,ofs,v. |
---|
| 984 | pointer_compat (block_space m b) psp' → |
---|
| 985 | load chunk m psp b ofs = Some ? v → |
---|
| 986 | load chunk m psp' b ofs = Some ? v. |
---|
| 987 | #chunk m psp psp' b ofs v Hcompat LOAD. |
---|
[124] | 988 | nlapply (load_valid_access … LOAD); #Hvalid; |
---|
[125] | 989 | ncut (valid_access m chunk psp' b ofs); |
---|
[124] | 990 | ##[ @;nelim Hvalid; //; |
---|
| 991 | ##| #Hvalid'; |
---|
| 992 | nrewrite < LOAD; nwhd in ⊢ (??%%); |
---|
| 993 | nrewrite > (in_bounds_true … (option val) ?? Hvalid); |
---|
| 994 | nrewrite > (in_bounds_true … (option val) ?? Hvalid'); |
---|
| 995 | // |
---|
| 996 | ##] nqed. |
---|
| 997 | |
---|
[391] | 998 | (* Nonessential properties that may require arithmetic |
---|
[3] | 999 | ntheorem load_store_similar: |
---|
[125] | 1000 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 1001 | ∀chunk'. |
---|
| 1002 | size_chunk chunk' = size_chunk chunk → |
---|
[125] | 1003 | load chunk' m2 psp b ofs = Some ? (load_result chunk' v). |
---|
| 1004 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1005 | #chunk';#Hsize;ncases (store_inv … STORE); |
---|
| 1006 | #Hv;#Heq; |
---|
| 1007 | nwhd in ⊢ (??%?); |
---|
[125] | 1008 | nrewrite > (in_bounds_true m2 chunk' psp b ofs ? (Some ? (load_result chunk' (getN (pred_size_chunk chunk') ofs (contents (blocks m2 b))))) |
---|
[3] | 1009 | (None ?) ?); |
---|
| 1010 | ##[nrewrite > Heq; |
---|
| 1011 | nwhd in ⊢ (??(??(? ? (? ? ? (? (? % ?)))))?); |
---|
| 1012 | nrewrite > (update_s ? b ? (blocks m1)); (* XXX too many metas for my taste *) |
---|
| 1013 | nrewrite > (? : pred_size_chunk chunk = pred_size_chunk chunk'); |
---|
| 1014 | ##[//; |
---|
| 1015 | ##|nrewrite > (size_chunk_pred …) in Hsize;#Hsize; |
---|
| 1016 | nrewrite > (size_chunk_pred …) in Hsize;#Hsize; |
---|
| 1017 | napply injective_Z_of_nat;napply (injective_Zplus_r 1);//;##] |
---|
[124] | 1018 | ##|napply (store_valid_access_1 … STORE); |
---|
| 1019 | ncases Hv;#H1;#H2;#H3;#H4;#H5;@;//; |
---|
[3] | 1020 | nrewrite > (align_chunk_compat … Hsize);//] |
---|
| 1021 | nqed. |
---|
| 1022 | |
---|
| 1023 | ntheorem load_store_same: |
---|
[125] | 1024 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 1025 | load chunk m2 psp b ofs = Some ? (load_result chunk v). |
---|
| 1026 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1027 | napply load_store_similar;//; |
---|
| 1028 | nqed. |
---|
| 1029 | |
---|
| 1030 | ntheorem load_store_other: |
---|
[125] | 1031 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 1032 | ∀chunk',psp',b',ofs'. |
---|
[3] | 1033 | b' ≠ b |
---|
| 1034 | ∨ ofs' + size_chunk chunk' ≤ ofs |
---|
| 1035 | ∨ ofs + size_chunk chunk ≤ ofs' → |
---|
[125] | 1036 | load chunk' m2 psp' b' ofs' = load chunk' m1 psp' b' ofs'. |
---|
| 1037 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
| 1038 | #chunk';#psp';#b';#ofs';#H; |
---|
[3] | 1039 | ncases (store_inv … STORE); |
---|
| 1040 | #Hvalid;#Heq;nwhd in ⊢ (??%%); |
---|
[125] | 1041 | ncases (in_bounds m1 chunk' psp' b' ofs'); |
---|
| 1042 | ##[#Hvalid1;nrewrite > (in_bounds_true m2 chunk' psp' b' ofs' ? (Some ? ?) ??); |
---|
[190] | 1043 | ##[nwhd in ⊢ (???%); napply (eq_f … (Some val));napply (eq_f … (load_result chunk')); |
---|
[3] | 1044 | nrewrite > Heq;nwhd in ⊢ (??(???(? (? % ?)))?); |
---|
| 1045 | nwhd in ⊢ (??(???(? %))?); |
---|
| 1046 | nlapply (eqZb_to_Prop b' b);ncases (eqZb b' b); |
---|
| 1047 | nwhd in ⊢ (% → ?); |
---|
| 1048 | ##[#Heq1;nrewrite > Heq1;nwhd in ⊢ (??(??? (? %))?); |
---|
| 1049 | nrewrite > (size_chunk_pred …) in H; |
---|
| 1050 | nrewrite > (size_chunk_pred …);#H; |
---|
| 1051 | napply (getN_setN_other …);ncases H |
---|
| 1052 | ##[* |
---|
| 1053 | ##[#Hfalse;nelim Hfalse;#H1;nelim (H1 Heq1) |
---|
| 1054 | ##|#H1;@2;(*H1*)napply daemon ##] |
---|
| 1055 | ##|#H1;@;(*H1*)napply daemon ##] |
---|
| 1056 | ##|#Hneq;@ ##] |
---|
| 1057 | ##|napply (store_valid_access_1 … STORE);//##] |
---|
[125] | 1058 | ##|nwhd in ⊢ (? → ???%);nlapply (in_bounds m2 chunk' psp' b' ofs'); |
---|
[3] | 1059 | #H1;ncases H1; |
---|
| 1060 | ##[#H2;#H3;nlapply (store_valid_access_2 … STORE … H2);#Hfalse; |
---|
| 1061 | ncases H3;#H4;nelim (H4 Hfalse) |
---|
| 1062 | ##|#H2;#H3;@] |
---|
| 1063 | ##] |
---|
| 1064 | nqed. |
---|
| 1065 | |
---|
| 1066 | |
---|
| 1067 | ntheorem load_store_overlap: |
---|
[125] | 1068 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 1069 | ∀chunk',ofs',v'. load chunk' m2 psp b ofs' = Some ? v' → |
---|
[3] | 1070 | ofs' ≠ ofs → |
---|
| 1071 | ofs < ofs' + size_chunk chunk' → |
---|
| 1072 | ofs' < ofs + size_chunk chunk → |
---|
| 1073 | v' = Vundef. |
---|
[125] | 1074 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1075 | #chunk';#ofs';#v';#H; |
---|
| 1076 | #H1;#H2;#H3; |
---|
| 1077 | ncases (store_inv … STORE); |
---|
| 1078 | ncases (load_inv … H); |
---|
| 1079 | #Hvalid;#Heq;#Hvalid1;#Heq1;nrewrite > Heq;nrewrite > Heq1; |
---|
| 1080 | nchange in ⊢ (??(??(???(?(?%?))))?) with (mk_mem ???); |
---|
| 1081 | nrewrite > (update_s block_contents …); |
---|
| 1082 | nrewrite > (getN_setN_overlap …); |
---|
| 1083 | ##[ncases chunk';// |
---|
| 1084 | ##|nrewrite > (size_chunk_pred …) in H2;(*arith*) napply daemon |
---|
| 1085 | ##|nrewrite > (size_chunk_pred …) in H3;(*arith*) napply daemon |
---|
| 1086 | ##|napply sym_neq;//] |
---|
| 1087 | nqed. |
---|
| 1088 | |
---|
| 1089 | ntheorem load_store_overlap': |
---|
[125] | 1090 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 1091 | ∀chunk',ofs'. |
---|
[125] | 1092 | valid_access m1 chunk' psp b ofs' → |
---|
[3] | 1093 | ofs' ≠ ofs → |
---|
| 1094 | ofs < ofs' + size_chunk chunk' → |
---|
| 1095 | ofs' < ofs + size_chunk chunk → |
---|
[125] | 1096 | load chunk' m2 psp b ofs' = Some ? Vundef. |
---|
| 1097 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1098 | #chunk';#ofs';#Hvalid;#H;#H1;#H2; |
---|
[125] | 1099 | ncut (∃v'.load chunk' m2 psp b ofs' = Some ? v') |
---|
[3] | 1100 | ##[napply valid_access_load; |
---|
| 1101 | napply (store_valid_access_1 … STORE);// |
---|
| 1102 | ##|#H3;ncases H3; |
---|
[190] | 1103 | #x;#Hx;nrewrite > Hx;napply (eq_f … (Some val)); |
---|
[3] | 1104 | napply (load_store_overlap … STORE … Hx);//;##] |
---|
| 1105 | nqed. |
---|
| 1106 | |
---|
| 1107 | ntheorem load_store_mismatch: |
---|
[125] | 1108 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 1109 | ∀chunk',v'. |
---|
[125] | 1110 | load chunk' m2 psp b ofs = Some ? v' → |
---|
[3] | 1111 | size_chunk chunk' ≠ size_chunk chunk → |
---|
| 1112 | v' = Vundef. |
---|
[125] | 1113 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1114 | #chunk';#v';#H;#H1; |
---|
| 1115 | ncases (store_inv … STORE); |
---|
| 1116 | ncases (load_inv … H); |
---|
| 1117 | #Hvalid;#H2;#Hvalid1;#H3; |
---|
| 1118 | nrewrite > H2; |
---|
| 1119 | nchange in H3:(???%) with (mk_mem ???); |
---|
| 1120 | nrewrite > H3;nrewrite > (update_s block_contents …); |
---|
| 1121 | nrewrite > (getN_setN_mismatch …); |
---|
| 1122 | ##[ncases chunk';//; |
---|
| 1123 | ##|nrewrite > (size_chunk_pred …) in H1;nrewrite > (size_chunk_pred …); |
---|
| 1124 | #H1;napply nmk;#Hfalse;nelim H1;#H4;napply H4; |
---|
[10] | 1125 | napply (eq_f ?? (λx.1 + x) ???);//##] |
---|
[3] | 1126 | nqed. |
---|
| 1127 | |
---|
| 1128 | ntheorem load_store_mismatch': |
---|
[125] | 1129 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
[3] | 1130 | ∀chunk'. |
---|
[125] | 1131 | valid_access m1 chunk' psp b ofs → |
---|
[3] | 1132 | size_chunk chunk' ≠ size_chunk chunk → |
---|
[125] | 1133 | load chunk' m2 psp b ofs = Some ? Vundef. |
---|
| 1134 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
[3] | 1135 | #chunk';#Hvalid;#Hsize; |
---|
[125] | 1136 | ncut (∃v'.load chunk' m2 psp b ofs = Some ? v') |
---|
[3] | 1137 | ##[napply (valid_access_load …); |
---|
[124] | 1138 | napply |
---|
| 1139 | (store_valid_access_1 … STORE);// |
---|
[190] | 1140 | ##|*;#x;#Hx;nrewrite > Hx;napply (eq_f … (Some val)); |
---|
[3] | 1141 | napply (load_store_mismatch … STORE … Hsize);//;##] |
---|
| 1142 | nqed. |
---|
| 1143 | |
---|
| 1144 | ninductive load_store_cases |
---|
| 1145 | (chunk1: memory_chunk) (b1: block) (ofs1: Z) |
---|
| 1146 | (chunk2: memory_chunk) (b2: block) (ofs2: Z) : Type ≝ |
---|
| 1147 | | lsc_similar: |
---|
| 1148 | b1 = b2 → ofs1 = ofs2 → size_chunk chunk1 = size_chunk chunk2 → |
---|
| 1149 | load_store_cases chunk1 b1 ofs1 chunk2 b2 ofs2 |
---|
| 1150 | | lsc_other: |
---|
| 1151 | b1 ≠ b2 ∨ ofs2 + size_chunk chunk2 ≤ ofs1 ∨ ofs1 + size_chunk chunk1 ≤ ofs2 → |
---|
| 1152 | load_store_cases chunk1 b1 ofs1 chunk2 b2 ofs2 |
---|
| 1153 | | lsc_overlap: |
---|
| 1154 | b1 = b2 -> ofs1 ≠ ofs2 -> ofs1 < ofs2 + size_chunk chunk2 → ofs2 < ofs1 + size_chunk chunk1 -> |
---|
| 1155 | load_store_cases chunk1 b1 ofs1 chunk2 b2 ofs2 |
---|
| 1156 | | lsc_mismatch: |
---|
| 1157 | b1 = b2 → ofs1 = ofs2 → size_chunk chunk1 ≠ size_chunk chunk2 -> |
---|
| 1158 | load_store_cases chunk1 b1 ofs1 chunk2 b2 ofs2. |
---|
| 1159 | |
---|
| 1160 | ndefinition load_store_classification: |
---|
| 1161 | ∀chunk1,b1,ofs1,chunk2,b2,ofs2. |
---|
| 1162 | load_store_cases chunk1 b1 ofs1 chunk2 b2 ofs2. |
---|
| 1163 | #chunk1;#b1;#ofs1;#chunk2;#b2;#ofs2; |
---|
| 1164 | ncases (decidable_eq_Z_Type b1 b2);#H; |
---|
| 1165 | ##[ncases (decidable_eq_Z_Type ofs1 ofs2);#H1; |
---|
| 1166 | ##[ncases (decidable_eq_Z_Type (size_chunk chunk1) (size_chunk chunk2));#H2 |
---|
| 1167 | ##[napply lsc_similar;//; |
---|
| 1168 | ##|napply lsc_mismatch;//;##] |
---|
| 1169 | ##|nlapply (Z_compare_to_Prop (ofs2 + size_chunk chunk2) ofs1); |
---|
| 1170 | ncases (Z_compare (ofs2+size_chunk chunk2) ofs1); |
---|
| 1171 | ##[nchange with (Zlt ? ? → ?);#H2; |
---|
| 1172 | napply lsc_other;@;@2;(*trivial Zlt_to_Zle BUT the statement is strange*) |
---|
| 1173 | napply daemon |
---|
| 1174 | ##|nchange with (? = ? → ?);#H2; |
---|
| 1175 | napply lsc_other;@;@2;(*trivial eq_to_Zle not defined *) napply daemon |
---|
| 1176 | ##|nchange with (Zlt ? ? → ?);#H2; |
---|
| 1177 | nlapply (Z_compare_to_Prop (ofs1 + size_chunk chunk1) ofs2); |
---|
| 1178 | ncases (Z_compare (ofs1 + size_chunk chunk1) ofs2); |
---|
| 1179 | ##[nchange with (Zlt ? ? → ?);#H3; |
---|
| 1180 | napply lsc_other;@2;(*trivial as previously*) napply daemon |
---|
| 1181 | ##|nchange with (? = ? → ?);#H3; |
---|
| 1182 | napply lsc_other;@2;(*trivial as previously*) napply daemon |
---|
| 1183 | ##|nchange with (Zlt ? ? → ?);#H3; |
---|
| 1184 | napply lsc_overlap;//;##] |
---|
| 1185 | ##] |
---|
| 1186 | ##] |
---|
| 1187 | ##|napply lsc_other;@;@; (* XXX // doesn't spot this! *) napply H ##] |
---|
| 1188 | nqed. |
---|
| 1189 | |
---|
| 1190 | ntheorem load_store_characterization: |
---|
[125] | 1191 | ∀chunk,m1,psp,b,ofs,v,m2.store chunk m1 psp b ofs v = Some ? m2 → |
---|
| 1192 | ∀chunk',psp',b',ofs'. |
---|
| 1193 | valid_access m1 chunk' psp' b' ofs' → |
---|
| 1194 | load chunk' m2 psp' b' ofs' = |
---|
[3] | 1195 | match load_store_classification chunk b ofs chunk' b' ofs' with |
---|
[124] | 1196 | [ lsc_similar _ _ _ ⇒ Some ? (load_result chunk' v) |
---|
[125] | 1197 | | lsc_other _ ⇒ load chunk' m1 psp' b' ofs' |
---|
[124] | 1198 | | lsc_overlap _ _ _ _ ⇒ Some ? Vundef |
---|
| 1199 | | lsc_mismatch _ _ _ ⇒ Some ? Vundef ]. |
---|
[125] | 1200 | #chunk;#m1;#psp b ofs;#v;#m2;#STORE; |
---|
| 1201 | #chunk';#psp';#b';#ofs';#Hvalid; |
---|
| 1202 | ncut (∃v'. load chunk' m2 psp' b' ofs' = Some ? v') |
---|
[3] | 1203 | ##[napply valid_access_load; |
---|
| 1204 | napply (store_valid_access_1 … STORE … Hvalid); |
---|
| 1205 | ##|*;#x;#Hx; |
---|
| 1206 | ncases (load_store_classification chunk b ofs chunk' b' ofs') |
---|
[124] | 1207 | ##[#H1;#H2;#H3;nwhd in ⊢ (???%);nrewrite < H1;nrewrite < H2; |
---|
[125] | 1208 | napply (load_compat_pointer … psp); |
---|
[480] | 1209 | ##[ nrewrite > (region_store … STORE b); |
---|
[124] | 1210 | ncases Hvalid; //; |
---|
| 1211 | ##| napply (load_store_similar … STORE);//; |
---|
| 1212 | ##] |
---|
[3] | 1213 | ##|#H1;napply (load_store_other … STORE); |
---|
| 1214 | ncases H1 |
---|
| 1215 | ##[* |
---|
| 1216 | ##[#H2;@;@;napply sym_neq;// |
---|
| 1217 | ##|#H2;@;@2;//] |
---|
| 1218 | ##|#H2;@2;//] |
---|
[125] | 1219 | ##|#H1;#H2;#H3;#H4; nlapply (load_compat_pointer … psp … Hx); |
---|
[480] | 1220 | ##[ nrewrite > (region_store … STORE b'); |
---|
[124] | 1221 | nrewrite > H1; nelim (store_valid_access_3 … STORE); // |
---|
| 1222 | ##| nrewrite < H1 in ⊢ (% → ?);#Hx'; |
---|
| 1223 | nrewrite < H1 in Hx;#Hx;nrewrite > Hx; |
---|
[190] | 1224 | napply (eq_f … (Some val));napply (load_store_overlap … STORE … Hx');/2/; |
---|
[124] | 1225 | ##] |
---|
| 1226 | ##|#H1;#H2;#H3; |
---|
[125] | 1227 | nlapply (load_compat_pointer … psp … Hx); |
---|
[480] | 1228 | ##[ nrewrite > (region_store … STORE b'); |
---|
[124] | 1229 | nrewrite > H1; nelim (store_valid_access_3 … STORE); // |
---|
| 1230 | ##| nrewrite < H1 in Hx ⊢ %; nrewrite < H2; #Hx Hx'; |
---|
[190] | 1231 | nrewrite > Hx;napply (eq_f … (Some val)); |
---|
[124] | 1232 | napply (load_store_mismatch … STORE … Hx');/2/ |
---|
| 1233 | ##] |
---|
| 1234 | ##] |
---|
| 1235 | |
---|
[3] | 1236 | ##] |
---|
| 1237 | nqed. |
---|
| 1238 | |
---|
| 1239 | (*nlemma d : ∀a,b,c,d:nat.∀e:〈a,b〉 = 〈c,d〉. ∀P:(∀a,b,c,d,e.Prop). |
---|
| 1240 | P a b c d e → P a b a b (refl ??). |
---|
| 1241 | #a;#b;#c;#d;#e;#P;#H1;ndestruct;*) |
---|
| 1242 | |
---|
| 1243 | (* |
---|
| 1244 | Section ALLOC. |
---|
| 1245 | |
---|
| 1246 | Variable m1: mem. |
---|
| 1247 | Variables lo hi: Z. |
---|
| 1248 | Variable m2: mem. |
---|
| 1249 | Variable b: block. |
---|
| 1250 | Hypothesis ALLOC: alloc m1 lo hi = (m2, b). |
---|
| 1251 | *) |
---|
| 1252 | |
---|
| 1253 | ndefinition pairdisc ≝ |
---|
| 1254 | λA,B.λx,y:pair A B. |
---|
| 1255 | match x with |
---|
| 1256 | [(mk_pair t0 t1) ⇒ |
---|
| 1257 | match y with |
---|
| 1258 | [(mk_pair u0 u1) ⇒ |
---|
| 1259 | ∀P: Type[1]. |
---|
| 1260 | (∀e0: (eq A (R0 ? t0) u0). |
---|
| 1261 | ∀e1: (eq (? u0 e0) (R1 ? t0 ? t1 u0 e0) u1).P) → P ] ]. |
---|
| 1262 | |
---|
| 1263 | nlemma pairdisc_elim : ∀A,B,x,y.x = y → pairdisc A B x y. |
---|
| 1264 | #A;#B;#x;#y;#e;nrewrite > e;ncases y; |
---|
| 1265 | #a;#b;nnormalize;#P;#PH;napply PH;@; |
---|
| 1266 | nqed. |
---|
| 1267 | |
---|
| 1268 | nlemma nextblock_alloc: |
---|
[124] | 1269 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1270 | nextblock m2 = Zsucc (nextblock m1). |
---|
[124] | 1271 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1272 | nwhd in ALLOC:(??%%); ndestruct; //; |
---|
| 1273 | nqed. |
---|
| 1274 | |
---|
| 1275 | nlemma alloc_result: |
---|
[124] | 1276 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1277 | b = nextblock m1. |
---|
[124] | 1278 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1279 | nwhd in ALLOC:(??%%); ndestruct; //; |
---|
| 1280 | nqed. |
---|
| 1281 | |
---|
| 1282 | |
---|
| 1283 | nlemma valid_block_alloc: |
---|
[124] | 1284 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1285 | ∀b'. valid_block m1 b' → valid_block m2 b'. |
---|
[124] | 1286 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1287 | #b'; nrewrite > (unfold_valid_block m1 b'); |
---|
| 1288 | nrewrite > (unfold_valid_block m2 b'); |
---|
| 1289 | nrewrite > (nextblock_alloc … ALLOC); |
---|
| 1290 | (* arith *) napply daemon; |
---|
| 1291 | nqed. |
---|
| 1292 | |
---|
| 1293 | nlemma fresh_block_alloc: |
---|
[124] | 1294 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1295 | ¬(valid_block m1 b). |
---|
[124] | 1296 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1297 | nrewrite > (unfold_valid_block m1 b); |
---|
| 1298 | nrewrite > (alloc_result … ALLOC); |
---|
| 1299 | (* arith *) napply daemon; |
---|
| 1300 | nqed. |
---|
| 1301 | |
---|
| 1302 | nlemma valid_new_block: |
---|
[124] | 1303 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1304 | valid_block m2 b. |
---|
[124] | 1305 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1306 | nrewrite > (unfold_valid_block m2 b); |
---|
| 1307 | nrewrite > (alloc_result … ALLOC); |
---|
| 1308 | nrewrite > (nextblock_alloc … ALLOC); |
---|
| 1309 | (* arith *) napply daemon; |
---|
| 1310 | nqed. |
---|
| 1311 | |
---|
| 1312 | (* |
---|
| 1313 | Hint Resolve valid_block_alloc fresh_block_alloc valid_new_block: mem. |
---|
| 1314 | *) |
---|
| 1315 | |
---|
| 1316 | nlemma valid_block_alloc_inv: |
---|
[124] | 1317 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1318 | ∀b'. valid_block m2 b' → b' = b ∨ valid_block m1 b'. |
---|
[124] | 1319 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1320 | #b'; |
---|
| 1321 | nrewrite > (unfold_valid_block m2 b'); |
---|
| 1322 | nrewrite > (unfold_valid_block m1 b'); |
---|
| 1323 | nrewrite > (nextblock_alloc … ALLOC); #H; |
---|
| 1324 | nrewrite > (alloc_result … ALLOC); |
---|
| 1325 | (* arith *) napply daemon; |
---|
| 1326 | nqed. |
---|
| 1327 | |
---|
| 1328 | nlemma low_bound_alloc: |
---|
[124] | 1329 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1330 | ∀b'. low_bound m2 b' = if eqZb b' b then lo else low_bound m1 b'. |
---|
[124] | 1331 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1332 | nwhd in ALLOC:(??%%); ndestruct; #b'; |
---|
| 1333 | nrewrite > (unfold_update block_contents ????); |
---|
| 1334 | ncases (eqZb b' (nextblock m1)); //; |
---|
| 1335 | nqed. |
---|
| 1336 | |
---|
| 1337 | nlemma low_bound_alloc_same: |
---|
[124] | 1338 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1339 | low_bound m2 b = lo. |
---|
[124] | 1340 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1341 | nrewrite > (low_bound_alloc … ALLOC b); |
---|
| 1342 | nrewrite > (eqZb_z_z …); |
---|
| 1343 | //; |
---|
| 1344 | nqed. |
---|
| 1345 | |
---|
| 1346 | nlemma low_bound_alloc_other: |
---|
[124] | 1347 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1348 | ∀b'. valid_block m1 b' → low_bound m2 b' = low_bound m1 b'. |
---|
[124] | 1349 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1350 | #b'; nrewrite > (low_bound_alloc … ALLOC b'); |
---|
| 1351 | napply eqZb_elim; #Hb; |
---|
| 1352 | ##[ nrewrite > Hb; #bad; napply False_ind; napply (absurd ? bad); |
