2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
8 \ / This file is distributed under the terms of the
9 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "basics/pts.ma".
13 include "hints_declaration.ma".
15 (* propositional equality *)
17 inductive eq (A:Type[1]) (x:A) : A → Prop ≝
20 interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
24 ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. x = a → Type[2]. P a (refl A a) → P x p.
25 #A #a #x #p (cases p) // qed.
28 ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl A a) →
30 #A #a #P #p #x0 #p0; @(eq_rect_r ? ? ? p0) //; qed.
32 lemma eq_rect_Type2_r:
33 ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) →
35 #A #a #P #H #x #p (generalize in match H) (generalize in match P)
38 theorem rewrite_l: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. x = y → P y.
39 #A #x #P #Hx #y #Heq (cases Heq); //; qed.
41 theorem sym_eq: ∀A.∀x,y:A. x = y → y = x.
42 #A #x #y #Heq @(rewrite_l A x (λz.z=x)); //; qed.
44 theorem rewrite_r: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. y = x → P y.
45 #A #x #P #Hx #y #Heq (cases (sym_eq ? ? ? Heq)); //; qed.
47 theorem eq_coerc: ∀A,B:Type[0].A→(A=B)→B.
48 #A #B #Ha #Heq (elim Heq); //; qed.
50 theorem trans_eq : ∀A.∀x,y,z:A. x = y → y = z → x = z.
51 #A #x #y #z #H1 #H2 >H1; //; qed.
53 theorem eq_f: ∀A,B.∀f:A→B.∀x,y:A. x=y → f x = f y.
54 #A #B #f #x #y #H >H; //; qed.
56 (* deleterio per auto? *)
57 theorem eq_f2: ∀A,B,C.∀f:A→B→C.
58 ∀x1,x2:A.∀y1,y2:B. x1=x2 → y1=y2 → f x1 y1 = f x2 y2.
59 #A #B #C #f #x1 #x2 #y1 #y2 #E1 #E2 >E1; >E2; //; qed.
61 (* hint to genereric equality
62 definition eq_equality: equality ≝
63 mk_equality eq refl rewrite_l rewrite_r.
66 unification hint 0 ≔ T,a,b;
68 (*------------------------------------*) ⊢
69 equal X T a b ≡ eq T a b.
72 (********** connectives ********)
74 inductive True: Prop ≝
77 inductive False: Prop ≝ .
79 (* ndefinition Not: Prop → Prop ≝
82 inductive Not (A:Prop): Prop ≝
83 nmk: (A → False) → Not A.
85 interpretation "logical not" 'not x = (Not x).
87 theorem absurd : ∀A:Prop. A → ¬A → False.
88 #A #H #Hn (elim Hn); /2/; qed.
91 ntheorem absurd : ∀ A,C:Prop. A → ¬A → C.
92 #A; #C; #H; #Hn; nelim (Hn H).
95 theorem not_to_not : ∀A,B:Prop. (A → B) → ¬B →¬A.
99 interpretation "leibnitz's non-equality" 'neq t x y = (Not (eq t x y)).
101 theorem sym_not_eq: ∀A.∀x,y:A. x ≠ y → y ≠ x.
105 inductive And (A,B:Prop) : Prop ≝
106 conj : A → B → And A B.
108 interpretation "logical and" 'and x y = (And x y).
110 theorem proj1: ∀A,B:Prop. A ∧ B → A.
111 #A #B #AB (elim AB) //; qed.
113 theorem proj2: ∀ A,B:Prop. A ∧ B → B.
114 #A #B #AB (elim AB) //; qed.
117 inductive Or (A,B:Prop) : Prop ≝
118 or_introl : A → (Or A B)
119 | or_intror : B → (Or A B).
121 interpretation "logical or" 'or x y = (Or x y).
123 definition decidable : Prop → Prop ≝
127 inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
128 ex_intro: ∀ x:A. P x → ex A P.
130 interpretation "exists" 'exists x = (ex ? x).
132 inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
133 ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
137 λ A,B. (A → B) ∧ (B → A).
139 interpretation "iff" 'iff a b = (iff a b).
141 (* cose per destruct: da rivedere *)
143 definition R0 ≝ λT:Type[0].λt:T.t.
145 definition R1 ≝ eq_rect_Type0.
151 ∀T1:∀x0:T0. a0=x0 → Type[0].
152 ∀a1:T1 a0 (refl ? a0).
153 ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
154 ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
158 ∀e1:R1 ?? T1 a1 ? e0 = b1.
160 #T0 #a0 #T1 #a1 #T2 #a2 #b0 #e0 #b1 #e1
161 @(eq_rect_Type0 ????? e1)
169 ∀T1:∀x0:T0. a0=x0 → Type[0].
170 ∀a1:T1 a0 (refl ? a0).
171 ∀T2:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0. R1 ?? T1 a1 ? p0 = x1 → Type[0].
172 ∀a2:T2 a0 (refl ? a0) a1 (refl ? a1).
173 ∀T3:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0.∀p1:R1 ?? T1 a1 ? p0 = x1.
174 ∀x2:T2 x0 p0 x1 p1.R2 ???? T2 a2 x0 p0 ? p1 = x2 → Type[0].
175 ∀a3:T3 a0 (refl ? a0) a1 (refl ? a1) a2 (refl ? a2).
179 ∀e1:R1 ?? T1 a1 ? e0 = b1.
181 ∀e2:R2 ???? T2 a2 b0 e0 ? e1 = b2.
182 T3 b0 e0 b1 e1 b2 e2.
183 #T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #b0 #e0 #b1 #e1 #b2 #e2
184 @(eq_rect_Type0 ????? e2)
185 @(R2 ?? ? ???? e0 ? e1)
192 ∀T1:∀x0:T0. eq T0 a0 x0 → Type[0].
193 ∀a1:T1 a0 (refl T0 a0).
194 ∀T2:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1 → Type[0].
195 ∀a2:T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1).
196 ∀T3:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
197 ∀x2:T2 x0 p0 x1 p1.eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2 → Type[0].
198 ∀a3:T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
199 a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2).
200 ∀T4:∀x0:T0. ∀p0:eq (T0 …) a0 x0. ∀x1:T1 x0 p0.∀p1:eq (T1 …) (R1 T0 a0 T1 a1 x0 p0) x1.
201 ∀x2:T2 x0 p0 x1 p1.∀p2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 x0 p0 x1 p1) x2.
202 ∀x3:T3 x0 p0 x1 p1 x2 p2.∀p3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 x0 p0 x1 p1 x2 p2) x3.
204 ∀a4:T4 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
205 a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2)
206 a3 (refl (T3 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)
207 a2 (refl (T2 a0 (refl T0 a0) a1 (refl (T1 a0 (refl T0 a0)) a1)) a2))
212 ∀e1:eq (T1 …) (R1 T0 a0 T1 a1 b0 e0) b1.
214 ∀e2:eq (T2 …) (R2 T0 a0 T1 a1 T2 a2 b0 e0 b1 e1) b2.
215 ∀b3: T3 b0 e0 b1 e1 b2 e2.
216 ∀e3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
217 T4 b0 e0 b1 e1 b2 e2 b3 e3.
218 #T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #T4 #a4 #b0 #e0 #b1 #e1 #b2 #e2 #b3 #e3
219 @(eq_rect_Type0 ????? e3)
220 @(R3 ????????? e0 ? e1 ? e2)
224 (* TODO concrete definition by means of proof irrelevance *)
225 axiom streicherK : ∀T:Type[1].∀t:T.∀P:t = t → Type[2].P (refl ? t) → ∀p.P p.