[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 | (* *) |
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| 7 | (* Copyright Institut National de Recherche en Informatique et en *) |
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| 8 | (* Automatique. All rights reserved. This file is distributed *) |
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| 9 | (* under the terms of the GNU General Public License as published by *) |
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| 10 | (* the Free Software Foundation, either version 2 of the License, or *) |
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| 11 | (* (at your option) any later version. This file is also distributed *) |
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| 12 | (* under the terms of the INRIA Non-Commercial License Agreement. *) |
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| 13 | (* *) |
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| 14 | (* *********************************************************************) |
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| 15 | |
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| 16 | (* * This file defines a number of data types and operations used in |
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| 17 | the abstract syntax trees of many of the intermediate languages. *) |
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| 18 | |
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| 19 | include "datatypes/sums.ma". |
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| 20 | include "extralib.ma". |
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| 21 | include "Integers.ma". |
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| 22 | include "Floats.ma". |
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[10] | 23 | include "binary/positive.ma". |
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[3] | 24 | |
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| 25 | (* * * Syntactic elements *) |
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| 26 | |
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| 27 | (* * Identifiers (names of local variables, of global symbols and functions, |
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| 28 | etc) are represented by the type [positive] of positive integers. *) |
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[10] | 29 | |
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| 30 | ndefinition ident ≝ Pos. |
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| 31 | |
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| 32 | ndefinition ident_eq : ∀x,y:ident. (x=y) + (x≠y). |
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| 33 | #x y; nlapply (pos_compare_to_Prop x y); ncases (pos_compare x y); |
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| 34 | ##[ #H; @2; /2/; ##| #H; @1; //; ##| #H; @2; /2/ ##] nqed. |
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| 35 | |
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[3] | 36 | (* |
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[9] | 37 | (* XXX: we use nats for now, but if in future we use binary like compcert |
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| 38 | then the maps will be easier to define. *) |
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| 39 | ndefinition ident ≝ nat. |
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| 40 | ndefinition ident_eq : ∀x,y:ident. (x=y) + (x≠y). |
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| 41 | #x; nelim x; |
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| 42 | ##[ #y; ncases y; /3/; |
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| 43 | ##| #x'; #IH; #y; ncases y; |
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| 44 | ##[ @2; @; #H; ndestruct |
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| 45 | ##| #y'; nelim (IH y'); |
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| 46 | ##[ #e; ndestruct; /2/ |
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| 47 | ##| #ne; @2; /2/; |
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| 48 | ##] |
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| 49 | ##] |
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| 50 | ##] nqed. |
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[10] | 51 | *) |
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[3] | 52 | (* * The intermediate languages are weakly typed, using only two types: |
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| 53 | [Tint] for integers and pointers, and [Tfloat] for floating-point |
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| 54 | numbers. *) |
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| 55 | |
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| 56 | ninductive typ : Type ≝ |
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| 57 | | Tint : typ |
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| 58 | | Tfloat : typ. |
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| 59 | |
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| 60 | ndefinition typesize : typ → Z ≝ λty. |
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| 61 | match ty return λ_.Z with [ Tint ⇒ 4 | Tfloat ⇒ 8 ]. |
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| 62 | |
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| 63 | nlemma typesize_pos: ∀ty. typesize ty > 0. |
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| 64 | #ty; ncases ty; //; nqed. |
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| 65 | |
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| 66 | nlemma typ_eq: ∀t1,t2: typ. (t1=t2) + (t1≠t2). |
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| 67 | #t1;#t2;ncases t1;ncases t2;/2/; @2; napply nmk; #H; ndestruct; nqed. |
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| 68 | |
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| 69 | nlemma opt_typ_eq: ∀t1,t2: option typ. (t1=t2) + (t1≠t2). |
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| 70 | #t1;#t2;ncases t1;ncases t2;/2/; |
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| 71 | ##[ ##1,2: #ty; @2; napply nmk; #H; ndestruct; |
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| 72 | ##| #ty1;#ty2; nelim (typ_eq ty1 ty2); /2/; #neq; @2; napply nmk; |
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| 73 | #H; ndestruct; /2/; |
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| 74 | ##] nqed. |
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| 75 | |
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| 76 | (* * Additionally, function definitions and function calls are annotated |
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| 77 | by function signatures indicating the number and types of arguments, |