---|
| 1353 | napply (fresh_block_alloc … ALLOC) |
---|
| 1354 | ##| // |
---|
| 1355 | ##] nqed. |
---|
| 1356 | |
---|
| 1357 | nlemma high_bound_alloc: |
---|
[124] | 1358 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1359 | ∀b'. high_bound m2 b' = if eqZb b' b then hi else high_bound m1 b'. |
---|
[124] | 1360 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1361 | nwhd in ALLOC:(??%%); ndestruct; #b'; |
---|
| 1362 | nrewrite > (unfold_update block_contents ????); |
---|
| 1363 | ncases (eqZb b' (nextblock m1)); //; |
---|
| 1364 | nqed. |
---|
| 1365 | |
---|
| 1366 | nlemma high_bound_alloc_same: |
---|
[124] | 1367 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1368 | high_bound m2 b = hi. |
---|
[124] | 1369 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1370 | nrewrite > (high_bound_alloc … ALLOC b); |
---|
| 1371 | nrewrite > (eqZb_z_z …); |
---|
| 1372 | //; |
---|
| 1373 | nqed. |
---|
| 1374 | |
---|
| 1375 | nlemma high_bound_alloc_other: |
---|
[124] | 1376 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[3] | 1377 | ∀b'. valid_block m1 b' → high_bound m2 b' = high_bound m1 b'. |
---|
[124] | 1378 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[3] | 1379 | #b'; nrewrite > (high_bound_alloc … ALLOC b'); |
---|
| 1380 | napply eqZb_elim; #Hb; |
---|
| 1381 | ##[ nrewrite > Hb; #bad; napply False_ind; napply (absurd ? bad); |
---|
| 1382 | napply (fresh_block_alloc … ALLOC) |
---|
| 1383 | ##| // |
---|
| 1384 | ##] nqed. |
---|
| 1385 | |
---|
[124] | 1386 | nlemma class_alloc: |
---|
| 1387 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1388 | ∀b'.block_space m2 b' = if eqZb b' b then bcl else block_space m1 b'. |
---|
[124] | 1389 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
| 1390 | nwhd in ALLOC:(??%%); ndestruct; #b'; |
---|
| 1391 | ncases (eqZb b' (nextblock m1)); //; |
---|
| 1392 | nqed. |
---|
| 1393 | |
---|
| 1394 | nlemma class_alloc_same: |
---|
| 1395 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1396 | block_space m2 b = bcl. |
---|
[124] | 1397 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
| 1398 | nwhd in ALLOC:(??%%); ndestruct; |
---|
| 1399 | nrewrite > (eqZb_z_z ?); //; |
---|
| 1400 | nqed. |
---|
| 1401 | |
---|
| 1402 | nlemma class_alloc_other: |
---|
| 1403 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1404 | ∀b'. valid_block m1 b' → block_space m2 b' = block_space m1 b'. |
---|
[124] | 1405 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
| 1406 | #b'; nrewrite > (class_alloc … ALLOC b'); |
---|
| 1407 | napply eqZb_elim; #Hb; |
---|
| 1408 | ##[ nrewrite > Hb; #bad; napply False_ind; napply (absurd ? bad); |
---|
| 1409 | napply (fresh_block_alloc … ALLOC) |
---|
| 1410 | ##| // |
---|
| 1411 | ##] nqed. |
---|
| 1412 | |
---|
[3] | 1413 | nlemma valid_access_alloc_other: |
---|
[124] | 1414 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1415 | ∀chunk,psp,b',ofs. |
---|
| 1416 | valid_access m1 chunk psp b' ofs → |
---|
| 1417 | valid_access m2 chunk psp b' ofs. |
---|
[124] | 1418 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1419 | #chunk;#psp;#b';#ofs;#H; |
---|
[124] | 1420 | ncases H; #Hvalid; #Hlow; #Hhigh;#Halign;#Hcompat; |
---|
[3] | 1421 | @; |
---|
| 1422 | ##[ napply (valid_block_alloc … ALLOC); // |
---|
| 1423 | ##| nrewrite > (low_bound_alloc_other … ALLOC b' Hvalid); // |
---|
| 1424 | ##| nrewrite > (high_bound_alloc_other … ALLOC b' Hvalid); // |
---|
| 1425 | ##| // |
---|
[124] | 1426 | ##| nrewrite > (class_alloc_other … ALLOC b' Hvalid); //; |
---|
[3] | 1427 | ##] nqed. |
---|
| 1428 | |
---|
| 1429 | nlemma valid_access_alloc_same: |
---|
[124] | 1430 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1431 | ∀chunk,psp,ofs. |
---|
[3] | 1432 | lo ≤ ofs → ofs + size_chunk chunk ≤ hi → (align_chunk chunk ∣ ofs) → |
---|
[125] | 1433 | pointer_compat bcl psp → |
---|
| 1434 | valid_access m2 chunk psp b ofs. |
---|
[124] | 1435 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1436 | #chunk;#psp;#ofs; #Hlo; #Hhi; #Halign; #Hcompat; |
---|
[3] | 1437 | @; |
---|
| 1438 | ##[ napply (valid_new_block … ALLOC) |
---|
| 1439 | ##| nrewrite > (low_bound_alloc_same … ALLOC); // |
---|
| 1440 | ##| nrewrite > (high_bound_alloc_same … ALLOC); // |
---|
| 1441 | ##| // |
---|
[124] | 1442 | ##| nrewrite > (class_alloc_same … ALLOC); // |
---|
[3] | 1443 | ##] nqed. |
---|
| 1444 | |
---|
| 1445 | (* |
---|
| 1446 | Hint Resolve valid_access_alloc_other valid_access_alloc_same: mem. |
---|
| 1447 | *) |
---|
| 1448 | |
---|
| 1449 | nlemma valid_access_alloc_inv: |
---|
[124] | 1450 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1451 | ∀chunk,psp,b',ofs. |
---|
| 1452 | valid_access m2 chunk psp b' ofs → |
---|
| 1453 | valid_access m1 chunk psp b' ofs ∨ |
---|
| 1454 | (b' = b ∧ lo ≤ ofs ∧ (ofs + size_chunk chunk ≤ hi ∧ (align_chunk chunk ∣ ofs) ∧ pointer_compat bcl psp)). |
---|
[124] | 1455 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1456 | #chunk;#psp;#b';#ofs;*;#Hblk;#Hlo;#Hhi;#Hal;#Hct; |
---|
[3] | 1457 | nelim (valid_block_alloc_inv … ALLOC ? Hblk);#H; |
---|
| 1458 | ##[ nrewrite < H in ALLOC ⊢ %; #ALLOC'; |
---|
| 1459 | nrewrite > (low_bound_alloc_same … ALLOC') in Hlo; #Hlo'; |
---|
| 1460 | nrewrite > (high_bound_alloc_same … ALLOC') in Hhi; #Hhi'; |
---|
[124] | 1461 | nrewrite > (class_alloc_same … ALLOC') in Hct; #Hct; |
---|
| 1462 | @2; /4/; |
---|
[3] | 1463 | ##| @1;@;//; |
---|
| 1464 | ##[ nrewrite > (low_bound_alloc_other … ALLOC ??) in Hlo; // |
---|
| 1465 | ##| nrewrite > (high_bound_alloc_other … ALLOC ??) in Hhi; // |
---|
[124] | 1466 | ##| nrewrite > (class_alloc_other … ALLOC ??) in Hct; // |
---|
[3] | 1467 | ##] |
---|
| 1468 | ##] nqed. |
---|
| 1469 | |
---|
| 1470 | ntheorem load_alloc_unchanged: |
---|
[124] | 1471 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo bcl hi = 〈m2,b〉 → |
---|
[125] | 1472 | ∀chunk,psp,b',ofs. |
---|
[3] | 1473 | valid_block m1 b' → |
---|
[125] | 1474 | load chunk m2 psp b' ofs = load chunk m1 psp b' ofs. |
---|
[124] | 1475 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1476 | #chunk;#psp;#b';#ofs;#H;nwhd in ⊢ (??%%); |
---|
| 1477 | ncases (in_bounds m2 chunk psp b' ofs); #H'; |
---|
[124] | 1478 | ##[ nelim (valid_access_alloc_inv … ALLOC ???? H'); |
---|
[3] | 1479 | ##[ #H''; (* XXX: if there's no hint that the result type is an option then the rewrite fails with an odd type error |
---|
| 1480 | nrewrite > (in_bounds_true ???? ??? H''); *) nrewrite > (in_bounds_true … ? (option val) ?? H''); |
---|
| 1481 | nwhd in ⊢ (??%?); (* XXX: if you do this at the point below the proof term is ill-typed. *) |
---|
[124] | 1482 | ncut (b' ≠ b); |
---|
| 1483 | ##[ napply (valid_not_valid_diff … H); napply (fresh_block_alloc … ALLOC); |
---|
| 1484 | ##| nwhd in ALLOC:(??%%); ndestruct; |
---|
| 1485 | nrewrite > (update_o block_contents ?????); /2/; |
---|
| 1486 | ##] |
---|
[3] | 1487 | ##| *;*;#A;#B;#C; nrewrite < A in ALLOC ⊢ %; #ALLOC; |
---|
| 1488 | napply False_ind; napply (absurd ? H); napply (fresh_block_alloc … ALLOC) |
---|
| 1489 | ##] |
---|
[125] | 1490 | ##| ncases (in_bounds m1 chunk psp b' ofs); #H''; nwhd in ⊢ (??%%); //; |
---|
[3] | 1491 | napply False_ind; napply (absurd ? ? H'); napply (valid_access_alloc_other … ALLOC); // |
---|
| 1492 | ##] nqed. |
---|
| 1493 | |
---|
| 1494 | ntheorem load_alloc_other: |
---|
[124] | 1495 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1496 | ∀chunk,psp,b',ofs,v. |
---|
| 1497 | load chunk m1 psp b' ofs = Some ? v → |
---|
| 1498 | load chunk m2 psp b' ofs = Some ? v. |
---|
[124] | 1499 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1500 | #chunk;#psp;#b';#ofs;#v;#H; |
---|
[124] | 1501 | nrewrite < H; napply (load_alloc_unchanged … ALLOC ???); ncases (load_valid_access … H); //; |
---|
[3] | 1502 | nqed. |
---|
| 1503 | |
---|
| 1504 | ntheorem load_alloc_same: |
---|
[124] | 1505 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1506 | ∀chunk,psp,ofs,v. |
---|
| 1507 | load chunk m2 psp b ofs = Some ? v → |
---|
[3] | 1508 | v = Vundef. |
---|
[124] | 1509 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1510 | #chunk;#psp;#ofs;#v;#H; |
---|
[3] | 1511 | nelim (load_inv … H); #H0; #H1; nrewrite > H1; |
---|
| 1512 | nwhd in ALLOC:(??%%); (* XXX ndestruct; ??? *) napply (pairdisc_elim … ALLOC); |
---|
| 1513 | nwhd in ⊢ (??%% → ?);#e0;nrewrite < e0 in ⊢ (%→?); |
---|
| 1514 | nwhd in ⊢ (??%% → ?);#e1; |
---|
[124] | 1515 | nrewrite < e1; nrewrite < e0; nrewrite > (update_s ? ? (empty_block lo hi bcl) ?); |
---|
[3] | 1516 | nnormalize; ncases chunk; //; |
---|
| 1517 | nqed. |
---|
| 1518 | |
---|
| 1519 | ntheorem load_alloc_same': |
---|
[124] | 1520 | ∀m1,lo,hi,bcl,m2,b.alloc m1 lo hi bcl = 〈m2,b〉 → |
---|
[125] | 1521 | ∀chunk,psp,ofs. |
---|
[3] | 1522 | lo ≤ ofs → ofs + size_chunk chunk ≤ hi → (align_chunk chunk ∣ ofs) → |
---|
[125] | 1523 | pointer_compat bcl psp → |
---|
| 1524 | load chunk m2 psp b ofs = Some ? Vundef. |
---|
[124] | 1525 | #m1;#lo;#hi;#bcl;#m2;#b;#ALLOC; |
---|
[125] | 1526 | #chunk;#psp;#ofs;#Hlo;#Hhi;#Hal;#Hct; |
---|
| 1527 | ncut (∃v. load chunk m2 psp b ofs = Some ? v); |
---|
[124] | 1528 | ##[ napply valid_access_load; @; //; |
---|
| 1529 | ##[ napply (valid_new_block … ALLOC); |
---|
| 1530 | ##| nrewrite > (low_bound_alloc_same … ALLOC); // |
---|
[3] | 1531 | ##| nrewrite > (high_bound_alloc_same … ALLOC); // |
---|
[124] | 1532 | ##| nrewrite > (class_alloc_same … ALLOC); // |
---|
[3] | 1533 | ##] |
---|
[190] | 1534 | ##| *; #v;#LOAD; nrewrite > LOAD; napply (eq_f … (Some val)); |
---|
[3] | 1535 | napply (load_alloc_same … ALLOC … LOAD); |
---|
| 1536 | ##] nqed. |
---|
| 1537 | |
---|
| 1538 | (* |
---|
| 1539 | End ALLOC. |
---|
| 1540 | |
---|
| 1541 | Hint Resolve valid_block_alloc fresh_block_alloc valid_new_block: mem. |
---|
| 1542 | Hint Resolve valid_access_alloc_other valid_access_alloc_same: mem. |
---|
| 1543 | Hint Resolve load_alloc_unchanged: mem. |
---|
| 1544 | |
---|
| 1545 | *) |
---|
| 1546 | |
---|
| 1547 | (* ** Properties related to [free]. *) |
---|
| 1548 | (* |
---|
| 1549 | Section FREE. |
---|
| 1550 | |
---|
| 1551 | Variable m: mem. |
---|
| 1552 | Variable bf: block. |
---|
| 1553 | *) |
---|
| 1554 | |
---|
| 1555 | nlemma valid_block_free_1: |
---|
| 1556 | ∀m,bf,b. valid_block m b → valid_block (free m bf) b. |
---|
| 1557 | nnormalize;//; |
---|
| 1558 | nqed. |
---|
| 1559 | |
---|
| 1560 | nlemma valid_block_free_2: |
---|
| 1561 | ∀m,bf,b. valid_block (free m bf) b → valid_block m b. |
---|
| 1562 | nnormalize;//; |
---|
| 1563 | nqed. |
---|
| 1564 | |
---|
| 1565 | (* |
---|
| 1566 | Hint Resolve valid_block_free_1 valid_block_free_2: mem. |
---|
| 1567 | *) |
---|
| 1568 | |
---|
| 1569 | nlemma low_bound_free: |
---|
| 1570 | ∀m,bf,b. b ≠ bf -> low_bound (free m bf) b = low_bound m b. |
---|
| 1571 | #m;#bf;#b;#H;nwhd in ⊢ (??%%); nwhd in ⊢ (??(?(?%?))?); |
---|
| 1572 | nrewrite > (update_o block_contents …); //; napply sym_neq; //; |
---|
| 1573 | nqed. |
---|
| 1574 | |
---|
| 1575 | nlemma high_bound_free: |
---|
| 1576 | ∀m,bf,b. b ≠ bf → high_bound (free m bf) b = high_bound m b. |
---|
| 1577 | #m;#bf;#b;#H;nwhd in ⊢ (??%%); nwhd in ⊢ (??(?(?%?))?); |
---|
| 1578 | nrewrite > (update_o block_contents …); //; napply sym_neq; //; |
---|
| 1579 | nqed. |
---|
| 1580 | |
---|
| 1581 | nlemma low_bound_free_same: |
---|
| 1582 | ∀m,b. low_bound (free m b) b = 0. |
---|
| 1583 | #m;#b;nwhd in ⊢ (??%%); nwhd in ⊢ (??(?(?%?))?); |
---|
| 1584 | nrewrite > (update_s block_contents …); //; |
---|
| 1585 | nqed. |
---|
| 1586 | |
---|
| 1587 | nlemma high_bound_free_same: |
---|
| 1588 | ∀m,b. high_bound (free m b) b = 0. |
---|
| 1589 | #m;#b;nwhd in ⊢ (??%%); nwhd in ⊢ (??(?(?%?))?); |
---|
| 1590 | nrewrite > (update_s block_contents …); //; |
---|
| 1591 | nqed. |
---|
| 1592 | |
---|
[124] | 1593 | nlemma class_free: |
---|
[125] | 1594 | ∀m,bf,b. b ≠ bf → block_space (free m bf) b = block_space m b. |
---|
[124] | 1595 | #m;#bf;#b;#H;nwhd in ⊢ (??%%); nwhd in ⊢ (??(?(?%?))?); |
---|
| 1596 | nrewrite > (update_o block_contents …); //; napply sym_neq; //; |
---|
| 1597 | nqed. |
---|
| 1598 | |
---|
[3] | 1599 | nlemma valid_access_free_1: |
---|
[125] | 1600 | ∀m,bf,chunk,psp,b,ofs. |
---|
| 1601 | valid_access m chunk psp b ofs → b ≠ bf → |
---|
| 1602 | valid_access (free m bf) chunk psp b ofs. |
---|
| 1603 | #m;#bf;#chunk;#psp b ofs;*;#Hval;#Hlo;#Hhi;#Hal;#Hcompat;#Hneq; |
---|
[3] | 1604 | @; //; |
---|
| 1605 | ##[ napply valid_block_free_1; // |
---|
| 1606 | ##| nrewrite > (low_bound_free … Hneq); // |
---|
| 1607 | ##| nrewrite > (high_bound_free … Hneq); // |
---|
[124] | 1608 | ##| nrewrite > (class_free … Hneq); // |
---|
[3] | 1609 | ##] nqed. |
---|
| 1610 | |
---|
| 1611 | nlemma valid_access_free_2: |
---|
[125] | 1612 | ∀m,psp,bf,chunk,ofs. ¬(valid_access (free m bf) chunk psp bf ofs). |
---|
| 1613 | #m;#psp;#bf;#chunk;#ofs; napply nmk; *; #Hval;#Hlo;#Hhi;#Hal;#Hct; |
---|
[3] | 1614 | nwhd in Hlo:(?%?);nwhd in Hlo:(?(?(?%?))?); nrewrite > (update_s block_contents …) in Hlo; |
---|
| 1615 | nwhd in Hhi:(??%);nwhd in Hhi:(??(?(?%?))); nrewrite > (update_s block_contents …) in Hhi; |
---|
| 1616 | nwhd in ⊢ ((??%)→(?%?)→?); (* arith *) napply daemon; |
---|
| 1617 | nqed. |
---|
| 1618 | |
---|
| 1619 | (* |
---|
| 1620 | Hint Resolve valid_access_free_1 valid_access_free_2: mem. |
---|
| 1621 | *) |
---|
| 1622 | |
---|
| 1623 | nlemma valid_access_free_inv: |
---|
[125] | 1624 | ∀m,bf,chunk,psp,b,ofs. |
---|
| 1625 | valid_access (free m bf) chunk psp b ofs → |
---|
| 1626 | valid_access m chunk psp b ofs ∧ b ≠ bf. |
---|
| 1627 | #m;#bf;#chunk;#psp b ofs; nelim (decidable_eq_Z_Type b bf); |
---|
[3] | 1628 | ##[ #e; nrewrite > e; |
---|
| 1629 | #H; napply False_ind; napply (absurd ? H (valid_access_free_2 …)); |
---|
| 1630 | ##| #ne; *; nrewrite > (low_bound_free … ne); |
---|
| 1631 | nrewrite > (high_bound_free … ne); |
---|
[124] | 1632 | nrewrite > (class_free … ne); |
---|
| 1633 | #Hval; #Hlo; #Hhi; #Hal; #Hct; |
---|
| 1634 | @; ##[ @; /2/; ##| /2/ ##] |
---|
[3] | 1635 | ##] nqed. |
---|
| 1636 | |
---|
| 1637 | ntheorem load_free: |
---|
[125] | 1638 | ∀m,bf,chunk,psp,b,ofs. |
---|
[3] | 1639 | b ≠ bf → |
---|
[125] | 1640 | load chunk (free m bf) psp b ofs = load chunk m psp b ofs. |
---|
| 1641 | #m;#bf;#chunk;#psp b ofs;#ne; nwhd in ⊢ (??%%); |
---|
| 1642 | nelim (in_bounds m chunk psp b ofs); |
---|
[124] | 1643 | ##[ #Hval; nwhd in ⊢ (???%); nrewrite > (in_bounds_true ????? (option val) ???); |
---|
[3] | 1644 | ##[ nwhd in ⊢ (??(??(??(???(?(?%?)))))?); nrewrite > (update_o block_contents …); //; |
---|
| 1645 | napply sym_neq; // |
---|
| 1646 | ##| napply valid_access_free_1; //; napply ne; |
---|
| 1647 | ##] |
---|
[125] | 1648 | ##| nelim (in_bounds (free m bf) chunk psp b ofs); (* XXX just // used to work *) ##[ ##2: nnormalize; //; ##] |
---|
[3] | 1649 | #H;#H'; napply False_ind; napply (absurd ? ? H'); |
---|
| 1650 | nelim (valid_access_free_inv …H); //; |
---|
| 1651 | ##] nqed. |
---|
| 1652 | |
---|
| 1653 | (* |
---|
| 1654 | End FREE. |
---|
| 1655 | *) |
---|
| 1656 | |
---|
| 1657 | (* ** Properties related to [free_list] *) |
---|
| 1658 | |
---|
| 1659 | nlemma valid_block_free_list_1: |
---|
| 1660 | ∀bl,m,b. valid_block m b → valid_block (free_list m bl) b. |
---|
| 1661 | #bl;nelim bl; |
---|
| 1662 | ##[ #m;#b;#H; nwhd in ⊢ (?%?); // |
---|
| 1663 | ##| #h;#t;#IH;#m;#b;#H; nrewrite > (unfold_free_list m h t); |
---|
| 1664 | napply valid_block_free_1; napply IH; // |
---|
| 1665 | ##] nqed. |
---|
| 1666 | |
---|
| 1667 | nlemma valid_block_free_list_2: |
---|
| 1668 | ∀bl,m,b. valid_block (free_list m bl) b → valid_block m b. |
---|
| 1669 | #bl; nelim bl; |
---|
| 1670 | ##[ #m;#b;#H; nwhd in H:(?%?); // |
---|
| 1671 | ##| #h;#t;#IH;#m;#b; nrewrite > (unfold_free_list m h t);#H; |
---|
| 1672 | napply IH; napply valid_block_free_2; // |
---|
| 1673 | ##] nqed. |
---|
| 1674 | |
---|
| 1675 | nlemma valid_access_free_list: |
---|
[125] | 1676 | ∀chunk,psp,b,ofs,m,bl. |
---|
| 1677 | valid_access m chunk psp b ofs → ¬in_list ? b bl → |
---|
| 1678 | valid_access (free_list m bl) chunk psp b ofs. |
---|
| 1679 | #chunk; #psp b ofs; #m; #bl; nelim bl; |
---|
[124] | 1680 | ##[ nwhd in ⊢ (?→?→(?%????)); // |
---|
[3] | 1681 | ##| #h;#t;#IH;#H;#notin; nrewrite > (unfold_free_list m h t); napply valid_access_free_1; |
---|
| 1682 | ##[ napply IH; //; napply (not_to_not ??? notin); #Ht; napply (in_list_cons … Ht) |
---|
| 1683 | ##| napply nmk; #e; napply (absurd ?? notin); nrewrite > e; // ##] |
---|
| 1684 | ##] nqed. |
---|
| 1685 | |
---|
| 1686 | nlemma valid_access_free_list_inv: |
---|
[125] | 1687 | ∀chunk,psp,b,ofs,m,bl. |
---|
| 1688 | valid_access (free_list m bl) chunk psp b ofs → |
---|
| 1689 | ¬in_list ? b bl ∧ valid_access m chunk psp b ofs. |
---|
| 1690 | #chunk; #psp b ofs; #m; #bl; nelim bl; |
---|
[124] | 1691 | ##[ nwhd in ⊢ ((?%????)→?); #H; @; // |
---|
[3] | 1692 | ##| #h;#t;#IH; nrewrite > (unfold_free_list m h t); #H; |
---|
| 1693 | nelim (valid_access_free_inv … H); #H';#ne; |
---|
| 1694 | nelim (IH H'); #notin;#H''; @; //; |
---|
| 1695 | napply (not_to_not ??? notin); #Ht; |
---|
| 1696 | (* WTF? this is specialised to nat! napply (in_list_tail t b h); *) napply daemon |
---|
| 1697 | ##] nqed. |
---|
| 1698 | |
---|
| 1699 | (* ** Properties related to pointer validity *) |
---|
| 1700 | |
---|
| 1701 | nlemma valid_pointer_valid_access: |
---|
[125] | 1702 | ∀m,psp,b,ofs. |
---|
| 1703 | valid_pointer m psp b ofs = true ↔ valid_access m Mint8unsigned psp b ofs. |
---|
| 1704 | #m;#psp b ofs;nwhd in ⊢ (?(??%?)?); @; |
---|
[3] | 1705 | ##[ #H; |
---|
| 1706 | nlapply (andb_true_l … H); #H'; |
---|
[124] | 1707 | nlapply (andb_true_l … H'); #H''; |
---|
| 1708 | nlapply (andb_true_l … H''); #H1; |
---|
| 1709 | nlapply (andb_true_r … H''); #H2; |
---|
| 1710 | nlapply (andb_true_r … H'); #H3; |
---|
| 1711 | nlapply (andb_true_r … H); #H4; |
---|
[3] | 1712 | @; |
---|
| 1713 | ##[ nrewrite > (unfold_valid_block m b); napply (Zltb_true_to_Zlt … H1) |
---|
