source: src/utilities/extralib.ma @ 2101

Last change on this file since 2101 was 2101, checked in by boender, 7 years ago
  • renamed medium to absolute jump
  • revised proofs of policy, some daemons removed
  • reverted earlier change in extralib, bounded quantification now again uses lt
File size: 6.5 KB
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2(*       ___                                                              *)
3(*      ||M||                                                             *)
4(*      ||A||       A project by Andrea Asperti                           *)
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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
44(* Paolo: already in library, generates ambiguity
45interpretation "logical iff" 'iff x y = (iff x y).*)
46
47(* bool *)
48
49definition xorb : bool → bool → bool ≝
50  λx,y. match x with [ false ⇒ y | true ⇒ notb y ].
51 
52 
53 
54
55(* should be proved in nat.ma, but it's not! *)
56(* Paolo: there is eqb_elim which does something very similar *)
57lemma eqb_to_Prop : ∀n,m:nat.match eqb n m with [ true ⇒ n = m | false ⇒ n ≠ m ].
58#n elim n;
59[ #m cases m; //;
60| #n' #IH #m cases m; [ /2/; | #m' whd in match (eqb (S n') (S m')) in ⊢ %;
61  lapply (IH m'); cases (eqb n' m'); /2/; ]
62] qed.
63
64(* datatypes/list.ma *)
65
66theorem nil_append_nil_both:
67  ∀A:Type[0]. ∀l1,l2:list A.
68    l1 @ l2 = [] → l1 = [] ∧ l2 = [].
69#A #l1 #l2 cases l1
70[ cases l2
71  [ /2/
72  | normalize #h #t #H destruct
73  ]
74| cases l2
75  [ normalize #h #t #H destruct
76  | normalize #h1 #t1 #h2 #h3 #H destruct
77  ]
78] qed.
79
80(* some useful stuff for quantifiers *)
81
82lemma dec_bounded_forall:
83  ∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ∀k.(∀n.n < k → P n) + ¬(∀n.n < k → P n).
84 #P #HP_dec #k elim k -k
85 [ %1 #n #Hn @⊥ @(absurd (n < 0) Hn) @not_le_Sn_O
86 | #k #Hind cases Hind
87   [ #Hk cases (HP_dec k)
88     [ #HPk %1 #n #Hn cases (le_to_or_lt_eq … Hn)
89       [ #H @(Hk … (le_S_S_to_le … H))
90       | #H >(injective_S … H) @HPk
91       ]
92     | #HPk %2 @nmk #Habs @(absurd (P k)) [ @(Habs … (le_n (S k))) | @HPk ]
93     ]
94   | #Hk %2 @nmk #Habs @(absurd (∀n.n<k→P n)) [ #n' #Hn' @(Habs … (le_S … Hn')) | @Hk ]
95   ]
96 ]
97qed.
98
99lemma dec_bounded_exists:
100  ∀P:ℕ→Prop.(∀n.(P n) + (¬P n)) → ∀k.(∃n.n < k ∧ P n) + ¬(∃n.n < k ∧ P n).
101 #P #HP_dec #k elim k -k
102 [ %2 @nmk #Habs elim Habs #n #Hn @(absurd (n < 0) (proj1 … Hn)) @not_le_Sn_O
103 | #k #Hind cases Hind
104   [ #Hk %1 elim Hk #n #Hn @(ex_intro … n) @conj [ @le_S @(proj1 … Hn) | @(proj2 … Hn) ]
105   | #Hk cases (HP_dec k)
106     [ #HPk %1 @(ex_intro … k) @conj [ @le_n | @HPk ]
107     | #HPk %2 @nmk #Habs elim Habs #n #Hn cases (le_to_or_lt_eq … (proj1 … Hn))
108       [ #H @(absurd (∃n.n < k ∧ P n)) [ @(ex_intro … n) @conj
109         [ @(le_S_S_to_le … H) | @(proj2 … Hn) ] | @Hk ]
110       | #H @(absurd (P k)) [ <(injective_S … H) @(proj2 … Hn) | @HPk ]
111       ] 
112     ]
113   ]
114 ]
115qed.