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| 78 | as well as the type of the returned value if any. These signatures |
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| 79 | are used in particular to determine appropriate calling conventions |
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| 80 | for the function. *) |
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| 81 | |
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| 82 | nrecord signature : Type ≝ { |
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| 83 | sig_args: list typ; |
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| 84 | sig_res: option typ |
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| 85 | }. |
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| 86 | |
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| 87 | ndefinition proj_sig_res : signature → typ ≝ λs. |
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| 88 | match sig_res s with |
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| 89 | [ None ⇒ Tint |
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| 90 | | Some t ⇒ t |
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| 91 | ]. |
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| 92 | |
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| 93 | (* * Memory accesses (load and store instructions) are annotated by |
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| 94 | a ``memory chunk'' indicating the type, size and signedness of the |
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| 95 | chunk of memory being accessed. *) |
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| 96 | |
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| 97 | ninductive memory_chunk : Type[0] ≝ |
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| 98 | | Mint8signed : memory_chunk (*r 8-bit signed integer *) |
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| 99 | | Mint8unsigned : memory_chunk (*r 8-bit unsigned integer *) |
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| 100 | | Mint16signed : memory_chunk (*r 16-bit signed integer *) |
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| 101 | | Mint16unsigned : memory_chunk (*r 16-bit unsigned integer *) |
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[153] | 102 | | Mint24 : memory_chunk |
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[3] | 103 | | Mint32 : memory_chunk (*r 32-bit integer, or pointer *) |
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| 104 | | Mfloat32 : memory_chunk (*r 32-bit single-precision float *) |
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| 105 | | Mfloat64 : memory_chunk. (*r 64-bit double-precision float *) |
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| 106 | |
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[125] | 107 | (* Memory spaces *) |
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| 108 | |
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| 109 | ninductive memory_space : Type ≝ |
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| 110 | | Any : memory_space |
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| 111 | | Data : memory_space |
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| 112 | | IData : memory_space |
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[156] | 113 | | PData : memory_space |
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[125] | 114 | | XData : memory_space |
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| 115 | | Code : memory_space. |
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| 116 | |
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[3] | 117 | (* * Initialization data for global variables. *) |
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| 118 | |
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| 119 | ninductive init_data: Type[0] ≝ |
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| 120 | | Init_int8: int → init_data |
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| 121 | | Init_int16: int → init_data |
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| 122 | | Init_int32: int → init_data |
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| 123 | | Init_float32: float → init_data |
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| 124 | | Init_float64: float → init_data |
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| 125 | | Init_space: Z → init_data |
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| 126 | | Init_addrof: ident → int → init_data (*r address of symbol + offset *) |
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| 127 | | Init_pointer: list init_data → init_data. |
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| 128 | |
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| 129 | (* * Whole programs consist of: |
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| 130 | - a collection of function definitions (name and description); |
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| 131 | - the name of the ``main'' function that serves as entry point in the program; |
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| 132 | - a collection of global variable declarations, consisting of |
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| 133 | a name, initialization data, and additional information. |
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| 134 | |
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| 135 | The type of function descriptions and that of additional information |
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| 136 | for variables vary among the various intermediate languages and are |
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| 137 | taken as parameters to the [program] type. The other parts of whole |
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| 138 | programs are common to all languages. *) |
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| 139 | |
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| 140 | nrecord program (F,V: Type) : Type := { |
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| 141 | prog_funct: list (ident × F); |
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| 142 | prog_main: ident; |
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[125] | 143 | prog_vars: list (ident × (list init_data) × memory_space × V) |
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[3] | 144 | }. |
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| 145 | |
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| 146 | ndefinition prog_funct_names ≝ λF,V: Type. λp: program F V. |
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| 147 | map ?? (fst ident F) (prog_funct ?? p). |
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| 148 | |
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| 149 | ndefinition prog_var_names ≝ λF,V: Type. λp: program F V. |