| 1714 | ##| napply (Zleb_true_to_Zle … H2) |
---|
| 1715 | ##| nwhd in ⊢ (?(??%)?); (* arith, Zleb_true_to_Zle *) napply daemon |
---|
[10] | 1716 | ##| ncases ofs; /2/; |
---|
[125] | 1717 | ##| nwhd in H4:(??%?); nelim (pointer_compat_dec (block_space m b) psp) in H4; |
---|
[124] | 1718 | ##[ //; ##| #Hn e; nwhd in e:(??%%); ndestruct ##] |
---|
[3] | 1719 | ##] |
---|
[124] | 1720 | ##| *; #Hval;#Hlo;#Hhi;#Hal;#Hct; |
---|
[3] | 1721 | nrewrite > (Zlt_to_Zltb_true … Hval); |
---|
| 1722 | nrewrite > (Zle_to_Zleb_true … Hlo); |
---|
| 1723 | nwhd in Hhi:(?(??%)?); nrewrite > (Zlt_to_Zltb_true … ?); |
---|
[125] | 1724 | ##[ nwhd in ⊢ (??%?); nelim (pointer_compat_dec (block_space m b) psp); |
---|
[124] | 1725 | ##[ //; |
---|
| 1726 | ##| #Hct'; napply False_ind; napply (absurd … Hct Hct'); |
---|
| 1727 | ##] |
---|
| 1728 | ##| (* arith *) napply daemon |
---|
| 1729 | ##] |
---|
| 1730 | ##] |
---|
| 1731 | nqed. |
---|
[3] | 1732 | |
---|
| 1733 | ntheorem valid_pointer_alloc: |
---|
[125] | 1734 | ∀m1,m2: mem. ∀lo,hi: Z. ∀bcl,psp. ∀b,b': block. ∀ofs: Z. |
---|
[124] | 1735 | alloc m1 lo hi bcl = 〈m2, b'〉 → |
---|
[125] | 1736 | valid_pointer m1 psp b ofs = true → |
---|
| 1737 | valid_pointer m2 psp b ofs = true. |
---|
| 1738 | #m1;#m2;#lo;#hi;#bcl psp;#b;#b';#ofs;#ALLOC;#VALID; |
---|
[124] | 1739 | nlapply ((proj1 ?? (valid_pointer_valid_access ????)) VALID); #Hval; |
---|
| 1740 | napply (proj2 ?? (valid_pointer_valid_access ????)); |
---|
[3] | 1741 | napply (valid_access_alloc_other … ALLOC … Hval); |
---|
| 1742 | nqed. |
---|
| 1743 | |
---|
| 1744 | ntheorem valid_pointer_store: |
---|
[124] | 1745 | ∀chunk: memory_chunk. ∀m1,m2: mem. |
---|
[480] | 1746 | ∀psp,psp': region. ∀b,b': block. ∀ofs,ofs': Z. ∀v: val. |
---|
[125] | 1747 | store chunk m1 psp' b' ofs' v = Some ? m2 → |
---|
| 1748 | valid_pointer m1 psp b ofs = true → valid_pointer m2 psp b ofs = true. |
---|
| 1749 | #chunk;#m1;#m2;#psp psp';#b;#b';#ofs;#ofs';#v;#STORE;#VALID; |
---|
[124] | 1750 | nlapply ((proj1 ?? (valid_pointer_valid_access ????)) VALID); #Hval; |
---|
| 1751 | napply (proj2 ?? (valid_pointer_valid_access ????)); |
---|
[3] | 1752 | napply (store_valid_access_1 … STORE … Hval); |
---|
| 1753 | nqed. |
---|
| 1754 | |
---|
| 1755 | (* * * Generic injections between memory states. *) |
---|
[124] | 1756 | (* |
---|
[3] | 1757 | (* Section GENERIC_INJECT. *) |
---|
| 1758 | |
---|
| 1759 | ndefinition meminj : Type ≝ block → option (block × Z). |
---|
| 1760 | (* |
---|
| 1761 | Variable val_inj: meminj -> val -> val -> Prop. |
---|
| 1762 | |
---|
| 1763 | Hypothesis val_inj_undef: |
---|
| 1764 | ∀mi. val_inj mi Vundef Vundef. |
---|
| 1765 | *) |
---|
| 1766 | |
---|
| 1767 | ndefinition mem_inj ≝ λval_inj.λmi: meminj. λm1,m2: mem. |
---|
| 1768 | ∀chunk, b1, ofs, v1, b2, delta. |
---|
| 1769 | mi b1 = Some ? 〈b2, delta〉 → |
---|
| 1770 | load chunk m1 b1 ofs = Some ? v1 → |
---|
| 1771 | ∃v2. load chunk m2 b2 (ofs + delta) = Some ? v2 ∧ val_inj mi v1 v2. |
---|
| 1772 | |
---|
| 1773 | (* FIXME: another nunfold hack*) |
---|
| 1774 | nlemma unfold_mem_inj : ∀val_inj.∀mi: meminj. ∀m1,m2: mem. |
---|
| 1775 | (mem_inj val_inj mi m1 m2) = |
---|
| 1776 | (∀chunk, b1, ofs, v1, b2, delta. |
---|
| 1777 | mi b1 = Some ? 〈b2, delta〉 → |
---|
| 1778 | load chunk m1 b1 ofs = Some ? v1 → |
---|
| 1779 | ∃v2. load chunk m2 b2 (ofs + delta) = Some ? v2 ∧ val_inj mi v1 v2). |
---|
| 1780 | //; nqed. |
---|
| 1781 | |
---|
| 1782 | nlemma valid_access_inj: ∀val_inj. |
---|
| 1783 | ∀mi,m1,m2,chunk,b1,ofs,b2,delta. |
---|
| 1784 | mi b1 = Some ? 〈b2, delta〉 → |
---|
| 1785 | mem_inj val_inj mi m1 m2 → |
---|
| 1786 | valid_access m1 chunk b1 ofs → |
---|
| 1787 | valid_access m2 chunk b2 (ofs + delta). |
---|
| 1788 | #val_inj; |
---|
| 1789 | #mi;#m1;#m2;#chunk;#b1;#ofs;#b2;#delta;#H;#Hinj;#Hval; |
---|
| 1790 | ncut (∃v1. load chunk m1 b1 ofs = Some ? v1); |
---|
| 1791 | ##[ /2/; |
---|
| 1792 | ##| *;#v1;#LOAD1; |
---|
| 1793 | nelim (Hinj … H LOAD1);#v2;*;#LOAD2;#VCP; |
---|
| 1794 | /2/ |
---|
| 1795 | ##] nqed. |
---|
| 1796 | |
---|
| 1797 | (*Hint Resolve valid_access_inj: mem.*) |
---|
[124] | 1798 | *) |
---|
[3] | 1799 | (* FIXME: can't use ndestruct below *) |
---|
| 1800 | nlemma grumpydestruct : ∀A,v. None A = Some A v → False. |
---|
| 1801 | #A;#v;#H;ndestruct; |
---|
| 1802 | nqed. |
---|
[124] | 1803 | (* |
---|
[3] | 1804 | nlemma store_unmapped_inj: ∀val_inj. |
---|
[125] | 1805 | ∀mi,m1,m2,psp,b,ofs,v,chunk,m1'. |
---|
[3] | 1806 | mem_inj val_inj mi m1 m2 → |
---|
| 1807 | mi b = None ? → |
---|
[125] | 1808 | store chunk m1 psp b ofs v = Some ? m1' → |
---|
[3] | 1809 | mem_inj val_inj mi m1' m2. |
---|
| 1810 | #val_inj; |
---|
[125] | 1811 | #mi;#m1;#m2;#psp b ofs;#v;#chunk;#m1';#Hinj;#Hmi;#Hstore; |
---|
[3] | 1812 | nwhd; #chunk0;#b1;#ofs0;#v1;#b2;#delta; #Hmi0; #Hload; |
---|
| 1813 | ncut (load chunk0 m1 b1 ofs0 = Some ? v1); |
---|
| 1814 | ##[ nrewrite < Hload; napply sym_eq; napply (load_store_other … Hstore); |
---|
[15] | 1815 | @1;@1; napply nmk; #eq; nrewrite > eq in Hmi0; nrewrite > Hmi; #H; ndestruct; |
---|
[3] | 1816 | ##| #Hload'; napply (Hinj … Hmi0 Hload'); |
---|
| 1817 | ##] nqed. |
---|
| 1818 | |
---|
| 1819 | nlemma store_outside_inj: ∀val_inj. |
---|
[125] | 1820 | ∀mi,m1,m2,chunk,psp,b,ofs,v,m2'. |
---|
[3] | 1821 | mem_inj val_inj mi m1 m2 → |
---|
| 1822 | (∀b',delta. |
---|
| 1823 | mi b' = Some ? 〈b, delta〉 → |
---|
| 1824 | high_bound m1 b' + delta ≤ ofs |
---|
| 1825 | ∨ ofs + size_chunk chunk ≤ low_bound m1 b' + delta) → |
---|
[125] | 1826 | store chunk m2 psp b ofs v = Some ? m2' → |
---|
[3] | 1827 | mem_inj val_inj mi m1 m2'. |
---|
| 1828 | #val_inj; |
---|
[125] | 1829 | #mi;#m1;#m2;#chunk;#psp b ofs;#v;#m2';#Hinj;#Hbounds;#Hstore; |
---|
[3] | 1830 | nwhd; #chunk0;#b1;#ofs0;#v1;#b2;#delta; #Hmi0; #Hload; |
---|
| 1831 | nlapply (Hinj … Hmi0 Hload);*;#v2;*;#LOAD2;#VINJ; |
---|
| 1832 | @ v2;@;//; |
---|
| 1833 | nrewrite < LOAD2; napply (load_store_other … Hstore); |
---|
| 1834 | nelim (decidable_eq_Z b2 b); |
---|
| 1835 | ##[ #Heq; nrewrite > Heq in Hmi0 LOAD2; #Hmi0;#LOAD2; |
---|
| 1836 | nlapply (Hbounds … Hmi0); #Hb; |
---|
| 1837 | ncut (valid_access m1 chunk0 b1 ofs0); /2/; |
---|
| 1838 | #Hv; nelim Hv; #Hv1; #Hlo1; #Hhi1; #Hal1; |
---|
| 1839 | nelim Hb; #Hb; ##[ @1;@2;(* arith *) napply daemon ##| @2;(* arith *) napply daemon ##] |
---|
| 1840 | ##| #ineq; @1; @1; napply ineq; |
---|
| 1841 | ##] nqed. |
---|
[14] | 1842 | |
---|
| 1843 | (* XXX: use Or rather than ∨ to bring resource usage under control. *) |
---|
[3] | 1844 | ndefinition meminj_no_overlap ≝ λmi: meminj. λm: mem. |
---|
| 1845 | ∀b1,b1',delta1,b2,b2',delta2. |
---|
| 1846 | b1 ≠ b2 → |
---|
| 1847 | mi b1 = Some ? 〈b1', delta1〉 → |
---|
| 1848 | mi b2 = Some ? 〈b2', delta2〉 → |
---|
[14] | 1849 | Or (Or (Or (b1' ≠ b2') |
---|
| 1850 | (low_bound m b1 ≥ high_bound m b1)) |
---|
| 1851 | (low_bound m b2 ≥ high_bound m b2)) |
---|
| 1852 | (Or (high_bound m b1 + delta1 ≤ low_bound m b2 + delta2) |
---|
| 1853 | (high_bound m b2 + delta2 ≤ low_bound m b1 + delta1)). |
---|
[124] | 1854 | *) |
---|
[3] | 1855 | (* FIXME *) |
---|
| 1856 | nlemma grumpydestruct1 : ∀A,x1,x2. Some A x1 = Some A x2 → x1 = x2. |
---|
| 1857 | #A;#x1;#x2;#H;ndestruct;//; |
---|
| 1858 | nqed. |
---|
| 1859 | nlemma grumpydestruct2 : ∀A,B,x1,y1,x2,y2. Some (A×B) 〈x1,y1〉 = Some (A×B) 〈x2,y2〉 → x1 = x2 ∧ y1 = y2. |
---|
| 1860 | #A;#B;#x1;#y1;#x2;#y2;#H;ndestruct;/2/; |
---|
| 1861 | nqed. |
---|
[124] | 1862 | (* |
---|
[3] | 1863 | nlemma store_mapped_inj: ∀val_inj.∀val_inj_undef:∀mi. val_inj mi Vundef Vundef. |
---|
| 1864 | ∀mi,m1,m2,b1,ofs,b2,delta,v1,v2,chunk,m1'. |
---|
| 1865 | mem_inj val_inj mi m1 m2 → |
---|
| 1866 | meminj_no_overlap mi m1 → |
---|
| 1867 | mi b1 = Some ? 〈b2, delta〉 → |
---|
| 1868 | store chunk m1 b1 ofs v1 = Some ? m1' → |
---|
| 1869 | (∀chunk'. size_chunk chunk' = size_chunk chunk → |
---|
| 1870 | val_inj mi (load_result chunk' v1) (load_result chunk' v2)) → |
---|
| 1871 | ∃m2'. |
---|
| 1872 | store chunk m2 b2 (ofs + delta) v2 = Some ? m2' ∧ mem_inj val_inj mi m1' m2'. |
---|
| 1873 | #val_inj;#val_inj_undef; |
---|
| 1874 | #mi;#m1;#m2;#b1;#ofs;#b2;#delta;#v1;#v2;#chunk;#m1'; |
---|
| 1875 | #Hinj;#Hnoover;#Hb1;#STORE;#Hvalinj; |
---|
| 1876 | ncut (∃m2'.store chunk m2 b2 (ofs + delta) v2 = Some ? m2'); |
---|
| 1877 | ##[ napply valid_access_store; napply (valid_access_inj ? mi ??????? Hb1 Hinj ?); (* XXX why do I have to give mi here? *) /2/ |
---|
| 1878 | ##| *;#m2';#STORE2; |
---|
| 1879 | @ m2'; @; //; |
---|
| 1880 | nwhd; #chunk';#b1';#ofs';#v;#b2';#delta';#CP;#LOAD1; |
---|
| 1881 | ncut (valid_access m1 chunk' b1' ofs'); |
---|
| 1882 | ##[ napply (store_valid_access_2 … STORE); napply (load_valid_access … LOAD1); |
---|
| 1883 | ##| #Hval; |
---|
| 1884 | nlapply (load_store_characterization … STORE … Hval); |
---|
| 1885 | nelim (load_store_classification chunk b1 ofs chunk' b1' ofs'); |
---|
| 1886 | ##[ #e;#e0;#e1;#H; (* similar *) |
---|
| 1887 | nrewrite > e in Hb1; #Hb1; |
---|
[15] | 1888 | nrewrite > CP in Hb1; #Hb1; (* XXX ndestruct expands proof state too much;*) |
---|
[3] | 1889 | nelim (grumpydestruct2 ?????? Hb1); |
---|
| 1890 | #e2;#e3; nrewrite < e0; nrewrite > e2; nrewrite > e3; |
---|
| 1891 | @ (load_result chunk' v2);@; |
---|
| 1892 | ##[ napply (load_store_similar … STORE2); // |
---|
[15] | 1893 | ##| nrewrite > LOAD1 in H; #H; nwhd in H:(??%%); ndestruct; |
---|
| 1894 | napply Hvalinj; //; |
---|
[3] | 1895 | ##] |
---|
| 1896 | ##| #Hdis; #H; (* disjoint *) |
---|
| 1897 | nrewrite > LOAD1 in H; #H; |
---|
| 1898 | nlapply (Hinj … CP ?); ##[ napply sym_eq; napply H; ##| ##*: ##] |
---|
| 1899 | *;#v2';*;#LOAD2;#VCP; |
---|
| 1900 | @ v2'; @; //; |
---|
| 1901 | nrewrite < LOAD2; napply (load_store_other … STORE2); |
---|
| 1902 | nelim (decidable_eq_Z b1 b1'); #eb1; |
---|
[15] | 1903 | ##[ nrewrite < eb1 in CP; #CP; nrewrite > CP in Hb1; #eb2d; |
---|
[3] | 1904 | nelim (grumpydestruct2 ?????? eb2d); #eb2; #ed; |
---|
| 1905 | nelim Hdis; ##[ #Hdis; nelim Hdis; |
---|
| 1906 | ##[ #eb1'; napply False_ind; napply (absurd ? eb1 eb1'); |
---|
| 1907 | ##| #Hdis;@1;@2;(* arith *) napply daemon |
---|
| 1908 | ##] ##| #Hdis;@2;(* arith *) napply daemon ##] |
---|
| 1909 | ##| ncut (valid_access m1 chunk b1 ofs); /2/; #Hval'; |
---|
| 1910 | nlapply (Hnoover … eb1 Hb1 CP); |
---|
| 1911 | #Ha; nelim Ha; #Ha; |
---|
| 1912 | ##[ nelim Ha; #Ha; |
---|
| 1913 | ##[ nelim Ha; #Ha; |
---|
| 1914 | ##[ @1;@1;/2/ |
---|
| 1915 | ##| nelim Hval'; nlapply (size_chunk_pos chunk); (* arith *) napply daemon ##] |
---|
| 1916 | ##| nelim Hval; nlapply (size_chunk_pos chunk'); (* arith *) napply daemon ##] |
---|
| 1917 | ##| nelim Hval'; nelim Hval; (* arith *) napply daemon ##] |
---|
| 1918 | ##] |
---|
| 1919 | ##| #eb1;#Hofs1;#Hofs2;#Hofs3;#H; (* overlapping *) |
---|
| 1920 | nrewrite < eb1 in CP; #CP; nrewrite > CP in Hb1; #eb2d; |
---|
| 1921 | nelim (grumpydestruct2 ?????? eb2d); #eb2; #ed; |
---|
| 1922 | ncut (∃v2'. load chunk' m2' b2 (ofs' + delta) = Some ? v2'); |
---|
| 1923 | ##[ napply valid_access_load; napply (store_valid_access_1 … STORE2); napply (valid_access_inj … Hinj Hval); nrewrite > eb1; // |
---|
| 1924 | ##| *;#v2';#LOAD2'; |
---|
| 1925 | ncut (v2' = Vundef); ##[ napply (load_store_overlap … STORE2 … LOAD2'); (* arith *) napply daemon ##| ##] |
---|
| 1926 | #ev2'; @ v2'; @;//; |
---|
| 1927 | nrewrite > LOAD1 in H;#H; nwhd in H:(??%%); nrewrite > (grumpydestruct1 … H); |
---|
| 1928 | nrewrite > ev2'; |
---|
| 1929 | napply val_inj_undef; ##] |
---|
| 1930 | ##| #eb1;#Hofs;nrewrite < Hofs in Hval LOAD1 ⊢ %;#Hval;#LOAD1;#Hsize;#H; (* overlapping *) |
---|
| 1931 | |
---|
| 1932 | nrewrite < eb1 in CP; #CP; nrewrite > CP in Hb1; #eb2d; |
---|
| 1933 | nelim (grumpydestruct2 ?????? eb2d); #eb2; #ed; |
---|
| 1934 | ncut (∃v2'. load chunk' m2' b2 (ofs + delta) = Some ? v2'); |
---|
| 1935 | ##[ napply valid_access_load; napply (store_valid_access_1 … STORE2); napply (valid_access_inj … Hinj Hval); nrewrite > eb1; // |
---|
| 1936 | ##| *;#v2';#LOAD2'; |
---|
| 1937 | ncut (v2' = Vundef); ##[ napply (load_store_mismatch … STORE2 … LOAD2' ?); napply sym_neq;// ##| ##] |
---|
| 1938 | #ev2'; @ v2'; @;//; |
---|
| 1939 | nrewrite > LOAD1 in H;#H; nwhd in H:(??%%); nrewrite > (grumpydestruct1 … H); |
---|
| 1940 | nrewrite > ev2'; |
---|
| 1941 | napply val_inj_undef; ##] |
---|
| 1942 | ##] |
---|
| 1943 | ##] |
---|
| 1944 | ##] nqed. |
---|
| 1945 | |
---|
| 1946 | ndefinition inj_offset_aligned ≝ λdelta: Z. λsize: Z. |
---|
| 1947 | ∀chunk. size_chunk chunk ≤ size → (align_chunk chunk ∣ delta). |
---|
| 1948 | |
---|
| 1949 | nlemma alloc_parallel_inj: ∀val_inj.∀val_inj_undef:∀mi. val_inj mi Vundef Vundef. |
---|
| 1950 | ∀mi,m1,m2,lo1,hi1,m1',b1,lo2,hi2,m2',b2,delta. |
---|
| 1951 | mem_inj val_inj mi m1 m2 → |
---|
| 1952 | alloc m1 lo1 hi1 = 〈m1', b1〉 → |
---|
| 1953 | alloc m2 lo2 hi2 = 〈m2', b2〉 → |
---|
| 1954 | mi b1 = Some ? 〈b2, delta〉 → |
---|
| 1955 | lo2 ≤ lo1 + delta → hi1 + delta ≤ hi2 → |
---|
| 1956 | inj_offset_aligned delta (hi1 - lo1) → |
---|
| 1957 | mem_inj val_inj mi m1' m2'. |
---|
| 1958 | #val_inj;#val_inj_undef; |
---|
| 1959 | #mi;#m1;#m2;#lo1;#hi1;#m1';#b1;#lo2;#hi2;#m2';#b2;#delta; |
---|
| 1960 | #Hinj;#ALLOC1;#ALLOC2;#Hbinj;#Hlo;#Hhi;#Hal; |
---|
| 1961 | nwhd; #chunk;#b1';#ofs;#v;#b2';#delta';#Hbinj';#LOAD; |
---|
| 1962 | nlapply (valid_access_alloc_inv … m1 … ALLOC1 chunk b1' ofs ?); /2/; |
---|
| 1963 | *; |
---|
| 1964 | ##[ #A; |
---|
| 1965 | ncut (load chunk m1 b1' ofs = Some ? v); |
---|
| 1966 | ##[ nrewrite < LOAD; napply sym_eq; napply (load_alloc_unchanged … ALLOC1); /2/; ##] |
---|
| 1967 | #LOAD0; nlapply (Hinj … Hbinj' LOAD0); *;#v2;*;#LOAD2;#VINJ; |
---|
| 1968 | @ v2; @; |
---|
| 1969 | ##[ nrewrite < LOAD2; napply (load_alloc_unchanged … ALLOC2); |
---|
| 1970 | napply valid_access_valid_block; ##[ ##3: napply load_valid_access; ##] |
---|
| 1971 | // |
---|
| 1972 | ##| // |
---|
| 1973 | ##] |
---|
| 1974 | ##| *;*;#A;#B;#C; |
---|
| 1975 | nrewrite > A in Hbinj' LOAD; #Hbinj';#LOAD; |
---|
| 1976 | nrewrite > Hbinj in Hbinj'; #Hbinj'; nelim (grumpydestruct2 ?????? Hbinj'); |
---|
| 1977 | #eb2;#edelta; nrewrite < eb2; nrewrite < edelta; |
---|
| 1978 | ncut (v = Vundef); ##[ napply (load_alloc_same … ALLOC1 … LOAD); ##] |
---|
| 1979 | #ev; nrewrite > ev; |
---|
| 1980 | ncut (∃v2. load chunk m2' b2 (ofs + delta) = Some ? v2); |
---|
| 1981 | ##[ napply valid_access_load; |
---|
| 1982 | napply (valid_access_alloc_same … ALLOC2); |
---|
| 1983 | ##[ ##1,2: (*arith*) napply daemon; |
---|
| 1984 | ##| (* arith using Hal *) napply daemon |
---|
| 1985 | ##] ##] |
---|
| 1986 | *;#v2;#LOAD2; |
---|
| 1987 | ncut (v2 = Vundef); ##[ napply (load_alloc_same … ALLOC2 … LOAD2) ##] |
---|
| 1988 | #ev2; nrewrite > ev2; |
---|
| 1989 | @ Vundef; @; //; |
---|
| 1990 | ##] nqed. |
---|
| 1991 | |
---|
| 1992 | nlemma alloc_right_inj: ∀val_inj. |
---|
| 1993 | ∀mi,m1,m2,lo,hi,b2,m2'. |
---|
| 1994 | mem_inj val_inj mi m1 m2 → |
---|
| 1995 | alloc m2 lo hi = 〈m2', b2〉 → |
---|
| 1996 | mem_inj val_inj mi m1 m2'. |
---|
| 1997 | #val_inj; |
---|
| 1998 | #mi;#m1;#m2;#lo;#hi;#b2;#m2'; |
---|
| 1999 | #Hinj;#ALLOC; |
---|
| 2000 | nwhd; #chunk; #b1; #ofs; #v1; #b2'; #delta; #Hbinj; #LOAD; |
---|
| 2001 | nlapply (Hinj … Hbinj LOAD); *; #v2;*;#LOAD2;#VINJ; |
---|
| 2002 | @ v2; @; //; |
---|
| 2003 | ncut (valid_block m2 b2'); |
---|
| 2004 | ##[ napply (valid_access_valid_block ? chunk ? (ofs + delta)); /2/ ##] |
---|
| 2005 | #Hval; |
---|
| 2006 | nrewrite < LOAD2; napply (load_alloc_unchanged … ALLOC … Hval); |
---|
| 2007 | nqed. |
---|
| 2008 | |
---|
| 2009 | (* |
---|
| 2010 | Hypothesis val_inj_undef_any: |
---|
| 2011 | ∀mi,v. val_inj mi Vundef v. |
---|
| 2012 | *) |
---|
| 2013 | |
---|
| 2014 | nlemma alloc_left_unmapped_inj: ∀val_inj. |
---|
| 2015 | ∀mi,m1,m2,lo,hi,b1,m1'. |
---|
| 2016 | mem_inj val_inj mi m1 m2 → |
---|
| 2017 | alloc m1 lo hi = 〈m1', b1〉 → |
---|
| 2018 | mi b1 = None ? → |
---|
| 2019 | mem_inj val_inj mi m1' m2. |
---|
| 2020 | #val_inj; |
---|
| 2021 | #mi;#m1;#m2;#lo;#hi;#b1;#m1'; |
---|
| 2022 | #Hinj;#ALLOC;#Hbinj; |
---|
| 2023 | nwhd; #chunk; #b1'; #ofs; #v1; #b2'; #delta; #Hbinj'; #LOAD; |
---|
| 2024 | nlapply (valid_access_alloc_inv … m1 … ALLOC chunk b1' ofs ?); /2/; |
---|
| 2025 | *; |
---|
| 2026 | ##[ #A; |
---|
| 2027 | napply (Hinj … Hbinj' ); |
---|
| 2028 | nrewrite < LOAD; napply sym_eq; napply (load_alloc_unchanged … ALLOC); /2/; |
---|
| 2029 | ##| *;*;#A;#B;#C; |
---|
| 2030 | nrewrite > A in Hbinj' LOAD; #Hbinj'; #LOAD; |
---|
[15] | 2031 | nrewrite > Hbinj in Hbinj'; #bad; ndestruct; |
---|
[3] | 2032 | ##] nqed. |
---|
| 2033 | |
---|
| 2034 | nlemma alloc_left_mapped_inj: ∀val_inj.∀val_inj_undef_any:∀mi,v. val_inj mi Vundef v. |
---|
| 2035 | ∀mi,m1,m2,lo,hi,b1,m1',b2,delta. |
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| 2036 | mem_inj val_inj mi m1 m2 → |
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| 2037 | alloc m1 lo hi = 〈m1', b1〉 → |
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| 2038 | mi b1 = Some ? 〈b2, delta〉 → |
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| 2039 | valid_block m2 b2 → |
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| 2040 | low_bound m2 b2 ≤ lo + delta → hi + delta ≤ high_bound m2 b2 → |
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| 2041 | inj_offset_aligned delta (hi - lo) → |
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| 2042 | mem_inj val_inj mi m1' m2. |
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| 2043 | #val_inj;#val_inj_undef_any; |
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| 2044 | #mi;#m1;#m2;#lo;#hi;#b1;#m1';#b2;#delta; |
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| 2045 | #Hinj;#ALLOC;#Hbinj;#Hval;#Hlo;#Hhi;#Hal; |
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| 2046 | nwhd; #chunk; #b1'; #ofs; #v1; #b2'; #delta'; #Hbinj'; #LOAD; |
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| 2047 | nlapply (valid_access_alloc_inv … m1 … ALLOC chunk b1' ofs ?); /2/; |
