author | oheimb |
Fri, 20 Feb 1998 16:00:18 +0100 | |
changeset 4637 | bac998af6ea2 |
parent 4477 | b3e5857d8d99 |
child 6255 | db63752140c7 |
permissions | -rw-r--r-- |
3807 | 1 |
(* |
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File: Action.ML |
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Author: Stephan Merz |
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Copyright: 1997 University of Munich |
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Lemmas and tactics for TLA actions. |
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*) |
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val act_rews = [pairSF_def RS eq_reflection,unl_before,unl_after,unchanged_def, |
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pr_con,pr_before,pr_lift,pr_lift2,pr_lift3,pr_all,pr_ex]; |
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val action_rews = act_rews @ intensional_rews; |
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qed_goal "actionI" Action.thy "(!!s t. ([[s,t]] |= A)) ==> A" |
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(fn [prem] => [REPEAT (resolve_tac [prem,intI,state2_ext] 1)]); |
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qed_goal "actionD" Action.thy "A ==> ([[s,t]] |= A)" |
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(fn [prem] => [REPEAT (resolve_tac [prem,intD] 1)]); |
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(* ================ Functions to "unlift" action theorems into HOL rules ================ *) |
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(* Basic unlifting introduces a world parameter and applies basic rewrites, e.g. |
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A .= B gets ([[s,t]] |= A) = ([[s,t]] |= B) |
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A .-> B gets ([[s,t]] |= A) --> ([[s,t]] |= B) |
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*) |
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fun action_unlift th = rewrite_rule action_rews (th RS actionD); |
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(* A .-> B becomes A [[s,t]] ==> B [[s,t]] *) |
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fun action_mp th = zero_var_indexes ((action_unlift th) RS mp); |
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(* A .-> B becomes [| A[[s,t]]; B[[s,t]] ==> R |] ==> R |
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so that it can be used as an elimination rule |
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*) |
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fun action_impE th = zero_var_indexes ((action_unlift th) RS impE); |
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(* A .& B .-> C becomes [| A[[s,t]]; B[[s,t]] |] ==> C[[s,t]] *) |
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fun action_conjmp th = zero_var_indexes (conjI RS (action_mp th)); |
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(* A .& B .-> C becomes [| A[[s,t]]; B[[s,t]]; (C[[s,t]] ==> R) |] ==> R *) |
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fun action_conjimpE th = zero_var_indexes (conjI RS (action_impE th)); |
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(* Turn A .= B into meta-level rewrite rule A == B *) |
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fun action_rewrite th = (rewrite_rule action_rews (th RS inteq_reflection)); |
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(* ===================== Update simpset and classical prover ============================= *) |
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(* Make the simplifier use action_unlift rather than int_unlift |
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when action simplifications are added. |
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*) |
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fun maybe_unlift th = |
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(case concl_of th of |
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Const("Intensional.TrueInt",_) $ p |
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=> (action_unlift th |
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handle _ => int_unlift th) |
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| _ => th); |
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simpset_ref() := simpset() setmksimps ((mksimps mksimps_pairs) o maybe_unlift); |
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(* make act_rews be always active -- intensional_rews has been added before *) |
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Addsimps act_rews; |
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use "cladata.ML"; (* local version! *) |
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(* ================================ action_simp_tac ================================== *) |
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(* A dumb simplification tactic with just a little first-order logic: |
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should plug in only "very safe" rules that can be applied blindly. |
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Note that it applies whatever simplifications are currently active. |
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*) |
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fun action_simp_tac ss intros elims i = |
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(asm_full_simp_tac |
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(ss setloop ((resolve_tac (intros @ [refl,impI,conjI,actionI,allI])) |
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ORELSE' (eresolve_tac (elims @ [conjE,disjE,exE_prop])))) |
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i); |
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(* default version without additional plug-in rules *) |
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fun Action_simp_tac i = (action_simp_tac (simpset()) [] [] i); |
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(* ==================== Simplification of abstractions ==================== *) |
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(* Somewhat obscure simplifications, rarely necessary to get rid |
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of abstractions that may be introduced by higher-order unification. |
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*) |
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qed_goal "pr_con_abs" Action.thy "(%w. c)` .= #c" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac (con_abs::action_rews), |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift_abs" Action.thy "(%w. f(x w))` .= f[x`]" |
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(fn _ => [rtac actionI 1, |
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(* give all rewrites to the engine and it loops! *) |
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rewrite_goals_tac intensional_rews, |
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rewtac lift_abs, |
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rewtac pr_lift, |
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rewtac unl_lift, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift2_abs" Action.thy "(%w. f(x w) (y w))` .= f[x`,y`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift2_abs, |
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rewtac pr_lift2, |