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[125] | 150 | map ?? (λx: ident × (list init_data) × memory_space × V. fst ?? (fst ?? x)) (prog_vars ?? p). |
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[3] | 151 | (* |
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| 152 | (** * Generic transformations over programs *) |
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| 153 | |
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| 154 | (** We now define a general iterator over programs that applies a given |
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| 155 | code transformation function to all function descriptions and leaves |
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| 156 | the other parts of the program unchanged. *) |
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| 157 | |
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| 158 | Section TRANSF_PROGRAM. |
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| 159 | |
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| 160 | Variable A B V: Type. |
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| 161 | Variable transf: A -> B. |
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[24] | 162 | *) |
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[3] | 163 | |
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[24] | 164 | ndefinition transf_program : ∀A,B. (A → B) → list (ident × A) → list (ident × B) ≝ |
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| 165 | λA,B,transf,l. |
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| 166 | map ?? (λid_fn. 〈fst ?? id_fn, transf (snd ?? id_fn)〉) l. |
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[3] | 167 | |
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[24] | 168 | ndefinition transform_program : ∀A,B,V. (A → B) → program A V → program B V ≝ |
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| 169 | λA,B,V,transf,p. |
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| 170 | mk_program B V |
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| 171 | (transf_program ?? transf (prog_funct A V p)) |
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| 172 | (prog_main A V p) |
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| 173 | (prog_vars A V p). |
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[3] | 174 | |
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[24] | 175 | nlemma transform_program_function: |
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| 176 | ∀A,B,V,transf,p,i,tf. |
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| 177 | in_list ? 〈i, tf〉 (prog_funct ?? (transform_program A B V transf p)) → |
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| 178 | ∃f. in_list ? 〈i, f〉 (prog_funct ?? p) ∧ transf f = tf. |
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| 179 | nnormalize; #A B V transf p i tf H; nelim (list_in_map_inv ????? H); |
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| 180 | #x; nelim x; #i' tf'; *; #e H; ndestruct; @tf'; /2/; |
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| 181 | nqed. |
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[3] | 182 | |
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[24] | 183 | (* |
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[3] | 184 | End TRANSF_PROGRAM. |
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| 185 | |
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| 186 | (** The following is a variant of [transform_program] where the |
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| 187 | code transformation function can fail and therefore returns an |
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| 188 | option type. *) |
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| 189 | |
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| 190 | Open Local Scope error_monad_scope. |
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| 191 | Open Local Scope string_scope. |
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| 192 | |
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| 193 | Section MAP_PARTIAL. |
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| 194 | |
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| 195 | Variable A B C: Type. |
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| 196 | Variable prefix_errmsg: A -> errmsg. |
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| 197 | Variable f: B -> res C. |
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| 198 | |
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| 199 | Fixpoint map_partial (l: list (A * B)) : res (list (A * C)) := |
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| 200 | match l with |
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| 201 | | nil => OK nil |
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| 202 | | (a, b) :: rem => |
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| 203 | match f b with |
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| 204 | | Error msg => Error (prefix_errmsg a ++ msg)%list |
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| 205 | | OK c => |
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| 206 | do rem' <- map_partial rem; |
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| 207 | OK ((a, c) :: rem') |
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| 208 | end |
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| 209 | end. |
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| 210 | |
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| 211 | Remark In_map_partial: |
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| 212 | forall l l' a c, |
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| 213 | map_partial l = OK l' -> |
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| 214 | In (a, c) l' -> |
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| 215 | exists b, In (a, b) l /\ f b = OK c. |
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| 216 | Proof. |
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| 217 | induction l; simpl. |
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| 218 | intros. inv H. elim H0. |
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| 219 | intros until c. destruct a as [a1 b1]. |
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| 220 | caseEq (f b1); try congruence. |
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| 221 | intro c1; intros. monadInv H0. |
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| 222 | elim H1; intro. inv H0. exists b1; auto. |
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| 223 | exploit IHl; eauto. intros [b [P Q]]. exists b; auto. |
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| 224 | Qed. |
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| 225 | |
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| 226 | Remark map_partial_forall2: |
