include "common/prod.ma".
include "num/bool_lemmas.ma".
-(* ***** *)
-(* TUPLE *)
-(* ***** *)
+(* ********* *)
+(* TUPLE x 2 *)
+(* ********* *)
nlemma pair_destruct_1 :
∀T1,T2.∀x1,x2:T1.∀y1,y2:T2.
nlemma eq_to_eqpair :
∀T1,T2.∀p1,p2:ProdT T1 T2.
∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
- (∀x1,x2:T1.x1 = x2 → f1 x1 x2 = true) →
- (∀y1,y2:T2.y1 = y2 → f2 y1 y2 = true) →
+ (∀x,y:T1.x = y → f1 x y = true) →
+ (∀x,y:T2.x = y → f2 x y = true) →
(p1 = p2 → eq_pair T1 T2 p1 p2 f1 f2 = true).
#T1; #T2; #p1; #p2; #f1; #f2; #H1; #H2;
nelim p1;
nlemma eqpair_to_eq :
∀T1,T2.∀p1,p2:ProdT T1 T2.
∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
- (∀x1,x2:T1.f1 x1 x2 = true → x1 = x2) →
- (∀y1,y2:T2.f2 y1 y2 = true → y1 = y2) →
+ (∀x,y:T1.f1 x y = true → x = y) →
+ (∀x,y:T2.f2 x y = true → x = y) →
(eq_pair T1 T2 p1 p2 f1 f2 = true → p1 = p2).
#T1; #T2; #p1; #p2; #f1; #f2; #H1; #H2;
nelim p1;
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) ?);
+ ##[ ##1: #H2; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H3; napply (H2 (pair_destruct_1 T1 T2 … H3))
+ ##| ##2: #H2; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##1: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H4; napply (H3 (pair_destruct_2 T1 T2 … H4))
+ ##| ##2: #H3; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H2; nrewrite > H3; napply refl_eq
+ ##]
+ ##]
+nqed.
+
+nlemma neqpair_to_neq :
+ ∀T1,T2.∀p1,p2:ProdT 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) →
+ (eq_pair T1 T2 p1 p2 f1 f2 = false → p1 ≠ p2).
+ #T1; #T2; #p1; #p2; #f1; #f2; #H1; #H2;
+ nelim p1;
+ #x1; #y1;
+ 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 (H2 y1 y2 H4); napply (pair_destruct_2 T1 T2 … H3)
+ ##| ##2: #H4; napply (H1 x1 x2 H4); napply (pair_destruct_1 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) ?);
+ ##[ ##1: #H3; napply (or2_intro1 … H3)
+ ##| ##2: #H3; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##1: #H4; napply (or2_intro2 … H4)
+ ##| ##2: #H4; nrewrite > H3 in H:(%);
+ nrewrite > H4; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqpair :
+ ∀T1,T2.∀p1,p2:ProdT 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 → eq_pair T1 T2 p1 p2 f1 f2 = false).
+ #T1; #T2; #p1; #p2; #f1; #f2; #H1; #H2; #H3; #H4;
+ nelim p1;
+ #x1; #y1;
+ 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) ?);
+ ##[ ##1: #H5; nrewrite > (H4 … H5); nrewrite > (andb_false2_2 (f1 x1 x2)); napply refl_eq
+ ##| ##2: #H5; nrewrite > (H3 … H5); nnormalize; napply refl_eq
+ ##]
+nqed.
+
+(* ********* *)
+(* 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.
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) ?);
+ ##[ ##1: #H3; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H4; napply (H3 (triple_destruct_1 T1 T2 T3 … H4))
+ ##| ##2: #H3; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##1: #H4; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H5; napply (H4 (triple_destruct_2 T1 T2 T3 … H5))
+ ##| ##2: #H4; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##1: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (triple_destruct_3 T1 T2 T3 … H6))
+ ##| ##2: #H5; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H3;
+ nrewrite > H4;
+ nrewrite > H5;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqtriple_to_neq :
+ ∀T1,T2,T3.∀p1,p2:Prod3T 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) →
+ (eq_triple T1 T2 T3 p1 p2 f1 f2 f3 = false → p1 ≠ p2).
