--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* ********************************************************************** *)
+(* Progetto FreeScale *)
+(* *)
+(* Sviluppato da: Ing. Cosimo Oliboni, oliboni@cs.unibo.it *)
+(* Sviluppo: 2008-2010 *)
+(* *)
+(* ********************************************************************** *)
+
+include "common/comp.ma".
+include "common/prod_base.ma".
+include "num/bool_lemmas.ma".
+
+(* ********* *)
+(* TUPLE x 2 *)
+(* ********* *)
+
+nlemma pair_destruct_1 :
+∀T1,T2.∀x1,x2:T1.∀y1,y2:T2.
+ pair T1 T2 x1 y1 = pair T1 T2 x2 y2 → x1 = x2.
+ #T1; #T2; #x1; #x2; #y1; #y2; #H;
+ nchange with (match pair T1 T2 x2 y2 with [ pair a _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma pair_destruct_2 :
+∀T1,T2.∀x1,x2:T1.∀y1,y2:T2.
+ pair T1 T2 x1 y1 = pair T1 T2 x2 y2 → y1 = y2.
+ #T1; #T2; #x1; #x2; #y1; #y2; #H;
+ nchange with (match pair T1 T2 x2 y2 with [ pair _ b ⇒ y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqpair :
+∀T1,T2:Type.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
+ (symmetricT T1 bool f1) →
+ (symmetricT T2 bool f2) →
+ (∀p1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = eq_pair T1 T2 f1 f2 p2 p1)).
+ #T1; #T2; #f1; #f2; #H; #H1;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqpair :
+∀T1,T2.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
+ (∀x,y:T1.x = y → f1 x y = true) →
+ (∀x,y:T2.x = y → f2 x y = true) →
+ (∀p1,p2:ProdT T1 T2.
+ (p1 = p2 → eq_pair T1 T2 f1 f2 p1 p2 = true)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nnormalize;
+ nrewrite > (H1 … (pair_destruct_1 … H));
+ nnormalize;
+ nrewrite > (H2 … (pair_destruct_2 … H));
+ napply refl_eq.
+nqed.
+
+nlemma eqpair_to_eq :
+∀T1,T2.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
+ (∀x,y:T1.f1 x y = true → x = y) →
+ (∀x,y:T2.f2 x y = true → x = y) →
+ (∀p1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = true → p1 = p2)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nnormalize in H:(%);
+ nletin K ≝ (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ #H3;
+ ##[ ##2: ndestruct (*napply (bool_destruct … H3)*) ##]
+ #H4;
+ nrewrite > (H4 (refl_eq …));
+ nrewrite > (H2 y1 y2 H3);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_pair :
+∀T1,T2.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:ProdT T1 T2.decidable (x = y)).
+ #T1; #T2; #H; #H1;
+ #x; nelim x; #xx1; #xx2;
+ #y; nelim y; #yy1; #yy2;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 ≠ yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H2; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H3; napply (H2 (pair_destruct_1 T1 T2 … H3))
+ ##| ##1: #H2; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H4; napply (H3 (pair_destruct_2 T1 T2 … H4))
+ ##| ##1: #H3; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H2; nrewrite > H3; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma neqpair_to_neq :
+∀T1,T2.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
+ (∀x,y:T1.f1 x y = false → x ≠ y) →
+ (∀x,y:T2.f2 x y = false → x ≠ y) →
+ (∀p1,p2:ProdT T1 T2.
+ (eq_pair T1 T2 f1 f2 p1 p2 = false → p1 ≠ p2)).
+ #T1; #T2; #f1; #f2; #H1; #H2;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2;
+ nchange with ((((f1 x1 x2) ⊗ (f2 y1 y2)) = false) → ?); #H;
+ nnormalize; #H3;
+ napply (or2_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ? (andb_false2 … H) ?);
+ ##[ ##1: #H4; napply (H1 x1 x2 H4); napply (pair_destruct_1 T1 T2 … H3)
+ ##| ##2: #H4; napply (H2 y1 y2 H4); napply (pair_destruct_2 T1 T2 … H3)
+ ##]
+nqed.
+
+nlemma pair_destruct :
+∀T1,T2.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x1,x2:T1.∀y1,y2:T2.
+ (pair T1 T2 x1 y1) ≠ (pair T1 T2 x2 y2) → x1 ≠ x2 ∨ y1 ≠ y2).
+ #T1; #T2; #H1; #H2; #x1; #x2; #y1; #y2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 ≠ x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H3; napply (or2_intro1 … H3)
+ ##| ##1: #H3; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H4; napply (or2_intro2 … H4)
+ ##| ##1: #H4; nrewrite > H3 in H:(%);
+ nrewrite > H4; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqpair :
+∀T1,T2.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T1.x ≠ y → f1 x y = false) →
+ (∀x,y:T2.x ≠ y → f2 x y = false) →
+ (∀p1,p2:ProdT T1 T2.
