napply I.
nqed.
-nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
- nnormalize;
- #n1;
- nelim n1;
- ##[ ##1: #n2;
- nelim n2;
- nnormalize;
- ##[ ##1: napply refl_eq
- ##| ##2: #n3; #H; napply refl_eq
- ##]
- ##| ##2: #n2; #H; #n3;
- nnormalize;
- ncases n3;
- nnormalize;
- ##[ ##1: napply refl_eq
- ##| ##2: #n4; napply (H n4)
- ##]
- ##]
-nqed.
-
nlemma eq_to_eqnat : ∀n1,n2:nat.n1 = n2 → eq_nat n1 n2 = true.
#n1;
nelim n1;
napply (H n4 (nat_destruct_S_S … H1))
##]
##]
-nqed.
+nqed.
+
+nlemma neqnat_to_neq : ∀n1,n2:nat.(eq_nat n1 n2 = false → n1 ≠ n2).
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_nat n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqnat n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
nlemma eqnat_to_eq : ∀n1,n2:nat.(eq_nat n1 n2 = true → n1 = n2).
#n1;
##]
nqed.
+nlemma neq_to_neqnat : ∀n1,n2:nat.n1 ≠ n2 → eq_nat n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_nat n1 n2));
+ napply (not_to_not (eq_nat n1 n2 = true) (n1 = n2) ? H);
+ napply (eqnat_to_eq n1 n2).
+nqed.
+
+nlemma decidable_nat : ∀x,y:nat.decidable (x = y).
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_nat x y = true) (eq_nat x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqnat_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqnat_to_neq … H))
+ ##]
+nqed.
+
+nlemma symmetric_eqnat : symmetricT nat bool eq_nat.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_nat n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqnat n1 n2 H);
+ napply (symmetric_eq ? (eq_nat n2 n1) false);
+ napply (neq_to_neqnat n2 n1 (symmetric_neq ? n1 n2 H))
+ ##]
+nqed.
+
nlemma Sn_p_n_to_S_npn : ∀n1,n2.(S n1) + n2 = S (n1 + n2).
#n1;
nelim n1;