+ #T1; #T2; #T3; #p1; #p2; #f1; #f2; #f3; #H1; #H2; #H3;
+ nelim p1;
+ #x1; #y1; #z1;
+ 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 (H2 y1 y2 H5); napply (triple_destruct_2 T1 T2 T3 … H4)
+ ##| ##2: #H5; napply (H3 z1 z2 H5); napply (triple_destruct_3 T1 T2 T3 … H4)
+ ##| ##3: #H5; napply (H1 x1 x2 H5); napply (triple_destruct_1 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) ?);
+ ##[ ##1: #H4; napply (or3_intro1 … H4)
+ ##| ##2: #H4; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##1: #H5; napply (or3_intro2 … H5)
+ ##| ##2: #H5; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##1: #H6; napply (or3_intro3 … H6)
+ ##| ##2: #H6; nrewrite > H4 in H:(%);
+ nrewrite > H5;
+ nrewrite > H6; #H; nelim (H (refl_eq …))
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqtriple :
+ ∀T1,T2,T3.∀p1,p2:Prod3T 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 → eq_triple T1 T2 T3 p1 p2 f1 f2 f3 = false).
+ #T1; #T2; #T3; #p1; #p2; #f1; #f2; #f3; #H1; #H2; #H3; #H4; #H5; #H6;
+ nelim p1;
+ #x1; #y1; #z1;
+ 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 > (H5 … H7); nrewrite > (andb_false3_2 (f1 x1 x2) (f3 z1 z2)); napply refl_eq
+ ##| ##2: #H7; nrewrite > (H6 … H7); nrewrite > (andb_false3_3 (f1 x1 x2) (f2 y1 y2)); napply refl_eq
+ ##| ##3: #H7; nrewrite > (H4 … H7); nrewrite > (andb_false3_1 (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##]
+nqed.
+
+(* ********* *)
+(* 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.
nlemma eq_to_eqquadruple :
∀T1,T2,T3,T4.∀p1,p2:Prod4T T1 T2 T3 T4.
∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → 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) →
- (∀v1,v2:T4.v1 = v2 → f4 v1 v2 = true) →
+ (∀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 → eq_quadruple T1 T2 T3 T4 p1 p2 f1 f2 f3 f4 = true).
#T1; #T2; #T3; #T4; #p1; #p2; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
nelim p1;
nlemma eqquadruple_to_eq :
∀T1,T2,T3,T4.∀p1,p2:Prod4T T1 T2 T3 T4.
∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.∀f4:T4 → T4 → 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) →
- (∀v1,v2:T4.f4 v1 v2 = true → v1 = v2) →
+ (∀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) →
(eq_quadruple T1 T2 T3 T4 p1 p2 f1 f2 f3 f4 = true → p1 = p2).
#T1; #T2; #T3; #T4; #p1; #p2; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
nelim p1;
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) ?);
+ ##[ ##1: #H4; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H5; napply (H4 (quadruple_destruct_1 T1 T2 T3 T4 … H5))
+ ##| ##2: #H4; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##1: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (quadruple_destruct_2 T1 T2 T3 T4 … H6))
+ ##| ##2: #H5; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##1: #H6; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H7; napply (H6 (quadruple_destruct_3 T1 T2 T3 T4 … H7))
+ ##| ##2: #H6; napply (or2_elim (xx4 = yy4) (xx4 ≠ yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##1: #H7; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H8; napply (H7 (quadruple_destruct_4 T1 T2 T3 T4 … H8))
+ ##| ##2: #H7; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+ nrewrite > H4;
+ nrewrite > H5;
+ nrewrite > H6;
+ nrewrite > H7;
+ napply refl_eq
+ ##]
+ ##]
+ ##]
+ ##]
+nqed.
+
+nlemma neqquadruple_to_neq :
+ ∀T1,T2,T3,T4.∀p1,p2:Prod4T 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) →
+ (eq_quadruple T1 T2 T3 T4 p1 p2 f1 f2 f3 f4 = false → p1 ≠ p2).
