-(*
+ (*
||M|| This file is part of HELM, an Hypertextual, Electronic
||A|| Library of Mathematics, developed at the Computer Science
||T|| Department of the University of Bologna, Italy.
(* propositional equality *)
-inductive eq (A:Type[1]) (x:A) : A → Prop ≝
+inductive eq (A:Type[2]) (x:A) : A → Prop ≝
refl: eq A x x.
interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
+interpretation "leibniz reflexivity" 'refl = refl.
lemma eq_rect_r:
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → P x p.
#A #a #x #p (cases p) // qed.
lemma eq_ind_r :
∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
#A #a #P #p #x0 #p0; @(eq_rect_r ? ? ? p0) //; qed.
+lemma eq_rect_Type0_r:
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ #A #a #P #H #x #p lapply H lapply P
+ cases p; //; qed.
+
+lemma eq_rect_Type1_r:
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ #A #a #P #H #x #p lapply H lapply P
+ cases p; //; qed.
+
lemma eq_rect_Type2_r:
∀A.∀a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
- #A #a #P #H #x #p (generalize in match H) (generalize in match P)
+ #A #a #P #H #x #p lapply H lapply P
cases p; //; qed.
-theorem rewrite_l: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. x = y → P y.
+lemma eq_rect_Type3_r:
+ ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+ #A #a #P #H #x #p lapply H lapply P
+ cases p; //; qed.
+
+theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x = y → P y.
#A #x #P #Hx #y #Heq (cases Heq); //; qed.
theorem sym_eq: ∀A.∀x,y:A. x = y → y = x.
-#A #x #y #Heq @(rewrite_l A x (λz.z=x)); //; qed.
+#A #x #y #Heq @(rewrite_l A x (λz.z=x)) // qed.
-theorem rewrite_r: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. y = x → P y.
+theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y = x → P y.
#A #x #P #Hx #y #Heq (cases (sym_eq ? ? ? Heq)); //; qed.
theorem eq_coerc: ∀A,B:Type[0].A→(A=B)→B.
∀x1,x2:A.∀y1,y2:B. x1=x2 → y1=y2 → f x1 y1 = f x2 y2.
#A #B #C #f #x1 #x2 #y1 #y2 #E1 #E2 >E1; >E2; //; qed.
+lemma eq_f3: ∀A,B,C,D.∀f:A→B→C->D.
+∀x1,x2:A.∀y1,y2:B. ∀z1,z2:C. x1=x2 → y1=y2 → z1=z2 → f x1 y1 z1 = f x2 y2 z2.
+#A #B #C #D #f #x1 #x2 #y1 #y2 #z1 #z2 #E1 #E2 #E3 >E1; >E2; >E3 //; qed.
+
(* hint to genereric equality
definition eq_equality: equality ≝
mk_equality eq refl rewrite_l rewrite_r.
interpretation "exists" 'exists x = (ex ? x).
inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
- ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
+ ex2_intro: ∀ x:A. P x → Q x → ex2 A P Q.
+
+interpretation "exists on two predicates" 'exists2 x1 x2 = (ex2 ? x1 x2).
+
+lemma ex2_commute: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
+#A0 #P0 #P1 * /2 width=3/
+qed-.
(* iff *)
definition iff :=
λ A,B. (A → B) ∧ (B → A).
-interpretation "iff" 'iff a b = (iff a b).
+interpretation "iff" 'iff a b = (iff a b).
+
+lemma iff_sym: ∀A,B. A ↔ B → B ↔ A.
+#A #B * /3/ qed.
+
+lemma iff_trans:∀A,B,C. A ↔ B → B ↔ C → A ↔ C.
+#A #B #C * #H1 #H2 * #H3 #H4 % /3/ qed.
+
+lemma iff_not: ∀A,B. A ↔ B → ¬A ↔ ¬B.
+#A #B * #H1 #H2 % /3/ qed.
+
+lemma iff_and_l: ∀A,B,C. A ↔ B → C ∧ A ↔ C ∧ B.
+#A #B #C * #H1 #H2 % * /3/ qed.