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| 2048 | *; |
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| 2049 | ##[ #A; |
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| 2050 | napply (Hinj … Hbinj'); |
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| 2051 | nrewrite < LOAD; napply sym_eq; napply (load_alloc_unchanged … ALLOC); /2/; |
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| 2052 | ##| *;*;#A;#B;*;#C;#D; |
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| 2053 | nrewrite > A in Hbinj' LOAD; #Hbinj'; #LOAD; nrewrite > Hbinj in Hbinj'; |
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[15] | 2054 | #Hbinj'; (* XXX ndestruct normalizes too much here *) nelim (grumpydestruct2 ?????? Hbinj'); #eb2; #edelta; |
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[3] | 2055 | nrewrite < eb2; nrewrite < edelta; |
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| 2056 | ncut (v1 = Vundef); ##[ napply (load_alloc_same … ALLOC … LOAD) ##] |
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| 2057 | #ev1; nrewrite > ev1; |
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| 2058 | ncut (∃v2. load chunk m2 b2 (ofs + delta) = Some ? v2); |
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| 2059 | ##[ napply valid_access_load; @; //; |
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| 2060 | ##[ (* arith *) napply daemon |
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| 2061 | ##| (* arith *) napply daemon |
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| 2062 | ##| (* arith *) napply daemon |
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| 2063 | ##] |
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| 2064 | ##] |
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| 2065 | *;#v2;#LOAD2; @ v2; @; //; |
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| 2066 | ##] nqed. |
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| 2067 | |
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| 2068 | nlemma free_parallel_inj: ∀val_inj. |
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| 2069 | ∀mi,m1,m2,b1,b2,delta. |
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| 2070 | mem_inj val_inj mi m1 m2 → |
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| 2071 | mi b1 = Some ? 〈b2, delta〉 → |
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| 2072 | (∀b,delta'. mi b = Some ? 〈b2, delta'〉 → b = b1) → |
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| 2073 | mem_inj val_inj mi (free m1 b1) (free m2 b2). |
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| 2074 | #val_inj; |
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| 2075 | #mi;#m1;#m2;#b1;#b2;#delta; #Hinj;#Hb1inj;#Hbinj; |
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| 2076 | nwhd; #chunk; #b1'; #ofs; #v1; #b2'; #delta'; #Hbinj'; #LOAD; |
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| 2077 | nlapply (valid_access_free_inv … (load_valid_access … LOAD)); *; #A;#B; |
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| 2078 | ncut (load chunk m1 b1' ofs = Some ? v1); |
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| 2079 | ##[ nrewrite < LOAD; napply sym_eq; napply load_free; napply B ##] #LOAD'; |
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| 2080 | nelim (Hinj … Hbinj' LOAD'); #v2;*;#LOAD2;#INJ; |
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| 2081 | @ v2;@; |
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| 2082 | ##[ nrewrite < LOAD2; napply load_free; |
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| 2083 | napply nmk; #e; napply (absurd ?? B); |
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| 2084 | nrewrite > e in Hbinj'; #Hbinj'; napply (Hbinj ?? Hbinj'); |
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| 2085 | ##| // |
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| 2086 | ##] nqed. |
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| 2087 | |
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| 2088 | nlemma free_left_inj: ∀val_inj. |
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| 2089 | ∀mi,m1,m2,b1. |
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| 2090 | mem_inj val_inj mi m1 m2 → |
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| 2091 | mem_inj val_inj mi (free m1 b1) m2. |
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| 2092 | #val_inj;#mi;#m1;#m2;#b1;#Hinj; |
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| 2093 | nwhd; #chunk; #b1'; #ofs; #v1; #b2'; #delta'; #Hbinj'; #LOAD; |
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| 2094 | nlapply (valid_access_free_inv … (load_valid_access … LOAD)); *; #A;#B; |
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| 2095 | napply (Hinj … Hbinj'); |
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| 2096 | nrewrite < LOAD; napply sym_eq; napply load_free; napply B; |
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| 2097 | nqed. |
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| 2098 | |
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| 2099 | nlemma free_list_left_inj: ∀val_inj. |
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| 2100 | ∀mi,bl,m1,m2. |
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| 2101 | mem_inj val_inj mi m1 m2 → |
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| 2102 | mem_inj val_inj mi (free_list m1 bl) m2. |
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| 2103 | #val_inj;#mi;#bl;nelim bl; |
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| 2104 | ##[ nwhd in ⊢ (?→?→?→???%?); // |
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| 2105 | ##| #h;#t;#IH; #m1;#m2;#H; nrewrite > (unfold_free_list m1 h t); |
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| 2106 | napply free_left_inj; napply IH; // |
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| 2107 | ##] nqed. |
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| 2108 | |
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| 2109 | nlemma free_right_inj: ∀val_inj. |
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| 2110 | ∀mi,m1,m2,b2. |
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| 2111 | mem_inj val_inj mi m1 m2 → |
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| 2112 | (∀b1,delta,chunk,ofs. |
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| 2113 | mi b1 = Some ? 〈b2, delta〉 → ¬(valid_access m1 chunk b1 ofs)) → |
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| 2114 | mem_inj val_inj mi m1 (free m2 b2). |
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| 2115 | #val_inj;#mi;#m1;#m2;#b2; #Hinj; #Hinval; |
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| 2116 | nwhd; #chunk; #b1'; #ofs; #v1; #b2'; #delta'; #Hbinj'; #LOAD; |
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| 2117 | ncut (b2' ≠ b2); |
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| 2118 | ##[ napply nmk; #e; nrewrite > e in Hbinj'; #Hbinj'; |
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| 2119 | napply (absurd ?? (Hinval … Hbinj')); napply (load_valid_access … LOAD); ##] |
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| 2120 | #ne; nlapply (Hinj … Hbinj' LOAD); *;#v2;*;#LOAD2;#INJ; |
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| 2121 | @ v2; @; //; |
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| 2122 | nrewrite < LOAD2; napply load_free; napply ne; |
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| 2123 | nqed. |
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| 2124 | |
---|
| 2125 | nlemma valid_pointer_inj: ∀val_inj. |
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| 2126 | ∀mi,m1,m2,b1,ofs,b2,delta. |
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| 2127 | mi b1 = Some ? 〈b2, delta〉 → |
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| 2128 | mem_inj val_inj mi m1 m2 → |
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| 2129 | valid_pointer m1 b1 ofs = true → |
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| 2130 | valid_pointer m2 b2 (ofs + delta) = true. |
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| 2131 | #val_inj;#mi;#m1;#m2;#b1;#ofs;#b2;#delta;#Hbinj;#Hinj;#VALID; |
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| 2132 | nlapply ((proj1 ?? (valid_pointer_valid_access ???)) VALID); #Hval; |
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| 2133 | napply (proj2 ?? (valid_pointer_valid_access ???)); |
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| 2134 | napply (valid_access_inj … Hval); //; |
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| 2135 | nqed. |
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| 2136 | |
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| 2137 | (* |
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| 2138 | End GENERIC_INJECT. |
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| 2139 | *) |
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| 2140 | (* ** Store extensions *) |
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| 2141 | |
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| 2142 | (* A store [m2] extends a store [m1] if [m2] can be obtained from [m1] |
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| 2143 | by increasing the sizes of the memory blocks of [m1] (decreasing |
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| 2144 | the low bounds, increasing the high bounds), while still keeping the |
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| 2145 | same contents for block offsets that are valid in [m1]. *) |
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| 2146 | |
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| 2147 | ndefinition inject_id : meminj ≝ λb. Some ? 〈b, OZ〉. |
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| 2148 | |
---|
| 2149 | ndefinition val_inj_id ≝ λmi: meminj. λv1,v2: val. v1 = v2. |
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| 2150 | |
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| 2151 | ndefinition extends ≝ λm1,m2: mem. |
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| 2152 | nextblock m1 = nextblock m2 ∧ mem_inj val_inj_id inject_id m1 m2. |
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| 2153 | |
---|
| 2154 | ntheorem extends_refl: |
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| 2155 | ∀m: mem. extends m m. |
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| 2156 | #m;@;//; |
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[15] | 2157 | nwhd; #chunk;#b1;#ofs;#v1;#b2;#delta;nnormalize in ⊢ (%→?);#H; |
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| 2158 | (* XXX: ndestruct makes the goal unreadable *) nelim (grumpydestruct2 ?????? H); #eb1;#edelta;#LOAD; |
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[3] | 2159 | @ v1; @; |
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| 2160 | ##[ nrewrite < edelta; nrewrite > (Zplus_z_OZ ofs); //; |
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| 2161 | ##| // |
---|
| 2162 | ##] nqed. |
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| 2163 | |
---|
| 2164 | ntheorem alloc_extends: |
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| 2165 | ∀m1,m2,m1',m2': mem. ∀lo1,hi1,lo2,hi2: Z. ∀b1,b2: block. |
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| 2166 | extends m1 m2 → |
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| 2167 | lo2 ≤ lo1 → hi1 ≤ hi2 → |
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| 2168 | alloc m1 lo1 hi1 = 〈m1', b1〉 → |
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| 2169 | alloc m2 lo2 hi2 = 〈m2', b2〉 → |
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| 2170 | b1 = b2 ∧ extends m1' m2'. |
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| 2171 | #m1;#m2;#m1';#m2';#lo1;#hi1;#lo2;#hi2;#b1;#b2; |
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| 2172 | *;#Hnext;#Hinj;#Hlo;#Hhi;#ALLOC1;#ALLOC2; |
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| 2173 | ncut (b1 = b2); |
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| 2174 | ##[ napply (transitive_eq … (nextblock m1)); ##[ napply (alloc_result … ALLOC1); |
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| 2175 | ##| napply sym_eq; nrewrite > Hnext; napply (alloc_result … ALLOC2) ##] ##] |
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| 2176 | #eb; nrewrite < eb in ALLOC2 ⊢ %; #ALLOC2; @; //; @; |
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| 2177 | ##[ nrewrite > (nextblock_alloc … ALLOC1); |
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| 2178 | nrewrite > (nextblock_alloc … ALLOC2); |
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| 2179 | //; |
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| 2180 | ##| napply (alloc_parallel_inj ??????????????? ALLOC1 ALLOC2 ????); |
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| 2181 | ##[ ##1,4: nnormalize; //; |
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| 2182 | ##| ##3,5,6: // |
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| 2183 | ##| ##7: nwhd; #chunk;#Hsize; (* divides 0 *) napply daemon |
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| 2184 | ##] |
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| 2185 | ##] nqed. |
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| 2186 | |
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| 2187 | ntheorem free_extends: |
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| 2188 | ∀m1,m2: mem. ∀b: block. |
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| 2189 | extends m1 m2 → |
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| 2190 | extends (free m1 b) (free m2 b). |
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| 2191 | #m1;#m2;#b;*;#Hnext;#Hinj; @; |
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| 2192 | ##[ nnormalize; //; |
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| 2193 | ##| napply (free_parallel_inj … Hinj); |
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| 2194 | ##[ ##2: //; |
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[15] | 2195 | ##| ##3: nnormalize; #b';#delta;#ee; ndestruct; // |
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[3] | 2196 | ##] |
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| 2197 | ##] nqed. |
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| 2198 | |
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| 2199 | ntheorem load_extends: |
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| 2200 | ∀chunk: memory_chunk. ∀m1,m2: mem. ∀b: block. ∀ofs: Z. ∀v: val. |
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| 2201 | extends m1 m2 → |
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[125] | 2202 | load chunk m1 psp b ofs = Some ? v → |
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| 2203 | load chunk m2 psp b ofs = Some ? v. |
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| 2204 | #chunk;#m1;#m2;#psp b ofs;#v; |
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[3] | 2205 | *;#Hnext;#Hinj;#LOAD; |
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| 2206 | nlapply (Hinj … LOAD); ##[ nnormalize; // ##| ##2,3: ##] |
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| 2207 | *;#v2;*; nrewrite > (Zplus_z_OZ ofs); #LOAD2;#EQ;nwhd in EQ; |
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| 2208 | //; |
---|
| 2209 | nqed. |
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| 2210 | |
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| 2211 | ntheorem store_within_extends: |
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| 2212 | ∀chunk: memory_chunk. ∀m1,m2,m1': mem. ∀b: block. ∀ofs: Z. ∀v: val. |
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| 2213 | extends m1 m2 → |
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[125] | 2214 | store chunk m1 psp b ofs v = Some ? m1' → |
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| 2215 | ∃m2'. store chunk m2 psp b ofs v = Some ? m2' ∧ extends m1' m2'. |
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[3] | 2216 | #chunk;#m1;#m2;#m1';#b;#ofs;#v;*;#Hnext;#Hinj;#STORE1; |
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| 2217 | nlapply (store_mapped_inj … Hinj ?? STORE1 ?); |
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| 2218 | ##[ ##1,2,7: nnormalize; // |
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| 2219 | ##| (* TODO: unfolding, etc ought to tidy this up *) |
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| 2220 | nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2;#neb;#Hinj1;#Hinj2; |
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| 2221 | nnormalize in Hinj1 Hinj2; @1; @1; @1; ndestruct; // |
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| 2222 | ##| ##4,5,6: ##skip |
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| 2223 | ##] |
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| 2224 | *;#m2';*;#STORE;#MINJ; |
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| 2225 | @ m2'; @; nrewrite > (Zplus_z_OZ ofs) in STORE; #STORE; //; |
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| 2226 | @; |
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| 2227 | ##[ nrewrite > (nextblock_store … STORE1); |
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| 2228 | nrewrite > (nextblock_store … STORE); |
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| 2229 | // |
---|
| 2230 | ##| // |
---|
| 2231 | ##] nqed. |
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| 2232 | |
---|
| 2233 | ntheorem store_outside_extends: |
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| 2234 | ∀chunk: memory_chunk. ∀m1,m2,m2': mem. ∀b: block. ∀ofs: Z. ∀v: val. |
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| 2235 | extends m1 m2 → |
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| 2236 | ofs + size_chunk chunk ≤ low_bound m1 b ∨ high_bound m1 b ≤ ofs → |
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[125] | 2237 | store chunk m2 psp b ofs v = Some ? m2' → |
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[3] | 2238 | extends m1 m2'. |
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| 2239 | #chunk;#m1;#m2;#m2';#b;#ofs;#v;*;#Hnext;#Hinj;#Houtside;#STORE; @; |