+ #T1; #T2; #T3; #T4; #p1; #p2; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4;
+ nelim p1;
+ #x1; #y1; #z1; #v1;
+ 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 (H2 y1 y2 H6); napply (quadruple_destruct_2 T1 T2 T3 T4 … H5)
+ ##| ##2: #H6; napply (H3 z1 z2 H6); napply (quadruple_destruct_3 T1 T2 T3 T4 … H5)
+ ##| ##3: #H6; napply (H4 v1 v2 H6); napply (quadruple_destruct_4 T1 T2 T3 T4 … H5)
+ ##| ##4: #H6; napply (H1 x1 x2 H6); napply (quadruple_destruct_1 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) ?);
+ ##[ ##1: #H5; napply (or4_intro1 … H5)
+ ##| ##2: #H5; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##1: #H6; napply (or4_intro2 … H6)
+ ##| ##2: #H6; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##1: #H7; napply (or4_intro3 … H7)
+ ##| ##2: #H7; napply (or2_elim (v1 = v2) (v1 ≠ v2) ? (H4 v1 v2) ?);
+ ##[ ##1: #H8; napply (or4_intro4 … H8)
+ ##| ##2: #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.∀p1,p2:Prod4T 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 → eq_quadruple T1 T2 T3 T4 p1 p2 f1 f2 f3 f4 = false).
+ #T1; #T2; #T3; #T4; #p1; #p2; #f1; #f2; #f3; #f4; #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8;
+ nelim p1;
+ #x1; #y1; #z1; #v1;
+ 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 > (H6 … H9); nrewrite > (andb_false4_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##2: #H9; nrewrite > (H7 … H9); nrewrite > (andb_false4_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2)); napply refl_eq
+ ##| ##3: #H9; nrewrite > (H8 … H9); nrewrite > (andb_false4_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2)); napply refl_eq
+ ##| ##4: #H9; nrewrite > (H5 … H9); nrewrite > (andb_false4_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##]
+nqed.
+
+(* ********* *)
+(* 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.
nlemma eq_to_eqquintuple :
∀T1,T2,T3,T4,T5.∀p1,p2:Prod5T 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.
- (∀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) →
- (∀v1,v2:T4.v1 = v2 → f4 v1 v2 = true) →
- (∀w1,w2:T5.w1 = w2 → f5 w1 w2 = true) →
+ (∀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 → eq_quintuple T1 T2 T3 T4 T5 p1 p2 f1 f2 f3 f4 f5 = true).
#T1; #T2; #T3; #T4; #T5; #p1; #p2; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
nelim p1;
nlemma eqquintuple_to_eq :
∀T1,T2,T3,T4,T5.∀p1,p2:Prod5T 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.
- (∀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) →
- (∀v1,v2:T4.f4 v1 v2 = true → v1 = v2) →
- (∀w1,w2:T5.f5 w1 w2 = true → w1 = w2) →
+ (∀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) →
(eq_quintuple T1 T2 T3 T4 T5 p1 p2 f1 f2 f3 f4 f5 = true → p1 = p2).
#T1; #T2; #T3; #T4; #T5; #p1; #p2; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
nelim p1;
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) ?);
+ ##[ ##1: #H5; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H6; napply (H5 (quintuple_destruct_1 T1 T2 T3 T4 T5 … H6))
+ ##| ##2: #H5; napply (or2_elim (xx2 = yy2) (xx2 ≠ yy2) ? (H1 xx2 yy2) ?);
+ ##[ ##1: #H6; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H7; napply (H6 (quintuple_destruct_2 T1 T2 T3 T4 T5 … H7))
+ ##| ##2: #H6; napply (or2_elim (xx3 = yy3) (xx3 ≠ yy3) ? (H2 xx3 yy3) ?);
+ ##[ ##1: #H7; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H8; napply (H7 (quintuple_destruct_3 T1 T2 T3 T4 T5 … H8))
+ ##| ##2: #H7; napply (or2_elim (xx4 = yy4) (xx4 ≠ yy4) ? (H3 xx4 yy4) ?);
+ ##[ ##1: #H8; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H9; napply (H8 (quintuple_destruct_4 T1 T2 T3 T4 T5 … H9))
+ ##| ##2: #H8; napply (or2_elim (xx5 = yy5) (xx5 ≠ yy5) ? (H4 xx5 yy5) ?);
+ ##[ ##1: #H9; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+ nnormalize; #H10; napply (H9 (quintuple_destruct_5 T1 T2 T3 T4 T5 … H10))
+ ##| ##2: #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.∀p1,p2:Prod5T 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) →
+ (eq_quintuple T1 T2 T3 T4 T5 p1 p2 f1 f2 f3 f4 f5 = false → p1 ≠ p2).
