freescale porting

[helm.git] / helm / software / matita / contribs / ng_assembly / common / prod_lemmas.ma

diff --git a/helm/software/matita/contribs/ng_assembly/common/prod_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/prod_lemmas.ma
index ecf12f382e46c38b06bba9759e78c564990f3223..79863d9a01586d7f6ca7f7effa1d542a785bd8d5 100644 (file)
--- a/helm/software/matita/contribs/ng_assembly/common/prod_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/common/prod_lemmas.ma
@@ -50,15 +50,13 @@ nqed.
 nlemma symmetric_eqpair :
 ∀T1,T2:Type.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
-∀p1,p2:ProdT T1 T2.
  (symmetricT T1 bool f1) →
  (symmetricT T2 bool f2) →
- (eq_pair T1 T2 f1 f2 p1 p2 = eq_pair T1 T2 f1 f2 p1 p2).
- #T1; #T2; #f1; #f2; #p1; #p2; #H; #H1;
- nelim p1;
- #x1; #y1;
- nelim p2;
- #x2; #y2;
+ (∀p1,p2:ProdT T1 T2. 
+  (eq_pair T1 T2 f1 f2 p1 p2 = eq_pair T1 T2 f1 f2 p1 p2)).
+ #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);
@@ -71,15 +69,13 @@ nqed.
 nlemma eq_to_eqpair :
 ∀T1,T2.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
-∀p1,p2:ProdT T1 T2.
  (∀x,y:T1.x = y → f1 x y = true) →
  (∀x,y:T2.x = y → f2 x y = true) →
- (p1 = p2 → eq_pair T1 T2 f1 f2 p1 p2 = true).
- #T1; #T2; #f1; #f2; #p1; #p2; #H1; #H2;
- nelim p1;
- #x1; #y1;
- nelim p2;
- #x2; #y2; #H;
+ (∀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;
@@ -90,15 +86,13 @@ nqed.
 nlemma eqpair_to_eq :
 ∀T1,T2.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
-∀p1,p2:ProdT T1 T2.
  (∀x,y:T1.f1 x y = true → x = y) →
  (∀x,y:T2.f2 x y = true → x = y) →
- (eq_pair T1 T2 f1 f2 p1 p2 = true → p1 = p2).
- #T1; #T2; #f1; #f2; #p1; #p2; #H1; #H2;
- nelim p1;
- #x1; #y1;
- nelim p2;
- #x2; #y2; #H;
+ (∀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:(%);
@@ -111,11 +105,11 @@ nlemma eqpair_to_eq :
  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).
+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;
@@ -135,15 +129,13 @@ nqed.
 nlemma neqpair_to_neq :
 ∀T1,T2.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
-∀p1,p2:ProdT T1 T2. 
  (∀x,y:T1.f1 x y = false → x ≠ y) →
  (∀x,y:T2.f2 x y = false → x ≠ y) →
- (eq_pair T1 T2 f1 f2 p1 p2 = false → p1 ≠ p2).
- #T1; #T2; #f1; #f2; #p1; #p2; #H1; #H2;
- nelim p1;
- #x1; #y1;
- nelim p2;
- #x2; #y2;
+ (∀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) ?);
@@ -152,11 +144,12 @@ nlemma neqpair_to_neq :
  ##]
 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.
+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) ?);
@@ -172,17 +165,15 @@ nqed.
 nlemma neq_to_neqpair :
 ∀T1,T2.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.
-∀p1,p2:ProdT T1 T2. 
  (∀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 f1 f2 p1 p2 = false).
- #T1; #T2; #f1; #f2; #p1; #p2; #H1; #H2; #H3; #H4;
- nelim p1;
- #x1; #y1;
- nelim p2;
- #x2; #y2; #H;
+ (∀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
@@ -227,16 +218,14 @@ nqed.
 nlemma symmetric_eqtriple :
 ∀T1,T2,T3:Type.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
-∀p1,p2:Prod3T T1 T2 T3.
