src/HOL/simpdata.ML
author paulson
Wed Dec 14 18:01:50 2005 +0100 (2005-12-14)
changeset 18407 fa075b606571
parent 18324 d1c4b1112e33
child 18529 540da2415751
permissions -rw-r--r--
deleted redundant (looping!) simprule
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(*  Title:      HOL/simpdata.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1991  University of Cambridge
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Instantiation of the generic simplifier for HOL.
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*)
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(* legacy ML bindings *)
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val Eq_FalseI = thm "Eq_FalseI";
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val Eq_TrueI = thm "Eq_TrueI";
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val all_conj_distrib = thm "all_conj_distrib";
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val all_simps = thms "all_simps";
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val cases_simp = thm "cases_simp";
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val conj_assoc = thm "conj_assoc";
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val conj_comms = thms "conj_comms";
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val conj_commute = thm "conj_commute";
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val conj_cong = thm "conj_cong";
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val conj_disj_distribL = thm "conj_disj_distribL";
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val conj_disj_distribR = thm "conj_disj_distribR";
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val conj_left_commute = thm "conj_left_commute";
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val de_Morgan_conj = thm "de_Morgan_conj";
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val de_Morgan_disj = thm "de_Morgan_disj";
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val disj_assoc = thm "disj_assoc";
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val disj_comms = thms "disj_comms";
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val disj_commute = thm "disj_commute";
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val disj_cong = thm "disj_cong";
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val disj_conj_distribL = thm "disj_conj_distribL";
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val disj_conj_distribR = thm "disj_conj_distribR";
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val disj_left_commute = thm "disj_left_commute";
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val disj_not1 = thm "disj_not1";
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val disj_not2 = thm "disj_not2";
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val eq_ac = thms "eq_ac";
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val eq_assoc = thm "eq_assoc";
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val eq_commute = thm "eq_commute";
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val eq_left_commute = thm "eq_left_commute";
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val eq_sym_conv = thm "eq_sym_conv";
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val eta_contract_eq = thm "eta_contract_eq";
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val ex_disj_distrib = thm "ex_disj_distrib";
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val ex_simps = thms "ex_simps";
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val if_False = thm "if_False";
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val if_P = thm "if_P";
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val if_True = thm "if_True";
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val if_bool_eq_conj = thm "if_bool_eq_conj";
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val if_bool_eq_disj = thm "if_bool_eq_disj";
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val if_cancel = thm "if_cancel";
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val if_def2 = thm "if_def2";
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val if_eq_cancel = thm "if_eq_cancel";
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val if_not_P = thm "if_not_P";
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val if_splits = thms "if_splits";
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val iff_conv_conj_imp = thm "iff_conv_conj_imp";
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val imp_all = thm "imp_all";
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val imp_cong = thm "imp_cong";
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val imp_conjL = thm "imp_conjL";
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val imp_conjR = thm "imp_conjR";
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val imp_conv_disj = thm "imp_conv_disj";
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val imp_disj1 = thm "imp_disj1";
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val imp_disj2 = thm "imp_disj2";
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val imp_disjL = thm "imp_disjL";
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val imp_disj_not1 = thm "imp_disj_not1";
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val imp_disj_not2 = thm "imp_disj_not2";
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val imp_ex = thm "imp_ex";
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val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
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val neq_commute = thm "neq_commute";
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val not_all = thm "not_all";
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val not_ex = thm "not_ex";
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val not_iff = thm "not_iff";
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val not_imp = thm "not_imp";
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val not_not = thm "not_not";
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val rev_conj_cong = thm "rev_conj_cong";
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val simp_impliesE = thm "simp_impliesI";
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val simp_impliesI = thm "simp_impliesI";
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val simp_implies_cong = thm "simp_implies_cong";
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val simp_implies_def = thm "simp_implies_def";
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val simp_thms = thms "simp_thms";
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val split_if = thm "split_if";
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val split_if_asm = thm "split_if_asm";
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val atomize_not = thm"atomize_not";
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local
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val uncurry = prove_goal (the_context()) "P --> Q --> R ==> P & Q --> R"
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              (fn prems => [cut_facts_tac prems 1, Blast_tac 1]);
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val iff_allI = allI RS
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    prove_goal (the_context()) "!x. P x = Q x ==> (!x. P x) = (!x. Q x)"
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               (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
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val iff_exI = allI RS
