+nlemma andb_false2
+ : ∀b1,b2.(b1 ⊗ b2) = false →
+ (b1 = false) ∨ (b2 = false).
+ #b1; #b2;
+ ncases b1;
+ ncases b2;
+ nnormalize;
+ ##[ ##1: #H; napply (bool_destruct … H)
+ ##| ##2,4: #H; napply (or2_intro2 … H)
+ ##| ##3: #H; napply (or2_intro1 … H)
+ ##]
+nqed.
+
+nlemma andb_false3
+ : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ b3) = false →
+ Or3 (b1 = false) (b2 = false) (b3 = false).
+ #b1; #b2; #b3;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ nnormalize;
+ ##[ ##1: #H; napply (bool_destruct … H)
+ ##| ##5,6,7,8: #H; napply (or3_intro1 … H)
+ ##| ##2,4: #H; napply (or3_intro3 … H)
+ ##| ##3: #H; napply (or3_intro2 … H)
+ ##]
+nqed.
+
+nlemma andb_false4
+ : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ b4) = false →
+ Or4 (b1 = false) (b2 = false) (b3 = false) (b4 = false).
+ #b1; #b2; #b3; #b4;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ ncases b4;
+ nnormalize;
+ ##[ ##1: #H; napply (bool_destruct … H)
+ ##| ##9,10,11,12,13,14,15,16: #H; napply (or4_intro1 … H)
+ ##| ##5,6,7,8: #H; napply (or4_intro2 … H)
+ ##| ##3,4: #H; napply (or4_intro3 … H)
+ ##| ##2: #H; napply (or4_intro4 … H)
+ ##]
+nqed.
+
+nlemma andb_false5
+ : ∀b1,b2,b3,b4,b5.(b1 ⊗ b2 ⊗ b3 ⊗ b4 ⊗ b5) = false →
+ Or5 (b1 = false) (b2 = false) (b3 = false) (b4 = false) (b5 = false).
+ #b1; #b2; #b3; #b4; #b5;
+ ncases b1;
+ ncases b2;
+ ncases b3;
+ ncases b4;
+ ncases b5;
+ nnormalize;
+ ##[ ##1: #H; napply (bool_destruct … H)
+ ##| ##17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32: #H; napply (or5_intro1 … H)
+ ##| ##9,10,11,12,13,14,15,16: #H; napply (or5_intro2 … H)
+ ##| ##5,6,7,8: #H; napply (or5_intro3 … H)
+ ##| ##3,4: #H; napply (or5_intro4 … H)
+ ##| ##2: #H; napply (or5_intro5 … H)
+ ##]
+nqed.
+
+nlemma andb_false2_1 : ∀b.(false ⊗ b) = false.
+ #b; nnormalize; napply refl_eq. nqed.
+nlemma andb_false2_2 : ∀b.(b ⊗ false) = false.
+ #b; nelim b; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false3_1 : ∀b1,b2.(false ⊗ b1 ⊗ b2) = false.
+ #b1; #b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_2 : ∀b1,b2.(b1 ⊗ false ⊗ b2) = false.
+ #b1; #b2; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false3_3 : ∀b1,b2.(b1 ⊗ b2 ⊗ false) = false.
+ #b1; #b2; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false4_1 : ∀b1,b2,b3.(false ⊗ b1 ⊗ b2 ⊗ b3) = false.
+ #b1; #b2; #b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_2 : ∀b1,b2,b3.(b1 ⊗ false ⊗ b2 ⊗ b3) = false.
+ #b1; #b2; #b3; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_3 : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ false ⊗ b3) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false4_4 : ∀b1,b2,b3.(b1 ⊗ b2 ⊗ b3 ⊗ false) = false.
+ #b1; #b2; #b3; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+
+nlemma andb_false5_1 : ∀b1,b2,b3,b4.(false ⊗ b1 ⊗ b2 ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_2 : ∀b1,b2,b3,b4.(b1 ⊗ false ⊗ b2 ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_3 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ false ⊗ b3 ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_4 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ false ⊗ b4) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nnormalize; napply refl_eq. nqed.
+nlemma andb_false5_5 : ∀b1,b2,b3,b4.(b1 ⊗ b2 ⊗ b3 ⊗ b4 ⊗ false) = false.
+ #b1; #b2; #b3; #b4; nelim b1; nelim b2; nelim b3; nelim b4; nnormalize; napply refl_eq. nqed.
+