refactor wp adequacy: add and use a dedicated lemma for "preserving wptp"

iris_core.v

-Require Import world_prop core_lang lang masks.
+Require Import world_prop core_lang masks.
 Require Import ModuRes.PCM ModuRes.UPred ModuRes.BI ModuRes.PreoMet ModuRes.Finmap.
 
 Module IrisRes (RL : PCM_T) (C : CORE_LANG) <: PCM_T.
@@ -10,7 +10,7 @@ Module IrisRes (RL : PCM_T) (C : CORE_LANG) <: PCM_T.
 End IrisRes.
   
 Module IrisCore (RL : PCM_T) (C : CORE_LANG).
-  Module Export L  := Lang C.
+  Export C.
   Module Export R  := IrisRes RL C.
   Module Export WP := WorldProp R.
 
@@ -310,6 +310,13 @@ Module IrisCore (RL : PCM_T) (C : CORE_LANG).
         | (x :: xs)%list => Some x · comp_list xs
       end.
 
+    Lemma comp_list_app rs1 rs2 :
+      comp_list (rs1 ++ rs2) == comp_list rs1 · comp_list rs2.
+    Proof.
+      induction rs1; simpl comp_list; [now rewrite pcm_op_unit by apply _ |].
+      now rewrite IHrs1, assoc.
+    Qed.
+
     Definition cod (m : nat -f> res) : list res := List.map snd (findom_t m).
     Definition comp_map (m : nat -f> res) : option res := comp_list (cod m).
 

lang.v

@@ -159,4 +159,31 @@ Module Lang (C : CORE_LANG).
     - symmetry in EQK; now apply fill_noinv in EQK.
   Qed.
 
+  (* Reflexive, transitive closure of the step relation *)
+  Inductive steps : cfg -> cfg -> Prop :=
+  | steps_refl ρ : steps ρ ρ
+  | stepn_trans ρ1 ρ2 ρ3 : step ρ1 ρ2 -> steps ρ2 ρ3 -> steps ρ1 ρ3.
+                     
+  
+  Inductive stepn : nat -> cfg -> cfg -> Prop :=
+  | stepn_O ρ : stepn O ρ ρ
+  | stepn_S ρ1 ρ2 ρ3 n
+            (HS  : step ρ1 ρ2)
+            (HSN : stepn n ρ2 ρ3) :
+      stepn (S n) ρ1 ρ3.
+
+  Lemma steps_stepn ρ1 ρ2:
+    steps ρ1 ρ2 -> exists n, stepn n ρ1 ρ2.
+  Proof.
+    induction 1.
+    - eexists. eauto using stepn.
+    - destruct IHsteps as [n IH]. eexists. eauto using stepn.
+  Qed.
+
+  Lemma stepn_steps n ρ1 ρ2:
+    stepn n ρ1 ρ2 -> steps ρ1 ρ2.
+  Proof.
+    induction 1; now eauto using steps.
+  Qed.
+
 End Lang.

lib/ModuRes/PCM.v

@@ -150,6 +150,11 @@ Section Order.
        erewrite pcm_op_zero in EQ by apply _; contradiction.
   Qed.
 
+  Lemma unit_min r : pcm_unit _ ⊑ r.
+  Proof.
+    exists r; now erewrite comm, pcm_op_unit by apply _.
+  Qed.
+
 End Order.
 
 Section Exclusive.