previously hidden simplifications (in old destruct) added

[helm.git] / matita / tests / coercions_russell.ma

diff --git a/matita/tests/coercions_russell.ma b/matita/tests/coercions_russell.ma
index 50bf81dfd44647c6970923381b708c168175d3ef..36217ad9dbfba62e8299b06789df498f38d47456 100644 (file)
--- a/matita/tests/coercions_russell.ma
+++ b/matita/tests/coercions_russell.ma
@@ -16,6 +16,7 @@ set "baseuri" "cic:/matita/test/russell/".
 
 include "nat/orders.ma".
 include "list/list.ma".
+include "datatypes/constructors.ma".
 
 inductive sigma (A:Type) (P:A → Prop) : Type ≝
  sig_intro: ∀a:A. P a → sigma A P.
@@ -51,3 +52,93 @@ letin program_spec ≝
     intros; cases (H H1); ]
 exact program_spec.
 qed.
+
+definition nat_return := λn:nat. Some ? n.
+
+coercion cic:/matita/test/russell/nat_return.con.
+
+definition raise_exn := None nat.
+
+definition try_with :=
+ λx,e. match x with [ None => e | Some (x : nat) => x].
+
+lemma hd : list nat → option nat :=
+  λl.match l with [ nil ⇒ raise_exn | cons x _ ⇒ nat_return x ].
+  
+axiom f : nat -> nat.
+
+definition bind ≝ λf:nat->nat.λx.
+  match x with [None ⇒ raise_exn| Some x ⇒ nat_return(f x)].
+
+include "datatypes/bool.ma".
+include "list/sort.ma".
+include "nat/compare.ma".
+
+definition inject_opt ≝ λP.λa:option nat.λp:P a. sig_intro ? P ? p.
+
+coercion cic:/matita/test/russell/inject_opt.con 0 1.
+
+definition eject_opt ≝ λP.λc: ∃n:option nat.P n. match c with [ sig_intro w _ ⇒ w].
+
+coercion cic:/matita/test/russell/eject_opt.con.
+
+(* we may define mem as in the following lemma and get rid of it *)
+definition find_spec ≝
+  λl,p.λres:option nat.
+   match res with
+    [ None ⇒ ∀y. mem ? eqb y l = true → p y = false
+    | Some x ⇒ mem ? eqb x l = true ∧ 
+               p x = true ∧ 
+               ∀y.mem ? eqb y l = true → p y = true → x ≠ y → 
+                    ∃l1,l2,l3.l = l1 @ [x] @ l2 @ [y] @ l3].
+
+lemma mem_x_to_ex_l1_l2 : ∀l,x.mem ? eqb x l = true → ∃l1,l2.l = l1 @ [x] @ l2.
+intros 2; elim l (H hd tl IH H); [simplify in H; destruct H]
+generalize in match H; clear H;
+simplify; apply (eqb_elim x hd); simplify; intros;
+[1:clear IH; rewrite < H; apply (ex_intro ? ? []); 
+|2:lapply(IH H1); clear H1 IH; decompose; rewrite > H2; clear H2]
+simplify; autobatch;
+qed.
+
+definition if : ∀A:Type.bool → A → A → A ≝ 
+ λA,b,t,f. match b with [ true ⇒ t | false ⇒ f].
+ 
+notation < "'If' \nbsp b \nbsp 'Then' \nbsp t \nbsp 'Else' \nbsp f" for @{ 'if $b $t $f }.
+notation > "'If' b 'Then' t 'Else' f" for @{ 'if $b $t $f }.
+interpretation "if" 'if a b c = (cic:/matita/test/russell/if.con _ a b c). 
+
+definition sigma_find_spec : ∀p,l. sigma ? (λres.find_spec l p res).
+(* define the find function *) 
+letin find ≝ (λp. 
+  let rec aux l ≝
+    match l with
+    [ nil ⇒ raise_exn
+    | cons x l ⇒ If p x Then nat_return x Else aux l]
+    in aux);
+(* manca una delta?! *) unfold if in find;
+apply (find: ∀p.∀l.sigma ? (λres.find_spec l p res)); clear find;
+(* l = x::tl ∧ p x = false *)
+[1: cases (aux l1); clear aux;
+    generalize in match H2; clear H2; cases a; clear a; simplify;
+    [1: intros 2; apply (eqb_elim y n); intros (Eyn); [rewrite > Eyn; assumption]
+        apply H3; simplify in H2; assumption;
+    |2: intros; decompose; repeat split; [2: assumption]; intros;
+        [1: cases (eqb n1 n); simplify; autobatch;
+        |2: generalize in match (refl_eq ? (eqb y n)); generalize in ⊢ (? ? ? %→?); 
+            intro; cases b; clear b; intro Eyn; rewrite > Eyn in H3; simplify in H3;
+            [1: rewrite > (eqb_true_to_eq ? ? Eyn) in H6; rewrite > H1 in H6; destruct H6;
+            |2: lapply H4; try assumption; decompose; clear H4; rewrite > H8;
+                simplify; autobatch depth = 4;]]]
+(* l = x::tl ∧ p x = true *)
+|2: unfold find_spec; unfold nat_return; simplify; repeat split; [2: assumption]
+    [1: rewrite > eqb_n_n; reflexivity
+    |2: intro; generalize in match (refl_eq ? (eqb y n)); generalize in ⊢ (? ? ? %→?);
+        intro; cases b; clear b; intro Eyn; rewrite > Eyn; 
+        [1: rewrite > (eqb_true_to_eq ? ? Eyn);] clear Eyn; simplify; intros;
+        [1: cases H4; reflexivity
+        |2: lapply (mem_x_to_ex_l1_l2 ? ? H2); decompose; rewrite > H6;
+            apply (ex_intro ? ? []); simplify; autobatch;]]
+(* l = [] *)
+|3: unfold raise_exn; simplify; intros; destruct H1;]
+qed.