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| 2240 | ##[ nrewrite > (nextblock_store … STORE); // |
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| 2241 | ##| napply (store_outside_inj … STORE); //; |
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| 2242 | #b';#delta;#einj;nnormalize in einj; ndestruct; |
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| 2243 | nelim Houtside; |
---|
[15] | 2244 | ##[ #lo;@ 2; nrewrite > (Zplus_z_OZ ?); /2/ |
---|
| 2245 | ##| #hi;@ 1; nrewrite > (Zplus_z_OZ ?); /2/ |
---|
[3] | 2246 | ##] |
---|
| 2247 | ##] nqed. |
---|
| 2248 | |
---|
| 2249 | ntheorem valid_pointer_extends: |
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| 2250 | ∀m1,m2,b,ofs. |
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[125] | 2251 | extends m1 m2 → valid_pointer m1 psp b ofs = true → |
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| 2252 | valid_pointer m2 psp b ofs = true. |
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[3] | 2253 | #m1;#m2;#b;#ofs;*;#Hnext;#Hinj;#VALID; |
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| 2254 | nrewrite < (Zplus_z_OZ ofs); |
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| 2255 | napply (valid_pointer_inj … Hinj VALID); //; |
---|
| 2256 | nqed. |
---|
| 2257 | |
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| 2258 | |
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| 2259 | (* * The ``less defined than'' relation over memory states *) |
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| 2260 | |
---|
| 2261 | (* A memory state [m1] is less defined than [m2] if, for all addresses, |
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| 2262 | the value [v1] read in [m1] at this address is less defined than |
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| 2263 | the value [v2] read in [m2], that is, either [v1 = v2] or [v1 = Vundef]. *) |
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| 2264 | |
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| 2265 | ndefinition val_inj_lessdef ≝ λmi: meminj. λv1,v2: val. |
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| 2266 | Val_lessdef v1 v2. |
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| 2267 | |
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| 2268 | ndefinition lessdef ≝ λm1,m2: mem. |
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| 2269 | nextblock m1 = nextblock m2 ∧ |
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| 2270 | mem_inj val_inj_lessdef inject_id m1 m2. |
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| 2271 | |
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| 2272 | nlemma lessdef_refl: |
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| 2273 | ∀m. lessdef m m. |
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| 2274 | #m; @; //; |
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| 2275 | nwhd; #chunk;#b1;#ofs;#v1;#b2;#delta;#H;#LOAD; |
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| 2276 | nwhd in H:(??%?); nelim (grumpydestruct2 ?????? H); #eb1; #edelta; |
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| 2277 | @ v1; @; //; |
---|
| 2278 | nqed. |
---|
| 2279 | |
---|
| 2280 | nlemma load_lessdef: |
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| 2281 | ∀m1,m2,chunk,b,ofs,v1. |
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[125] | 2282 | lessdef m1 m2 → load chunk m1 psp b ofs = Some ? v1 → |
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| 2283 | ∃v2. load chunk m2 psp b ofs = Some ? v2 ∧ Val_lessdef v1 v2. |
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[3] | 2284 | #m1;#m2;#chunk;#b;#ofs;#v1;*;#Hnext;#Hinj;#LOAD0; |
---|
| 2285 | nlapply (Hinj … LOAD0); ##[ nwhd in ⊢ (??%?); // ##| ##2,3:##skip ##] |
---|
| 2286 | *;#v2;*;#LOAD;#INJ; @ v2; @;//; |
---|
| 2287 | nqed. |
---|
| 2288 | |
---|
| 2289 | nlemma loadv_lessdef: |
---|
| 2290 | ∀m1,m2,chunk,addr1,addr2,v1. |
---|
| 2291 | lessdef m1 m2 → Val_lessdef addr1 addr2 → |
---|
| 2292 | loadv chunk m1 addr1 = Some ? v1 → |
---|
| 2293 | ∃v2. loadv chunk m2 addr2 = Some ? v2 ∧ Val_lessdef v1 v2. |
---|
| 2294 | #m1;#m2;#chunk;#addr1;#addr2;#v1;#H;#H0;#LOAD; |
---|
| 2295 | ninversion H0; |
---|
| 2296 | ##[ #v; #e1;#e2; nrewrite > e1 in LOAD; ncases v; |
---|
| 2297 | ##[ nwhd in ⊢ ((??%?)→?); #H'; ndestruct; |
---|
| 2298 | ##| ##2,3: #v'; nwhd in ⊢ ((??%?)→?); #H'; ndestruct; |
---|
| 2299 | ##| #b';#off; napply load_lessdef; // |
---|
| 2300 | ##] |
---|
[15] | 2301 | ##| #v;#e;nrewrite > e in LOAD; #LOAD; nwhd in LOAD:(??%?); ndestruct; |
---|
[3] | 2302 | ##] nqed. |
---|
| 2303 | |
---|
| 2304 | nlemma store_lessdef: |
---|
| 2305 | ∀m1,m2,chunk,b,ofs,v1,v2,m1'. |
---|
| 2306 | lessdef m1 m2 → Val_lessdef v1 v2 → |
---|
[125] | 2307 | store chunk m1 psp b ofs v1 = Some ? m1' → |
---|
| 2308 | ∃m2'. store chunk m2 psp b ofs v2 = Some ? m2' ∧ lessdef m1' m2'. |
---|
[3] | 2309 | #m1;#m2;#chunk;#b;#ofs;#v1;#v2;#m1'; |
---|
| 2310 | *;#Hnext;#Hinj;#Hvless;#STORE0; |
---|
| 2311 | nlapply (store_mapped_inj … Hinj … STORE0 ?); |
---|
| 2312 | ##[ #chunk';#Hsize;nwhd;napply load_result_lessdef;napply Hvless |
---|
| 2313 | ##| nwhd in ⊢ (??%?); // |
---|
| 2314 | ##| nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2;#neq; |
---|
| 2315 | nwhd in ⊢ ((??%?)→(??%?)→?); #e1; #e2; ndestruct; |
---|
| 2316 | @;@;@;// |
---|
| 2317 | ##| ##7: #mi; nwhd; //; |
---|
| 2318 | ##| ##8: *;#m2';*;#STORE;#MINJ; |
---|
| 2319 | @ m2';@; /2/; |
---|
| 2320 | @; |
---|
| 2321 | nrewrite > (nextblock_store … STORE0); |
---|
| 2322 | nrewrite > (nextblock_store … STORE); |
---|
| 2323 | //; |
---|
| 2324 | ##] nqed. |
---|
| 2325 | |
---|
| 2326 | nlemma storev_lessdef: |
---|
| 2327 | ∀m1,m2,chunk,addr1,v1,addr2,v2,m1'. |
---|
| 2328 | lessdef m1 m2 → Val_lessdef addr1 addr2 → Val_lessdef v1 v2 → |
---|
| 2329 | storev chunk m1 addr1 v1 = Some ? m1' → |
---|
| 2330 | ∃m2'. storev chunk m2 addr2 v2 = Some ? m2' ∧ lessdef m1' m2'. |
---|
| 2331 | #m1;#m2;#chunk;#addr1;#v1;#addr2;#v2;#m1'; |
---|
| 2332 | #Hmless;#Haless;#Hvless;#STORE; |
---|
| 2333 | ninversion Haless; |
---|
| 2334 | ##[ #v; #e1;#e2; nrewrite > e1 in STORE; ncases v; |
---|
| 2335 | ##[ nwhd in ⊢ ((??%?)→?); #H'; napply False_ind; ndestruct; |
---|
[15] | 2336 | ##| ##2,3: #v'; nwhd in ⊢ ((??%?)→?); #H'; ndestruct; |
---|
[3] | 2337 | ##| #b';#off; napply store_lessdef; // |
---|
| 2338 | ##] |
---|
[15] | 2339 | ##| #v;#e;nrewrite > e in STORE; #STORE; nwhd in STORE:(??%?); ndestruct |
---|
[3] | 2340 | ##] nqed. |
---|
| 2341 | |
---|
| 2342 | nlemma alloc_lessdef: |
---|
| 2343 | ∀m1,m2,lo,hi,b1,m1',b2,m2'. |
---|
| 2344 | lessdef m1 m2 → alloc m1 lo hi = 〈m1', b1〉 → alloc m2 lo hi = 〈m2', b2〉 → |
---|
| 2345 | b1 = b2 ∧ lessdef m1' m2'. |
---|
| 2346 | #m1;#m2;#lo;#hi;#b1;#m1';#b2;#m2'; |
---|
| 2347 | *;#Hnext;#Hinj;#ALLOC1;#ALLOC2; |
---|
| 2348 | ncut (b1 = b2); |
---|
| 2349 | ##[ nrewrite > (alloc_result … ALLOC1); nrewrite > (alloc_result … ALLOC2); // |
---|
| 2350 | ##] |
---|
| 2351 | #e; nrewrite < e in ALLOC2 ⊢ %; #ALLOC2; @; //; |
---|
| 2352 | @; |
---|
| 2353 | ##[ nrewrite > (nextblock_alloc … ALLOC1); |
---|
| 2354 | nrewrite > (nextblock_alloc … ALLOC2); |
---|
| 2355 | // |
---|
| 2356 | ##| napply (alloc_parallel_inj … Hinj ALLOC1 ALLOC2); |
---|
| 2357 | ##[ // |
---|
| 2358 | ##| ##3: nwhd in ⊢ (??%?); // |
---|
| 2359 | ##| ##4,5: //; |
---|
[14] | 2360 | ##| ##6: nwhd; #chunk;#_; ncases chunk;//; |
---|
[3] | 2361 | ##] nqed. |
---|
| 2362 | |
---|
| 2363 | nlemma free_lessdef: |
---|
| 2364 | ∀m1,m2,b. lessdef m1 m2 → lessdef (free m1 b) (free m2 b). |
---|
| 2365 | #m1;#m2;#b;*;#Hnext;#Hinj; @; |
---|
| 2366 | ##[ nwhd in ⊢ (??%%); // |
---|
| 2367 | ##| napply (free_parallel_inj … Hinj); //; |
---|
| 2368 | #b';#delta;#H; nwhd in H:(??%?); nelim (grumpydestruct2 ?????? H); // |
---|
| 2369 | ##] nqed. |
---|
| 2370 | |
---|
| 2371 | nlemma free_left_lessdef: |
---|
| 2372 | ∀m1,m2,b. |
---|
| 2373 | lessdef m1 m2 → lessdef (free m1 b) m2. |
---|
| 2374 | #m1;#m2;#b;*;#Hnext;#Hinj;@; |
---|
| 2375 | nrewrite < Hnext; //; |
---|
| 2376 | napply free_left_inj; //; |
---|
| 2377 | nqed. |
---|
| 2378 | |
---|
| 2379 | nlemma free_right_lessdef: |
---|
| 2380 | ∀m1,m2,b. |
---|
| 2381 | lessdef m1 m2 → low_bound m1 b ≥ high_bound m1 b → |
---|
| 2382 | lessdef m1 (free m2 b). |
---|
| 2383 | #m1;#m2;#b;*;#Hnext;#Hinj;#Hbounds; |
---|
| 2384 | @; nrewrite > Hnext; //; |
---|
| 2385 | napply free_right_inj; //; |
---|
| 2386 | #b1;#delta;#chunk;#ofs;#H; nwhd in H:(??%?); ndestruct; |
---|
| 2387 | napply nmk; *; #H1;#H2;#H3;#H4; |
---|
| 2388 | (* arith H2 and H3 contradict Hbounds. *) napply daemon; |
---|
| 2389 | nqed. |
---|
| 2390 | |
---|
| 2391 | nlemma valid_block_lessdef: |
---|
| 2392 | ∀m1,m2,b. lessdef m1 m2 → valid_block m1 b → valid_block m2 b. |
---|
| 2393 | #m1;#m2;#b;*;#Hnext;#Hinj; |
---|
| 2394 | nrewrite > (unfold_valid_block …); nrewrite > (unfold_valid_block m2 b); |
---|
| 2395 | //; nqed. |
---|
| 2396 | |
---|
| 2397 | nlemma valid_pointer_lessdef: |
---|
| 2398 | ∀m1,m2,b,ofs. |
---|
[125] | 2399 | lessdef m1 m2 → valid_pointer m1 psp b ofs = true → valid_pointer m2 psp b ofs = true. |
---|
[3] | 2400 | #m1;#m2;#b;#ofs;*;#Hnext;#Hinj;#VALID; |
---|
| 2401 | nrewrite < (Zplus_z_OZ ofs); napply (valid_pointer_inj … Hinj VALID); //; |
---|
| 2402 | nqed. |
---|
| 2403 | |
---|
| 2404 | |
---|
| 2405 | (* ** Memory injections *) |
---|
| 2406 | |
---|
| 2407 | (* A memory injection [f] is a function from addresses to either [None] |
---|
| 2408 | or [Some] of an address and an offset. It defines a correspondence |
---|
| 2409 | between the blocks of two memory states [m1] and [m2]: |
---|
| 2410 | - if [f b = None], the block [b] of [m1] has no equivalent in [m2]; |
---|
| 2411 | - if [f b = Some〈b', ofs〉], the block [b] of [m2] corresponds to |
---|
| 2412 | a sub-block at offset [ofs] of the block [b'] in [m2]. |
---|
| 2413 | *) |
---|
| 2414 | |
---|
| 2415 | (* A memory injection defines a relation between values that is the |
---|
| 2416 | identity relation, except for pointer values which are shifted |
---|
| 2417 | as prescribed by the memory injection. *) |
---|
| 2418 | |
---|
| 2419 | ninductive val_inject (mi: meminj): val → val → Prop := |
---|
| 2420 | | val_inject_int: |
---|
| 2421 | ∀i. val_inject mi (Vint i) (Vint i) |
---|
| 2422 | | val_inject_float: |
---|
| 2423 | ∀f. val_inject mi (Vfloat f) (Vfloat f) |
---|
| 2424 | | val_inject_ptr: |
---|
| 2425 | ∀b1,ofs1,b2,ofs2,x. |
---|
| 2426 | mi b1 = Some ? 〈b2, x〉 → |
---|
| 2427 | ofs2 = add ofs1 (repr x) → |
---|
| 2428 | val_inject mi (Vptr b1 ofs1) (Vptr b2 ofs2) |
---|
| 2429 | | val_inject_undef: ∀v. |
---|
| 2430 | val_inject mi Vundef v. |
---|
| 2431 | (* |
---|
| 2432 | Hint Resolve val_inject_int val_inject_float val_inject_ptr |
---|
| 2433 | val_inject_undef. |
---|
| 2434 | *) |
---|
| 2435 | ninductive val_list_inject (mi: meminj): list val → list val→ Prop:= |
---|
| 2436 | | val_nil_inject : |
---|
| 2437 | val_list_inject mi (nil ?) (nil ?) |
---|
| 2438 | | val_cons_inject : ∀v,v',vl,vl'. |
---|
| 2439 | val_inject mi v v' → val_list_inject mi vl vl'→ |
---|
| 2440 | val_list_inject mi (v :: vl) (v' :: vl'). |
---|
| 2441 | (* |
---|
| 2442 | Hint Resolve val_nil_inject val_cons_inject. |
---|
| 2443 | *) |
---|
| 2444 | (* A memory state [m1] injects into another memory state [m2] via the |
---|
| 2445 | memory injection [f] if the following conditions hold: |
---|
| 2446 | - loads in [m1] must have matching loads in [m2] in the sense |
---|
| 2447 | of the [mem_inj] predicate; |
---|
| 2448 | - unallocated blocks in [m1] must be mapped to [None] by [f]; |
---|
| 2449 | - if [f b = Some〈b', delta〉], [b'] must be valid in [m2]; |
---|
| 2450 | - distinct blocks in [m1] are mapped to non-overlapping sub-blocks in [m2]; |
---|
| 2451 | - the sizes of [m2]'s blocks are representable with signed machine integers; |
---|
| 2452 | - the offsets [delta] are representable with signed machine integers. |
---|
| 2453 | *) |
---|
| 2454 | |
---|
| 2455 | nrecord mem_inject (f: meminj) (m1,m2: mem) : Prop ≝ |
---|
| 2456 | { |
---|
| 2457 | mi_inj: |
---|
| 2458 | mem_inj val_inject f m1 m2; |
---|
| 2459 | mi_freeblocks: |
---|
| 2460 | ∀b. ¬(valid_block m1 b) → f b = None ?; |
---|
| 2461 | mi_mappedblocks: |
---|
| 2462 | ∀b,b',delta. f b = Some ? 〈b', delta〉 → valid_block m2 b'; |
---|
| 2463 | mi_no_overlap: |
---|
| 2464 | meminj_no_overlap f m1; |
---|
| 2465 | mi_range_1: |
---|
| 2466 | ∀b,b',delta. |
---|
| 2467 | f b = Some ? 〈b', delta〉 → |
---|
| 2468 | min_signed ≤ delta ∧ delta ≤ max_signed; |
---|
| 2469 | mi_range_2: |
---|
| 2470 | ∀b,b',delta. |
---|
| 2471 | f b = Some ? 〈b', delta〉 → |
---|
| 2472 | delta = 0 ∨ (min_signed ≤ low_bound m2 b' ∧ high_bound m2 b' ≤ max_signed) |
---|
| 2473 | }. |
---|
| 2474 | |
---|
| 2475 | |
---|
| 2476 | (* The following lemmas establish the absence of machine integer overflow |
---|
| 2477 | during address computations. *) |
---|
| 2478 | |
---|
| 2479 | nlemma address_inject: |
---|
| 2480 | ∀f,m1,m2,chunk,b1,ofs1,b2,delta. |
---|
| 2481 | mem_inject f m1 m2 → |
---|
| 2482 | valid_access m1 chunk b1 (signed ofs1) → |
---|
| 2483 | f b1 = Some ? 〈b2, delta〉 → |
---|
| 2484 | signed (add ofs1 (repr delta)) = signed ofs1 + delta. |
---|
| 2485 | #f;#m1;#m2;#chunk;#b1;#ofs1;#b2;#delta; |
---|
| 2486 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2487 | #Hvalid;#Hb1inj; |
---|
| 2488 | nelim (mi_range_2 ??? Hb1inj); |
---|
| 2489 | ##[ (* delta = 0 *) |
---|
| 2490 | #edelta; nrewrite > edelta; |
---|
| 2491 | nrewrite > (?:repr O = zero); ##[ ##2: // ##] |
---|
| 2492 | nrewrite > (add_zero ?); |
---|
[14] | 2493 | nrewrite > (Zplus_z_OZ …); |
---|
[3] | 2494 | //; |
---|
| 2495 | ##| (* delta ≠ 0 *) |
---|
| 2496 | #Hrange; nrewrite > (add_signed ??); |
---|
| 2497 | nrewrite > (signed_repr delta ?); |
---|
| 2498 | ##[ nrewrite > (signed_repr ??); |
---|
| 2499 | ##[ // |
---|
| 2500 | ##| ncut (valid_access m2 chunk b2 (signed ofs1 + delta)); |
---|
| 2501 | ##[ napply (valid_access_inj … Hvalid); //; |
---|
| 2502 | ##| *; #_; #Hlo; #Hhi; #_; (* arith *) napply daemon |
---|
| 2503 | ##] |
---|
| 2504 | ##] |
---|
| 2505 | ##| /2/ |
---|
| 2506 | ##] |
---|
| 2507 | ##] nqed. |
---|
[10] | 2508 | |
---|
[3] | 2509 | nlemma valid_pointer_inject_no_overflow: |
---|
| 2510 | ∀f,m1,m2,b,ofs,b',x. |
---|
| 2511 | mem_inject f m1 m2 → |
---|
| 2512 | valid_pointer m1 b (signed ofs) = true → |
---|
| 2513 | f b = Some ? 〈b', x〉 → |
---|
| 2514 | min_signed ≤ signed ofs + signed (repr x) ∧ signed ofs + signed (repr x) ≤ max_signed. |
---|
| 2515 | #f;#m1;#m2;#b;#ofs;#b';#x; |
---|
| 2516 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2517 | #Hvalid;#Hb1inj; |
---|
| 2518 | nlapply ((proj1 ?? (valid_pointer_valid_access ???)) Hvalid); #Hvalid'; |
---|
| 2519 | ncut (valid_access m2 Mint8unsigned b' (signed ofs + x)); |
---|
| 2520 | ##[ napply (valid_access_inj … Hvalid'); // ##] |
---|
| 2521 | *; nrewrite > (?:size_chunk Mint8unsigned = 1); ##[ ##2: // ##] #_; #Hlo; #Hhi; #_; |
---|
| 2522 | nrewrite > (signed_repr ??); ##[ ##2: /2/; ##] |
---|
| 2523 | nlapply (mi_range_2 … Hb1inj); *; |
---|
| 2524 | ##[ #ex; nrewrite > ex; nrewrite > (Zplus_z_OZ ?); napply signed_range; |
---|
| 2525 | ##| (* arith *) napply daemon |
---|
| 2526 | ##] nqed. |
---|
| 2527 | |
---|
[14] | 2528 | (* XXX: should use ndestruct, but reduces large definitions *) |
---|
| 2529 | nremark vptr_eq: ∀b,b',i,i'. Vptr b i = Vptr b' i' → b = b' ∧ i = i'. |
---|
| 2530 | #b b' i i' e; ndestruct; /2/; nqed. |
---|
| 2531 | |
---|
[3] | 2532 | nlemma valid_pointer_inject: |
---|
| 2533 | ∀f,m1,m2,b,ofs,b',ofs'. |
---|
| 2534 | mem_inject f m1 m2 → |
---|
| 2535 | valid_pointer m1 b (signed ofs) = true → |
---|
[125] | 2536 | val_inject f (Vptr psp b ofs) (Vptr b' ofs') → |
---|
[3] | 2537 | valid_pointer m2 b' (signed ofs') = true. |
---|
| 2538 | #f;#m1;#m2;#b;#ofs;#b';#ofs'; |
---|
| 2539 | #Hinj; #Hvalid; #Hvinj; ninversion Hvinj; |
---|
| 2540 | ##[ ##1,2,4: #x;#H;ndestruct; ##] |
---|
| 2541 | #b0;#i;#b0';#i';#delta;#Hb;#Hi';#eptr;#eptr'; |
---|
[14] | 2542 | nrewrite < (proj1 … (vptr_eq ???? eptr)) in Hb; nrewrite < (proj1 … (vptr_eq ???? eptr')); |
---|
| 2543 | nrewrite < (proj2 … (vptr_eq ???? eptr)) in Hi'; nrewrite < (proj2 … (vptr_eq ???? eptr')); |
---|
[3] | 2544 | #Hofs; #Hbinj; |
---|
| 2545 | nrewrite > Hofs; |
---|
| 2546 | nlapply (valid_pointer_inject_no_overflow … Hinj Hvalid Hbinj); #NOOV; |
---|
| 2547 | nelim Hinj;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2548 | nrewrite > (add_signed ??); nrewrite > (signed_repr ??); //; |
---|
| 2549 | nrewrite > (signed_repr ??); /2/; |
---|
| 2550 | napply (valid_pointer_inj … mi_inj Hvalid); //; |
---|
| 2551 | nqed. |
---|
| 2552 | |
---|
| 2553 | nlemma different_pointers_inject: |
---|
| 2554 | ∀f,m,m',b1,ofs1,b2,ofs2,b1',delta1,b2',delta2. |
---|
| 2555 | mem_inject f m m' → |
---|
| 2556 | b1 ≠ b2 → |
---|
| 2557 | valid_pointer m b1 (signed ofs1) = true → |
---|
| 2558 | valid_pointer m b2 (signed ofs2) = true → |
---|
| 2559 | f b1 = Some ? 〈b1', delta1〉 → |
---|
| 2560 | f b2 = Some ? 〈b2', delta2〉 → |
---|
| 2561 | b1' ≠ b2' ∨ |
---|
| 2562 | signed (add ofs1 (repr delta1)) ≠ |
---|
| 2563 | signed (add ofs2 (repr delta2)). |
---|
| 2564 | #f;#m;#m';#b1;#ofs1;#b2;#ofs2;#b1';#delta1;#b2';#delta2; |
---|
| 2565 | #Hinj;#neb;#Hval1;#Hval2;#Hf1;#Hf2; |
---|
| 2566 | nlapply ((proj1 ?? (valid_pointer_valid_access …)) Hval1); #Hval1'; |
---|
| 2567 | nlapply ((proj1 ?? (valid_pointer_valid_access …)) Hval2); #Hval2'; |
---|
| 2568 | nrewrite > (address_inject … Hinj Hval1' Hf1); |
---|
| 2569 | nrewrite > (address_inject … Hinj Hval2' Hf2); |
---|
| 2570 | nelim Hval1'; #Hbval; #Hlo; #Hhi;#Hal; nwhd in Hhi:(?(??%)?); |
---|
| 2571 | nelim Hval2'; #Hbval2; #Hlo2; #Hhi2;#Hal2; nwhd in Hhi2:(?(??%)?); |
---|
| 2572 | nlapply (mi_no_overlap ??? Hinj … Hf1 Hf2 …); //; |
---|
| 2573 | *; ##[ |
---|
| 2574 | *; ##[ |
---|
| 2575 | *; ##[ /2/; |
---|
| 2576 | ##| (* arith contradiction *) napply daemon ##] |
---|
| 2577 | ##| (* arith contradiction *) napply daemon ##] |
---|