+ (p1 ≠ p2 → eq_pair T1 T2 f1 f2 p1 p2 = false)).
+ #T1; #T2; #f1; #f2; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1;
+ #p2; nelim p2; #x2; #y2; #H;
+ nchange with (((f1 x1 x2) ⊗ (f2 y1 y2)) = false);
+ napply (or2_elim (x1 ≠ x2) (y1 ≠ y2) ? (pair_destruct T1 T2 H1 H2 … H) ?);
+ ##[ ##2: #H5; nrewrite > (H4 … H5); nrewrite > (andb_false2_2 (f1 x1 x2)); napply refl_eq
+ ##| ##1: #H5; nrewrite > (H3 … H5); nnormalize; napply refl_eq
+ ##]
+nqed.
+
+nlemma pair_is_comparable :
+ comparable → comparable → comparable.
+ #T1; #T2; napply (mk_comparable (ProdT T1 T2));
+ ##[ napply (pair … (zeroc T1) (zeroc T2))
+ ##| napply (λx.false)
+ ##| napply (eq_pair … (eqc T1) (eqc T2))
+ ##| napply (eqpair_to_eq … (eqc T1) (eqc T2));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##]
+ ##| napply (eq_to_eqpair … (eqc T1) (eqc T2));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##]
+ ##| napply (neqpair_to_neq … (eqc T1) (eqc T2));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##]
+ ##| napply (neq_to_neqpair … (eqc T1) (eqc T2));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##]
+ ##| napply decidable_pair;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##]
+ ##| napply symmetric_eqpair;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ S1: comparable, S2: comparable;
+ T1 ≟ (carr S1), T2 ≟ (carr S2),
+ X ≟ (pair_is_comparable S1 S2)
+ (*********************************************) ⊢
+ carr X ≡ ProdT T1 T2.
+
+(* ********* *)
+(* TUPLE x 3 *)
+(* ********* *)
+
+nlemma triple_destruct_1 :
+∀T1,T2,T3.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 → x1 = x2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple a _ _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma triple_destruct_2 :
+∀T1,T2,T3.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 → y1 = y2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple _ b _ ⇒ y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma triple_destruct_3 :
+∀T1,T2,T3.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.
+ triple T1 T2 T3 x1 y1 z1 = triple T1 T2 T3 x2 y2 z2 → z1 = z2.
+ #T1; #T2; #T3; #x1; #x2; #y1; #y2; #z1; #z2; #H;
+ nchange with (match triple T1 T2 T3 x2 y2 z2 with [ triple _ _ c ⇒ z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqtriple :
+∀T1,T2,T3:Type.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
+ (symmetricT T1 bool f1) →
+ (symmetricT T2 bool f2) →
+ (symmetricT T3 bool f3) →
+ (∀p1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = eq_triple T1 T2 T3 f1 f2 f3 p2 p1)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H; #H1; #H2;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqtriple :
+∀T1,T2,T3.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
+ (∀x1,x2:T1.x1 = x2 → f1 x1 x2 = true) →
+ (∀y1,y2:T2.y1 = y2 → f2 y1 y2 = true) →
+ (∀z1,z2:T3.z1 = z2 → f3 z1 z2 = true) →
+ (∀p1,p2:Prod3T T1 T2 T3.
+ (p1 = p2 → eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nnormalize;
+ nrewrite > (H1 … (triple_destruct_1 … H));
+ nnormalize;
+ nrewrite > (H2 … (triple_destruct_2 … H));
+ nnormalize;
+ nrewrite > (H3 … (triple_destruct_3 … H));
+ napply refl_eq.
+nqed.
+
+nlemma eqtriple_to_eq :
+∀T1,T2,T3.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
+ (∀x1,x2:T1.f1 x1 x2 = true → x1 = x2) →
+ (∀y1,y2:T2.f2 y1 y2 = true → y1 = y2) →
+ (∀z1,z2:T3.f3 z1 z2 = true → z1 = z2) →
+ (∀p1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true → p1 = p2)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nnormalize in H:(%);
+ nletin K ≝ (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H4; ndestruct (*napply (bool_destruct … H4)*) ##]
+ nletin K1 ≝ (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H4; #H5; ndestruct (*napply (bool_destruct … H5)*) ##]
+ #H4; #H5; #H6;
+ nrewrite > (H4 (refl_eq …));
+ nrewrite > (H6 (refl_eq …));
+ nrewrite > (H3 z1 z2 H5);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_triple :
+∀T1,T2,T3.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:Prod3T T1 T2 T3.decidable (x = y)).
