example(wip): 2wp_gen, cont'd.

examples/in_progress/2wp_gen/base.mlw

@@ -95,6 +95,16 @@ module Set
 
 end
 
+(* Some usual big operators on sets. *)
+module SetBigOps
+
+  use import Set
+
+  function bigunion (c:set (set 'a)) : set 'a = \x. exists s. c s /\ s x
+  function biginter (c:set (set 'a)) : set 'a = \x. forall s. c s -> s x
+
+end
+
 (* Basic definition on relations-as-arrows *)
 module Rel
 

examples/in_progress/2wp_gen/base/why3session.xml

@@ -2,13 +2,13 @@
 <!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
 "http://why3.lri.fr/why3session.dtd">
 <why3session shape_version="4">
-<prover id="1" name="Alt-Ergo" version="1.01" timelimit="5" steplimit="1" memlimit="1000"/>
+<prover id="1" name="Alt-Ergo" version="1.01" timelimit="5" steplimit="1000" memlimit="1"/>
 <file name="../base.mlw">
 <theory name="Fun" sum="d41d8cd98f00b204e9800998ecf8427e">
 </theory>
 <theory name="FunExt" sum="74d4af5538daa9eb9ddee6284a05e5bf">
  <goal name="WP_parameter extensional" expl="VC for extensional">
- <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
  </goal>
 </theory>
 <theory name="FunCategory" sum="d41d8cd98f00b204e9800998ecf8427e">
@@ -19,36 +19,38 @@
   <goal name="ext.1" expl="1.">
   <transf name="inline_goal">
    <goal name="ext.1.1" expl="1.">
-   <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+   <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
    </goal>
   </transf>
   </goal>
   <goal name="ext.2" expl="2.">
-  <proof prover="1"><result status="valid" time="0.00" steps="3"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="3"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="assoc">
- <proof prover="1"><result status="valid" time="0.00" steps="4"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="4"/></proof>
  </goal>
  <goal name="neutral">
- <proof prover="1"><result status="valid" time="0.00" steps="9"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="9"/></proof>
  </goal>
  <goal name="FunExt.extensionality">
- <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
  </goal>
  <goal name="FunCategory.assoc">
- <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
  </goal>
  <goal name="FunCategory.neutral">
- <proof prover="1"><result status="valid" time="0.01" steps="10"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="10"/></proof>
  </goal>
 </theory>
 <theory name="Set" sum="8dc39e2ae96fd0498b0d50bb47f235eb">
  <goal name="sext_is_ext">
- <proof prover="1"><result status="valid" time="0.00" steps="4"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="4"/></proof>
  </goal>
 </theory>
+<theory name="SetBigOps" sum="d41d8cd98f00b204e9800998ecf8427e">
+</theory>
 <theory name="Rel" sum="d41d8cd98f00b204e9800998ecf8427e">
 </theory>
 <theory name="RelExt" sum="d41d8cd98f00b204e9800998ecf8427e">
@@ -63,66 +65,66 @@
  <goal name="extensionality">
  <transf name="split_goal_wp">
   <goal name="extensionality.1" expl="1.">
-  <proof prover="1"><result status="valid" time="0.01" steps="4"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="4"/></proof>
   </goal>
   <goal name="extensionality.2" expl="2.">
-  <proof prover="1"><result status="valid" time="0.01" steps="4"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="4"/></proof>
   </goal>
   <goal name="extensionality.3" expl="3.">
-  <proof prover="1"><result status="valid" time="0.00" steps="3"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="3"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="assoc">
  <transf name="compute_specified">
   <goal name="assoc.1" expl="1.">
-  <proof prover="1"><result status="valid" time="0.01" steps="12"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="12"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="reverse_antimorphism">
- <proof prover="1"><result status="valid" time="0.01" steps="27"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="27"/></proof>
  </goal>
  <goal name="reverse_antimorphism_id">
- <proof prover="1"><result status="valid" time="0.01" steps="9"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="9"/></proof>
  </goal>
  <goal name="reverse_involution">
- <proof prover="1"><result status="valid" time="0.01" steps="5"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="5"/></proof>
  </goal>
  <goal name="neutral">
  <transf name="compute_specified">
   <goal name="neutral.1" expl="1.">
-  <proof prover="1"><result status="valid" time="0.00" steps="14"/></proof>
+  <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="14"/></proof>
   </goal>
  </transf>
  </goal>
  <goal name="RelExt.extensionality">
- <proof prover="1"><result status="valid" time="0.01" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="2"/></proof>
  </goal>
  <goal name="RelCategory.assoc">
- <proof prover="1"><result status="valid" time="0.01" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="2"/></proof>
  </goal>
  <goal name="RelCategory.reverse_antimorphism">
- <proof prover="1"><result status="valid" time="0.01" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="2"/></proof>
  </goal>
  <goal name="RelCategory.reverse_antimorphism_id">
- <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
  </goal>
  <goal name="RelCategory.reverse_involution">
- <proof prover="1"><result status="valid" time="0.01" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="2"/></proof>
  </goal>
  <goal name="RelCategory.neutral">
- <proof prover="1"><result status="valid" time="0.01" steps="11"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.01" steps="11"/></proof>
  </goal>
 </theory>
 <theory name="SubsetOrder" sum="d41d8cd98f00b204e9800998ecf8427e">
 </theory>
 <theory name="SetProofs" sum="0be5aa0fe01aa313571b5c5ef55cb77b">
  <goal name="anti_subset">
- <proof prover="1"><result status="valid" time="0.00" steps="7"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="7"/></proof>
  </goal>
  <goal name="SubsetOrder.subset_order">
- <proof prover="1"><result status="valid" time="0.02" steps="65"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.02" steps="65"/></proof>
  </goal>
 </theory>
 <theory name="Image" sum="d41d8cd98f00b204e9800998ecf8427e">

