For some obscure reason, more universes are now needed (but not used in

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

diff --git a/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma b/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
index 303d11ad38dfa8600b813d0ccff86ab3244b60bc..c8942ae6c277cc5c628fab86cac4902daf072c3b 100644 (file)
--- a/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
+++ b/helm/software/matita/contribs/ng_assembly/common/option_lemmas.ma
@@ -99,3 +99,58 @@ nlemma eqoption_to_eq :
           napply refl_eq
  ##]
 nqed.
+
+nlemma decidable_option : ∀T.∀H:(Πx,y:T.decidable (x = y)).∀x,y:option T.decidable (x = y).
+ #T; #H; #x; nelim x;
+ ##[ ##1: #y; ncases y;
+          ##[ ##1: nnormalize; napply (or2_intro1 (? = ?) (? ≠ ?) (refl_eq …))
+          ##| ##2: #yy; nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                   nnormalize; #H1; napply (option_destruct_none_some T … H1)
+          ##]
+ ##| ##2: #xx; #y; ncases y;
+          ##[ ##1: nnormalize; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                   nnormalize; #H2; napply (option_destruct_some_none T … H2)
+          ##| ##2: #yy; nnormalize; napply (or2_elim (xx = yy) (xx ≠ yy) ? (H …));
+                   ##[ ##2: #H1; napply (or2_intro2 (? = ?) (? ≠ ?) ?);
+                            nnormalize; #H2; napply (H1 (option_destruct_some_some T … H2))
+                   ##| ##1: #H1; napply (or2_intro1 (? = ?) (? ≠ ?) ?);
+                            nrewrite > H1; napply refl_eq
+                   ##]
+          ##]
+ ##]
+nqed.
+
+nlemma neq_to_neqoption :
+∀T.∀op1,op2:option T.∀f:T → T → bool.
+ (∀x1,x2:T.x1 ≠ x2 → f x1 x2 = false) →
+ (op1 ≠ op2 → eq_option T op1 op2 f = false).
+ #T; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+          ##[ ##1: nnormalize; #f; #H; #H1; nelim (H1 (refl_eq …))
+          ##| ##2: #yy; #f; #H; nnormalize; #H1; napply refl_eq
+          ##]
+ ##| ##2: #xx; #op2; ncases op2;
+          ##[ ##1: #f; #H; nnormalize; #H1; napply refl_eq
+          ##| ##2: #yy; #f; #H; nnormalize; #H1; napply (H xx yy …);
+                   nnormalize; #H2; nrewrite > H2 in H1:(%); #H1;
+                   napply (H1 (refl_eq …))
+          ##]
+ ##]
+nqed.
+
+nlemma neqoption_to_neq :
+∀T.∀op1,op2:option T.∀f:T → T → bool.
+ (∀x1,x2:T.f x1 x2 = false → x1 ≠ x2) →
+ (eq_option T op1 op2 f = false → op1 ≠ op2).
+ #T; #op1; nelim op1;
+ ##[ ##1: #op2; ncases op2;
+          ##[ ##1: nnormalize; #f; #H; #H1; napply (bool_destruct … H1)
+          ##| ##2: #yy; #f; #H; nnormalize; #H1; #H2; napply (option_destruct_none_some T … H2)
+          ##]
+ ##| ##2: #xx; #op2; ncases op2;
+          ##[ ##1: nnormalize; #f; #H; #H1; #H2; napply (option_destruct_some_none T … H2)
+          ##| ##2: #yy; #f; #H; nnormalize; #H1; #H2; napply (H xx yy H1 ?);
+                   napply (option_destruct_some_some T … H2)
+          ##]
+ ##]
+nqed.