+ #T1; #T2; #T3; #H; #H1; #H2;
+ #x; nelim x; #xx1; #xx2; #xx3;
+ #y; nelim y; #yy1; #yy2; #yy3;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 ≠ yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H4; napply (H3 (triple_destruct_1 T1 T2 T3 … H4))
+ ##| ##1: #H3; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H4; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H5; napply (H4 (triple_destruct_2 T1 T2 T3 … H5))
+ ##| ##1: #H4; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (triple_destruct_3 T1 T2 T3 … H6))
+ ##| ##1: #H5; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H3;
+ nrewrite > H4;
+ nrewrite > H5;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqtriple_to_neq :
+∀T1,T2,T3.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
+ (∀x,y:T1.f1 x y = false → x ≠ y) →
+ (∀x,y:T2.f2 x y = false → x ≠ y) →
+ (∀x,y:T3.f3 x y = false → x ≠ y) →
+ (∀p1,p2:Prod3T T1 T2 T3.
+ (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = false → p1 ≠ p2)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2;
+ nchange with ((((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2)) = false) → ?); #H;
+ nnormalize; #H4;
+ napply (or3_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ? (andb_false3 … H) ?);
+ ##[ ##1: #H5; napply (H1 x1 x2 H5); napply (triple_destruct_1 T1 T2 T3 … H4)
+ ##| ##2: #H5; napply (H2 y1 y2 H5); napply (triple_destruct_2 T1 T2 T3 … H4)
+ ##| ##3: #H5; napply (H3 z1 z2 H5); napply (triple_destruct_3 T1 T2 T3 … H4)
+ ##]
+nqed.
+
+nlemma triple_destruct :
+∀T1,T2,T3.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.
+ (triple T1 T2 T3 x1 y1 z1) ≠ (triple T1 T2 T3 x2 y2 z2) →
+ Or3 (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2)).
+ #T1; #T2; #T3; #H1; #H2; #H3; #x1; #x2; #y1; #y2; #z1; #z2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 ≠ x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H4; napply (or3_intro1 … H4)
+ ##| ##1: #H4; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H5; napply (or3_intro2 … H5)
+ ##| ##1: #H5; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H6; napply (or3_intro3 … H6)
+ ##| ##1: #H6; nrewrite > H4 in H:(%);
+ nrewrite > H5;
+ nrewrite > H6; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqtriple :
+∀T1,T2,T3.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T1.x ≠ y → f1 x y = false) →
+ (∀x,y:T2.x ≠ y → f2 x y = false) →
+ (∀x,y:T3.x ≠ y → f3 x y = false) →
+ (∀p1,p2:Prod3T T1 T2 T3.
+ (p1 ≠ p2 → eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = false)).
+ #T1; #T2; #T3; #f1; #f2; #f3; #H1; #H2; #H3; #H4; #H5; #H6;
+ #p1; nelim p1; #x1; #y1; #z1;
+ #p2; nelim p2; #x2; #y2; #z2; #H;
+ nchange with (((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2)) = false);
+ napply (or3_elim (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2) ? (triple_destruct T1 T2 T3 H1 H2 H3 … H) ?);
+ ##[ ##1: #H7; nrewrite > (H4 … H7); nrewrite > (andb_false3_1 (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##| ##2: #H7; nrewrite > (H5 … H7); nrewrite > (andb_false3_2 (f1 x1 x2) (f3 z1 z2)); napply refl_eq
+ ##| ##3: #H7; nrewrite > (H6 … H7); nrewrite > (andb_false3_3 (f1 x1 x2) (f2 y1 y2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma triple_is_comparable :
+ comparable → comparable → comparable → comparable.
+ #T1; #T2; #T3; napply (mk_comparable (Prod3T T1 T2 T3));
+ ##[ napply (triple … (zeroc T1) (zeroc T2) (zeroc T3))
+ ##| napply (λx.false)
+ ##| napply (eq_triple … (eqc T1) (eqc T2) (eqc T3))
+ ##| napply (eqtriple_to_eq … (eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##]
+ ##| napply (eq_to_eqtriple … (eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##]
+ ##| napply (neqtriple_to_neq … (eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##]
+ ##| napply (neq_to_neqtriple … (eqc T1) (eqc T2) (eqc T3));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##]
+ ##| napply decidable_triple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##]
+ ##| napply symmetric_eqtriple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ S1: comparable, S2: comparable, S3: comparable;
+ T1 ≟ (carr S1), T2 ≟ (carr S2), T3 ≟ (carr S3),
+ X ≟ (triple_is_comparable S1 S2 S3)
+ (*********************************************) ⊢
+ carr X ≡ Prod3T T1 T2 T3.
+
+(* ********* *)
+(* TUPLE x 4 *)
+(* ********* *)
+
+nlemma quadruple_destruct_1 :
+∀T1,T2,T3,T4.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 → x1 = x2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple a _ _ _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_2 :
+∀T1,T2,T3,T4.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 → y1 = y2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ b _ _ ⇒ y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_3 :
+∀T1,T2,T3,T4.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 → z1 = z2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ _ c _ ⇒ z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quadruple_destruct_4 :
+∀T1,T2,T3,T4.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.
+ quadruple T1 T2 T3 T4 x1 y1 z1 v1 = quadruple T1 T2 T3 T4 x2 y2 z2 v2 → v1 = v2.