examples/in_progress/2wp_gen/choice/why3session.xml

@@ -2,14 +2,14 @@
 <!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
 "http://why3.lri.fr/why3session.dtd">
 <why3session shape_version="4">
-<prover id="1" name="Alt-Ergo" version="1.01" timelimit="5" steplimit="1" memlimit="1000"/>
+<prover id="1" name="Alt-Ergo" version="1.01" timelimit="5" steplimit="1000" memlimit="1"/>
 <file name="../choice.mlw">
 <theory name="Choice" sum="7c1ca533a0ae37f3bd6e59347ceef0bd">
  <goal name="WP_parameter choose" expl="VC for choose">
- <proof prover="1"><result status="valid" time="0.00" steps="3"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="3"/></proof>
  </goal>
  <goal name="WP_parameter choose_if" expl="VC for choose_if">
- <proof prover="1"><result status="valid" time="0.00" steps="2"/></proof>
+ <proof prover="1" steplimit="1" memlimit="1000"><result status="valid" time="0.00" steps="2"/></proof>
  </goal>
 </theory>
 </file>

examples/in_progress/2wp_gen/ho_set.mlw

 
 module Set use export base.Set end
-module SubsetOrder use export base.SubsetOrder end
\ No newline at end of file
+module SubsetOrder use export base.SubsetOrder end
+module SetBigOps use export base.SetBigOps end
\ No newline at end of file

examples/in_progress/2wp_gen/ho_set/why3session.xml

@@ -7,5 +7,7 @@
 </theory>
 <theory name="SubsetOrder" sum="d41d8cd98f00b204e9800998ecf8427e" expanded="true">
 </theory>
+<theory name="SetBigOps" sum="d41d8cd98f00b204e9800998ecf8427e" expanded="true">
+</theory>
 </file>
 </why3session>

examples/in_progress/2wp_gen/order.mlw

@@ -92,13 +92,14 @@ module Chain
   predicate chain_complete (o:erel 'a) =
     forall s. chain o s -> exists y. supremum o s y
 
+  (* quasi-chain-complete = all non-empty chains admit supremum. *)
+  predicate q_chain_complete (o:erel 'a) =
+    forall s inh. chain o s /\ s inh -> exists y. supremum o s y
+
   predicate subchain (o:erel 'a) (s1 s2:set 'a) =
     subset s1 s2 /\
     forall x y. s1 x /\ s2 y /\ not s1 y -> o x y
 