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rewtac unl_lift2, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift2_abs_con1" Action.thy "(%w. f x (y w))` .= f[#x, y`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift2_abs_con1, |
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rewtac pr_lift2, |
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rewtac unl_lift2, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift2_abs_con2" Action.thy "(%w. f (x w) y)` .= f[x`, #y]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift2_abs_con2, |
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rewtac pr_lift2, |
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rewtac unl_lift2, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs" Action.thy "(%w. f(x w) (y w) (z w))` .= f[x`,y`,z`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con1" Action.thy "(%w. f x (y w) (z w))` .= f[#x, y`, z`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con1, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con2" Action.thy "(%w. f (x w) y (z w))` .= f[x`, #y, z`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con2, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con3" Action.thy "(%w. f (x w) (y w) z)` .= f[x`, y`, #z]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con3, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con12" Action.thy "(%w. f x y (z w))` .= f[#x, #y, z`]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con12, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con13" Action.thy "(%w. f x (y w) z)` .= f[#x, y`, #z]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con13, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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qed_goal "pr_lift3_abs_con23" Action.thy "(%w. f (x w) y z)` .= f[x`, #y, #z]" |
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(fn _ => [rtac actionI 1, |
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rewrite_goals_tac intensional_rews, |
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rewtac lift3_abs_con23, |
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rewtac pr_lift3, |
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rewtac unl_lift3, |
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rewtac pr_con, |
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rewtac unl_con, |
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rtac refl 1 |
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]); |
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(* We don't add these as default rewrite rules, because they are |
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rarely needed and may slow down automatic proofs. |
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*) |
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val pr_abs_rews = map (fn th => th RS inteq_reflection) |
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[pr_con_abs, |
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pr_lift_abs,pr_lift2_abs,pr_lift2_abs_con1,pr_lift2_abs_con2, |
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pr_lift3_abs,pr_lift3_abs_con1,pr_lift3_abs_con2,pr_lift3_abs_con3, |
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pr_lift3_abs_con12,pr_lift3_abs_con13,pr_lift3_abs_con23]; |
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(* =========================== square / angle brackets =========================== *) |
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qed_goalw "idle_squareI" Action.thy [square_def] |
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"!!s t. ([[s,t]] |= unchanged v) ==> ([[s,t]] |= [A]_v)" |
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parents:
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(fn _ => [ Auto_tac ]); |
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qed_goalw "busy_squareI" Action.thy [square_def] |
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"!!s t. ([[s,t]] |= A) ==> ([[s,t]] |= [A]_v)" |
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(fn _ => [ Auto_tac ]); |
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qed_goalw "square_simulation" Action.thy [square_def] |
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"[| unchanged f .& .~B .-> unchanged g; \ |
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\ A .& .~unchanged g .-> B \ |
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\ |] ==> [A]_f .-> [B]_g" |
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parents:
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(fn [p1,p2] => [Auto_tac, |
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etac (action_conjimpE p2) 1, |
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etac swap 3, etac (action_conjimpE p1) 3, |
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ALLGOALS atac |
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]); |
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qed_goalw "not_square" Action.thy [square_def,angle_def] |
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"(.~ [A]_v) .= <.~A>_v" |
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New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4089
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changeset
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(fn _ => [ Auto_tac ]); |
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qed_goalw "not_angle" Action.thy [square_def,angle_def] |
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"(.~ <A>_v) .= [.~A]_v" |
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New Auto_tac (by Oheimb), and new syntax (without parens), and expandshort
paulson
parents:
4089
diff
changeset
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(fn _ => [ Auto_tac ]); |
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(* ============================== Facts about ENABLED ============================== *) |
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qed_goalw "enabledI" Action.thy [enabled_def] |
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"A [[s,t]] ==> (Enabled A) s" |
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(fn prems => [ REPEAT (resolve_tac (exI::prems) 1) ]); |
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qed_goalw "enabledE" Action.thy [enabled_def] |
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"[| (Enabled A) s; !!u. A[[s,u]] ==> PROP R |] ==> PROP R" |
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(fn prems => [cut_facts_tac prems 1, |
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etac exE_prop 1, |
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resolve_tac prems 1, atac 1 |
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]); |
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qed_goal "notEnabledD" Action.thy |
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"!!G. ~ (Enabled G s) ==> ~ G [[s,t]]" |
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(fn _ => [ auto_tac (action_css addsimps2 [enabled_def]) ]); |
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(* Monotonicity *) |
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qed_goal "enabled_mono" Action.thy |
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"[| (Enabled F) s; F .-> G |] ==> (Enabled G) s" |
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(fn [min,maj] => [rtac (min RS enabledE) 1, |