+ #T1; #T2; #T3; #T4; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #H;
+ nchange with (match quadruple T1 T2 T3 T4 x2 y2 z2 v2 with [ quadruple _ _ _ d ⇒ v1 = d ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqquadruple :
+∀T1,T2,T3,T4:Type.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.
+ (symmetricT T1 bool f1) →
+ (symmetricT T2 bool f2) →
+ (symmetricT T3 bool f3) →
+ (symmetricT T4 bool f4) →
+ (∀p1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p2 p1)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H; #H1; #H2; #H3;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2);
+ ncases (f3 z2 z1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H3 v1 v2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqquadruple :
+∀T1,T2,T3,T4.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.
+ (∀x,y:T1.x = y → f1 x y = true) →
+ (∀x,y:T2.x = y → f2 x y = true) →
+ (∀x,y:T3.x = y → f3 x y = true) →
+ (∀x,y:T4.x = y → f4 x y = true) →
+ (∀p1,p2:Prod4T T1 T2 T3 T4.
+ (p1 = p2 → eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = true)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nnormalize;
+ nrewrite > (H1 … (quadruple_destruct_1 … H));
+ nnormalize;
+ nrewrite > (H2 … (quadruple_destruct_2 … H));
+ nnormalize;
+ nrewrite > (H3 … (quadruple_destruct_3 … H));
+ nnormalize;
+ nrewrite > (H4 … (quadruple_destruct_4 … H));
+ napply refl_eq.
+nqed.
+
+nlemma eqquadruple_to_eq :
+∀T1,T2,T3,T4.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.
+ (∀x,y:T1.f1 x y = true → x = y) →
+ (∀x,y:T2.f2 x y = true → x = y) →
+ (∀x,y:T3.f3 x y = true → x = y) →
+ (∀x,y:T4.f4 x y = true → x = y) →
+ (∀p1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = true → p1 = p2)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nnormalize in H:(%);
+ nletin K ≝ (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H5; ndestruct (*napply (bool_destruct … H5)*) ##]
+ nletin K1 ≝ (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H5; #H6; ndestruct (*napply (bool_destruct … H6)*) ##]
+ nletin K2 ≝ (H3 z1 z2);
+ ncases (f3 z1 z2) in K2:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H5; #H6; #H7; ndestruct (*napply (bool_destruct … H7)*) ##]
+ #H5; #H6; #H7; #H8;
+ nrewrite > (H5 (refl_eq …));
+ nrewrite > (H6 (refl_eq …));
+ nrewrite > (H8 (refl_eq …));
+ nrewrite > (H4 v1 v2 H7);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_quadruple :
+∀T1,T2,T3,T4.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x,y:Prod4T T1 T2 T3 T4.decidable (x = y)).
+ #T1; #T2; #T3; #T4; #H; #H1; #H2; #H3;
+ #x; nelim x; #xx1; #xx2; #xx3; #xx4;
+ #y; nelim y; #yy1; #yy2; #yy3; #yy4;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 ≠ yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H4; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H5; napply (H4 (quadruple_destruct_1 T1 T2 T3 T4 … H5))
+ ##| ##1: #H4; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (quadruple_destruct_2 T1 T2 T3 T4 … H6))
+ ##| ##1: #H5; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H6; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H7; napply (H6 (quadruple_destruct_3 T1 T2 T3 T4 … H7))
+ ##| ##1: #H6; napply (or2_elim (xx4 = yy4) (xx4 ≠ yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##2: #H7; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H8; napply (H7 (quadruple_destruct_4 T1 T2 T3 T4 … H8))
+ ##| ##1: #H7; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H4;
+ nrewrite > H5;
+ nrewrite > H6;
+ nrewrite > H7;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqquadruple_to_neq :
+∀T1,T2,T3,T4.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.
+ (∀x,y:T1.f1 x y = false → x ≠ y) →
+ (∀x,y:T2.f2 x y = false → x ≠ y) →
+ (∀x,y:T3.f3 x y = false → x ≠ y) →
+ (∀x,y:T4.f4 x y = false → x ≠ y) →
+ (∀p1,p2:Prod4T T1 T2 T3 T4.
+ (eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = false → p1 ≠ p2)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2;
+ nchange with ((((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2) ⊗ (f4 v1 v2)) = false) → ?); #H;
+ nnormalize; #H5;
+ napply (or4_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ((f4 v1 v2) = false) ? (andb_false4 … H) ?);
+ ##[ ##1: #H6; napply (H1 x1 x2 H6); napply (quadruple_destruct_1 T1 T2 T3 T4 … H5)
+ ##| ##2: #H6; napply (H2 y1 y2 H6); napply (quadruple_destruct_2 T1 T2 T3 T4 … H5)
+ ##| ##3: #H6; napply (H3 z1 z2 H6); napply (quadruple_destruct_3 T1 T2 T3 T4 … H5)
+ ##| ##4: #H6; napply (H4 v1 v2 H6); napply (quadruple_destruct_4 T1 T2 T3 T4 … H5)
+ ##]
+nqed.
