rename Recursion

[coinductive-monad.git] / Recursion / Termination.v

diff --git a/Recursion/Termination.v b/Recursion/Termination.v
deleted file mode 100644 (file)
index 4d47a4a..0000000
--- a/Recursion/Termination.v
+++ /dev/null
@@ -1,187 +0,0 @@
-Require Import Recursion.Monad.
-Require Import Coq.Logic.JMeq.
-
-Section Termination.
-
-     (* an inductive predicate proving that a given computation terminates with a particular value *)  
-  Inductive TerminatesWith (A:Set) : #A -> A -> Prop :=
-  | TerminatesReturnWith :
-    forall (a:A),
-      TerminatesWith A (Return a) a
-      
-  | TerminatesBindWith :
-    forall (B:Set) (b:B) (a:A) (f:B->#A) (c:#B),
-      (TerminatesWith B c b)
-      -> TerminatesWith A (f b) a
-      -> TerminatesWith A (@Bind A B f c) a
-      .
-  
-     (* an inductive predicate proving that a given computation terminates with /some/ value *)  
-  Inductive Terminates (A:Set) : #A -> Prop :=
-  | TerminatesReturn :
-    forall (a:A),
-      Terminates A (Return a)
-      
-  | TerminatesBind :
-    forall (B:Set) (f:B->#A) (c:#B),
-      Terminates B c
-      -> (forall (b:B), (TerminatesWith B c b) -> Terminates A (f b))
-      -> Terminates A (@Bind A B f c)
-      .
-
-  Lemma terminating_computations_must_produce_a_value
-    : forall (A:Set) (c:#A), Terminates A c -> ex (fun a:A => TerminatesWith A c a).
-    induction 1.
-    
-    exists a.
-    constructor.
-    
-    elim IHTerminates.
-    intros.
-    clear IHTerminates.
-    
-    assert (exists a : A, TerminatesWith A (f x) a).
-    apply H1.
-    assumption.
-    
-    elim H3.
-    intros.
-    
-    exists x0.
-    apply (TerminatesBindWith A B x x0 f c).
-    assumption.
-    assumption.
-  Defined.
-
-  Theorem coerce : forall (A B:Set)(pf:A=B)(a:#A), #B.
-    intros.
-    subst.
-    exact a.
-  Defined.
-  
-  Inductive InvokedBy (A B C:Set) : #A -> #B -> #C -> Prop :=
-  | invokesPrev : forall
-    (z:#C)
-    (c:#B)
-    (f:B->#A),
-    InvokedBy A B C (@Bind A B f c) c z
-  | invokesFunc : forall
-    (c:#C)
-    (b:C)
-    (f:C->#A)
-    (pf:#A->#B)
-    (eqpf:A=B)
-    (_:JMeq (pf (f b)) (f b))
-    (_:TerminatesWith C c b),
-    InvokedBy A B C (@Bind A C f c) (pf (f b)) c.
-
-  Inductive Safe : forall (A:Set) (c:#A), Prop :=
-    Safe_intro :
-    forall (A:Set) (c:#A),
-      (forall (B C:Set) (c':#B)(z:#C), InvokedBy A B C c c' z -> Safe B c')
-      -> Safe A c.
-
-  Definition Safe_inv
-    : forall (A B C:Set)(c:#A)(_:Safe A c)(c':#B)(z:#C)(_:InvokedBy A B C c c' z), Safe B c'.
-    destruct 1.
-    apply (H B).
-  Defined.
-
-  Notation "! c :: A" := {a:A|TerminatesWith A c a} (at level 5).
-  Notation "'!Let' x := y 'in' z" := ((fun x => z)y)(at level 100).
-  Definition eval' CC cc (Z:Set) (z:#Z) (s:Safe CC cc) : (!cc::CC).
-    refine(
-      !Let eval_one_step :=
-        fun C c Z z =>
-          match c return (forall PRED pred Z z, InvokedBy C PRED Z c pred z -> !pred::PRED) -> !c::C with
-            | Return x => fun _ => exist _ x _
-            | Bind CN f cn =>
-              fun eval_pred =>
-                match eval_pred CN cn Z z (invokesPrev C CN Z z cn f) with
-                  | exist b pf =>
-                    match eval_pred C (f b) CN cn _ with
-                      | exist a' pf' => exist _ a' _
-                    end
-                end
-          end
-          in
-          fix eval_all C c Z z (s:Safe C c) {struct s} : !c::C :=
-          eval_one_step C c Z z (fun C' c' Z z icc => eval_all C' c' Z z (Safe_inv C C' Z c s c' z icc))
-    ).
-    
-    constructor.
-
-    refine (invokesFunc C C CN cn b f (fun x:#C=>x) _ _ _).
-    auto.
-    auto.
-    assumption.
-    
-    apply (TerminatesBindWith C CN b a' f cn).
-    assumption.
-    assumption.
-  Defined.
-
-  Theorem jmeq_lemma : forall (A1 A2 B:Set)(c1:#A1)(c2:#A2)(f1:A1->#B)(f2:A2->#B),
-    ((c1>>=f1)=(c2>>=f2))
-    -> (JMeq c1 c2) /\ (JMeq f1 f2) /\ (A1=A2).
-    intros.
-    inversion H.
-    split.
-    dependent rewrite H3.
-    simpl.
-    auto.
-    split.
-    dependent rewrite H2.
-    simpl.
-    auto.
-    auto.
-  Qed.
-
-  Theorem termination_is_safe : forall (A:Set) (c:#A) (t:Terminates A c), Safe A c.
-    intros.
-    induction t.
-    apply Safe_intro.
-    intros.
-    inversion H.
-
-    apply Safe_intro.
-    intros.
-    simple inversion H1.
-    subst.
-
-    apply jmeq_lemma in H2.
-    destruct H2.
-    destruct H3.
-    subst.
-    apply JMeq_eq in H2.
-    subst.
-    auto.
-
-    intros.
-    apply jmeq_lemma in H4.
-    destruct H4.
-    destruct H7.
-    rewrite <- H5.
-    clear H5.
-    generalize H2.
-    clear H2.
-    subst.
-    apply JMeq_eq in H7.
-    apply JMeq_eq in H4.
-    subst.
-    intros.
-    apply JMeq_eq in H2.
-    rewrite H2.
-    apply H0.
-    auto.
-  Defined.
-
-End Termination.
-
-Implicit Arguments Terminates [A].
-Implicit Arguments TerminatesReturn [A].
-Implicit Arguments TerminatesBind [A].
-Implicit Arguments TerminatesReturnWith [A].
-Implicit Arguments TerminatesBindWith [A].
-Implicit Arguments eval' [CC].
-