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    prove_goal (the_context()) "!x. P x = Q x ==> (? x. P x) = (? x. Q x)"
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               (fn prems => [cut_facts_tac prems 1, Blast_tac 1])
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val all_comm = prove_goal (the_context()) "(!x y. P x y) = (!y x. P x y)"
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               (fn _ => [Blast_tac 1])
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val ex_comm = prove_goal (the_context()) "(? x y. P x y) = (? y x. P x y)"
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               (fn _ => [Blast_tac 1])
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in
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(*** make simplification procedures for quantifier elimination ***)
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structure Quantifier1 = Quantifier1Fun
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(struct
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  (*abstract syntax*)
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  fun dest_eq((c as Const("op =",_)) $ s $ t) = SOME(c,s,t)
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    | dest_eq _ = NONE;
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  fun dest_conj((c as Const("op &",_)) $ s $ t) = SOME(c,s,t)
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    | dest_conj _ = NONE;
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  fun dest_imp((c as Const("op -->",_)) $ s $ t) = SOME(c,s,t)
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    | dest_imp _ = NONE;
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  val conj = HOLogic.conj
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  val imp  = HOLogic.imp
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  (*rules*)
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  val iff_reflection = eq_reflection
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  val iffI = iffI
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  val iff_trans = trans
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  val conjI= conjI
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  val conjE= conjE
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  val impI = impI
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  val mp   = mp
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  val uncurry = uncurry
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  val exI  = exI
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  val exE  = exE
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  val iff_allI = iff_allI
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  val iff_exI = iff_exI
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  val all_comm = all_comm
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  val ex_comm = ex_comm
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end);
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end;
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val defEX_regroup =
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  Simplifier.simproc (Theory.sign_of (the_context ()))
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    "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
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val defALL_regroup =
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  Simplifier.simproc (Theory.sign_of (the_context ()))
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    "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
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(*** simproc for proving "(y = x) == False" from prmise "~(x = y)" ***)
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val use_neq_simproc = ref true;
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local
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val neq_to_EQ_False = thm "not_sym" RS Eq_FalseI;
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fun neq_prover sg ss (eq $ lhs $ rhs) =
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let
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  fun test thm = (case #prop(rep_thm thm) of
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                    _ $ (Not $ (eq' $ l' $ r')) =>
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                      Not = HOLogic.Not andalso eq' = eq andalso
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                      r' aconv lhs andalso l' aconv rhs
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                  | _ => false)
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in
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  if !use_neq_simproc then
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    case Library.find_first test (prems_of_ss ss) of NONE => NONE
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    | SOME thm => SOME (thm RS neq_to_EQ_False)
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  else NONE
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end
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in
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val neq_simproc =
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  Simplifier.simproc (the_context ()) "neq_simproc" ["x = y"] neq_prover;
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end;
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(*** Simproc for Let ***)
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val use_let_simproc = ref true;
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local
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val Let_folded = thm "Let_folded";
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val Let_unfold = thm "Let_unfold";
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val f_Let_unfold = 
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      let val [(_$(f$_)$_)] = prems_of Let_unfold in cterm_of (sign_of (the_context ())) f end
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val f_Let_folded = 
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      let val [(_$(f$_)$_)] = prems_of Let_folded in cterm_of (sign_of (the_context ())) f end;
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val g_Let_folded = 
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      let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of (sign_of (the_context ())) g end;
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in
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val let_simproc =
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  Simplifier.simproc (Theory.sign_of (the_context ())) "let_simp" ["Let x f"] 
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   (fn sg => fn ss => fn t =>
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      (case t of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
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         if not (!use_let_simproc) then NONE
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         else if is_Free x orelse is_Bound x orelse is_Const x 
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         then SOME Let_def  
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         else
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          let
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             val n = case f of (Abs (x,_,_)) => x | _ => "x";