+
+nlemma quadruple_destruct :
+∀T1,T2,T3,T4.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.
+ (quadruple T1 T2 T3 T4 x1 y1 z1 v1) ≠ (quadruple T1 T2 T3 T4 x2 y2 z2 v2) →
+ Or4 (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2) (v1 ≠ v2)).
+ #T1; #T2; #T3; #T4; #H1; #H2; #H3; #H4;
+ #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 ≠ x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H5; napply (or4_intro1 … H5)
+ ##| ##1: #H5; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H6; napply (or4_intro2 … H6)
+ ##| ##1: #H6; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H7; napply (or4_intro3 … H7)
+ ##| ##1: #H7; napply (or2_elim (v1 = v2) (v1 ≠ v2) ? (H4 v1 v2) ?);
+ ##[ ##2: #H8; napply (or4_intro4 … H8)
+ ##| ##1: #H8; nrewrite > H5 in H:(%);
+ nrewrite > H6;
+ nrewrite > H7;
+ nrewrite > H8; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqquadruple :
+∀T1,T2,T3,T4.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x,y:T1.x ≠ y → f1 x y = false) →
+ (∀x,y:T2.x ≠ y → f2 x y = false) →
+ (∀x,y:T3.x ≠ y → f3 x y = false) →
+ (∀x,y:T4.x ≠ y → f4 x y = false) →
+ (∀p1,p2:Prod4T T1 T2 T3 T4.
+ (p1 ≠ p2 → eq_quadruple T1 T2 T3 T4 f1 f2 f3 f4 p1 p2 = false)).
+ #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8;
+ #p1; nelim p1; #x1; #y1; #z1; #v1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #H;
+ nchange with (((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2) ⊗ (f4 v1 v2)) = false);
+ napply (or4_elim (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2) (v1 ≠ v2) ? (quadruple_destruct T1 T2 T3 T4 H1 H2 H3 H4 … H) ?);
+ ##[ ##1: #H9; nrewrite > (H5 … H9); nrewrite > (andb_false4_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##2: #H9; nrewrite > (H6 … H9); nrewrite > (andb_false4_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##3: #H9; nrewrite > (H7 … H9); nrewrite > (andb_false4_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2)); napply refl_eq
+ ##| ##4: #H9; nrewrite > (H8 … H9); nrewrite > (andb_false4_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma quadruple_is_comparable :
+ comparable → comparable → comparable → comparable → comparable.
+ #T1; #T2; #T3; #T4; napply (mk_comparable (Prod4T T1 T2 T3 T4));
+ ##[ napply (quadruple … (zeroc T1) (zeroc T2) (zeroc T3) (zeroc T4))
+ ##| napply (λx.false)
+ ##| napply (eq_quadruple … (eqc T1) (eqc T2) (eqc T3) (eqc T4))
+ ##| napply (eqquadruple_to_eq … (eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##| napply (eqc_to_eq T4)
+ ##]
+ ##| napply (eq_to_eqquadruple … (eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##| napply (eq_to_eqc T4)
+ ##]
+ ##| napply (neqquadruple_to_neq … (eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##| napply (neqc_to_neq T4)
+ ##]
+ ##| napply (neq_to_neqquadruple … (eqc T1) (eqc T2) (eqc T3) (eqc T4));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##| napply (neq_to_neqc T4)
+ ##]
+ ##| napply decidable_quadruple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##]
+ ##| napply symmetric_eqquadruple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##| napply (symmetric_eqc T4)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ S1: comparable, S2: comparable, S3: comparable, S4: comparable;
+ T1 ≟ (carr S1), T2 ≟ (carr S2), T3 ≟ (carr S3), T4 ≟ (carr S4),
+ X ≟ (quadruple_is_comparable S1 S2 S3 S4)
+ (*********************************************) ⊢
+ carr X ≡ Prod4T T1 T2 T3 T4.
+
+(* ********* *)
+(* TUPLE x 5 *)
+(* ********* *)
+
+nlemma quintuple_destruct_1 :
+∀T1,T2,T3,T4,T5.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 → x1 = x2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple a _ _ _ _ ⇒ x1 = a ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_2 :
+∀T1,T2,T3,T4,T5.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 → y1 = y2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ b _ _ _ ⇒ y1 = b ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_3 :
+∀T1,T2,T3,T4,T5.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 → z1 = z2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ c _ _ ⇒ z1 = c ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_4 :
+∀T1,T2,T3,T4,T5.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 → v1 = v2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ _ d _ ⇒ v1 = d ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma quintuple_destruct_5 :
+∀T1,T2,T3,T4,T5.∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1 = quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 → w1 = w2.
