source: src/utilities/extralib.ma @ 1651

Last change on this file since 1651 was 1599, checked in by sacerdot, 8 years ago

Start of merging of stuff into the standard library of Matita.

File size: 4.7 KB
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1(**************************************************************************)
2(*       ___                                                              *)
3(*      ||M||                                                             *)
4(*      ||A||       A project by Andrea Asperti                           *)
5(*      ||T||                                                             *)
6(*      ||I||       Developers:                                           *)
7(*      ||T||         The HELM team.                                      *)
8(*      ||A||         http://helm.cs.unibo.it                             *)
9(*      \   /                                                             *)
10(*       \ /        This file is distributed under the terms of the       *)
11(*        v         GNU General Public License Version 2                  *)
12(*                                                                        *)
13(**************************************************************************)
14
15include "basics/types.ma".
16include "basics/lists/list.ma".
17include "basics/logic.ma".
18include "ASM/Util.ma".
19
20lemma eq_rect_Type0_r:
21 ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
22 #A #a #P #p #x0 #p0 @(eq_rect_r ??? p0) assumption.
23qed.
24
25lemma eq_rect_r2:
26 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → P x p.
27 #A #a #x #p cases p; #P #H assumption.
28qed.
29
30lemma eq_rect_Type2_r:
31 ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
32 #A #a #P #p #x0 #p0 @(eq_rect_r2 ??? p0) assumption.
33qed.
34
35lemma eq_rect_CProp0_r:
36 ∀A.∀a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
37 #A #a #P #p #x0 #p0 @(eq_rect_r2 ??? p0) assumption.
38qed.
39
40lemma sym_neq : ∀A.∀x,y:A. x ≠ y → y ≠ x.
41#A #x #y *;#H @nmk #H' /2/;
42qed.
43
44interpretation "logical iff" 'iff x y = (iff x y).
45
46(* bool *)
47
48definition xorb : bool → bool → bool ≝
49  λx,y. match x with [ false ⇒ y | true ⇒ notb y ].
50 
51 
52 
53
54(* should be proved in nat.ma, but it's not! *)
55lemma eqb_to_Prop : ∀n,m:nat.match eqb n m with [ true ⇒ n = m | false ⇒ n ≠ m ].
56#n elim n;
57[ #m cases m; //;
58| #n' #IH #m cases m; [ /2/; | #m' whd in match (eqb (S n') (S m')) in ⊢ %;
59  lapply (IH m'); cases (eqb n' m'); /2/; ]
60] qed.
61
62(* datatypes/list.ma *)
63
64theorem nil_append_nil_both:
65  ∀A:Type[0]. ∀l1,l2:list A.
66    l1 @ l2 = [] → l1 = [] ∧ l2 = [].
67#A #l1 #l2 cases l1
68[ cases l2
69  [ /2/
70  | normalize #h #t #H destruct
71  ]
72| cases l2
73  [ normalize #h #t #H destruct
74  | normalize #h1 #t1 #h2 #h3 #H destruct
75  ]
76] qed.
77
78(* some useful stuff for quantifiers *)
79
80lemma dec_bounded_forall:
81  ∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ∀k.(∀n.n < k → P n) + ¬(∀n.n < k → P n).
82 #P #HP_dec #k elim k -k
83 [ %1 #n #Hn @⊥ @(absurd (n < 0) Hn) @not_le_Sn_O
84 | #k #Hind cases Hind
85   [ #Hk cases (HP_dec k)
86     [ #HPk %1 #n #Hn cases (le_to_or_lt_eq … Hn)
87       [ #H @(Hk … (le_S_S_to_le … H))
88       | #H >(injective_S … H) @HPk
89       ]
90     | #HPk %2 @nmk #Habs @(absurd (P k)) [ @(Habs … (le_n (S k))) | @HPk ]
91     ]
92   | #Hk %2 @nmk #Habs @(absurd (∀n.n<k→P n)) [ #n' #Hn' @(Habs … (le_S … Hn')) | @Hk ]
93   ]
94 ]
95qed.
96
97lemma dec_bounded_exists:
98  ∀P:ℕ→Prop.(∀n.(P n) + (¬P n)) → ∀k.(∃n.n < k ∧ P n) + ¬(∃n.n < k ∧ P n).
99 #P #HP_dec #k elim k -k
100 [ %2 @nmk #Habs elim Habs #n #Hn @(absurd (n < 0) (proj1 … Hn)) @not_le_Sn_O
101 | #k #Hind cases Hind
102   [ #Hk %1 elim Hk #n #Hn @(ex_intro … n) @conj [ @le_S @(proj1 … Hn) | @(proj2 … Hn) ]
103   | #Hk cases (HP_dec k)
104     [ #HPk %1 @(ex_intro … k) @conj [ @le_n | @HPk ]
105     | #HPk %2 @nmk #Habs elim Habs #n #Hn cases (le_to_or_lt_eq … (proj1 … Hn))
106       [ #H @(absurd (∃n.n < k ∧ P n)) [ @(ex_intro … n) @conj
107         [ @(le_S_S_to_le … H) | @(proj2 … Hn) ] | @Hk ]
108       | #H @(absurd (P k)) [ <(injective_S … H) @(proj2 … Hn) | @HPk ]
109       ] 
110     ]
111   ]
112 ]
113qed.
114
115lemma not_exists_forall:
116  ∀k:ℕ.∀P:ℕ → Prop.¬(∃x.x < k ∧ P x) → ∀x.x < k → ¬P x.
117 #k #P #Hex #x #Hx @nmk #Habs @(absurd ? ? Hex) @(ex_intro … x)
118 @conj [ @Hx | @Habs ]
119qed.
120
121lemma not_forall_exists:
122  ∀k:ℕ.∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ¬(∀x.x < k → P x) → ∃x.x < k ∧ ¬P x.
123 #k #P #Hdec elim k
124 [ #Hfa @⊥ @(absurd ?? Hfa) #z #Hz @⊥ @(absurd ? Hz) @not_le_Sn_O
125 | -k #k #Hind #Hfa cases (Hdec k)
126   [ #HP elim (Hind ?)
127     [ -Hind; #x #Hx @(ex_intro ?? x) @conj [ @le_S @(proj1 ?? Hx) | @(proj2 ?? Hx) ]
128     | @nmk #H @(absurd ?? Hfa) #x #Hx cases (le_to_or_lt_eq ?? Hx)
129       [ #H2 @H @(le_S_S_to_le … H2)
130       | #H2 >(injective_S … H2) @HP
131       ]
132     ]
133   | #HP @(ex_intro … k) @conj [ @le_n | @HP ]
134   ]
135 ]
136qed.
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