116
117(* Replace decision functions by result. *)
118
119lemma dec_true: ∀P:Prop.∀f:P + ¬P.∀p:P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inl ?? p')) → Q f.
120#P #f #p #Q #H cases f;
121[ @H
122| #np cut False [ @(absurd ? p np) | * ]
123] qed.
124
125lemma dec_false: ∀P:Prop.∀f:P + ¬P.∀p:¬P.∀Q:(P + ¬P) → Type[0]. (∀p'.Q (inr ?? p')) → Q f.
126#P #f #p #Q #H cases f;
127[ #np cut False [ @(absurd ? np p) | * ]
128| @H
129] qed.
130
131
132lemma not_exists_forall:
133  ∀k:ℕ.∀P:ℕ → Prop.¬(∃x.x < k ∧ P x) → ∀x.x < k → ¬P x.
134 #k #P #Hex #x #Hx @nmk #Habs @(absurd ? ? Hex) @(ex_intro … x)
135 @conj [ @Hx | @Habs ]
136qed.
137
138lemma not_forall_exists:
139  ∀k:ℕ.∀P:ℕ → Prop.(∀n.(P n) + (¬P n)) → ¬(∀x.x < k → P x) → ∃x.x < k ∧ ¬P x.
140 #k #P #Hdec elim k
141 [ #Hfa @⊥ @(absurd ?? Hfa) #z #Hz @⊥ @(absurd ? Hz) @not_le_Sn_O
142 | -k #k #Hind #Hfa cases (Hdec k)
143   [ #HP elim (Hind ?)
144     [ -Hind; #x #Hx @(ex_intro ?? x) @conj [ @le_S @(proj1 ?? Hx) | @(proj2 ?? Hx) ]
145     | @nmk #H @(absurd ?? Hfa) #x #Hx cases (le_to_or_lt_eq ?? Hx)
146       [ #H2 @H @(le_S_S_to_le … H2)
147       | #H2 >(injective_S … H2) @HP
148       ]
149     ]
150   | #HP @(ex_intro … k) @conj [ @le_n | @HP ]
151   ]
152 ]
153qed.
154
155lemma associative_orb : associative ? orb.
156*** // qed.
157
158lemma commutative_orb : commutative ? orb.
159** // qed.
160
161lemma associative_andb : associative ? andb.
162*** // qed.
163
164lemma commutative_andb : commutative ? andb.
165** // qed.
166
167
168lemma notb_false : ∀b.(¬b) = false → b = true.
169* [#_ % | normalize #EQ destruct]
170qed.
171
172lemma notb_true : ∀b.(¬b) = true → b = false.
173* [normalize #EQ destruct | #_ %]
174qed.
175
176
177
178notation > "Σ 〈 ident x : tyx, ident y : tyy 〉 . P" with precedence 20 for
179  @{'sigma (λ${fresh p}.
180    match ${fresh p} with [mk_Prod (${ident x} : $tyx) (${ident y} : $tyy) ⇒ $P])}.
181notation > "Σ 〈 ident x, ident y 〉 . P" with precedence 20 for
182  @{'sigma (λ${fresh p}.
183    match ${fresh p} with [mk_Prod ${ident x} ${ident y} ⇒ $P])}.
184notation > "Σ 〈 ident x : tyx, ident y : tyy, ident z : tyz 〉 . P" with precedence 20 for
185  @{'sigma (λ${fresh p1}.
186    match ${fresh p1} with [mk_Prod ${fresh p2} (${ident z} : $tyz) ⇒
187      match ${fresh p2} with [mk_Prod (${ident x} : $tyx) (${ident y} : $tyy) ⇒ $P]])}.
188notation > "Σ 〈 ident x , ident y , ident z 〉 . P" with precedence 20 for
189  @{'sigma (λ${fresh p1}.
190    match ${fresh p1} with [mk_Prod ${fresh p2} ${ident z} ⇒
191      match ${fresh p2} with [mk_Prod ${ident x} ${ident y} ⇒ $P]])}.
192
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