+ #T1; #T2; #T3; #T4; #T5; #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2; #H;
+ nchange with (match quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2 with [ quintuple _ _ _ _ e ⇒ w1 = e ]);
+ nrewrite < H;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma symmetric_eqquintuple :
+∀T1,T2,T3,T4,T5:Type.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.∀f5:T5 → T5 → bool.
+ (symmetricT T1 bool f1) →
+ (symmetricT T2 bool f2) →
+ (symmetricT T3 bool f3) →
+ (symmetricT T4 bool f4) →
+ (symmetricT T5 bool f5) →
+ (∀p1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p2 p1)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H; #H1; #H2; #H3; #H4;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2;
+ nnormalize;
+ nrewrite > (H x1 x2);
+ ncases (f1 x2 x1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H1 y1 y2);
+ ncases (f2 y2 y1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H2 z1 z2);
+ ncases (f3 z2 z1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H3 v1 v2);
+ ncases (f4 v2 v1);
+ nnormalize;
+ ##[ ##1: nrewrite > (H4 w1 w2); napply refl_eq
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+ ##| ##2: napply refl_eq
+ ##]
+nqed.
+
+nlemma eq_to_eqquintuple :
+∀T1,T2,T3,T4,T5.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.∀f5:T5 → T5 → bool.
+ (∀x,y:T1.x = y → f1 x y = true) →
+ (∀x,y:T2.x = y → f2 x y = true) →
+ (∀x,y:T3.x = y → f3 x y = true) →
+ (∀x,y:T4.x = y → f4 x y = true) →
+ (∀x,y:T5.x = y → f5 x y = true) →
+ (∀p1,p2:Prod5T T1 T2 T3 T4 T5.
+ (p1 = p2 → eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = true)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nnormalize;
+ nrewrite > (H1 … (quintuple_destruct_1 … H));
+ nnormalize;
+ nrewrite > (H2 … (quintuple_destruct_2 … H));
+ nnormalize;
+ nrewrite > (H3 … (quintuple_destruct_3 … H));
+ nnormalize;
+ nrewrite > (H4 … (quintuple_destruct_4 … H));
+ nnormalize;
+ nrewrite > (H5 … (quintuple_destruct_5 … H));
+ napply refl_eq.
+nqed.
+
+nlemma eqquintuple_to_eq :
+∀T1,T2,T3,T4,T5.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.∀f5:T5 → T5 → bool.
+ (∀x,y:T1.f1 x y = true → x = y) →
+ (∀x,y:T2.f2 x y = true → x = y) →
+ (∀x,y:T3.f3 x y = true → x = y) →
+ (∀x,y:T4.f4 x y = true → x = y) →
+ (∀x,y:T5.f5 x y = true → x = y) →
+ (∀p1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = true → p1 = p2)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nnormalize in H:(%);
+ nletin K ≝ (H1 x1 x2);
+ ncases (f1 x1 x2) in H:(%) K:(%);
+ nnormalize;
+ ##[ ##2: #H6; ndestruct (*napply (bool_destruct … H6)*) ##]
+ nletin K1 ≝ (H2 y1 y2);
+ ncases (f2 y1 y2) in K1:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H6; #H7; ndestruct (*napply (bool_destruct … H7)*) ##]
+ nletin K2 ≝ (H3 z1 z2);
+ ncases (f3 z1 z2) in K2:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H6; #H7; #H8; ndestruct (*napply (bool_destruct … H8)*) ##]
+ nletin K3 ≝ (H4 v1 v2);
+ ncases (f4 v1 v2) in K3:(%) ⊢ %;
+ nnormalize;
+ ##[ ##2: #H6; #H7; #H8; #H9; ndestruct (*napply (bool_destruct … H9)*) ##]
+ #H6; #H7; #H8; #H9; #H10;
+ nrewrite > (H6 (refl_eq …));
+ nrewrite > (H7 (refl_eq …));
+ nrewrite > (H8 (refl_eq …));
+ nrewrite > (H10 (refl_eq …));
+ nrewrite > (H5 w1 w2 H9);
+ napply refl_eq.
+nqed.
+
+nlemma decidable_quintuple :
+∀T1,T2,T3,T4,T5.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x,y:T5.decidable (x = y)) →
+ (∀x,y:Prod5T T1 T2 T3 T4 T5.decidable (x = y)).