| 2578 | ##| *; ##[ #H;@2; (* arith *) napply daemon |
---|
| 2579 | ##| #H;@2; (* arith *) napply daemon ##] ##] |
---|
| 2580 | nqed. |
---|
| 2581 | |
---|
| 2582 | (* Relation between injections and loads. *) |
---|
| 2583 | |
---|
| 2584 | nlemma load_inject: |
---|
| 2585 | ∀f,m1,m2,chunk,b1,ofs,b2,delta,v1. |
---|
| 2586 | mem_inject f m1 m2 → |
---|
| 2587 | load chunk m1 b1 ofs = Some ? v1 → |
---|
| 2588 | f b1 = Some ? 〈b2, delta〉 → |
---|
| 2589 | ∃v2. load chunk m2 b2 (ofs + delta) = Some ? v2 ∧ val_inject f v1 v2. |
---|
| 2590 | #f;#m1;#m2;#chunk;#b1;#ofs;#b2;#delta;#v1; |
---|
| 2591 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2592 | #LOAD;#Hbinj; |
---|
| 2593 | napply mi_inj; //; |
---|
| 2594 | nqed. |
---|
| 2595 | |
---|
| 2596 | nlemma loadv_inject: |
---|
| 2597 | ∀f,m1,m2,chunk,a1,a2,v1. |
---|
| 2598 | mem_inject f m1 m2 → |
---|
| 2599 | loadv chunk m1 a1 = Some ? v1 → |
---|
| 2600 | val_inject f a1 a2 → |
---|
| 2601 | ∃v2. loadv chunk m2 a2 = Some ? v2 ∧ val_inject f v1 v2. |
---|
| 2602 | #f;#m1;#m2;#chunk;#a1;#a2;#v1; |
---|
| 2603 | #Hinj;#LOADV;#Hvinj; ninversion Hvinj; |
---|
| 2604 | ##[ ##1,2,4: #x;#ex;#ex'; napply False_ind; ndestruct; ##] |
---|
| 2605 | #b;#ofs;#b';#ofs';#delta;#Hbinj;#Hofs;#ea1;#ea2; |
---|
| 2606 | nrewrite > ea1 in LOADV; #LOADV; |
---|
| 2607 | nlapply (load_inject … Hinj LOADV … Hbinj); *; #v2; *; #LOAD; #INJ; |
---|
| 2608 | @ v2; @; //; nrewrite > Hofs; |
---|
| 2609 | nrewrite < (?:signed (add ofs (repr delta)) = signed ofs + delta) in LOAD; |
---|
| 2610 | ##[ #H; napply H; (* XXX: used to work with /2/ *) |
---|
| 2611 | ##| napply (address_inject … chunk … Hinj ? Hbinj); napply (load_valid_access …); |
---|
| 2612 | ##[ ##2: napply LOADV; ##] |
---|
| 2613 | ##] nqed. |
---|
| 2614 | |
---|
| 2615 | (* Relation between injections and stores. *) |
---|
| 2616 | |
---|
| 2617 | ninductive val_content_inject (f: meminj): memory_chunk → val → val → Prop ≝ |
---|
| 2618 | | val_content_inject_base: |
---|
| 2619 | ∀chunk,v1,v2. |
---|
| 2620 | val_inject f v1 v2 → |
---|
| 2621 | val_content_inject f chunk v1 v2 |
---|
| 2622 | | val_content_inject_8: |
---|
| 2623 | ∀chunk,n1,n2. |
---|
| 2624 | chunk = Mint8unsigned ∨ chunk = Mint8signed → |
---|
| 2625 | zero_ext 8 n1 = zero_ext 8 n2 → |
---|
| 2626 | val_content_inject f chunk (Vint n1) (Vint n2) |
---|
| 2627 | | val_content_inject_16: |
---|
| 2628 | ∀chunk,n1,n2. |
---|
| 2629 | chunk = Mint16unsigned ∨ chunk = Mint16signed → |
---|
| 2630 | zero_ext 16 n1 = zero_ext 16 n2 → |
---|
| 2631 | val_content_inject f chunk (Vint n1) (Vint n2) |
---|
| 2632 | | val_content_inject_32: |
---|
| 2633 | ∀f1,f2. |
---|
| 2634 | singleoffloat f1 = singleoffloat f2 → |
---|
| 2635 | val_content_inject f Mfloat32 (Vfloat f1) (Vfloat f2). |
---|
| 2636 | |
---|
| 2637 | (*Hint Resolve val_content_inject_base.*) |
---|
| 2638 | |
---|
| 2639 | nlemma load_result_inject: |
---|
| 2640 | ∀f,chunk,v1,v2,chunk'. |
---|
| 2641 | val_content_inject f chunk v1 v2 → |
---|
| 2642 | size_chunk chunk = size_chunk chunk' → |
---|
| 2643 | val_inject f (load_result chunk' v1) (load_result chunk' v2). |
---|
| 2644 | #f;#chunk;#v1;#v2;#chunk'; |
---|
| 2645 | #Hvci; ninversion Hvci; |
---|
| 2646 | ##[ #chunk'';#v1';#v2';#Hvinj;#_;#_;#_;#Hsize; ninversion Hvinj; |
---|
| 2647 | ##[ ncases chunk'; #i;#_;#_; ##[ ##1,2,3,4,5: @ ##| ##6,7: @4 ##] |
---|
| 2648 | ##| ncases chunk'; #f;#_;#_; ##[ ##1,2,3,4,5: @4 ##| ##6,7: @2 ##] |
---|
| 2649 | ##| ncases chunk'; #b1;#ofs1;#b2;#ofs2;#delta;#Hmap;#Hofs;#_;#_; ##[ ##5: @3; // ##| ##*: @4 ##] |
---|
| 2650 | ##| ncases chunk'; #v;#_;#_; @4; |
---|
| 2651 | ##] |
---|
[14] | 2652 | (* FIXME: the next two cases are very similar *) |
---|
[3] | 2653 | ##| #chunk'';#i;#i';*;#echunk;nrewrite > echunk;#Hz;#_;#_;#_; |
---|
[15] | 2654 | nelim chunk'; nwhd in ⊢ ((??%%)→?); #Hsize; ndestruct; |
---|
| 2655 | ##[ ##2,4: nwhd in ⊢ (??%%); nrewrite > Hz; @ |
---|
| 2656 | ##| ##1,3: nwhd in ⊢ (??%%); nrewrite > (sign_ext_equal_if_zero_equal … Hz); |
---|
| 2657 | @; ##[ ##1,3: napply I; ##| ##2,4: napply leb_true_to_le; @; ##] |
---|
[3] | 2658 | ##] |
---|
| 2659 | ##| #chunk'';#i;#i';*;#echunk;nrewrite > echunk;#Hz;#_;#_;#_; |
---|
[15] | 2660 | nelim chunk'; nwhd in ⊢ ((??%%)→?); #Hsize; ndestruct; |
---|
| 2661 | ##[ ##2,4: nwhd in ⊢ (??%%); nrewrite > Hz; @ |
---|
| 2662 | ##| ##1,3: nwhd in ⊢ (??%%); nrewrite > (sign_ext_equal_if_zero_equal … Hz); |
---|
| 2663 | @; ##[ ##1,3: napply I; ##| ##2,4: napply leb_true_to_le; @; ##] |
---|
[3] | 2664 | ##] |
---|
| 2665 | |
---|
| 2666 | ##| #f;#f';#float;#echunk;nrewrite > echunk;#_;#_; |
---|
| 2667 | nelim chunk'; nwhd in ⊢ ((??%%)→?); #Hsize; ndestruct; |
---|
| 2668 | ##[ @4; ##| nrewrite > float; @2 ##] |
---|
| 2669 | ##] nqed. |
---|
| 2670 | |
---|
| 2671 | nlemma store_mapped_inject_1 : |
---|
| 2672 | ∀f,chunk,m1,b1,ofs,v1,n1,m2,b2,delta,v2. |
---|
| 2673 | mem_inject f m1 m2 → |
---|
| 2674 | store chunk m1 b1 ofs v1 = Some ? n1 → |
---|
| 2675 | f b1 = Some ? 〈b2, delta〉 → |
---|
| 2676 | val_content_inject f chunk v1 v2 → |
---|
| 2677 | ∃n2. |
---|
| 2678 | store chunk m2 b2 (ofs + delta) v2 = Some ? n2 |
---|
| 2679 | ∧ mem_inject f n1 n2. |
---|
| 2680 | #f;#chunk;#m1;#b1;#ofs;#v1;#n1;#m2;#b2;#delta;#v2; |
---|
| 2681 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2682 | #STORE1; #INJb1; #Hvcinj; |
---|
| 2683 | nlapply (store_mapped_inj … mi_inj mi_no_overlap INJb1 STORE1 ?); //; |
---|
| 2684 | ##[ #chunk';#Hchunksize;napply (load_result_inject … chunk …);//; |
---|
| 2685 | ##| ##skip ##] |
---|
| 2686 | *;#n2;*;#STORE;#MINJ; |
---|
| 2687 | @ n2; @; //; @; |
---|
| 2688 | ##[ (* inj *) // |
---|
| 2689 | ##| (* freeblocks *) #b;#notvalid; napply mi_freeblocks; |
---|
| 2690 | napply (not_to_not ??? notvalid); napply (store_valid_block_1 … STORE1); |
---|
| 2691 | ##| (* mappedblocks *) #b;#b';#delta';#INJb';napply (store_valid_block_1 … STORE); |
---|
| 2692 | /2/; |
---|
| 2693 | ##| (* no_overlap *) nwhd; #b1';#b1'';#delta1';#b2';#b2'';#delta2';#neqb'; |
---|
| 2694 | #fb1';#fb2'; |
---|
| 2695 | nrewrite > (low_bound_store … STORE1 ?); nrewrite > (low_bound_store … STORE1 ?); |
---|
| 2696 | nrewrite > (high_bound_store … STORE1 ?); nrewrite > (high_bound_store … STORE1 ?); |
---|
| 2697 | napply mi_no_overlap; //; |
---|
| 2698 | ##| (* range *) /2/; |
---|
| 2699 | ##| (* range 2 *) #b;#b';#delta';#INJb; |
---|
| 2700 | nrewrite > (low_bound_store … STORE ?); |
---|
| 2701 | nrewrite > (high_bound_store … STORE ?); |
---|
| 2702 | napply mi_range_2; //; |
---|
| 2703 | ##] nqed. |
---|
| 2704 | |
---|
| 2705 | nlemma store_mapped_inject: |
---|
| 2706 | ∀f,chunk,m1,b1,ofs,v1,n1,m2,b2,delta,v2. |
---|
| 2707 | mem_inject f m1 m2 → |
---|
| 2708 | store chunk m1 b1 ofs v1 = Some ? n1 → |
---|
| 2709 | f b1 = Some ? 〈b2, delta〉 → |
---|
| 2710 | val_inject f v1 v2 → |
---|
| 2711 | ∃n2. |
---|
| 2712 | store chunk m2 b2 (ofs + delta) v2 = Some ? n2 |
---|
| 2713 | ∧ mem_inject f n1 n2. |
---|
| 2714 | #f;#chunk;#m1;#b1;#ofs;#v1;#n1;#m2;#b2;#delta;#v2; |
---|
| 2715 | #MINJ;#STORE;#INJb1;#Hvalinj;napply (store_mapped_inject_1 … STORE);//; |
---|
| 2716 | napply val_content_inject_base;//; |
---|
| 2717 | nqed. |
---|
| 2718 | |
---|
| 2719 | nlemma store_unmapped_inject: |
---|
| 2720 | ∀f,chunk,m1,b1,ofs,v1,n1,m2. |
---|
| 2721 | mem_inject f m1 m2 → |
---|
| 2722 | store chunk m1 b1 ofs v1 = Some ? n1 → |
---|
| 2723 | f b1 = None ? → |
---|
| 2724 | mem_inject f n1 m2. |
---|
| 2725 | #f;#chunk;#m1;#b1;#ofs;#v1;#n1;#m2; |
---|
| 2726 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2727 | #STORE;#INJb1;@; |
---|
| 2728 | ##[ (* inj *) napply (store_unmapped_inj … STORE); // |
---|
| 2729 | ##| (* freeblocks *) #b;#notvalid; napply mi_freeblocks; |
---|
| 2730 | napply (not_to_not ??? notvalid); napply (store_valid_block_1 … STORE); |
---|
| 2731 | ##| (* mappedblocks *) #b;#b';#delta;#INJb; napply mi_mappedblock; //; |
---|
| 2732 | ##| (* no_overlap *) nwhd; #b1';#b1'';#delta1';#b2';#b2'';#delta2';#neqb'; |
---|
| 2733 | #fb1';#fb2'; |
---|
| 2734 | nrewrite > (low_bound_store … STORE ?); nrewrite > (low_bound_store … STORE ?); |
---|
| 2735 | nrewrite > (high_bound_store … STORE ?); nrewrite > (high_bound_store … STORE ?); |
---|
| 2736 | napply mi_no_overlap; //; |
---|
| 2737 | ##| (* range *) /2/ |
---|
| 2738 | ##| /2/ |
---|
| 2739 | ##] nqed. |
---|
| 2740 | |
---|
| 2741 | nlemma storev_mapped_inject_1: |
---|
| 2742 | ∀f,chunk,m1,a1,v1,n1,m2,a2,v2. |
---|
| 2743 | mem_inject f m1 m2 → |
---|
| 2744 | storev chunk m1 a1 v1 = Some ? n1 → |
---|
| 2745 | val_inject f a1 a2 → |
---|
| 2746 | val_content_inject f chunk v1 v2 → |
---|
| 2747 | ∃n2. |
---|
| 2748 | storev chunk m2 a2 v2 = Some ? n2 ∧ mem_inject f n1 n2. |
---|
| 2749 | #f;#chunk;#m1;#a1;#v1;#n1;#m2;#a2;#v2; |
---|
| 2750 | #MINJ;#STORE;#Hvinj;#Hvcinj; |
---|
| 2751 | ninversion Hvinj; |
---|
| 2752 | ##[ ##1,2,4:#x;#ex1;#ex2;nrewrite > ex1 in STORE; nwhd in ⊢ ((??%?)→?); #H; |
---|
| 2753 | napply False_ind; ndestruct; ##] |
---|
| 2754 | #b;#ofs;#b';#ofs';#delta;#INJb;#Hofs;#ea1;#ea2; |
---|
| 2755 | nrewrite > Hofs; nrewrite > ea1 in STORE; #STORE; |
---|
| 2756 | nlapply (store_mapped_inject_1 … MINJ STORE … INJb Hvcinj); |
---|
| 2757 | nrewrite < (?:signed (add ofs (repr delta)) = signed ofs + delta); //; |
---|
| 2758 | napply (address_inject … chunk … MINJ ? INJb); |
---|
| 2759 | napply (store_valid_access_3 … STORE); |
---|
| 2760 | nqed. |
---|
| 2761 | |
---|
| 2762 | nlemma storev_mapped_inject: |
---|
| 2763 | ∀f,chunk,m1,a1,v1,n1,m2,a2,v2. |
---|
| 2764 | mem_inject f m1 m2 → |
---|
| 2765 | storev chunk m1 a1 v1 = Some ? n1 → |
---|
| 2766 | val_inject f a1 a2 → |
---|
| 2767 | val_inject f v1 v2 → |
---|
| 2768 | ∃n2. |
---|
| 2769 | storev chunk m2 a2 v2 = Some ? n2 ∧ mem_inject f n1 n2. |
---|
| 2770 | #f;#chunk;#m1;#a1;#v1;#n1;#m2;#a2;#v2; #MINJ;#STOREV;#Hvinj;#Hvinj'; |
---|
| 2771 | napply (storev_mapped_inject_1 … STOREV); /2/; |
---|
| 2772 | nqed. |
---|
| 2773 | |
---|
| 2774 | (* Relation between injections and [free] *) |
---|
| 2775 | |
---|
| 2776 | nlemma meminj_no_overlap_free: |
---|
| 2777 | ∀mi,m,b. |
---|
| 2778 | meminj_no_overlap mi m → |
---|
| 2779 | meminj_no_overlap mi (free m b). |
---|
| 2780 | #mi;#m;#b;#H;nwhd;#b1;#b1';#delta1;#b2;#b2';#delta2;#Hne;#mi1;#mi2; |
---|
| 2781 | ncut (low_bound (free m b) b ≥ high_bound (free m b) b); |
---|
| 2782 | ##[ nrewrite > (low_bound_free_same …); nrewrite > (high_bound_free_same …);// ##] |
---|
| 2783 | #Hbounds; |
---|
| 2784 | ncases (decidable_eq_Z b1 b);#e1; ##[ nrewrite > e1 in Hne mi1 ⊢ %;#Hne;#mi1;##] |
---|
| 2785 | ncases (decidable_eq_Z b2 b);#e2; ##[ ##1,3: nrewrite > e2 in Hne mi2 ⊢ %;#Hne;#mi2;##] |
---|
| 2786 | ##[ napply False_ind; nelim Hne; /2/ |
---|
| 2787 | ##| @;@2;//; |
---|
| 2788 | ##| @;@;@2;// |
---|
| 2789 | ##| nrewrite > (low_bound_free …);//; nrewrite > (low_bound_free …);//; |
---|
| 2790 | nrewrite > (high_bound_free …);//; nrewrite > (high_bound_free …);//; |
---|
| 2791 | /2/; |
---|
| 2792 | ##] nqed. |
---|
| 2793 | |
---|
| 2794 | nlemma meminj_no_overlap_free_list: |
---|
| 2795 | ∀mi,m,bl. |
---|
| 2796 | meminj_no_overlap mi m → |
---|
| 2797 | meminj_no_overlap mi (free_list m bl). |
---|
| 2798 | #mi;#m;#bl; nelim bl; |
---|
| 2799 | ##[ #H; nwhd in ⊢ (??%); // |
---|
| 2800 | ##| #h;#t; #IH; #H; napply meminj_no_overlap_free; napply IH; // |
---|
| 2801 | ##] nqed. |
---|
| 2802 | |
---|
| 2803 | nlemma free_inject: |
---|
| 2804 | ∀f,m1,m2,l,b. |
---|
| 2805 | (∀b1,delta. f b1 = Some ? 〈b, delta〉 → in_list ? b1 l) → |
---|
| 2806 | mem_inject f m1 m2 → |
---|
| 2807 | mem_inject f (free_list m1 l) (free m2 b). |
---|
| 2808 | #f;#m1;#m2;#l;#b;#mappedin; |
---|
| 2809 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2810 | @; |
---|
| 2811 | ##[ (* inj *) |
---|
| 2812 | napply free_right_inj; ##[ napply free_list_left_inj; //; ##] |
---|
| 2813 | #b1;#delta;#chunk;#ofs;#INJb1; napply nmk; #Hvalid; |
---|
| 2814 | nelim (valid_access_free_list_inv … Hvalid); #b1ni; #Haccess; |
---|
| 2815 | napply (absurd ? (mappedin ?? INJb1) b1ni); |
---|
| 2816 | ##| (* freeblocks *) |
---|
| 2817 | #b';#notvalid; napply mi_freeblocks; napply (not_to_not ??? notvalid); |
---|
| 2818 | #H; napply valid_block_free_list_1; // |
---|
| 2819 | ##| (* mappedblocks *) |
---|
| 2820 | #b1;#b1';#delta;#INJb1; napply valid_block_free_1; /2/ |
---|
| 2821 | ##| (* overlap *) |
---|
| 2822 | napply meminj_no_overlap_free_list; // |
---|
| 2823 | ##| (* range *) |
---|
| 2824 | /2/ |
---|
| 2825 | ##| #b1;#b2;#delta;#INJb1; ncases (decidable_eq_Z b2 b); #eb; |
---|
| 2826 | ##[ nrewrite > eb; |
---|
| 2827 | nrewrite > (low_bound_free_same ??); nrewrite > (high_bound_free_same ??); |
---|
| 2828 | @2; (* arith *) napply daemon |
---|
| 2829 | ##| nrewrite > (low_bound_free …); //; nrewrite > (high_bound_free …); /2/; |
---|
| 2830 | ##] |
---|
| 2831 | ##] nqed. |
---|
| 2832 | |
---|
| 2833 | (* Monotonicity properties of memory injections. *) |
---|
| 2834 | |
---|
| 2835 | ndefinition inject_incr : meminj → meminj → Prop ≝ λf1,f2. |
---|
| 2836 | ∀b. f1 b = f2 b ∨ f1 b = None ?. |
---|
| 2837 | |
---|
| 2838 | nlemma inject_incr_refl : |
---|
| 2839 | ∀f. inject_incr f f . |
---|
| 2840 | #f;nwhd;#b;@;//; nqed. |
---|
| 2841 | |
---|
| 2842 | nlemma inject_incr_trans : |
---|
| 2843 | ∀f1,f2,f3. |
---|
| 2844 | inject_incr f1 f2 → inject_incr f2 f3 → inject_incr f1 f3 . |
---|
| 2845 | #f1;#f2;#f3;nwhd in ⊢ (%→%→%);#H1;#H2;#b; |
---|
| 2846 | nelim (H1 b); nelim (H2 b); /2/; nqed. |
---|
| 2847 | |
---|
| 2848 | nlemma val_inject_incr: |
---|
| 2849 | ∀f1,f2,v,v'. |
---|
| 2850 | inject_incr f1 f2 → |
---|
| 2851 | val_inject f1 v v' → |
---|
| 2852 | val_inject f2 v v'. |
---|
| 2853 | #f1;#f2;#v;#v';#Hincr;#Hvinj; |
---|
| 2854 | ninversion Hvinj; |
---|
| 2855 | ##[ ##1,2,4: #x;#_;#_; //; |
---|
| 2856 | ##|#b;#ofs;#b';#ofs';#delta; #INJb; #Hofs; #_;#_; |
---|
| 2857 | nelim (Hincr b); #H; |
---|
| 2858 | ##[ napply (val_inject_ptr ??????? Hofs); /2/; |
---|
| 2859 | ##| napply False_ind; nrewrite > INJb in H; #H; ndestruct; |
---|
| 2860 | ##] nqed. |
---|
| 2861 | |
---|
| 2862 | nlemma val_list_inject_incr: |
---|
| 2863 | ∀f1,f2,vl,vl'. |
---|
| 2864 | inject_incr f1 f2 → val_list_inject f1 vl vl' → |
---|
| 2865 | val_list_inject f2 vl vl'. |
---|
| 2866 | #f1;#f2;#vl;nelim vl; |
---|
| 2867 | ##[ #vl'; #Hincr; #H; ninversion H; //; #v;#v';#l;#l0;#_;#_;#_; #H; ndestruct; |
---|
| 2868 | ##| #h;#t;#IH;#vl';#Hincr;#H1; ninversion H1; |
---|
| 2869 | ##[ #H; ndestruct |
---|
| 2870 | ##| #h';#h'';#t';#t''; #Hinj1; #Hintt; #_; #e1; #e2; ndestruct; |
---|
| 2871 | @2;/2/; napply IH; //; napply Hincr; |
---|
| 2872 | ##] |
---|
| 2873 | ##] nqed. |
---|
| 2874 | |
---|
| 2875 | (* |
---|
| 2876 | Hint Resolve inject_incr_refl val_inject_incr val_list_inject_incr. |
---|
| 2877 | *) |
---|
| 2878 | |
---|
| 2879 | (* Properties of injections and allocations. *) |
---|
| 2880 | |
---|
| 2881 | ndefinition extend_inject ≝ |
---|
| 2882 | λb: block. λx: option (block × Z). λf: meminj. |
---|
| 2883 | λb': block. if eqZb b' b then x else f b'. |
---|
| 2884 | |
---|
| 2885 | nlemma extend_inject_incr: |
---|
| 2886 | ∀f,b,x. |
---|
| 2887 | f b = None ? → |
---|
| 2888 | inject_incr f (extend_inject b x f). |
---|
| 2889 | #f;#b;#x;#INJb;nwhd;#b'; nwhd in ⊢ (?(???%)?); |
---|
| 2890 | napply (eqZb_elim b' b); #eb; /2/; |
---|
| 2891 | nqed. |
---|
| 2892 | |
---|
| 2893 | nlemma alloc_right_inject: |
---|
| 2894 | ∀f,m1,m2,lo,hi,m2',b. |
---|
| 2895 | mem_inject f m1 m2 → |
---|
| 2896 | alloc m2 lo hi = 〈m2', b〉 → |
---|
| 2897 | mem_inject f m1 m2'. |
---|
| 2898 | #f;#m1;#m2;#lo;#hi;#m2';#b; |
---|
| 2899 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2900 | #ALLOC; @; |
---|
| 2901 | ##[ napply (alloc_right_inj … ALLOC); //; |
---|
| 2902 | ##| /2/; |
---|
| 2903 | ##| #b1;#b2;#delta;#INJb1; napply (valid_block_alloc … ALLOC); /2/; |
---|
| 2904 | ##| //; |
---|
| 2905 | ##| /2/; |
---|
| 2906 | ##|#b1;#b2;#delta;#INJb1; nrewrite > (?:low_bound m2' b2 = low_bound m2 b2); |
---|
| 2907 | ##[ nrewrite > (?:high_bound m2' b2 = high_bound m2 b2); /2/; |
---|
| 2908 | napply high_bound_alloc_other; /2/; |
---|
| 2909 | ##| napply low_bound_alloc_other; /2/ |
---|
| 2910 | ##] |
---|
| 2911 | ##] nqed. |
---|
| 2912 | |
---|
| 2913 | nlemma alloc_unmapped_inject: |
---|
| 2914 | ∀f,m1,m2,lo,hi,m1',b. |
---|
| 2915 | mem_inject f m1 m2 → |
---|
| 2916 | alloc m1 lo hi = 〈m1', b〉 → |
---|
| 2917 | mem_inject (extend_inject b (None ?) f) m1' m2 ∧ |
---|
| 2918 | inject_incr f (extend_inject b (None ?) f). |
---|
| 2919 | #f;#m1;#m2;#lo;#hi;#m1';#b; |
---|
| 2920 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2921 | #ALLOC; |
---|
| 2922 | ncut (inject_incr f (extend_inject b (None ?) f)); |
---|
| 2923 | ##[ napply extend_inject_incr; napply mi_freeblocks; /2/; ##] |
---|
| 2924 | #Hinject_incr; @; //; @; |
---|
| 2925 | ##[ (* inj *) |
---|
| 2926 | napply (alloc_left_unmapped_inj … ALLOC); |
---|
| 2927 | ##[ ##2: nwhd in ⊢ (??%?); nrewrite > (eqZb_z_z …); /2/; ##] |
---|
| 2928 | nwhd; #chunk;#b1;#ofs;#v1;#b2;#delta; |
---|
| 2929 | nwhd in ⊢ ((??%?)→?→?); napply eqZb_elim; #e; nwhd in ⊢ ((??%?)→?→?); |
---|
| 2930 | #Hextend;#LOAD; |
---|
[15] | 2931 | ##[ ndestruct; |
---|
[3] | 2932 | ##| nlapply (mi_inj … Hextend LOAD); *; #v2; *; #LOAD2; #VINJ; |
---|
| 2933 | @ v2; @; //; |
---|
| 2934 | napply val_inject_incr; //; |
---|
| 2935 | ##] |
---|
| 2936 | ##| (* freeblocks *) |
---|
| 2937 | #b';#Hinvalid; nwhd in ⊢ (??%?); napply (eqZb_elim b' b); //; |
---|
| 2938 | #neb; napply mi_freeblocks; napply (not_to_not ??? Hinvalid); |
---|
| 2939 | napply valid_block_alloc; //; |
---|
| 2940 | ##| (* mappedblocks *) |
---|
| 2941 | #b1;#b2;#delta; nwhd in ⊢ (??%?→?); napply (eqZb_elim b1 b); #eb; |
---|
| 2942 | ##[ #H; ndestruct; |
---|
| 2943 | ##| #H; napply (mi_mappedblock … H); |
---|
| 2944 | ##] |
---|
| 2945 | ##| (* overlap *) |
---|
| 2946 | nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2; #neb1; nwhd in ⊢ (??%?→??%?→?); |
---|
| 2947 | nrewrite > (low_bound_alloc … ALLOC ?); nrewrite > (low_bound_alloc … ALLOC ?); |