+ #T1; #T2; #T3; #T4; #T5; #p1; #p2; #f1; #f2; #f3; #f4; #f5; #H1; #H2; #H3; #H4; #H5;
+ nelim p1;
+ #x1; #y1; #z1; #v1; #w1;
+ 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 (H2 y1 y2 H7); napply (quintuple_destruct_2 T1 T2 T3 T4 T5 … H6)
+ ##| ##2: #H7; napply (H3 z1 z2 H7); napply (quintuple_destruct_3 T1 T2 T3 T4 T5 … H6)
+ ##| ##3: #H7; napply (H4 v1 v2 H7); napply (quintuple_destruct_4 T1 T2 T3 T4 T5 … H6)
+ ##| ##4: #H7; napply (H5 w1 w2 H7); napply (quintuple_destruct_5 T1 T2 T3 T4 T5 … H6)
+ ##| ##5: #H7; napply (H1 x1 x2 H7); napply (quintuple_destruct_1 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) ?);
+ ##[ ##1: #H6; napply (or5_intro1 … H6)
+ ##| ##2: #H6; napply (or2_elim (y1 = y2) (y1 ≠ y2) ? (H2 y1 y2) ?);
+ ##[ ##1: #H7; napply (or5_intro2 … H7)
+ ##| ##2: #H7; napply (or2_elim (z1 = z2) (z1 ≠ z2) ? (H3 z1 z2) ?);
+ ##[ ##1: #H8; napply (or5_intro3 … H8)
+ ##| ##2: #H8; napply (or2_elim (v1 = v2) (v1 ≠ v2) ? (H4 v1 v2) ?);
+ ##[ ##1: #H9; napply (or5_intro4 … H9)
+ ##| ##2: #H9; napply (or2_elim (w1 = w2) (w1 ≠ w2) ? (H5 w1 w2) ?);
+ ##[ ##1: #H10; napply (or5_intro5 … H10)
+ ##| ##2: #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.∀p1,p2:Prod5T 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 → eq_quintuple T1 T2 T3 T4 T5 p1 p2 f1 f2 f3 f4 f5 = false).
+ #T1; #T2; #T3; #T4; #T5; #p1; #p2;
+ #f1; #f2; #f3; #f4; #f5;
+ #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8; #H9; #H10;
+ nelim p1;
+ #x1; #y1; #z1; #v1; #w1;
+ 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 > (H7 … H11); nrewrite > (andb_false5_2 (f1 x1 x2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##2: #H11; nrewrite > (H8 … H11); nrewrite > (andb_false5_3 (f1 x1 x2) (f2 y1 y2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##| ##3: #H11; nrewrite > (H9 … H11); nrewrite > (andb_false5_4 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f5 w1 w2)); napply refl_eq
+ ##| ##4: #H11; nrewrite > (H10 … H11); nrewrite > (andb_false5_5 (f1 x1 x2) (f2 y1 y2) (f3 z1 z2) (f4 v1 v2)); napply refl_eq
+ ##| ##5: #H11; nrewrite > (H6 … H11); nrewrite > (andb_false5_1 (f2 y1 y2) (f3 z1 z2) (f4 v1 v2) (f5 w1 w2)); napply refl_eq
+ ##]
+nqed.
projects / helm.git / blobdiff