  (symmetricT T1 bool f1) →
  (symmetricT T2 bool f2) →
  (symmetricT T3 bool f3) →
- (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; #p1; #p2; #H; #H1; #H2;
- nelim p1;
- #x1; #y1; #z1;
- nelim p2;
- #x2; #y2; #z2;
+ (∀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);
@@ -254,16 +243,14 @@ nqed.
 nlemma eq_to_eqtriple :
 ∀T1,T2,T3.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
-∀p1,p2:Prod3T T1 T2 T3.
  (∀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 → eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true).
- #T1; #T2; #T3; #f1; #f2; #f3; #p1; #p2; #H1; #H2; #H3;
- nelim p1;
- #x1; #y1; #z1;
- nelim p2;
- #x2; #y2; #z2; #H;
+ (∀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;
@@ -276,16 +263,14 @@ nqed.
 nlemma eqtriple_to_eq :
 ∀T1,T2,T3.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
-∀p1,p2:Prod3T T1 T2 T3.
  (∀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) →
- (eq_triple T1 T2 T3 f1 f2 f3 p1 p2 = true → p1 = p2).
- #T1; #T2; #T3; #f1; #f2; #f3; #p1; #p2; #H1; #H2; #H3;
- nelim p1;
- #x1; #y1; #z1;
- nelim p2;
- #x2; #y2; #z2; #H;
+ (∀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:(%);
@@ -302,12 +287,12 @@ nlemma eqtriple_to_eq :
  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).
+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;
@@ -334,16 +319,14 @@ nqed.
 nlemma neqtriple_to_neq :
 ∀T1,T2,T3.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
-∀p1,p2:Prod3T T1 T2 T3. 
  (∀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 f1 f2 f3 p1 p2 = false → p1 ≠ p2).
- #T1; #T2; #T3; #f1; #f2; #f3; #p1; #p2; #H1; #H2; #H3;
- nelim p1;
- #x1; #y1; #z1;
- nelim p2;
- #x2; #y2; #z2;
+ (∀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) ?);
@@ -353,13 +336,14 @@ nlemma neqtriple_to_neq :
  ##]
 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).
+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) ?);
@@ -379,19 +363,17 @@ nqed.
 nlemma neq_to_neqtriple :
 ∀T1,T2,T3.
 ∀f1:T1 → T1 → bool.∀f2:T2 → T2 → bool.∀f3:T3 → T3 → bool.
-∀p1,p2:Prod3T T1 T2 T3. 
  (∀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 f1 f2 f3 p1 p2 = false).
- #T1; #T2; #T3; #f1; #f2; #f3; #p1; #p2; #H1; #H2; #H3; #H4; #H5; #H6;
- nelim p1;
- #x1; #y1; #z1;
- nelim p2;
- #x2; #y2; #z2; #H;
+ (∀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
@@ -447,17 +429,15 @@ 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.
-∀p1,p2:Prod4T T1 T2 T3 T4.
  (symmetricT T1 bool f1) →
  (symmetricT T2 bool f2) →
  (symmetricT T3 bool f3) →
  (symmetricT T4 bool f4) →
- (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; #p1; #p2; #H; #H1; #H2; #H3;
- nelim p1;
- #x1; #y1; #z1; #v1;
- nelim p2;
- #x2; #y2; #z2; #v2;
+ (∀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);
@@ -480,17 +460,15 @@ 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.
-∀p1,p2:Prod4T T1 T2 T3 T4.
  (∀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 f1 f2 f3 f4 p1 p2 = true).
- #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #p1; #p2; #H1; #H2; #H3; #H4;
- nelim p1;
- #x1; #y1; #z1; #v1;
- nelim p2;
- #x2; #y2; #z2; #v2; #H;
+ (∀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;
@@ -505,17 +483,15 @@ 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.
-∀p1,p2:Prod4T T1 T2 T3 T4.
  (∀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 f1 f2 f3 f4 p1 p2 = true → p1 = p2).
- #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #p1; #p2; #H1; #H2; #H3; #H4;
- nelim p1;
- #x1; #y1; #z1; #v1;
- nelim p2;
- #x2; #y2; #z2; #v2; #H;
+ (∀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:(%);
@@ -537,13 +513,13 @@ nlemma eqquadruple_to_eq :
  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).
+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;
@@ -575,17 +551,15 @@ 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.
-∀p1,p2:Prod4T T1 T2 T3 T4. 