-  function subchain_completion (c:set (set 'a)) : set 'a =
-    \x. exists s. c s /\ s x
-
   predicate wf_chain (o:erel 'a) (s:set 'a) =
     forall s2. subset s2 s /\ (exists x. s2 x) ->
       exists y. minimum o s2 y
@@ -162,13 +163,14 @@ end
 module SubChain "W:non_conservative_extension:N" (* => SubChainProof *)
 
   use export Chain
+  use import ho_set.SetBigOps
 
   axiom subchain_complete_order : forall o:erel 'a.
     transitive o -> order (subchain o) /\ chain_complete (subchain o)
 
   axiom subchain_completion : forall o:erel 'a,ch.
     chain (subchain o) ch ->
-    supremum (subchain o) ch (subchain_completion ch)
+    supremum (subchain o) ch (bigunion ch)
 
   axiom chain_subchain_completion : forall o:erel 'a,ch x.
     chain (subchain o) ch /\ (forall x. ch x -> chain o x) /\
@@ -183,6 +185,7 @@ end
 module SubChainProof
 
   use import ho_set.SubsetOrder
+  use import ho_set.SetBigOps
   use import Chain
 
   lemma subchain_antisymetric : forall o:erel 'a.
@@ -192,7 +195,7 @@ module SubChainProof
     transitive o -> order (subchain o)
 
   lemma subchain_completion : forall o:erel 'a,ch.
-    let u = subchain_completion ch in
+    let u = bigunion ch in
     let sb = subchain o in
     chain sb ch ->
     supremum sb ch u
@@ -211,7 +214,7 @@ module SubChainProof
     chain (subchain o) ch /\
     (forall x. ch x -> chain o x) /\ supremum (subchain o) ch x ->
     chain o x
-    by x = subchain_completion ch
+    by x = bigunion ch
     so forall u v. x u /\ x v -> o u v \/ o v u
     by exists wu. ch wu /\ wu u
     so exists wv. ch wv /\ wv v
@@ -222,7 +225,7 @@ module SubChainProof
     chain (subchain o) ch /\
     (forall x. ch x -> wf_chain o x) /\ supremum (subchain o) ch x ->
     wf_chain o x
-    by x = subchain_completion ch
+    by x = bigunion ch
     so forall s2 y. s2 y /\ subset s2 x ->
       (exists z. minimum o s2 z)
     by exists wy. ch wy /\ wy y
@@ -347,4 +350,13 @@ module ProductChain
     so exists y. supremum o2 (image snd ch) y
     so supremum (rprod o1 o2) ch (x,y)
 
+  lemma prod_q_chain_complete : forall o1:erel 'a,o2:erel 'b.
+    order o1 /\ order o2 /\ q_chain_complete o1 /\ q_chain_complete o2 ->
+    q_chain_complete (rprod o1 o2)
+    by forall ch inh. chain (rprod o1 o2) ch /\ ch inh ->
+      let (a,b) = inh in image fst ch a /\ image snd ch b
+    so exists x. supremum o1 (image fst ch) x
+    so exists y. supremum o2 (image snd ch) y
+    so supremum (rprod o1 o2) ch (x,y)
+
 end
\ No newline at end of file

examples/in_progress/2wp_gen/transfinite.mlw

@@ -37,7 +37,7 @@ module Iterates "W:non_conservative_extension:N" (* => IterateProof *)
   function fixpoint_above (o:'a -> 'a -> bool) (f:'a -> 'a) 'a : 'a
 
   axiom tr_reach_maximum : forall o f,x:'a.
-    order o /\ inflationary o f /\ chain_complete o ->
+    order o /\ inflationary o f /\ q_chain_complete o ->
     f (fixpoint_above o f x) = fixpoint_above o f x /\
     maximum o (tr_reach o f x) (fixpoint_above o f x)
 
@@ -109,11 +109,12 @@ module IterateProof
     sup o (tr_reach o f u)
 
   lemma fixpoint_max_proof : forall o f,u:'a.
-    order o /\ chain_complete o /\ inflationary o f ->
+    order o /\ q_chain_complete o /\ inflationary o f ->
     (forall x y. tr_reach o f u x /\ tr_reach o f u y ->
         separator o f x y && (o x y \/ o y x)) &&
       chain o (tr_reach o f u) && let y = fixpoint_above o f u in
-      maximum o (tr_reach o f u) y && (f y = y by o (f y) y /\ o y (f y))
+      (maximum o (tr_reach o f u) y by tr_reach o f u u)
+      && (f y = y by o (f y) y /\ o y (f y))
 
   lemma fixpoint_is_max_proof : forall o f,x y:'a.
     order o /\ inflationary o f /\ tr_reach o f x y /\ f y = y ->