+ #T1; #T2; #T3; #T4; #T5; #H; #H1; #H2; #H3; #H4;
+ #x; nelim x; #xx1; #xx2; #xx3; #xx4; #xx5;
+ #y; nelim y; #yy1; #yy2; #yy3; #yy4; #yy5;
+ nnormalize;
+ napply (or2_elim (xx1 = yy1) (xx1 ≠ yy1) ? (H xx1 yy1) ?);
+ ##[ ##2: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (quintuple_destruct_1 T1 T2 T3 T4 T5 … H6))
+ ##| ##1: #H5; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##2: #H6; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H7; napply (H6 (quintuple_destruct_2 T1 T2 T3 T4 T5 … H7))
+ ##| ##1: #H6; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##2: #H7; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H8; napply (H7 (quintuple_destruct_3 T1 T2 T3 T4 T5 … H8))
+ ##| ##1: #H7; napply (or2_elim (xx4 = yy4) (xx4 ≠ yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##2: #H8; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H9; napply (H8 (quintuple_destruct_4 T1 T2 T3 T4 T5 … H9))
+ ##| ##1: #H8; napply (or2_elim (xx5 = yy5) (xx5 ≠ yy5) ? (H4 xx5 yy5) ?);
+ ##[ ##2: #H9; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H10; napply (H9 (quintuple_destruct_5 T1 T2 T3 T4 T5 … H10))
+ ##| ##1: #H9; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H5;
+ nrewrite > H6;
+ nrewrite > H7;
+ nrewrite > H8;
+ nrewrite > H9;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqquintuple_to_neq :
+∀T1,T2,T3,T4,T5.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.∀f5:T5 → T5 → bool.
+ (∀x,y:T1.f1 x y = false → x ≠ y) →
+ (∀x,y:T2.f2 x y = false → x ≠ y) →
+ (∀x,y:T3.f3 x y = false → x ≠ y) →
+ (∀x,y:T4.f4 x y = false → x ≠ y) →
+ (∀x,y:T5.f5 x y = false → x ≠ y) →
+ (∀p1,p2:Prod5T T1 T2 T3 T4 T5.
+ (eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = false → p1 ≠ p2)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2;
+ nchange with ((((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2) ⊗ (f4 v1 v2) ⊗ (f5 w1 w2)) = false) → ?); #H;
+ nnormalize; #H6;
+ napply (or5_elim ((f1 x1 x2) = false) ((f2 y1 y2) = false) ((f3 z1 z2) = false) ((f4 v1 v2) = false) ((f5 w1 w2) = false) ? (andb_false5 … H) ?);
+ ##[ ##1: #H7; napply (H1 x1 x2 H7); napply (quintuple_destruct_1 T1 T2 T3 T4 T5 … H6)
+ ##| ##2: #H7; napply (H2 y1 y2 H7); napply (quintuple_destruct_2 T1 T2 T3 T4 T5 … H6)
+ ##| ##3: #H7; napply (H3 z1 z2 H7); napply (quintuple_destruct_3 T1 T2 T3 T4 T5 … H6)
+ ##| ##4: #H7; napply (H4 v1 v2 H7); napply (quintuple_destruct_4 T1 T2 T3 T4 T5 … H6)
+ ##| ##5: #H7; napply (H5 w1 w2 H7); napply (quintuple_destruct_5 T1 T2 T3 T4 T5 … H6)
+ ##]
+nqed.
+
+nlemma quintuple_destruct :
+∀T1,T2,T3,T4,T5.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x,y:T5.decidable (x = y)) →
+ (∀x1,x2:T1.∀y1,y2:T2.∀z1,z2:T3.∀v1,v2:T4.∀w1,w2:T5.
+ (quintuple T1 T2 T3 T4 T5 x1 y1 z1 v1 w1) ≠ (quintuple T1 T2 T3 T4 T5 x2 y2 z2 v2 w2) →
+ Or5 (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2) (v1 ≠ v2) (w1 ≠ w2)).
+ #T1; #T2; #T3; #T4; #T5; #H1; #H2; #H3; #H4; #H5;
+ #x1; #x2; #y1; #y2; #z1; #z2; #v1; #v2; #w1; #w2;
+ nnormalize; #H;
+ napply (or2_elim (x1 = x2) (x1 ≠ x2) ? (H1 x1 x2) ?);
+ ##[ ##2: #H6; napply (or5_intro1 … H6)
+ ##| ##1: #H6; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##2: #H7; napply (or5_intro2 … H7)
+ ##| ##1: #H7; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##2: #H8; napply (or5_intro3 … H8)
+ ##| ##1: #H8; napply (or2_elim (v1 = v2) (v1 ≠ v2) ? (H4 v1 v2) ?);
+ ##[ ##2: #H9; napply (or5_intro4 … H9)
+ ##| ##1: #H9; napply (or2_elim (w1 = w2) (w1 ≠ w2) ? (H5 w1 w2) ?);
+ ##[ ##2: #H10; napply (or5_intro5 … H10)
+ ##| ##1: #H10; nrewrite > H6 in H:(%);
+ nrewrite > H7;
+ nrewrite > H8;
+ nrewrite > H9;
+ nrewrite > H10; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqquintuple :
+∀T1,T2,T3,T4,T5.
+∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → bool.∀f5:T5 → T5 → bool.