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| 227 | forall l l', |
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| 228 | map_partial l = OK l' -> |
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| 229 | list_forall2 |
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| 230 | (fun (a_b: A * B) (a_c: A * C) => |
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| 231 | fst a_b = fst a_c /\ f (snd a_b) = OK (snd a_c)) |
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| 232 | l l'. |
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| 233 | Proof. |
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| 234 | induction l; simpl. |
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| 235 | intros. inv H. constructor. |
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| 236 | intro l'. destruct a as [a b]. |
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| 237 | caseEq (f b). 2: congruence. intro c; intros. monadInv H0. |
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| 238 | constructor. simpl. auto. auto. |
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| 239 | Qed. |
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| 240 | |
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| 241 | End MAP_PARTIAL. |
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| 242 | |
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| 243 | Remark map_partial_total: |
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| 244 | forall (A B C: Type) (prefix: A -> errmsg) (f: B -> C) (l: list (A * B)), |
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| 245 | map_partial prefix (fun b => OK (f b)) l = |
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| 246 | OK (List.map (fun a_b => (fst a_b, f (snd a_b))) l). |
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| 247 | Proof. |
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| 248 | induction l; simpl. |
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| 249 | auto. |
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| 250 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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| 251 | Qed. |
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| 252 | |
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| 253 | Remark map_partial_identity: |
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| 254 | forall (A B: Type) (prefix: A -> errmsg) (l: list (A * B)), |
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| 255 | map_partial prefix (fun b => OK b) l = OK l. |
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| 256 | Proof. |
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| 257 | induction l; simpl. |
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| 258 | auto. |
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| 259 | destruct a as [a1 b1]. rewrite IHl. reflexivity. |
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| 260 | Qed. |
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| 261 | |
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| 262 | Section TRANSF_PARTIAL_PROGRAM. |
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| 263 | |
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| 264 | Variable A B V: Type. |
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| 265 | Variable transf_partial: A -> res B. |
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| 266 | |
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| 267 | Definition prefix_funct_name (id: ident) : errmsg := |
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| 268 | MSG "In function " :: CTX id :: MSG ": " :: nil. |
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| 269 | |
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| 270 | Definition transform_partial_program (p: program A V) : res (program B V) := |
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| 271 | do fl <- map_partial prefix_funct_name transf_partial p.(prog_funct); |
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| 272 | OK (mkprogram fl p.(prog_main) p.(prog_vars)). |
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| 273 | |
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| 274 | Lemma transform_partial_program_function: |
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| 275 | forall p tp i tf, |
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| 276 | transform_partial_program p = OK tp -> |
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| 277 | In (i, tf) tp.(prog_funct) -> |
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| 278 | exists f, In (i, f) p.(prog_funct) /\ transf_partial f = OK tf. |
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| 279 | Proof. |
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| 280 | intros. monadInv H. simpl in H0. |
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| 281 | eapply In_map_partial; eauto. |
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| 282 | Qed. |
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| 283 | |
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| 284 | Lemma transform_partial_program_main: |
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| 285 | forall p tp, |
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| 286 | transform_partial_program p = OK tp -> |
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| 287 | tp.(prog_main) = p.(prog_main). |
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| 288 | Proof. |
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| 289 | intros. monadInv H. reflexivity. |
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| 290 | Qed. |
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| 291 | |
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| 292 | Lemma transform_partial_program_vars: |
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| 293 | forall p tp, |
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| 294 | transform_partial_program p = OK tp -> |
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| 295 | tp.(prog_vars) = p.(prog_vars). |
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| 296 | Proof. |
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| 297 | intros. monadInv H. reflexivity. |
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| 298 | Qed. |
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| 299 | |
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| 300 | End TRANSF_PARTIAL_PROGRAM. |
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| 301 | |
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| 302 | (** The following is a variant of [transform_program_partial] where |
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| 303 | both the program functions and the additional variable information |
