--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+
+
+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.
+
+interpretation "sigma" 'exists \eta.x =
+ (cic:/matita/tests/coercions_russell/sigma.ind#xpointer(1/1) _ x).
+
+definition inject ≝ λP.λa:list nat.λp:P a. sig_intro ? P ? p.
+
+coercion cic:/matita/tests/coercions_russell/inject.con 0 1.
+
+definition eject ≝ λP.λc: ∃n:list nat.P n. match c with [ sig_intro w _ ⇒ w].
+
+coercion cic:/matita/tests/coercions_russell/eject.con.
+
+alias symbol "exists" (instance 2) = "exists".
+lemma tl : ∀l:list nat. l ≠ [] → ∃l1.∃a.a :: l1 = l.
+letin program ≝
+ (λl:list nat. λH:l ≠ [].match l with [ nil ⇒ λH.[] | cons x l1 ⇒ λH.l1] H);
+letin program_spec ≝ (program : ∀l:list nat. l ≠ [] → ∃l1.∃a.a :: l1 = l);
+ [ generalize in match H; cases l; [intros (h1); cases (h1 ?); reflexivity]
+ intros; apply (ex_intro ? ? n); apply eq_f; reflexivity; ]
+exact program_spec;
+qed.
+
+alias symbol "exists" (instance 3) = "exists".
+lemma tl2 : ∀l:∃l:list nat. l ≠ []. ∃l1.∃a.a :: l1 = l.
+apply
+ (λl:list nat. match l with [ nil ⇒ [] | cons x l1 ⇒ l1]
+ :
+ ∀l:∃l:list nat. l ≠ []. ∃l1.∃a.a :: l1 = l);
+[ autobatch;
+| generalize in match H; clear H; cases s; simplify;
+ intros; cases (H H1);]
+qed.
+
+definition nat_return := λn:nat. Some ? n.
+
+coercion cic:/matita/tests/coercions_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/tests/coercions_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/tests/coercions_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.
+
+notation > "'If' b 'Then' t 'Else' f"
+for @{ match $b with [ true ⇒ $t | _ ⇒ $f ] }.
+
+definition sigma_find_spec : ∀p,l. sigma ? (λres.find_spec l p res).
+(* define the find function *)
+apply (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
+ :
+ ∀p.∀l.sigma ? (λres.find_spec l p res));
+(* 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.
projects / helm.git / blobdiff