+
+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.