+ (∀x,y:T1.decidable (x = y)) →
+ (∀x,y:T2.decidable (x = y)) →
+ (∀x,y:T3.decidable (x = y)) →
+ (∀x,y:T4.decidable (x = y)) →
+ (∀x,y:T5.decidable (x = y)) →
+ (∀x,y:T1.x ≠ y → f1 x y = false) →
+ (∀x,y:T2.x ≠ y → f2 x y = false) →
+ (∀x,y:T3.x ≠ y → f3 x y = false) →
+ (∀x,y:T4.x ≠ y → f4 x y = false) →
+ (∀x,y:T5.x ≠ y → f5 x y = false) →
+ (∀p1,p2:Prod5T T1 T2 T3 T4 T5.
+ (p1 ≠ p2 → eq_quintuple T1 T2 T3 T4 T5 f1 f2 f3 f4 f5 p1 p2 = false)).
+ #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5;
+ #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8; #H9; #H10;
+ #p1; nelim p1; #x1; #y1; #z1; #v1; #w1;
+ #p2; nelim p2; #x2; #y2; #z2; #v2; #w2; #H;
+ nchange with (((f1 x1 x2) ⊗ (f2 y1 y2) ⊗ (f3 z1 z2) ⊗ (f4 v1 v2) ⊗ (f5 w1 w2)) = false);
+ napply (or5_elim (x1 ≠ x2) (y1 ≠ y2) (z1 ≠ z2) (v1 ≠ v2) (w1 ≠ w2) ? (quintuple_destruct T1 T2 T3 T4 T5 H1 H2 H3 H4 H5 … H) ?);
+ ##[ ##1: #H11; nrewrite > (H6 … H11); nrewrite > (andb_false5_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##2: #H11; nrewrite > (H7 … H11); nrewrite > (andb_false5_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##3: #H11; nrewrite > (H8 … H11); nrewrite > (andb_false5_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##4: #H11; nrewrite > (H9 … H11); nrewrite > (andb_false5_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f5 w1 w2)); napply refl_eq
+ ##| ##5: #H11; nrewrite > (H10 … H11); nrewrite > (andb_false5_5 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##]
+nqed.
+
+nlemma quintuple_is_comparable :
+ comparable → comparable → comparable → comparable → comparable → comparable.
+ #T1; #T2; #T3; #T4; #T5; napply (mk_comparable (Prod5T T1 T2 T3 T4 T5));
+ ##[ napply (quintuple … (zeroc T1) (zeroc T2) (zeroc T3) (zeroc T4) (zeroc T5))
+ ##| napply (λx.false)
+ ##| napply (eq_quintuple … (eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5))
+ ##| napply (eqquintuple_to_eq … (eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (eqc_to_eq T1)
+ ##| napply (eqc_to_eq T2)
+ ##| napply (eqc_to_eq T3)
+ ##| napply (eqc_to_eq T4)
+ ##| napply (eqc_to_eq T5)
+ ##]
+ ##| napply (eq_to_eqquintuple … (eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (eq_to_eqc T1)
+ ##| napply (eq_to_eqc T2)
+ ##| napply (eq_to_eqc T3)
+ ##| napply (eq_to_eqc T4)
+ ##| napply (eq_to_eqc T5)
+ ##]
+ ##| napply (neqquintuple_to_neq … (eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (neqc_to_neq T1)
+ ##| napply (neqc_to_neq T2)
+ ##| napply (neqc_to_neq T3)
+ ##| napply (neqc_to_neq T4)
+ ##| napply (neqc_to_neq T5)
+ ##]
+ ##| napply (neq_to_neqquintuple … (eqc T1) (eqc T2) (eqc T3) (eqc T4) (eqc T5));
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (decidable_c T5)
+ ##| napply (neq_to_neqc T1)
+ ##| napply (neq_to_neqc T2)
+ ##| napply (neq_to_neqc T3)
+ ##| napply (neq_to_neqc T4)
+ ##| napply (neq_to_neqc T5)
+ ##]
+ ##| napply decidable_quintuple;
+ ##[ napply (decidable_c T1)
+ ##| napply (decidable_c T2)
+ ##| napply (decidable_c T3)
+ ##| napply (decidable_c T4)
+ ##| napply (decidable_c T5)
+ ##]
+ ##| napply symmetric_eqquintuple;
+ ##[ napply (symmetric_eqc T1)
+ ##| napply (symmetric_eqc T2)
+ ##| napply (symmetric_eqc T3)
+ ##| napply (symmetric_eqc T4)
+ ##| napply (symmetric_eqc T5)
+ ##]
+ ##]
+nqed.
+
+unification hint 0 ≔ S1: comparable, S2: comparable, S3: comparable, S4: comparable, S5: comparable;
+ T1 ≟ (carr S1), T2 ≟ (carr S2), T3 ≟ (carr S3), T4 ≟ (carr S4), T5 ≟ (carr S5),
+ X ≟ (quintuple_is_comparable S1 S2 S3 S4 S5)
+ (*********************************************) ⊢
+ carr X ≡ Prod5T T1 T2 T3 T4 T5.
projects / helm.git / blobdiff