Port environment simulation relation and abstract bounds

hol4/EnvironmentsScript.sml

 open preamble
-open simpLib realTheory realLib RealArith
-open AbbrevsTheory ExpressionAbbrevsTheory CommandsTheory
+open simpLib realTheory realLib RealArith sptreeTheory
+open AbbrevsTheory ExpressionAbbrevsTheory RealSimpsTheory CommandsTheory
 
 val _ = new_theory "Environments";
 
 val (approxEnv_rules, approxEnv_ind, approxEnv_cases) = Hol_reln `
-  (!(E:env) (A:analysisResult). approxEnv E A E) /\
-  (!(E1:env) (E2:env) (A:analysisResult) v1 v2 x.
-    approxEnv E1 A E2 /\
-    (abs (v1 - v2) <= SND (A (Var x))) ==>
-    approxEnv (updEnv x v1 E1) A (updEnv x v2 E2))`;
-    
+  (!(A:analysisResult).
+      approxEnv emptyEnv A LN LN emptyEnv) /\
+  (!(E1:env) (E2:env) (A:analysisResult) (fVars:num_set) (dVars:num_set) v1 v2 x.
+      approxEnv E1 A fVars dVars E2 /\
+      (abs (v1 - v2) <= abs v1 * machineEpsilon) /\
+      (lookup x (union fVars dVars) = NONE) ==>
+      approxEnv (updEnv x v1 E1) A (Insert x fVars) dVars (updEnv x v2 E2)) /\
+  (!(E1:env) (E2:env) (A:analysisResult) (fVars:num_set) (dVars:num_set) v1 v2 x.
+      approxEnv E1 A fVars dVars E2 /\
+      (abs (v1 - v2) <= SND (A (Var x))) /\
+      (lookup x (union fVars dVars) = NONE) ==>
+      approxEnv (updEnv x v1 E1) A fVars (Insert x dVars) (updEnv x v2 E2))`;
+
 val _ = export_theory ();;