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| 304 | are transformed by functions that can fail. *) |
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| 305 | |
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| 306 | Section TRANSF_PARTIAL_PROGRAM2. |
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| 307 | |
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| 308 | Variable A B V W: Type. |
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| 309 | Variable transf_partial_function: A -> res B. |
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| 310 | Variable transf_partial_variable: V -> res W. |
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| 311 | |
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| 312 | Definition prefix_var_name (id_init: ident * list init_data) : errmsg := |
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| 313 | MSG "In global variable " :: CTX (fst id_init) :: MSG ": " :: nil. |
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| 314 | |
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| 315 | Definition transform_partial_program2 (p: program A V) : res (program B W) := |
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| 316 | do fl <- map_partial prefix_funct_name transf_partial_function p.(prog_funct); |
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| 317 | do vl <- map_partial prefix_var_name transf_partial_variable p.(prog_vars); |
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| 318 | OK (mkprogram fl p.(prog_main) vl). |
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| 319 | |
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| 320 | Lemma transform_partial_program2_function: |
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| 321 | forall p tp i tf, |
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| 322 | transform_partial_program2 p = OK tp -> |
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| 323 | In (i, tf) tp.(prog_funct) -> |
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| 324 | exists f, In (i, f) p.(prog_funct) /\ transf_partial_function f = OK tf. |
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| 325 | Proof. |
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| 326 | intros. monadInv H. |
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| 327 | eapply In_map_partial; eauto. |
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| 328 | Qed. |
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| 329 | |
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| 330 | Lemma transform_partial_program2_variable: |
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| 331 | forall p tp i tv, |
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| 332 | transform_partial_program2 p = OK tp -> |
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| 333 | In (i, tv) tp.(prog_vars) -> |
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| 334 | exists v, In (i, v) p.(prog_vars) /\ transf_partial_variable v = OK tv. |
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| 335 | Proof. |
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| 336 | intros. monadInv H. |
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| 337 | eapply In_map_partial; eauto. |
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| 338 | Qed. |
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| 339 | |
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| 340 | Lemma transform_partial_program2_main: |
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| 341 | forall p tp, |
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| 342 | transform_partial_program2 p = OK tp -> |
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| 343 | tp.(prog_main) = p.(prog_main). |
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| 344 | Proof. |
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| 345 | intros. monadInv H. reflexivity. |
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| 346 | Qed. |
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| 347 | |
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| 348 | End TRANSF_PARTIAL_PROGRAM2. |
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| 349 | |
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| 350 | (** The following is a relational presentation of |
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| 351 | [transform_program_partial2]. Given relations between function |
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| 352 | definitions and between variable information, it defines a relation |
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| 353 | between programs stating that the two programs have the same shape |
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| 354 | (same global names, etc) and that identically-named function definitions |
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| 355 | are variable information are related. *) |
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| 356 | |
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| 357 | Section MATCH_PROGRAM. |
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| 358 | |
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| 359 | Variable A B V W: Type. |
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| 360 | Variable match_fundef: A -> B -> Prop. |
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| 361 | Variable match_varinfo: V -> W -> Prop. |
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| 362 | |
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| 363 | Definition match_funct_entry (x1: ident * A) (x2: ident * B) := |
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| 364 | match x1, x2 with |
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| 365 | | (id1, fn1), (id2, fn2) => id1 = id2 /\ match_fundef fn1 fn2 |
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| 366 | end. |
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| 367 | |
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| 368 | Definition match_var_entry (x1: ident * list init_data * V) (x2: ident * list init_data * W) := |
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| 369 | match x1, x2 with |
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| 370 | | (id1, init1, info1), (id2, init2, info2) => id1 = id2 /\ init1 = init2 /\ match_varinfo info1 info2 |
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| 371 | end. |
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| 372 | |
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| 373 | Definition match_program (p1: program A V) (p2: program B W) : Prop := |
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| 374 | list_forall2 match_funct_entry p1.(prog_funct) p2.(prog_funct) |
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| 375 | /\ p1.(prog_main) = p2.(prog_main) |