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             val cx = cterm_of sg x;
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             val {T=xT,...} = rep_cterm cx;
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             val cf = cterm_of sg f;
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             val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
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             val (_$_$g) = prop_of fx_g;
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             val g' = abstract_over (x,g);
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           in (if (g aconv g') 
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               then
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                  let
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                    val rl = cterm_instantiate [(f_Let_unfold,cf)] Let_unfold;
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                  in SOME (rl OF [fx_g]) end 
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               else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
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               else let 
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                     val abs_g'= Abs (n,xT,g');
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                     val g'x = abs_g'$x;
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                     val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
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                     val rl = cterm_instantiate
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                               [(f_Let_folded,cterm_of sg f),
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                                (g_Let_folded,cterm_of sg abs_g')]
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                               Let_folded; 
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                   in SOME (rl OF [transitive fx_g g_g'x]) end)
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           end
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        | _ => NONE))
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end
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(*** Case splitting ***)
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(*Make meta-equalities.  The operator below is Trueprop*)
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fun mk_meta_eq r = r RS eq_reflection;
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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fun mk_eq th = case concl_of th of
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        Const("==",_)$_$_       => th
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    |   _$(Const("op =",_)$_$_) => mk_meta_eq th
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    |   _$(Const("Not",_)$_)    => th RS Eq_FalseI
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    |   _                       => th RS Eq_TrueI;
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(* Expects Trueprop(.) if not == *)
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fun mk_eq_True r =
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  SOME (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
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(* Produce theorems of the form
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  (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
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*)
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fun lift_meta_eq_to_obj_eq i st =
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  let
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    val {sign, ...} = rep_thm st;
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    fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
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      | count_imp _ = 0;
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    val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
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  in if j = 0 then meta_eq_to_obj_eq
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    else
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      let
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        val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
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        fun mk_simp_implies Q = foldr (fn (R, S) =>
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          Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
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        val aT = TFree ("'a", HOLogic.typeS);
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        val x = Free ("x", aT);
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        val y = Free ("y", aT)
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      in standard (Goal.prove (Thm.theory_of_thm st) []
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        [mk_simp_implies (Logic.mk_equals (x, y))]
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        (mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))
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        (fn prems => EVERY
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         [rewrite_goals_tac [simp_implies_def],
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          REPEAT (ares_tac (meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)]))
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      end
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  end;
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(*Congruence rules for = (instead of ==)*)
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fun mk_meta_cong rl = zero_var_indexes
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  (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
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     rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
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   in mk_meta_eq rl' handle THM _ =>
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     if Logic.is_equals (concl_of rl') then rl'
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     else error "Conclusion of congruence rules must be =-equality"
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   end);
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(* Elimination of True from asumptions: *)
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local fun rd s = read_cterm (sign_of (the_context())) (s, propT);
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in val True_implies_equals = standard' (equal_intr
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  (implies_intr_hyps (implies_elim (assume (rd "True ==> PROP P")) TrueI))
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  (implies_intr_hyps (implies_intr (rd "True") (assume (rd "PROP P")))));
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end;
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structure SplitterData =
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  struct
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  structure Simplifier = Simplifier
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  val mk_eq          = mk_eq
