From e33dad7c431b28a73469f657abcb8ddadc35b90a Mon Sep 17 00:00:00 2001
From: adelaett <90894311+adelaett@users.noreply.github.com>
Date: Tue, 20 Feb 2024 12:20:11 +0100
Subject: [PATCH] lifting the result

---
 theories/simulation_cred_to_sred.v | 80 +++++++++++++++++++++++++++++-
 theories/syntax.v                  | 31 ++++++++++++
 2 files changed, 109 insertions(+), 2 deletions(-)

diff --git a/theories/simulation_cred_to_sred.v b/theories/simulation_cred_to_sred.v
index 2a30a04..064275c 100644
--- a/theories/simulation_cred_to_sred.v
+++ b/theories/simulation_cred_to_sred.v
@@ -372,7 +372,7 @@ Proof.
   all: try eapply IHkappa; eauto.
   all: repeat rewrite snd_apply_conts_last.
   all: try reflexivity.
-  { induction o; unfold apply_CDefault. admit "need to show that sim_term Empty v => sim_term Empty". }
+  { admit "need to show that sim_term Empty v => sim_term Empty". }
   { induction ts; asimpl; econstructor; try reflexivity; eauto. }
 Admitted.
 
@@ -400,7 +400,7 @@ Proof.
   { eapply star_sred_apply_conts. eauto. }
 Qed.
 
-Theorem simulation_cred_sred:
+Theorem simulation_cred_sred_base:
   forall s1 s2,
   cred s1 s2 ->
   exists t,
@@ -441,3 +441,79 @@ Proof.
     reflexivity.
   }
 Admitted.
+
+
+(*** Lifting of this result ***)
+
+
+
+Lemma proper_inv_state_sred:
+  forall t1 t2,
+    sred t1 t2 ->
+    forall u1,
+      sim_term t1 u1 ->
+      exists u2,
+        sim_term t2 u2 /\ sred u1 u2.
+Proof.
+  induction 1; intros; repeat sinv_sim_term; subst.
+  (* induction hypothesis *)
+  all: repeat match goal with
+    | [ih: forall x, ?P x -> _, h: ?P _ |- _] => 
+      learn (ih _ h); unpack; subst end.
+
+  (* Most cases are trivial. *)
+  all: try solve[repeat econstructor; eauto].
+
+  { repeat econstructor.
+    rewrite soe_nil.
+    asimpl.
+    eauto.
+  }
+  { repeat econstructor.
+    admit "need to replace the substitution into two substitutions". 
+  }
+  { admit "binop". }
+  { repeat econstructor.
+    eapply sim_term_subst; eauto.
+    induction x; simpl; repeat econstructor; eauto.
+  }
+Admitted.
+
+Lemma proper_inv_state_star_sred:
+  forall t1 t2,
+    star sred t1 t2 ->
+    forall u1,
+      sim_term t1 u1 ->
+      exists u2,
+        sim_term t2 u2 /\ star sred u1 u2.
+Proof.
+  induction 1 using star_ind_n1.
+  { repeat econstructor; eauto. }
+  { intros ? Ht1.
+    learn (IHstar _ Ht1); unpack.
+    learn (proper_inv_state_sred _ _ H _ H1); unpack.
+    eexists; split; eauto.
+    eapply star_step_n1; eauto.
+  }
+Qed.
+
+Theorem simulation_cred_sred:
+  forall s1 s2,
+    cred s1 s2 ->
+    forall t1,
+      inv_state s1 t1 ->
+      exists t2,
+        inv_state s2 t2 /\ star sred t1 t2.
+Proof.
+  intros ? ? Hs1s2 ? Hs1t1.
+  learn (simulation_cred_sred_base _ _ Hs1s2); unpack; subst.
+  repeat match goal with
+  | [h: inv_state  _ _ |- _] =>
+    learn (inv_state_inversion _ _ h); unpack; subst
+  end.
+  symmetry in H4.
+  learn (proper_inv_state_star_sred _ _ H1 _ H4); unpack.
+  eexists; split; [|eauto].
+  eapply inv_state_from_equiv.
+  etransitivity; [symmetry|]; eauto.
+Qed.
diff --git a/theories/syntax.v b/theories/syntax.v
index 9c98aef..54315b3 100644
--- a/theories/syntax.v
+++ b/theories/syntax.v
@@ -2,6 +2,9 @@ Require Export Autosubst.Autosubst.
 Require Export AutosubstExtra.
 Require Import String.
 Require Import Coq.ZArith.ZArith.
+Require Export Coq.Classes.RelationClasses.
+Require Import Ltac2.Ltac2.
+Set Default Proof Mode "Classic".
 
 Require Import tactics.
 
@@ -562,3 +565,31 @@ Instance Transtive_sim_term : Transitive sim_term. Admitted.
 Instance Reflexive_sim_value : Reflexive sim_value. eapply sim_term_refl. Qed. 
 Instance Symmetric_sim_value : Symmetric sim_value. Admitted.
 Instance Transtive_sim_value : Transitive sim_value. Admitted.
+
+
+Local Ltac2 sinv_sim_term () :=
+  match! goal with
+  | [ h: sim_term _ ?c |- _ ] => smart_inversion c h
+  | [ h: sim_term ?c _ |- _ ] => smart_inversion c h
+  | [ h: sim_value _ ?c |- _ ] => smart_inversion c h
+  | [ h: sim_value ?c _ |- _ ] => smart_inversion c h
+  | [ h: List.Forall2 _ (_ :: _) _ |- _ ] => 
+    Std.inversion Std.FullInversion (Std.ElimOnIdent h) None None;
+    Std.subst_all ();
+    Std.clear [h]
+  | [ h: List.Forall2 _ _ (_ :: _) |- _ ] => 
+    Std.inversion Std.FullInversion (Std.ElimOnIdent h) None None;
+    Std.subst_all ();
+    Std.clear [h]
+
+  | [ h: List.Forall2 _ [] _ |- _ ] => 
+    Std.inversion Std.FullInversion (Std.ElimOnIdent h) None None;
+    Std.subst_all ();
+    Std.clear [h]
+  | [ h: List.Forall2 _ _ [] |- _ ] => 
+    Std.inversion Std.FullInversion (Std.ElimOnIdent h) None None;
+    Std.subst_all ();
+    Std.clear [h]
+  end.
+
+Ltac sinv_sim_term := ltac2:(sinv_sim_term ()).