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| 376 | /\ list_forall2 match_var_entry p1.(prog_vars) p2.(prog_vars). |
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| 377 | |
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| 378 | End MATCH_PROGRAM. |
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| 379 | |
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| 380 | Remark transform_partial_program2_match: |
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| 381 | forall (A B V W: Type) |
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| 382 | (transf_partial_function: A -> res B) |
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| 383 | (transf_partial_variable: V -> res W) |
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| 384 | (p: program A V) (tp: program B W), |
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| 385 | transform_partial_program2 transf_partial_function transf_partial_variable p = OK tp -> |
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| 386 | match_program |
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| 387 | (fun fd tfd => transf_partial_function fd = OK tfd) |
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| 388 | (fun info tinfo => transf_partial_variable info = OK tinfo) |
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| 389 | p tp. |
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| 390 | Proof. |
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| 391 | intros. monadInv H. split. |
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| 392 | apply list_forall2_imply with |
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| 393 | (fun (ab: ident * A) (ac: ident * B) => |
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| 394 | fst ab = fst ac /\ transf_partial_function (snd ab) = OK (snd ac)). |
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| 395 | eapply map_partial_forall2. eauto. |
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| 396 | intros. destruct v1; destruct v2; simpl in *. auto. |
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| 397 | split. auto. |
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| 398 | apply list_forall2_imply with |
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| 399 | (fun (ab: ident * list init_data * V) (ac: ident * list init_data * W) => |
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| 400 | fst ab = fst ac /\ transf_partial_variable (snd ab) = OK (snd ac)). |
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| 401 | eapply map_partial_forall2. eauto. |
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| 402 | intros. destruct v1; destruct v2; simpl in *. destruct p0; destruct p1. intuition congruence. |
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| 403 | Qed. |
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| 404 | *) |
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| 405 | (* * * External functions *) |
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| 406 | |
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| 407 | (* * For most languages, the functions composing the program are either |
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| 408 | internal functions, defined within the language, or external functions |
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| 409 | (a.k.a. system calls) that emit an event when applied. We define |
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| 410 | a type for such functions and some generic transformation functions. *) |
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| 411 | |
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| 412 | nrecord external_function : Type ≝ { |
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| 413 | ef_id: ident; |
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| 414 | ef_sig: signature |
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| 415 | }. |
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| 416 | |
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| 417 | ninductive fundef (F: Type): Type ≝ |
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| 418 | | Internal: F → fundef F |
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| 419 | | External: external_function → fundef F. |
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| 420 | |
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| 421 | (* Implicit Arguments External [F]. *) |
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| 422 | (* |
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| 423 | Section TRANSF_FUNDEF. |
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| 424 | |
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| 425 | Variable A B: Type. |
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| 426 | Variable transf: A -> B. |
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[24] | 427 | *) |
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| 428 | ndefinition transf_fundef : ∀A,B. (A→B) → fundef A → fundef B ≝ |
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| 429 | λA,B,transf,fd. |
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[3] | 430 | match fd with |
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[24] | 431 | [ Internal f ⇒ Internal ? (transf f) |
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| 432 | | External ef ⇒ External ? ef |
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| 433 | ]. |
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[3] | 434 | |
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[24] | 435 | (* |
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[3] | 436 | End TRANSF_FUNDEF. |
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| 437 | |
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| 438 | Section TRANSF_PARTIAL_FUNDEF. |
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| 439 | |
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| 440 | Variable A B: Type. |
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| 441 | Variable transf_partial: A -> res B. |
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| 442 | |
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| 443 | Definition transf_partial_fundef (fd: fundef A): res (fundef B) := |
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| 444 | match fd with |
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| 445 | | Internal f => do f' <- transf_partial f; OK (Internal f') |
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| 446 | | External ef => OK (External ef) |
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| 447 | end. |
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| 448 | |
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| 449 | End TRANSF_PARTIAL_FUNDEF. |
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| 450 | *) |
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