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  val meta_eq_to_iff = meta_eq_to_obj_eq
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  val iffD           = iffD2
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  val disjE          = disjE
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  val conjE          = conjE
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  val exE            = exE
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  val contrapos      = contrapos_nn
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  val contrapos2     = contrapos_pp
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  val notnotD        = notnotD
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  end;
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structure Splitter = SplitterFun(SplitterData);
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val split_tac        = Splitter.split_tac;
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val split_inside_tac = Splitter.split_inside_tac;
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val split_asm_tac    = Splitter.split_asm_tac;
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val op addsplits     = Splitter.addsplits;
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val op delsplits     = Splitter.delsplits;
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val Addsplits        = Splitter.Addsplits;
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val Delsplits        = Splitter.Delsplits;
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val mksimps_pairs =
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  [("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
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   ("All", [spec]), ("True", []), ("False", []),
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   ("HOL.If", [if_bool_eq_conj RS iffD1])];
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(*
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val mk_atomize:      (string * thm list) list -> thm -> thm list
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looks too specific to move it somewhere else
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*)
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fun mk_atomize pairs =
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  let fun atoms th =
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        (case concl_of th of
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           Const("Trueprop",_) $ p =>
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             (case head_of p of
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                Const(a,_) =>
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                  (case AList.lookup (op =) pairs a of
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                     SOME(rls) => List.concat (map atoms ([th] RL rls))
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                   | NONE => [th])
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              | _ => [th])
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         | _ => [th])
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  in atoms end;
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fun mksimps pairs =
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  (List.mapPartial (try mk_eq) o mk_atomize pairs o gen_all);
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fun unsafe_solver_tac prems =
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  (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
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  FIRST'[resolve_tac(reflexive_thm::TrueI::refl::prems), atac, etac FalseE];
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val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
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(*No premature instantiation of variables during simplification*)
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fun safe_solver_tac prems =
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  (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
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  FIRST'[match_tac(reflexive_thm::TrueI::refl::prems),
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         eq_assume_tac, ematch_tac [FalseE]];
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val safe_solver = mk_solver "HOL safe" safe_solver_tac;
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val HOL_basic_ss =
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  Simplifier.theory_context (the_context ()) empty_ss
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    setsubgoaler asm_simp_tac
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    setSSolver safe_solver
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    setSolver unsafe_solver
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    setmksimps (mksimps mksimps_pairs)
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    setmkeqTrue mk_eq_True
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    setmkcong mk_meta_cong;
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fun unfold_tac ths =
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  let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths
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  in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
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(*In general it seems wrong to add distributive laws by default: they
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  might cause exponential blow-up.  But imp_disjL has been in for a while
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  and cannot be removed without affecting existing proofs.  Moreover,
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  rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the
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  grounds that it allows simplification of R in the two cases.*)
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val HOL_ss =
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    HOL_basic_ss addsimps
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     ([triv_forall_equality, (* prunes params *)
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       True_implies_equals, (* prune asms `True' *)
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       if_True, if_False, if_cancel, if_eq_cancel,
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       imp_disjL, conj_assoc, disj_assoc,
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       de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp,
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       disj_not1, not_all, not_ex, cases_simp,
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       thm "the_eq_trivial", the_sym_eq_trivial]
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     @ ex_simps @ all_simps @ simp_thms)
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     addsimprocs [defALL_regroup,defEX_regroup,neq_simproc,let_simproc]
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     addcongs [imp_cong, simp_implies_cong]
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     addsplits [split_if];
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   376
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   377
fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews);
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   379
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(*Simplifies x assuming c and y assuming ~c*)
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val prems = Goalw [if_def]
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  "[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
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\  (if b then x else y) = (if c then u else v)";
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by (asm_simp_tac (HOL_ss addsimps prems) 1);
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qed "if_cong";
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   386
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   387
(*Prevents simplification of x and y: faster and allows the execution
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   388
  of functional programs. NOW THE DEFAULT.*)
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   389
Goal "b=c ==> (if b then x else y) = (if c then x else y)";
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by (etac arg_cong 1);
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qed "if_weak_cong";
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   392
paulson@6293
   393
(*Prevents simplification of t: much faster*)
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Goal "a = b ==> (let x=a in t(x)) = (let x=b in t(x))";
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by (etac arg_cong 1);
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qed "let_weak_cong";
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   397
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   398
(*To tidy up the result of a simproc.  Only the RHS will be simplified.*)
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   399
Goal "u = u' ==> (t==u) == (t==u')";
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   400
by (asm_simp_tac HOL_ss 1);
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   401
qed "eq_cong2";
paulson@12975
   402
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   403
Goal "f(if c then x else y) = (if c then f x else f y)";
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   404
by (simp_tac (HOL_ss setloop (split_tac [split_if])) 1);
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   405
qed "if_distrib";
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   406
paulson@4327
   407
(*For expand_case_tac*)
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   408
val prems = Goal "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)";
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   409
by (case_tac "P" 1);
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   410
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems)));
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   411
qed "expand_case";
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   412
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   413
(*Used in Auth proofs.  Typically P contains Vars that become instantiated
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   414
  during unification.*)
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   415
fun expand_case_tac P i =
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   416
    res_inst_tac [("P",P)] expand_case i THEN
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   417
    Simp_tac (i+1) THEN
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   418
    Simp_tac i;
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   419
paulson@7584
   420
(*This lemma restricts the effect of the rewrite rule u=v to the left-hand
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   421
  side of an equality.  Used in {Integ,Real}/simproc.ML*)
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   422
Goal "x=y ==> (x=z) = (y=z)";
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   423
by (asm_simp_tac HOL_ss 1);
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   424
qed "restrict_to_left";
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   425
wenzelm@7357
   426
(* default simpset *)
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   427
val simpsetup =
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   428
  [fn thy => (change_simpset_of thy (fn _ => HOL_ss addcongs [if_weak_cong]); thy)];
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   429
oheimb@4652
   430
wenzelm@5219
   431
(*** integration of simplifier with classical reasoner ***)
oheimb@2636
   432
wenzelm@5219
   433
structure Clasimp = ClasimpFun
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   434
 (structure Simplifier = Simplifier and Splitter = Splitter
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   435
  and Classical  = Classical and Blast = Blast
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   436
  val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE
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   437
  val cla_make_elim = cla_make_elim);
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   438
open Clasimp;
oheimb@2636
   439
oheimb@2636
   440
val HOL_css = (HOL_cs, HOL_ss);
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   441
nipkow@5975
   442
wenzelm@8641
   443
nipkow@5975
   444
(*** A general refutation procedure ***)
wenzelm@9713
   445
nipkow@5975
   446
(* Parameters:
nipkow@5975
   447
nipkow@5975
   448
   test: term -> bool
nipkow@5975
   449
   tests if a term is at all relevant to the refutation proof;
nipkow@5975
   450
   if not, then it can be discarded. Can improve performance,
nipkow@5975
   451
   esp. if disjunctions can be discarded (no case distinction needed!).
nipkow@5975
   452
nipkow@5975
   453
   prep_tac: int -> tactic
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   454
   A preparation tactic to be applied to the goal once all relevant premises
nipkow@5975
   455
   have been moved to the conclusion.
nipkow@5975
   456
nipkow@5975
   457
   ref_tac: int -> tactic
nipkow@5975
   458
   the actual refutation tactic. Should be able to deal with goals
nipkow@5975
   459
   [| A1; ...; An |] ==> False
wenzelm@9876
   460
   where the Ai are atomic, i.e. no top-level &, | or EX
nipkow@5975
   461
*)
nipkow@5975
   462
nipkow@15184
   463
local
nipkow@15184
   464
  val nnf_simpset =
wenzelm@17892
   465
    empty_ss setmkeqTrue mk_eq_True
wenzelm@17892
   466
    setmksimps (mksimps mksimps_pairs)
wenzelm@17892
   467
    addsimps [imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
wenzelm@17892
   468
      not_all,not_ex,not_not];
wenzelm@17892
   469
  fun prem_nnf_tac i st =
wenzelm@17892
   470
    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
nipkow@15184
   471
in
nipkow@15184
   472
fun refute_tac test prep_tac ref_tac =
nipkow@15184
   473
  let val refute_prems_tac =
nipkow@12475
   474
        REPEAT_DETERM
nipkow@12475
   475
              (eresolve_tac [conjE, exE] 1 ORELSE
nipkow@5975
   476
               filter_prems_tac test 1 ORELSE
paulson@6301
   477
               etac disjE 1) THEN
nipkow@11200
   478
        ((etac notE 1 THEN eq_assume_tac 1) ORELSE
nipkow@11200
   479
         ref_tac 1);
nipkow@5975
   480
  in EVERY'[TRY o filter_prems_tac test,
nipkow@12475
   481
            REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
nipkow@5975
   482
            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
nipkow@5975
   483
  end;
wenzelm@17003
   484
end;