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rtac enabledI 1, |
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etac (action_mp maj) 1 |
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]); |
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(* stronger variant *) |
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qed_goal "enabled_mono2" Action.thy |
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"[| (Enabled F) s; !!t. (F [[s,t]] ==> G[[s,t]] ) |] ==> (Enabled G) s" |
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(fn [min,maj] => [rtac (min RS enabledE) 1, |
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rtac enabledI 1, |
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etac maj 1 |
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]); |
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qed_goal "enabled_disj1" Action.thy |
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"!!s. (Enabled F) s ==> (Enabled (F .| G)) s" |
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paulson
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4089
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changeset
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(fn _ => [etac enabled_mono 1, Auto_tac |
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]); |
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qed_goal "enabled_disj2" Action.thy |
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"!!s. (Enabled G) s ==> (Enabled (F .| G)) s" |
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paulson
parents:
4089
diff
changeset
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(fn _ => [etac enabled_mono 1, Auto_tac |
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]); |
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qed_goal "enabled_conj1" Action.thy |
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"!!s. (Enabled (F .& G)) s ==> (Enabled F) s" |
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paulson
parents:
4089
diff
changeset
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(fn _ => [etac enabled_mono 1, Auto_tac |
3807 | 292 |
]); |
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qed_goal "enabled_conj2" Action.thy |
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"!!s. (Enabled (F .& G)) s ==> (Enabled G) s" |
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paulson
parents:
4089
diff
changeset
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(fn _ => [etac enabled_mono 1, Auto_tac |
3807 | 297 |
]); |
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qed_goal "enabled_conjE" Action.thy |
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"[| (Enabled (F .& G)) s; [| (Enabled F) s; (Enabled G) s |] ==> PROP R |] ==> PROP R" |
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(fn prems => [cut_facts_tac prems 1, resolve_tac prems 1, |
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etac enabled_conj1 1, etac enabled_conj2 1]); |
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qed_goal "enabled_disjD" Action.thy |
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"!!s. (Enabled (F .| G)) s ==> ((Enabled F) s) | ((Enabled G) s)" |
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(fn _ => [etac enabledE 1, |
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auto_tac (action_css addSDs2 [notEnabledD] addSEs2 [enabledI]) |
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]); |
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qed_goal "enabled_disj" Action.thy |
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"(Enabled (F .| G)) s = ( (Enabled F) s | (Enabled G) s )" |
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(fn _ => [rtac iffI 1, |
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etac enabled_disjD 1, |
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REPEAT (eresolve_tac [disjE,enabled_disj1,enabled_disj2] 1) |
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]); |
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qed_goal "enabled_ex" Action.thy |
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"(Enabled (REX x. F x)) s = (EX x. (Enabled (F x)) s)" |
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(fn _ => [ auto_tac (action_css addsimps2 [enabled_def]) ]); |
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(* A rule that combines enabledI and baseE, but generates fewer possible instantiations *) |
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qed_goal "base_enabled" Action.thy |
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"[| base_var(v); !!u. v u = c s ==> A [[s,u]] |] ==> Enabled A s" |
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(fn prems => [cut_facts_tac prems 1, |
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etac baseE 1, rtac enabledI 1, |
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REPEAT (ares_tac prems 1)]); |
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(* ---------------- enabled_tac: tactic to prove (Enabled A) -------------------- *) |
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(* "Enabled A" can be proven as follows: |
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- Assume that we know which state variables are "base variables"; |
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this should be expressed by a theorem of the form "base_var <x,y,z,...>". |
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- Resolve this theorem with baseE to introduce a constant for the value of the |
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variables in the successor state, and resolve the goal with the result. |
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- E-resolve with PairVarE so that we have one constant per variable. |
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- Resolve with enabledI and do some rewriting. |
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- Solve for the unknowns using standard HOL reasoning. |
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The following tactic combines these steps except the final one. |
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*) |
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fun enabled_tac base_vars i = |
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EVERY [(* apply actionI (plus rewriting) if the goal is of the form $(Enabled A), |
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do nothing if it is of the form (Enabled A) s *) |
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TRY ((rtac actionI i) THEN (SELECT_GOAL (rewrite_goals_tac action_rews) i)), |
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rtac (base_vars RS base_enabled) i, |
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REPEAT_DETERM (etac PairVarE i), |
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(SELECT_GOAL (rewrite_goals_tac action_rews) i) |
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]; |
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(* Example of use: |
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val [prem] = goal Action.thy "base_var <x,y,z> ==> $x .-> $Enabled ($x .& (y$ .= #False))"; |
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by (REPEAT ((CHANGED (Action_simp_tac 1)) ORELSE (enabled_tac prem 1))); |
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*) |