---|
| 2948 | nrewrite > (high_bound_alloc … ALLOC ?); nrewrite > (high_bound_alloc … ALLOC ?); |
---|
| 2949 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 2950 | ##[ ndestruct |
---|
| 2951 | ##| nlapply (eqZb_to_Prop b2 b); nelim (eqZb b2 b); #e2; #INJb2; |
---|
| 2952 | ##[ ndestruct |
---|
| 2953 | ##| napply mi_no_overlap; /2/; |
---|
| 2954 | ##] |
---|
| 2955 | ##] |
---|
| 2956 | ##| (* range *) |
---|
| 2957 | #b1;#b2;#delta; nwhd in ⊢ (??%?→?); |
---|
| 2958 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 2959 | ##[ ndestruct |
---|
| 2960 | ##| napply (mi_range_1 … INJb1); |
---|
| 2961 | ##] |
---|
| 2962 | ##| #b1;#b2;#delta; nwhd in ⊢ (??%?→?); |
---|
| 2963 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 2964 | ##[ ndestruct |
---|
| 2965 | ##| napply (mi_range_2 … INJb1); |
---|
| 2966 | ##] |
---|
| 2967 | ##] nqed. |
---|
| 2968 | |
---|
| 2969 | nlemma alloc_mapped_inject: |
---|
| 2970 | ∀f,m1,m2,lo,hi,m1',b,b',ofs. |
---|
| 2971 | mem_inject f m1 m2 → |
---|
| 2972 | alloc m1 lo hi = 〈m1', b〉 → |
---|
| 2973 | valid_block m2 b' → |
---|
| 2974 | min_signed ≤ ofs ∧ ofs ≤ max_signed → |
---|
| 2975 | min_signed ≤ low_bound m2 b' → |
---|
| 2976 | high_bound m2 b' ≤ max_signed → |
---|
| 2977 | low_bound m2 b' ≤ lo + ofs → |
---|
| 2978 | hi + ofs ≤ high_bound m2 b' → |
---|
| 2979 | inj_offset_aligned ofs (hi-lo) → |
---|
| 2980 | (∀b0,ofs0. |
---|
| 2981 | f b0 = Some ? 〈b', ofs0〉 → |
---|
| 2982 | high_bound m1 b0 + ofs0 ≤ lo + ofs ∨ |
---|
| 2983 | hi + ofs ≤ low_bound m1 b0 + ofs0) → |
---|
| 2984 | mem_inject (extend_inject b (Some ? 〈b', ofs〉) f) m1' m2 ∧ |
---|
| 2985 | inject_incr f (extend_inject b (Some ? 〈b', ofs〉) f). |
---|
| 2986 | #f;#m1;#m2;#lo;#hi;#m1';#b;#b';#ofs; |
---|
| 2987 | *;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 2988 | #ALLOC;#validb';#rangeofs;#rangelo;#rangehi;#boundlo;#boundhi;#injaligned;#boundmapped; |
---|
| 2989 | ncut (inject_incr f (extend_inject b (Some ? 〈b', ofs〉) f)); |
---|
| 2990 | ##[ napply extend_inject_incr; napply mi_freeblocks; /2/; ##] |
---|
| 2991 | #Hincr; @; //; @; |
---|
| 2992 | ##[ (* inj *) |
---|
| 2993 | napply (alloc_left_mapped_inj … ALLOC … validb' boundlo boundhi); /2/; |
---|
| 2994 | ##[ ##2:nwhd in ⊢ (??%?); nrewrite > (eqZb_z_z …); /2/; ##] |
---|
| 2995 | nwhd; #chunk;#b1;#ofs';#v1;#b2;#delta;#Hextend;#LOAD; |
---|
| 2996 | nwhd in Hextend:(??%?); nrewrite > (eqZb_false b1 b ?) in Hextend; |
---|
| 2997 | ##[ #Hextend; nlapply (mi_inj … Hextend LOAD); |
---|
| 2998 | *; #v2; *; #LOAD2; #VINJ; |
---|
| 2999 | @ v2; @; //; |
---|
| 3000 | napply val_inject_incr; //; |
---|
| 3001 | ##| napply (valid_not_valid_diff m1); /2/; |
---|
| 3002 | napply (valid_access_valid_block … chunk … ofs'); /2/; |
---|
| 3003 | ##] |
---|
| 3004 | ##| (* freeblocks *) |
---|
| 3005 | #b';#Hinvalid; nwhd in ⊢ (??%?); nrewrite > (eqZb_false b' b ?); |
---|
| 3006 | ##[ napply mi_freeblocks; napply (not_to_not ??? Hinvalid); |
---|
| 3007 | napply valid_block_alloc; //; |
---|
| 3008 | ##| napply sym_neq; napply (valid_not_valid_diff m1'); //; |
---|
| 3009 | napply (valid_new_block … ALLOC); |
---|
| 3010 | ##] |
---|
| 3011 | ##| (* mappedblocks *) |
---|
| 3012 | #b1;#b2;#delta; nwhd in ⊢ (??%?→?); napply (eqZb_elim b1 b); #eb;#einj; |
---|
| 3013 | ##[ ndestruct; //; |
---|
| 3014 | ##| napply (mi_mappedblock … einj); |
---|
| 3015 | ##] |
---|
| 3016 | ##| (* overlap *) |
---|
| 3017 | nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2; #neb1; nwhd in ⊢ (??%?→??%?→?); |
---|
| 3018 | nrewrite > (low_bound_alloc … ALLOC ?); nrewrite > (low_bound_alloc … ALLOC ?); |
---|
| 3019 | nrewrite > (high_bound_alloc … ALLOC ?); nrewrite > (high_bound_alloc … ALLOC ?); |
---|
| 3020 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 3021 | ##[ nelim (grumpydestruct2 ?????? INJb1); #eb1';#eofs1 ##] |
---|
| 3022 | nlapply (eqZb_to_Prop b2 b); nelim (eqZb b2 b); #e2; #INJb2; |
---|
| 3023 | ##[ nelim (grumpydestruct2 ?????? INJb2); #eb2';#eofs2 ##] |
---|
| 3024 | ##[ napply False_ind; nrewrite > e in neb1; nrewrite > e2; /2/; |
---|
| 3025 | ##| nelim (decidable_eq_Z b1' b2'); #e'; |
---|
| 3026 | ##[ nrewrite < e' in INJb2 ⊢ %; nrewrite < eb1'; nrewrite < eofs1; #INJb2; nlapply (boundmapped … INJb2); |
---|
[15] | 3027 | *; #H; @2; ##[ @2 ##| @1 ##] napply H; |
---|
[3] | 3028 | ##| @1;@1;@1; napply e' |
---|
| 3029 | ##] |
---|
| 3030 | ##| nelim (decidable_eq_Z b1' b2'); #e'; |
---|
| 3031 | ##[ nrewrite < e' in INJb2 ⊢ %; #INJb2; nelim (grumpydestruct2 ?????? INJb2); #eb'; #eofs; nrewrite < eb' in INJb1; nrewrite < eofs; #INJb1; nlapply (boundmapped … INJb1); |
---|
[15] | 3032 | *; #H; @2; ##[ @1; /2/ ##| @2; napply H; ##] |
---|
[3] | 3033 | ##| @1;@1;@1; napply e' |
---|
| 3034 | ##] |
---|
| 3035 | ##| napply mi_no_overlap; /2/; |
---|
| 3036 | ##] |
---|
| 3037 | ##| (* range *) |
---|
| 3038 | #b1;#b2;#delta; nwhd in ⊢ (??%?→?); |
---|
| 3039 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 3040 | ##[ ndestruct; /2/; |
---|
| 3041 | ##| napply (mi_range_1 … INJb1); |
---|
| 3042 | ##] |
---|
| 3043 | ##| #b1;#b2;#delta; nwhd in ⊢ (??%?→?); |
---|
| 3044 | nlapply (eqZb_to_Prop b1 b); nelim (eqZb b1 b); #e; #INJb1; |
---|
| 3045 | ##[ ndestruct; @2;@;/2/; |
---|
| 3046 | ##| napply (mi_range_2 … INJb1); |
---|
| 3047 | ##] |
---|
| 3048 | ##] nqed. |
---|
| 3049 | |
---|
| 3050 | nlemma alloc_parallel_inject: |
---|
| 3051 | ∀f,m1,m2,lo,hi,m1',m2',b1,b2. |
---|
| 3052 | mem_inject f m1 m2 → |
---|
| 3053 | alloc m1 lo hi = 〈m1', b1〉 → |
---|
| 3054 | alloc m2 lo hi = 〈m2', b2〉 → |
---|
| 3055 | min_signed ≤ lo → hi ≤ max_signed → |
---|
| 3056 | mem_inject (extend_inject b1 (Some ? 〈b2, OZ〉) f) m1' m2' ∧ |
---|
| 3057 | inject_incr f (extend_inject b1 (Some ? 〈b2, OZ〉) f). |
---|
| 3058 | #f;#m1;#m2;#lo;#hi;#m1';#m2';#b1;#b2; |
---|
| 3059 | #Hminj;#ALLOC1;#ALLOC2;#Hlo;#Hhi; |
---|
| 3060 | napply (alloc_mapped_inject … ALLOC1); /2/; |
---|
[14] | 3061 | ##[ napply (alloc_right_inject … Hminj ALLOC2); |
---|
[3] | 3062 | ##| nrewrite > (low_bound_alloc_same … ALLOC2); // |
---|
| 3063 | ##| nrewrite > (high_bound_alloc_same … ALLOC2); // |
---|
| 3064 | ##| nrewrite > (low_bound_alloc_same … ALLOC2); // |
---|
| 3065 | ##| nrewrite > (high_bound_alloc_same … ALLOC2); // |
---|
| 3066 | ##| nwhd; (* arith *) napply daemon |
---|
| 3067 | ##| #b;#ofs;#INJb0; napply False_ind; |
---|
| 3068 | nelim Hminj;#mi_inj;#mi_freeblocks;#mi_mappedblock;#mi_no_overlap;#mi_range_1;#mi_range_2; |
---|
| 3069 | nlapply (mi_mappedblock … INJb0); |
---|
| 3070 | #H; napply (absurd ? H ?); /2/; |
---|
| 3071 | ##] nqed. |
---|
| 3072 | |
---|
| 3073 | ndefinition meminj_init ≝ λm: mem. |
---|
| 3074 | λb: block. if Zltb b (nextblock m) then Some ? 〈b, OZ〉 else None ?. |
---|
| 3075 | |
---|
| 3076 | ndefinition mem_inject_neutral ≝ λm: mem. |
---|
| 3077 | ∀f,chunk,b,ofs,v. |
---|
| 3078 | load chunk m b ofs = Some ? v → val_inject f v v. |
---|
| 3079 | |
---|
| 3080 | nlemma init_inject: |
---|
| 3081 | ∀m. |
---|
| 3082 | mem_inject_neutral m → |
---|
| 3083 | mem_inject (meminj_init m) m m. |
---|
| 3084 | #m;#neutral;@; |
---|
| 3085 | ##[ (* inj *) |
---|
| 3086 | nwhd; #chunk;#b1;#ofs;#v1;#b2;#delta; nwhd in ⊢ (??%?→?→?); |
---|
| 3087 | napply Zltb_elim_Type0; #ltb1; ##[ |
---|
| 3088 | #H; nelim (grumpydestruct2 ?????? H); #eb1; #edelta; |
---|
| 3089 | nrewrite < eb1; nrewrite < edelta; #LOAD; @v1; @; //; |
---|
| 3090 | napply neutral; //; |
---|
| 3091 | ##| #H;nwhd in H:(??%?); ndestruct; |
---|
| 3092 | ##] |
---|
| 3093 | ##| (* free blocks *) |
---|
| 3094 | #b;nrewrite > (unfold_valid_block …); nwhd in ⊢ (?→??%?); #notvalid; |
---|
| 3095 | napply Zltb_elim_Type0; #ltb1; |
---|
| 3096 | ##[ napply False_ind; napply (absurd ? ltb1 notvalid) |
---|
| 3097 | ##| // |
---|
| 3098 | ##] |
---|
| 3099 | ##| (* mapped blocks *) |
---|
| 3100 | #b;#b';#delta;nwhd in ⊢ (??%?→?); nrewrite > (unfold_valid_block …); |
---|
| 3101 | napply Zltb_elim_Type0; #ltb; |
---|
| 3102 | #H; nwhd in H:(??%?); ndestruct; // |
---|
| 3103 | ##| (* overlap *) |
---|
| 3104 | nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2;#neb1; nwhd in ⊢(??%?→??%?→?); |
---|
| 3105 | napply Zltb_elim_Type0; #ltb1; |
---|
| 3106 | ##[ #H; nwhd in H:(??%?); ndestruct; |
---|
| 3107 | napply Zltb_elim_Type0; #ltb2; |
---|
| 3108 | #H2; nwhd in H2:(??%?); ndestruct; @;@;@;/2/; |
---|
| 3109 | ##| #H; nwhd in H:(??%?); ndestruct; |
---|
| 3110 | ##] |
---|
| 3111 | ##| (* range *) |
---|
| 3112 | #b;#b';#delta;nwhd in ⊢ (??%?→?); |
---|
| 3113 | napply Zltb_elim_Type0; #ltb; |
---|
| 3114 | ##[ #H; nelim (grumpydestruct2 ?????? H); #eb; #edelta; nrewrite < edelta; |
---|
| 3115 | (* FIXME: should be in integers.ma *) napply daemon |
---|
| 3116 | ##| #H; nwhd in H:(??%?); ndestruct; |
---|
| 3117 | ##] |
---|
| 3118 | ##| (* range *) |
---|
| 3119 | #b;#b';#delta;nwhd in ⊢ (??%?→?); |
---|
| 3120 | napply Zltb_elim_Type0; #ltb; |
---|
| 3121 | ##[ #H; nelim (grumpydestruct2 ?????? H); #eb; #edelta; nrewrite < edelta; |
---|
| 3122 | (* FIXME: should be in integers.ma *) napply daemon |
---|
| 3123 | ##| #H; nwhd in H:(??%?); ndestruct; |
---|
| 3124 | ##] |
---|
| 3125 | ##] nqed. |
---|
| 3126 | |
---|
| 3127 | nremark getN_setN_inject: |
---|
| 3128 | ∀f,m,v,n1,p1,n2,p2. |
---|
| 3129 | val_inject f (getN n2 p2 m) (getN n2 p2 m) → |
---|
| 3130 | val_inject f v v → |
---|
| 3131 | val_inject f (getN n2 p2 (setN n1 p1 v m)) |
---|
| 3132 | (getN n2 p2 (setN n1 p1 v m)). |
---|
| 3133 | #f;#m;#v;#n1;#p1;#n2;#p2;#injget;#injv; |
---|
| 3134 | ncases (getN_setN_characterization m v n1 p1 n2 p2);##[ * ##] #A; |
---|
| 3135 | nrewrite > A; //; |
---|
| 3136 | nqed. |
---|
| 3137 | |
---|
| 3138 | nremark getN_contents_init_data_inject: |
---|
| 3139 | ∀f,n,ofs,id,pos. |
---|
| 3140 | val_inject f (getN n ofs (contents_init_data pos id)) |
---|
| 3141 | (getN n ofs (contents_init_data pos id)). |
---|
| 3142 | #f;#n;#ofs;#id;nelim id; |
---|
| 3143 | ##[ #pos; nrewrite > (getN_init …); // |
---|
| 3144 | ##| #h;#t;#IH;#pos; ncases h; |
---|
| 3145 | ##[ ##1,2,3,4,5: #x; napply getN_setN_inject; // |
---|
| 3146 | ##| ##6,8: #x; napply IH ##| #x;#y;napply IH ##] (* XXX // doesn't work? *) |
---|
| 3147 | nqed. |
---|
| 3148 | |
---|
| 3149 | nlemma alloc_init_data_neutral: |
---|
| 3150 | ∀m,id,m',b. |
---|
| 3151 | mem_inject_neutral m → |
---|
| 3152 | alloc_init_data m id = 〈m', b〉 → |
---|
| 3153 | mem_inject_neutral m'. |
---|
[15] | 3154 | #m;#id;#m';#b;#Hneutral;#INIT; nwhd in INIT:(??%?); (* XXX: ndestruct makes a bit of a mess *) |
---|
[3] | 3155 | napply (pairdisc_elim … INIT); |
---|
| 3156 | nwhd in ⊢ (??%% → ?);#B;nrewrite < B in ⊢ (??%% → ?); |
---|
| 3157 | nwhd in ⊢ (??%% → ?);#A; |
---|
| 3158 | nwhd; #f;#chunk;#b';#ofs;#v; #LOAD; |
---|
| 3159 | nlapply (load_inv … LOAD); *; #C; #D; |
---|
| 3160 | nrewrite < B in D; nrewrite > A; |
---|
| 3161 | nrewrite > (unfold_update block_contents …); napply eqZb_elim; |
---|
| 3162 | ##[ #eb'; #D; nwhd in D:(???(??(???%))); nrewrite > D; |
---|
| 3163 | napply (load_result_inject … chunk); //; @; |
---|
| 3164 | napply getN_contents_init_data_inject; |
---|
| 3165 | ##| #neb'; #D; napply (Hneutral ? chunk b' ofs ??); nwhd in ⊢ (??%?); |
---|
| 3166 | nrewrite > (in_bounds_true m chunk b' ofs (option ?) …); |
---|
| 3167 | ##[ nrewrite < D; // |
---|
| 3168 | ##| nelim C; #Cval;#Clo;#Chi;#Cal; @; |
---|
| 3169 | ##[ nrewrite > (unfold_valid_block …); |
---|
| 3170 | nrewrite > (unfold_valid_block …) in Cval; nrewrite < B; |
---|
| 3171 | (* arith using neb' *) napply daemon |
---|
| 3172 | ##| nrewrite > (?:low_bound m b' = low_bound m' b'); //; |
---|
| 3173 | nwhd in ⊢ (??%%); nrewrite < B; nrewrite > A; |
---|
| 3174 | nrewrite > (update_o block_contents …); //; napply sym_neq; //; |
---|
| 3175 | ##| nrewrite > (?:high_bound m b' = high_bound m' b'); //; |
---|
| 3176 | nwhd in ⊢ (??%%); nrewrite < B; nrewrite > A; |
---|
| 3177 | nrewrite > (update_o block_contents …); //; napply sym_neq; //; |
---|
| 3178 | ##| //; |
---|
| 3179 | ##] |
---|
| 3180 | ##] nqed. |
---|
| 3181 | |
---|
| 3182 | |
---|
| 3183 | (* ** Memory shifting *) |
---|
| 3184 | |
---|
| 3185 | (* A special case of memory injection where blocks are not coalesced: |
---|
| 3186 | each source block injects in a distinct target block. *) |
---|
| 3187 | |
---|
| 3188 | ndefinition memshift ≝ block → option Z. |
---|
| 3189 | |
---|
| 3190 | ndefinition meminj_of_shift : memshift → meminj ≝ λmi: memshift. |
---|
| 3191 | λb. match mi b with [ None ⇒ None ? | Some x ⇒ Some ? 〈b, x〉 ]. |
---|
| 3192 | |
---|
| 3193 | ndefinition val_shift ≝ λmi: memshift. λv1,v2: val. |
---|
| 3194 | val_inject (meminj_of_shift mi) v1 v2. |
---|
| 3195 | |
---|
| 3196 | nrecord mem_shift (f: memshift) (m1,m2: mem) : Prop := |
---|
| 3197 | { |
---|
| 3198 | ms_inj: |
---|
| 3199 | mem_inj val_inject (meminj_of_shift f) m1 m2; |
---|
| 3200 | ms_samedomain: |
---|
| 3201 | nextblock m1 = nextblock m2; |
---|
| 3202 | ms_domain: |
---|
| 3203 | ∀b. match f b with [ Some _ ⇒ b < nextblock m1 | None ⇒ b ≥ nextblock m1 ]; |
---|
| 3204 | ms_range_1: |
---|
| 3205 | ∀b,delta. |
---|
| 3206 | f b = Some ? delta → |
---|
| 3207 | min_signed ≤ delta ∧ delta ≤ max_signed; |
---|
| 3208 | ms_range_2: |
---|
| 3209 | ∀b,delta. |
---|
| 3210 | f b = Some ? delta → |
---|
| 3211 | min_signed ≤ low_bound m2 b ∧ high_bound m2 b ≤ max_signed |
---|
| 3212 | }. |
---|
| 3213 | |
---|
| 3214 | (* * The following lemmas establish the absence of machine integer overflow |
---|
| 3215 | during address computations. *) |
---|
| 3216 | |
---|
| 3217 | nlemma address_shift: |
---|
| 3218 | ∀f,m1,m2,chunk,b,ofs1,delta. |
---|
| 3219 | mem_shift f m1 m2 → |
---|
| 3220 | valid_access m1 chunk b (signed ofs1) → |
---|
| 3221 | f b = Some ? delta → |
---|
| 3222 | signed (add ofs1 (repr delta)) = signed ofs1 + delta. |
---|
| 3223 | #f;#m1;#m2;#chunk;#b;#ofs1;#delta; |
---|
| 3224 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2;#Hvalid_access;#INJb; |
---|
| 3225 | nelim (ms_range_2 … INJb); #Hlo;#Hhi; |
---|
| 3226 | nrewrite > (add_signed …); |
---|
| 3227 | nrewrite > (signed_repr …); nrewrite > (signed_repr …); /2/; |
---|
| 3228 | ncut (valid_access m2 chunk b (signed ofs1 + delta)); |
---|
| 3229 | ##[ napply (valid_access_inj ? (meminj_of_shift f) … ms_inj Hvalid_access); |
---|
| 3230 | nwhd in ⊢ (??%?); nrewrite > INJb; // ##] |
---|
| 3231 | *; (* arith *) napply daemon; |
---|
| 3232 | nqed. |
---|
| 3233 | |
---|
| 3234 | nlemma valid_pointer_shift_no_overflow: |
---|
| 3235 | ∀f,m1,m2,b,ofs,x. |
---|
| 3236 | mem_shift f m1 m2 → |
---|
| 3237 | valid_pointer m1 b (signed ofs) = true → |
---|
| 3238 | f b = Some ? x → |
---|
| 3239 | min_signed ≤ signed ofs + signed (repr x) ∧ signed ofs + signed (repr x) ≤ max_signed. |
---|
| 3240 | #f;#m1;#m2;#b;#ofs;#x; |
---|
| 3241 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3242 | #VALID;#INJb; |
---|
| 3243 | nlapply (proj1 ?? (valid_pointer_valid_access …) VALID); #Hvalid_access; |
---|
| 3244 | ncut (valid_access m2 Mint8unsigned b (signed ofs + x)); |
---|
| 3245 | ##[ napply (valid_access_inj … ms_inj Hvalid_access); |
---|
| 3246 | nwhd in ⊢ (??%?); nrewrite > INJb; // ##] |
---|
| 3247 | *;#Hvalid_block;#Hlo;#Hhi;#Hal; nwhd in Hhi:(?(??%)?); |
---|
| 3248 | nrewrite > (signed_repr …); /2/; |
---|
| 3249 | nlapply (ms_range_2 … INJb);*;#A;#B; |
---|
| 3250 | (* arith *) napply daemon; |
---|
| 3251 | nqed. |
---|
| 3252 | |
---|
| 3253 | (* FIXME to get around ndestruct problems *) |
---|
| 3254 | nlemma vptr_eq_1 : ∀b,b',ofs,ofs'. Vptr b ofs = Vptr b' ofs' → b = b'. |
---|
| 3255 | #b;#b';#ofs;#ofs';#H;ndestruct;//; |
---|
| 3256 | nqed. |
---|
| 3257 | nlemma vptr_eq_2 : ∀b,b',ofs,ofs'. Vptr b ofs = Vptr b' ofs' → ofs = ofs'. |
---|
| 3258 | #b;#b';#ofs;#ofs';#H;ndestruct;//; |
---|
| 3259 | nqed. |
---|
| 3260 | |
---|
| 3261 | nlemma valid_pointer_shift: |
---|
| 3262 | ∀f,m1,m2,b,ofs,b',ofs'. |
---|
| 3263 | mem_shift f m1 m2 → |
---|
| 3264 | valid_pointer m1 b (signed ofs) = true → |
---|
| 3265 | val_shift f (Vptr b ofs) (Vptr b' ofs') → |
---|
| 3266 | valid_pointer m2 b' (signed ofs') = true. |
---|
| 3267 | #f;#m1;#m2;#b;#ofs;#b';#ofs';#Hmem_shift;#VALID;#Hval_shift; |
---|
| 3268 | nwhd in Hval_shift; ninversion Hval_shift; |
---|
| 3269 | ##[ ##1,2,4: #a; #H; ndestruct; ##] |
---|
| 3270 | #b1;#ofs1;#b2;#ofs2;#delta;#INJb1;#Hofs;#eb1;#eb2; |
---|
| 3271 | nrewrite < (vptr_eq_1 … eb1) in INJb1; nrewrite < (vptr_eq_1 … eb2); #INJb'; |
---|
| 3272 | nrewrite < (vptr_eq_2 … eb1) in Hofs; nrewrite < (vptr_eq_2 … eb2); #Hofs; nrewrite > Hofs; |
---|
| 3273 | ncut (f b = Some ? delta); |
---|
| 3274 | ##[ nwhd in INJb':(??%?); ncases (f b) in INJb' ⊢ %; |
---|
[14] | 3275 | ##[ #H; napply (False_ind … (grumpydestruct … H)); ##| #delta'; #H; nelim (grumpydestruct2 ?????? H); // ##] |
---|
[3] | 3276 | ##] #INJb; |
---|
| 3277 | nlapply (valid_pointer_shift_no_overflow … VALID INJb); //; #NOOV; |
---|
| 3278 | nelim Hmem_shift;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3279 | nrewrite > (add_signed …); nrewrite > (signed_repr …); //; |
---|
| 3280 | nrewrite > (signed_repr …); /2/; |
---|
| 3281 | napply (valid_pointer_inj … VALID); /2/; |
---|
| 3282 | nqed. |
---|
| 3283 | |
---|
| 3284 | (* Relation between shifts and loads. *) |
---|
| 3285 | |
---|
| 3286 | nlemma load_shift: |
---|
| 3287 | ∀f,m1,m2,chunk,b,ofs,delta,v1. |
---|
| 3288 | mem_shift f m1 m2 → |
---|
| 3289 | load chunk m1 b ofs = Some ? v1 → |
---|
| 3290 | f b = Some ? delta → |
---|
| 3291 | ∃v2. load chunk m2 b (ofs + delta) = Some ? v2 ∧ val_shift f v1 v2. |
---|
| 3292 | #f;#m1;#m2;#chunk;#b;#ofs;#delta;#v1; |
---|
| 3293 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3294 | #LOAD; #INJb; |
---|
| 3295 | nwhd in ⊢ (??(λ_.??%)); napply (ms_inj … LOAD); |
---|
| 3296 | nwhd in ⊢ (??%?); nrewrite > INJb; //; |
---|
| 3297 | nqed. |
---|
| 3298 | |
---|
| 3299 | nlemma loadv_shift: |
---|
| 3300 | ∀f,m1,m2,chunk,a1,a2,v1. |
---|
| 3301 | mem_shift f m1 m2 → |
---|
| 3302 | loadv chunk m1 a1 = Some ? v1 → |
---|
| 3303 | val_shift f a1 a2 → |
---|