+
+lemma iff_and_r: ∀A,B,C. A ↔ B → A ∧ C ↔ B ∧ C.
+#A #B #C * #H1 #H2 % * /3/ qed.
+
+lemma iff_or_l: ∀A,B,C. A ↔ B → C ∨ A ↔ C ∨ B.
+#A #B #C * #H1 #H2 % * /3/ qed.
+
+lemma iff_or_r: ∀A,B,C. A ↔ B → A ∨ C ↔ B ∨ C.
+#A #B #C * #H1 #H2 % * /3/ qed.
(* cose per destruct: da rivedere *)
definition R1 ≝ eq_rect_Type0.
-(*
+(* used for lambda-delta *)
definition R2 :
∀T0:Type[0].
∀a0:T0.
∀b1: T1 b0 e0.
∀e1:R1 ?? T1 a1 ? e0 = b1.
T2 b0 e0 b1 e1.
-#T0;#a0;#T1;#a1;#T2;#a2;#b0;#e0;#b1;#e1;
-napply (eq_rect_Type0 ????? e1);
-napply (R1 ?? ? ?? e0);
-napply a2;
-nqed.
+#T0 #a0 #T1 #a1 #T2 #a2 #b0 #e0 #b1 #e1
+@(eq_rect_Type0 ????? e1)
+@(R1 ?? ? ?? e0)
+@a2
+qed.
-ndefinition R3 :
+definition R3 :
∀T0:Type[0].
∀a0:T0.
∀T1:∀x0:T0. a0=x0 → Type[0].
∀b2: T2 b0 e0 b1 e1.
∀e2:R2 ???? T2 a2 b0 e0 ? e1 = b2.
T3 b0 e0 b1 e1 b2 e2.
-#T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#b0;#e0;#b1;#e1;#b2;#e2;
-napply (eq_rect_Type0 ????? e2);
-napply (R2 ?? ? ???? e0 ? e1);
-napply a3;
-nqed.
+#T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #b0 #e0 #b1 #e1 #b2 #e2
+@(eq_rect_Type0 ????? e2)
+@(R2 ?? ? ???? e0 ? e1)
+@a3
+qed.
-ndefinition R4 :
+definition R4 :
∀T0:Type[0].
∀a0:T0.
∀T1:∀x0:T0. eq T0 a0 x0 → Type[0].
∀b3: T3 b0 e0 b1 e1 b2 e2.
∀e3:eq (T3 …) (R3 T0 a0 T1 a1 T2 a2 T3 a3 b0 e0 b1 e1 b2 e2) b3.
T4 b0 e0 b1 e1 b2 e2 b3 e3.
-#T0;#a0;#T1;#a1;#T2;#a2;#T3;#a3;#T4;#a4;#b0;#e0;#b1;#e1;#b2;#e2;#b3;#e3;
-napply (eq_rect_Type0 ????? e3);
-napply (R3 ????????? e0 ? e1 ? e2);
-napply a4;
-nqed.
-
-naxiom streicherK : ∀T:Type[2].∀t:T.∀P:t = t → Type[2].P (refl ? t) → ∀p.P p. *)
+#T0 #a0 #T1 #a1 #T2 #a2 #T3 #a3 #T4 #a4 #b0 #e0 #b1 #e1 #b2 #e2 #b3 #e3
+@(eq_rect_Type0 ????? e3)
+@(R3 ????????? e0 ? e1 ? e2)
+@a4
+qed.
+
+definition eqProp ≝ λA:Prop.eq A.
+
+(* Example to avoid indexing and the consequential creation of ill typed
+ terms during paramodulation *)
+lemma lemmaK : ∀A.∀x:A.∀h:x=x. eqProp ? h (refl A x).
+#A #x #h @(refl ? h: eqProp ? ? ?).
+qed-.
+
+theorem streicherK : ∀T:Type[2].∀t:T.∀P:t = t → Type[3].P (refl ? t) → ∀p.P p.
+ #T #t #P #H #p >(lemmaK T t p) @H
+qed-.
projects / helm.git / blobdiff