  (∀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 f1 f2 f3 f4 p1 p2 = false → p1 ≠ p2).
- #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #p1; #p2; #H1; #H2; #H3; #H4;
- nelim p1;
- #x1; #y1; #z1; #v1;
- nelim p2;
- #x2; #y2; #z2; #v2;
+ (∀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) ?);
@@ -596,15 +570,15 @@ nlemma neqquadruple_to_neq :
  ##]
 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).
+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;
@@ -629,7 +603,6 @@ 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.
-∀p1,p2:Prod4T T1 T2 T3 T4. 
  (∀x,y:T1.decidable (x = y)) →
  (∀x,y:T2.decidable (x = y)) →
  (∀x,y:T3.decidable (x = y)) →
@@ -638,12 +611,11 @@ nlemma neq_to_neqquadruple :
  (∀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 f1 f2 f3 f4 p1 p2 = false).
- #T1; #T2; #T3; #T4; #f1; #f2; #f3; #f4; #p1; #p2; #H1; #H2; #H3; #H4; #H5; #H6; #H7; #H8;
- nelim p1;
- #x1; #y1; #z1; #v1;
- nelim p2;
- #x2; #y2; #z2; #v2; #H;
+ (∀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
@@ -710,18 +682,16 @@ 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.
-∀p1,p2:Prod5T T1 T2 T3 T4 T5.
  (symmetricT T1 bool f1) →
  (symmetricT T2 bool f2) →
  (symmetricT T3 bool f3) →
  (symmetricT T4 bool f4) →
  (symmetricT T5 bool f5) →
- (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; #p1; #p2; #H; #H1; #H2; #H3; #H4;
- nelim p1;
- #x1; #y1; #z1; #v1; #w1;
- nelim p2;
- #x2; #y2; #z2; #v2; #w2;
+ (∀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);
@@ -749,18 +719,16 @@ 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.
-∀p1,p2:Prod5T T1 T2 T3 T4 T5.
  (∀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 f1 f2 f3 f4 f5 p1 p2 = true).
- #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #p1; #p2; #H1; #H2; #H3; #H4; #H5;
- nelim p1;
- #x1; #y1; #z1; #v1; #w1;
- nelim p2;
- #x2; #y2; #z2; #v2; #w2; #H;
+ (∀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;
@@ -777,18 +745,16 @@ 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.
-∀p1,p2:Prod5T T1 T2 T3 T4 T5.
  (∀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 f1 f2 f3 f4 f5 p1 p2 = true → p1 = p2).
- #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #p1; #p2; #H1; #H2; #H3; #H4; #H5;
- nelim p1;
- #x1; #y1; #z1; #v1; #w1;
- nelim p2;
- #x2; #y2; #z2; #v2; #w2; #H;
+ (∀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:(%);
@@ -815,14 +781,14 @@ nlemma eqquintuple_to_eq :
  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).
+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;
@@ -859,18 +825,16 @@ 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.
-∀p1,p2:Prod5T T1 T2 T3 T4 T5. 
  (∀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 f1 f2 f3 f4 f5 p1 p2 = false → p1 ≠ p2).
- #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #p1; #p2; #H1; #H2; #H3; #H4; #H5;
- nelim p1;
- #x1; #y1; #z1; #v1; #w1;
- nelim p2;
- #x2; #y2; #z2; #v2; #w2;
+ (∀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) ?);
@@ -882,16 +846,16 @@ nlemma neqquintuple_to_neq :
  ##]
 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).
+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;
@@ -920,7 +884,6 @@ 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.
-∀p1,p2:Prod5T T1 T2 T3 T4 T5. 
  (∀x,y:T1.decidable (x = y)) →
  (∀x,y:T2.decidable (x = y)) →
  (∀x,y:T3.decidable (x = y)) →
@@ -931,13 +894,12 @@ nlemma neq_to_neqquintuple :
  (∀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 f1 f2 f3 f4 f5 p1 p2 = false).
- #T1; #T2; #T3; #T4; #T5; #f1; #f2; #f3; #f4; #f5; #p1; #p2; 
+ (∀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;
- nelim p1;
- #x1; #y1; #z1; #v1; #w1;
- nelim p2;
- #x2; #y2; #z2; #v2; #w2; #H;
+ #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