| 3304 | ∃v2. loadv chunk m2 a2 = Some ? v2 ∧ val_shift f v1 v2. |
---|
| 3305 | #f;#m1;#m2;#chunk;#a1;#a2;#v1;#Hmem_shift;#LOADV;#Hval_shift; |
---|
| 3306 | ninversion Hval_shift; |
---|
| 3307 | ##[ ##1,2,4: #x;#H;nrewrite > H in LOADV;#H';nwhd in H':(??%?);napply False_ind; ndestruct; ##] |
---|
| 3308 | #b1;#ofs1;#b2;#ofs2;#delta;#INJb1;#Hofs;#ea1;#ea2; nrewrite > ea1 in LOADV; #LOADV; |
---|
| 3309 | nlapply INJb1; nwhd in ⊢ (??%? → ?); |
---|
| 3310 | nlapply (refl ? (f b1)); ncases (f b1) in ⊢ (???% → %); |
---|
[14] | 3311 | ##[ #_; nwhd in ⊢ (??%? → ?); #H; napply False_ind; napply (False_ind … (grumpydestruct … H)); |
---|
[3] | 3312 | ##| #delta'; #DELTA; nwhd in ⊢ (??%? → ?); #H; nelim (grumpydestruct2 ?????? H); |
---|
| 3313 | #eb;#edelta; |
---|
| 3314 | ##] nlapply (load_shift … Hmem_shift LOADV DELTA); *; #v2; *;#LOAD;#INJ; |
---|
| 3315 | @ v2; @; //; nrewrite > Hofs; nrewrite > eb in LOAD; nrewrite > edelta; |
---|
| 3316 | nrewrite < (?:signed (add ofs1 (repr delta)) = signed ofs1 + delta); |
---|
| 3317 | ##[#H'; napply H'; (* XXX // doesn't work *) |
---|
| 3318 | ##| nrewrite < edelta; napply (address_shift … chunk … Hmem_shift … DELTA); |
---|
| 3319 | napply (load_valid_access … LOADV); |
---|
| 3320 | ##] |
---|
| 3321 | nqed. |
---|
| 3322 | |
---|
| 3323 | (* Relation between shifts and stores. *) |
---|
| 3324 | |
---|
| 3325 | nlemma store_within_shift: |
---|
| 3326 | ∀f,chunk,m1,b,ofs,v1,n1,m2,delta,v2. |
---|
| 3327 | mem_shift f m1 m2 → |
---|
| 3328 | store chunk m1 b ofs v1 = Some ? n1 → |
---|
| 3329 | f b = Some ? delta → |
---|
| 3330 | val_shift f v1 v2 → |
---|
| 3331 | ∃n2. |
---|
| 3332 | store chunk m2 b (ofs + delta) v2 = Some ? n2 |
---|
| 3333 | ∧ mem_shift f n1 n2. |
---|
| 3334 | #f;#chunk;#m1;#b;#ofs;#v1;#n1;#m2;#delta;#v2; |
---|
| 3335 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3336 | #STORE1;#INJb;#Hval_shift; |
---|
| 3337 | nlapply (store_mapped_inj … ms_inj ?? STORE1 ?); |
---|
| 3338 | ##[ #chunk'; #echunk; napply (load_result_inject … chunk); /2/; |
---|
| 3339 | ##| nwhd in ⊢ (??%?); nrewrite > INJb; (* XXX: // has stopped working *) napply refl |
---|
| 3340 | ##| nwhd; #b1;#b1';#delta1;#b2;#b2';#delta2; |
---|
| 3341 | nwhd in ⊢ (? → ??%? → ??%? → ?); |
---|
| 3342 | nelim (f b1); nelim (f b2); |
---|
| 3343 | ##[#_;#e1;nwhd in e1:(??%?);ndestruct; |
---|
| 3344 | ##|#z;#_;#e1;nwhd in e1:(??%?);ndestruct; |
---|
| 3345 | ##|#z;#_;#_;#e2;nwhd in e2:(??%?);ndestruct; |
---|
| 3346 | ##|#delta1';#delta2';#neb;#e1;#e2; |
---|
| 3347 | nwhd in e1:(??%?) e2:(??%?); |
---|
| 3348 | ndestruct; |
---|
| 3349 | @1;@1;@1;//; |
---|
| 3350 | ##] |
---|
| 3351 | ##| ##7: //; |
---|
| 3352 | ##| ##4,5,6: ##skip |
---|
| 3353 | ##] |
---|
| 3354 | *;#n2;*;#STORE;#MINJ; |
---|
| 3355 | @ n2; @; //; @; |
---|
| 3356 | ##[ (* inj *) // |
---|
| 3357 | ##| (* samedomain *) |
---|
| 3358 | nrewrite > (nextblock_store … STORE1); |
---|
| 3359 | nrewrite > (nextblock_store … STORE); |
---|
| 3360 | //; |
---|
| 3361 | ##| (* domain *) |
---|
| 3362 | nrewrite > (nextblock_store … STORE1); |
---|
| 3363 | // |
---|
| 3364 | ##| (* range *) |
---|
| 3365 | /2/ |
---|
| 3366 | ##| #b1;#delta1;#INJb1; |
---|
| 3367 | nrewrite > (low_bound_store … STORE b1); |
---|
| 3368 | nrewrite > (high_bound_store … STORE b1); |
---|
| 3369 | napply ms_range_2;//; |
---|
| 3370 | ##] nqed. |
---|
| 3371 | |
---|
| 3372 | nlemma store_outside_shift: |
---|
| 3373 | ∀f,chunk,m1,b,ofs,m2,v,m2',delta. |
---|
| 3374 | mem_shift f m1 m2 → |
---|
| 3375 | f b = Some ? delta → |
---|
| 3376 | high_bound m1 b + delta ≤ ofs |
---|
| 3377 | ∨ ofs + size_chunk chunk ≤ low_bound m1 b + delta → |
---|
| 3378 | store chunk m2 b ofs v = Some ? m2' → |
---|
| 3379 | mem_shift f m1 m2'. |
---|
| 3380 | #f;#chunk;#m1;#b;#ofs;#m2;#v;#m2';#delta; |
---|
| 3381 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3382 | #INJb;#Hbounds;#STORE;@; |
---|
| 3383 | ##[ (* inj *) |
---|
| 3384 | napply (store_outside_inj … STORE); //; |
---|
| 3385 | #b';#d'; nwhd in ⊢ (??%? → ?); nlapply (refl ? (f b')); nelim (f b') in ⊢ (???% → %); |
---|
| 3386 | ##[ #_; #e; nwhd in e:(??%?); ndestruct; |
---|
| 3387 | ##| #d''; #ef; #e; nelim (grumpydestruct2 ?????? e); #eb; #ed; |
---|
| 3388 | nrewrite > eb in ef ⊢ %; nrewrite > ed; nrewrite > INJb; #ed'; |
---|
| 3389 | nrewrite < (grumpydestruct1 ??? ed'); // |
---|
| 3390 | ##] |
---|
| 3391 | ##| (* samedomain *) nrewrite > (nextblock_store … STORE); // |
---|
| 3392 | ##| (* domain *) // |
---|
| 3393 | ##| (* range *) /2/ |
---|
| 3394 | ##| #b1;#delta1;#INJb1; |
---|
| 3395 | nrewrite > (low_bound_store … STORE b1); |
---|
| 3396 | nrewrite > (high_bound_store … STORE b1); |
---|
| 3397 | napply ms_range_2;//; |
---|
| 3398 | ##] nqed. |
---|
| 3399 | |
---|
| 3400 | nlemma storev_shift: |
---|
| 3401 | ∀f,chunk,m1,a1,v1,n1,m2,a2,v2. |
---|
| 3402 | mem_shift f m1 m2 → |
---|
| 3403 | storev chunk m1 a1 v1 = Some ? n1 → |
---|
| 3404 | val_shift f a1 a2 → |
---|
| 3405 | val_shift f v1 v2 → |
---|
| 3406 | ∃n2. |
---|
| 3407 | storev chunk m2 a2 v2 = Some ? n2 ∧ mem_shift f n1 n2. |
---|
| 3408 | #f;#chunk;#m1;#a1;#v1;#n1;#m2;#a2;#v2; |
---|
| 3409 | #Hmem_shift;#STOREV;#Hval_shift_a;#Hval_shift_v; |
---|
| 3410 | ninversion Hval_shift_a; |
---|
| 3411 | ##[ ##1,2,4: #x;#H;nrewrite > H in STOREV;#H';nwhd in H':(??%?); napply False_ind; ndestruct; ##] |
---|
| 3412 | #b1;#ofs1;#b2;#ofs2;#delta; |
---|
| 3413 | nwhd in ⊢ (??%? → ?); nlapply (refl ? (f b1)); nelim (f b1) in ⊢ (???% → %); |
---|
| 3414 | ##[#_; #e; nwhd in e:(??%?); ndestruct; ##] |
---|
| 3415 | #x; #INJb1; #e; nelim (grumpydestruct2 ?????? e); #eb;#ex; |
---|
| 3416 | nrewrite > ex in INJb1; #INJb1; |
---|
| 3417 | #OFS; #ea1;#ea2; nrewrite > ea1 in STOREV; #STOREV; |
---|
| 3418 | nlapply (store_within_shift … Hmem_shift STOREV INJb1 Hval_shift_v); |
---|
| 3419 | *; #n2; *; #A;#B; |
---|
| 3420 | @ n2; @; //; |
---|
| 3421 | nrewrite > OFS; nrewrite > eb in A; |
---|
| 3422 | nrewrite < (?:signed (add ofs1 (repr delta)) = signed ofs1 + delta); |
---|
| 3423 | ##[ #H; napply H; (* XXX /2/ *) |
---|
| 3424 | ##| napply (address_shift … chunk … Hmem_shift ? INJb1); |
---|
| 3425 | napply (store_valid_access_3 … STOREV); |
---|
| 3426 | ##] |
---|
| 3427 | nqed. |
---|
| 3428 | |
---|
| 3429 | (* Relation between shifts and [free]. *) |
---|
| 3430 | |
---|
| 3431 | nlemma free_shift: |
---|
| 3432 | ∀f,m1,m2,b. |
---|
| 3433 | mem_shift f m1 m2 → |
---|
| 3434 | mem_shift f (free m1 b) (free m2 b). |
---|
| 3435 | #f;#m1;#m2;#b; |
---|
| 3436 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; @; |
---|
| 3437 | ##[ (* inj *) |
---|
| 3438 | napply free_right_inj; ##[ napply free_left_inj; // |
---|
| 3439 | ##| #b1;#delta; #chunk;#ofs; nwhd in ⊢ (??%? → ?); |
---|
| 3440 | nlapply (refl ? (f b1)); nelim (f b1) in ⊢ (???% → %); |
---|
| 3441 | ##[ #_; #e; nwhd in e:(??%?); ndestruct; |
---|
| 3442 | ##| #delta'; #INJb1; #e; nwhd in e:(??%?); ndestruct; |
---|
| 3443 | napply valid_access_free_2 |
---|
| 3444 | ##] |
---|
| 3445 | ##] |
---|
| 3446 | ##| (* samedomain *) nwhd in ⊢ (??%%); // |
---|
| 3447 | ##| (* domain *) nrewrite > (?:nextblock (free m1 b) = nextblock m1); // |
---|
| 3448 | ##| (* range *) /2/ |
---|
| 3449 | ##| #b';#delta;#INJb'; ncases (decidable_eq_Z b' b); #eb; |
---|
| 3450 | ##[ nrewrite > eb; |
---|
| 3451 | nrewrite > (low_bound_free_same ??); nrewrite > (high_bound_free_same ??); |
---|
| 3452 | (* arith *) napply daemon |
---|
| 3453 | ##| nrewrite > (low_bound_free …); //; nrewrite > (high_bound_free …); /2/; |
---|
| 3454 | ##] |
---|
| 3455 | ##] nqed. |
---|
| 3456 | |
---|
| 3457 | (* Relation between shifts and allocation. *) |
---|
| 3458 | |
---|
| 3459 | ndefinition shift_incr : memshift → memshift → Prop ≝ λf1,f2: memshift. |
---|
| 3460 | ∀b. f1 b = f2 b ∨ f1 b = None ?. |
---|
| 3461 | |
---|
| 3462 | nremark shift_incr_inject_incr: |
---|
| 3463 | ∀f1,f2. |
---|
| 3464 | shift_incr f1 f2 → inject_incr (meminj_of_shift f1) (meminj_of_shift f2). |
---|
| 3465 | #f1;#f2;#Hshift; nwhd in ⊢ (?%%); nwhd; #b; |
---|
| 3466 | nelim (Hshift b); #INJ; nrewrite > INJ; /2/; |
---|
| 3467 | nqed. |
---|
| 3468 | |
---|
| 3469 | nlemma val_shift_incr: |
---|
| 3470 | ∀f1,f2,v1,v2. |
---|
| 3471 | shift_incr f1 f2 → val_shift f1 v1 v2 → val_shift f2 v1 v2. |
---|
| 3472 | #f1;#f2;#v1;#v2;#Hshift_incr; nwhd in ⊢ (% → %); |
---|
| 3473 | napply val_inject_incr; |
---|
| 3474 | napply shift_incr_inject_incr; //; |
---|
| 3475 | nqed. |
---|
| 3476 | |
---|
| 3477 | (* * |
---|
| 3478 | Remark mem_inj_incr: |
---|
| 3479 | ∀f1,f2,m1,m2. |
---|
| 3480 | inject_incr f1 f2 → mem_inj val_inject f1 m1 m2 → mem_inj val_inject f2 m1 m2. |
---|
| 3481 | Proof. |
---|
| 3482 | intros; red; intros. |
---|
| 3483 | destruct (H b1). rewrite <- H3 in H1. |
---|
| 3484 | exploit H0; eauto. intros [v2 [A B]]. |
---|
| 3485 | exists v2; split. auto. apply val_inject_incr with f1; auto. |
---|
| 3486 | congruence. |
---|
| 3487 | ***) |
---|
| 3488 | |
---|
| 3489 | nlemma alloc_shift: |
---|
| 3490 | ∀f,m1,m2,lo1,hi1,m1',b,delta,lo2,hi2. |
---|
| 3491 | mem_shift f m1 m2 → |
---|
| 3492 | alloc m1 lo1 hi1 = 〈m1', b〉 → |
---|
| 3493 | lo2 ≤ lo1 + delta → hi1 + delta ≤ hi2 → |
---|
| 3494 | min_signed ≤ delta ∧ delta ≤ max_signed → |
---|
| 3495 | min_signed ≤ lo2 → hi2 ≤ max_signed → |
---|
| 3496 | inj_offset_aligned delta (hi1-lo1) → |
---|
| 3497 | ∃f'. ∃m2'. |
---|
| 3498 | alloc m2 lo2 hi2 = 〈m2', b〉 |
---|
| 3499 | ∧ mem_shift f' m1' m2' |
---|
| 3500 | ∧ shift_incr f f' |
---|
| 3501 | ∧ f' b = Some ? delta. |
---|
| 3502 | #f;#m1;#m2;#lo1;#hi1;#m1';#b;#delta;#lo2;#hi2; |
---|
| 3503 | *;#ms_inj;#ms_samedomain;#ms_domain;#ms_range_1;#ms_range_2; |
---|
| 3504 | #ALLOC;#Hlo_delta;#Hhi_delta;#Hdelta_range;#Hlo_range;#Hhi_range;#Hinj_aligned; |
---|
| 3505 | nlapply (refl ? (alloc m2 lo2 hi2)); nelim (alloc m2 lo2 hi2) in ⊢ (???% → %); |
---|
| 3506 | #m2';#b';#ALLOC2; |
---|
| 3507 | ncut (b' = b); |
---|
| 3508 | ##[ nrewrite > (alloc_result … ALLOC); nrewrite > (alloc_result … ALLOC2); // ##] |
---|
| 3509 | #eb; nrewrite > eb; |
---|
| 3510 | ncut (f b = None ?); |
---|
| 3511 | ##[ nlapply (ms_domain b); nrewrite > (alloc_result … ALLOC); |
---|
| 3512 | nelim (f (nextblock m1)); //; |
---|
| 3513 | #z; (* arith *) napply daemon |
---|
| 3514 | ##] |
---|
| 3515 | #FB; |
---|
| 3516 | nletin f' ≝ (λb':block. if eqZb b' b then Some ? delta else f b'); |
---|
| 3517 | ncut (shift_incr f f'); |
---|
| 3518 | ##[ nwhd; #b0; nwhd in ⊢ (?(???%)?); |
---|
| 3519 | napply eqZb_elim; /2/; ##] |
---|
| 3520 | #Hshift_incr; |
---|
| 3521 | ncut (f' b = Some ? delta); |
---|
| 3522 | ##[ nwhd in ⊢ (??%?); nrewrite > (eqZb_z_z …); // ##] #efb'; |
---|
| 3523 | @ f'; @ m2'; @; //; @; //; @; //; @; |
---|
| 3524 | ##[ (* inj *) |
---|
| 3525 | ncut (mem_inj val_inject (meminj_of_shift f') m1 m2); |
---|
| 3526 | ##[ nwhd; #chunk;#b1;#ofs;#v1;#b2;#delta2; #MINJf'; #LOAD; |
---|
| 3527 | ncut (meminj_of_shift f b1 = Some ? 〈b2, delta2〉); |
---|
| 3528 | ##[ nrewrite < MINJf'; nwhd in ⊢ (???(?%?)); nwhd in ⊢ (??%%); |
---|
| 3529 | napply eqZb_elim; //; #eb; |
---|
| 3530 | nrewrite > eb; |
---|
| 3531 | ncut (valid_block m1 b); |
---|
| 3532 | ##[ napply valid_access_valid_block; |
---|
| 3533 | ##[ ##3: napply load_valid_access; // ##] |
---|
| 3534 | ##] |
---|
| 3535 | ncut (¬valid_block m1 b); ##[ /2/ ##] |
---|
| 3536 | #H;#H'; napply False_ind; napply (absurd ? H' H) |
---|
| 3537 | ##] #MINJf; |
---|
| 3538 | nlapply (ms_inj … MINJf LOAD); *; #v2; *; #A; #B; |
---|
| 3539 | @ v2; @; //; |
---|
| 3540 | napply (val_inject_incr … B); |
---|
| 3541 | napply shift_incr_inject_incr; // |
---|
| 3542 | ##] #Hmem_inj; |
---|
| 3543 | napply (alloc_parallel_inj … delta Hmem_inj ALLOC ALLOC2 ?); /2/; |
---|
| 3544 | nwhd in ⊢ (??%?); nrewrite > efb'; /2/; |
---|
| 3545 | ##| (* samedomain *) |
---|
| 3546 | nrewrite > (nextblock_alloc … ALLOC); |
---|
| 3547 | nrewrite > (nextblock_alloc … ALLOC2); |
---|
| 3548 | //; |
---|
| 3549 | ##| (* domain *) |
---|
| 3550 | #b0; (* FIXME: unfold *) nrewrite > (refl ? (f' b0):f' b0 = if eqZb b0 b then Some ? delta else f b0); |
---|
| 3551 | nrewrite > (nextblock_alloc … ALLOC); |
---|
| 3552 | nrewrite > (alloc_result … ALLOC); |
---|
| 3553 | napply eqZb_elim; #eb0; |
---|
| 3554 | ##[ nrewrite > eb0; (* arith *) napply daemon |
---|
| 3555 | ##| nlapply (ms_domain b0); nelim (f b0); |
---|
| 3556 | (* arith *) napply daemon |
---|
| 3557 | ##] |
---|
| 3558 | ##| (* range *) |
---|
| 3559 | #b0;#delta0; nwhd in ⊢ (??%? → ?); |
---|
| 3560 | napply eqZb_elim; |
---|
| 3561 | ##[ #_; #e; nrewrite < (grumpydestruct1 ??? e); // |
---|
| 3562 | ##| #neb; #e; napply (ms_range_1 … b0); napply e; |
---|
| 3563 | ##] |
---|
| 3564 | ##| #b0;#delta0; nwhd in ⊢ (??%? → ?); |
---|
| 3565 | nrewrite > (low_bound_alloc … ALLOC2 ?); |
---|
| 3566 | nrewrite > (high_bound_alloc … ALLOC2 ?); |
---|
| 3567 | napply eqZb_elim; #eb0; nrewrite > eb; |
---|
| 3568 | ##[ nrewrite > eb0; #ed; nrewrite < (grumpydestruct1 ??? ed); |
---|
[15] | 3569 | nrewrite > (eqZb_z_z ?); /3/; |
---|
[3] | 3570 | ##| #edelta0; nrewrite > (eqZb_false … eb0); |
---|
| 3571 | napply ms_range_2; nwhd in edelta0:(??%?); //; |
---|
| 3572 | ##] |
---|
| 3573 | ##] |
---|
| 3574 | nqed. |
---|
[391] | 3575 | *)*) |
---|
[3] | 3576 | (* ** Relation between signed and unsigned loads and stores *) |
---|
| 3577 | |
---|
| 3578 | (* Target processors do not distinguish between signed and unsigned |
---|
| 3579 | stores of 8- and 16-bit quantities. We show these are equivalent. *) |
---|
| 3580 | |
---|
| 3581 | (* Signed 8- and 16-bit stores can be performed like unsigned stores. *) |
---|
| 3582 | |
---|
| 3583 | nremark in_bounds_equiv: |
---|
[125] | 3584 | ∀chunk1,chunk2,m,psp,b,ofs.∀A:Type.∀a1,a2: A. |
---|
[3] | 3585 | size_chunk chunk1 = size_chunk chunk2 → |
---|
[125] | 3586 | (match in_bounds m chunk1 psp b ofs with [ inl _ ⇒ a1 | inr _ ⇒ a2]) = |
---|
| 3587 | (match in_bounds m chunk2 psp b ofs with [ inl _ ⇒ a1 | inr _ ⇒ a2]). |
---|
| 3588 | #chunk1;#chunk2;#m;#psp b ofs;#A;#a1;#a2;#Hsize; |
---|
| 3589 | nelim (in_bounds m chunk1 psp b ofs); |
---|
[3] | 3590 | ##[ #H; nwhd in ⊢ (??%?); nrewrite > (in_bounds_true … A a1 a2 ?); //; |
---|
| 3591 | napply valid_access_compat; //; |
---|
[125] | 3592 | ##| #H; nwhd in ⊢ (??%?); nelim (in_bounds m chunk2 psp b ofs); //; |
---|
[3] | 3593 | #H'; napply False_ind; napply (absurd ?? H); |
---|
| 3594 | napply valid_access_compat; //; |
---|
| 3595 | ##] nqed. |
---|
| 3596 | |
---|
| 3597 | nlemma storev_8_signed_unsigned: |
---|
| 3598 | ∀m,a,v. |
---|
| 3599 | storev Mint8signed m a v = storev Mint8unsigned m a v. |
---|
| 3600 | #m;#a;#v; nwhd in ⊢ (??%%); nelim a; //; |
---|
[125] | 3601 | #psp b ofs; nwhd in ⊢ (??%%); |
---|
[3] | 3602 | nrewrite > (in_bounds_equiv Mint8signed Mint8unsigned … (option mem) ???); //; |
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| 3603 | nqed. |
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| 3604 | |
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| 3605 | nlemma storev_16_signed_unsigned: |
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| 3606 | ∀m,a,v. |
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| 3607 | storev Mint16signed m a v = storev Mint16unsigned m a v. |
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| 3608 | #m;#a;#v; nwhd in ⊢ (??%%); nelim a; //; |
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[125] | 3609 | #psp b ofs; nwhd in ⊢ (??%%); |
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[3] | 3610 | nrewrite > (in_bounds_equiv Mint16signed Mint16unsigned … (option mem) ???); //; |
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| 3611 | nqed. |
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| 3612 | |
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| 3613 | (* Likewise, some target processors (e.g. the PowerPC) do not have |
---|
| 3614 | a ``load 8-bit signed integer'' instruction. |
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| 3615 | We show that it can be synthesized as a ``load 8-bit unsigned integer'' |
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| 3616 | followed by a sign extension. *) |
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| 3617 | |
---|
[391] | 3618 | (* Nonessential properties that may require arithmetic |
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[3] | 3619 | nlemma loadv_8_signed_unsigned: |
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| 3620 | ∀m,a. |
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| 3621 | loadv Mint8signed m a = option_map ?? (sign_ext 8) (loadv Mint8unsigned m a). |
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| 3622 | #m;#a; nwhd in ⊢ (??%(????(%))); nelim a; //; |
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[125] | 3623 | #psp b ofs; nwhd in ⊢ (??%(????%)); |
---|
[3] | 3624 | nrewrite > (in_bounds_equiv Mint8signed Mint8unsigned … (option val) ???); //; |
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[125] | 3625 | nelim (in_bounds m Mint8unsigned psp b (signed ofs)); /2/; |
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[3] | 3626 | #H; nwhd in ⊢ (??%%); |
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[14] | 3627 | nelim (getN 0 (signed ofs) (contents (blocks m b))); |
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[153] | 3628 | ##[ ##1,3: //; |
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[14] | 3629 | ##| #i; nwhd in ⊢ (??(??%)(??%)); |
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[15] | 3630 | nrewrite > (sign_ext_zero_ext ?? i); ##[ napply refl; (* XXX: // ? *) |
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[14] | 3631 | ##| (* arith *) napply daemon; |
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| 3632 | ##] |
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[153] | 3633 | ##| #sp; ncases sp; //; |
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[14] | 3634 | ##] |
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[3] | 3635 | nqed. |
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[474] | 3636 | *) |
---|