author  krauss 
Tue, 30 Mar 2010 15:25:30 +0200  
changeset 36042  85efdadee8ae 
parent 35845  e5980f0ad025 
child 37233  b78f31ca4675 
permissions  rwrr 
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(* Title: HOL/Tools/rewrite_hol_proof.ML 
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Author: Stefan Berghofer, TU Muenchen 

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Rewrite rules for HOL proofs 

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*) 

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signature REWRITE_HOL_PROOF = 

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sig 

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val rews: (Proofterm.proof * Proofterm.proof) list 

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val elim_cong: typ list > Proofterm.proof > (Proofterm.proof * Proofterm.proof) option 
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end; 
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structure RewriteHOLProof : REWRITE_HOL_PROOF = 

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struct 

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open Proofterm; 

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val rews = map (pairself (Proof_Syntax.proof_of_term @{theory} true) o 
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Logic.dest_equals o Logic.varify_global o Proof_Syntax.read_term @{theory} propT) 
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(** eliminate metaequality rules **) 

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["(equal_elim % x1 % x2 %% \ 

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\ (combination % TYPE('T1) % TYPE('T2) % Trueprop % x3 % A % B %% \ 

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\ (axm.reflexive % TYPE('T3) % x4) %% prf1)) == \ 
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\ (iffD1 % A % B %% \ 
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\ (meta_eq_to_obj_eq % TYPE(bool) % A % B %% prf1))", 
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"(equal_elim % x1 % x2 %% (axm.symmetric % TYPE('T1) % x3 % x4 %% \ 

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\ (combination % TYPE('T2) % TYPE('T3) % Trueprop % x5 % A % B %% \ 

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\ (axm.reflexive % TYPE('T4) % x6) %% prf1))) == \ 
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\ (iffD2 % A % B %% \ 
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\ (meta_eq_to_obj_eq % TYPE(bool) % A % B %% prf1))", 
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"(meta_eq_to_obj_eq % TYPE('U) % x1 % x2 %% \ 

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\ (combination % TYPE('T) % TYPE('U) % f % g % x % y %% prf1 %% prf2)) == \ 
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\ (cong % TYPE('T) % TYPE('U) % f % g % x % y %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T => 'U) % f % g %% prf1) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % x % y %% prf2))", 

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"(meta_eq_to_obj_eq % TYPE('T) % x1 % x2 %% \ 

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\ (axm.transitive % TYPE('T) % x % y % z %% prf1 %% prf2)) == \ 

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\ (HOL.trans % TYPE('T) % x % y % z %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % x % y %% prf1) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % y % z %% prf2))", 

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"(meta_eq_to_obj_eq % TYPE('T) % x % x %% (axm.reflexive % TYPE('T) % x)) == \ 

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\ (HOL.refl % TYPE('T) % x)", 
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"(meta_eq_to_obj_eq % TYPE('T) % x % y %% \ 

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\ (axm.symmetric % TYPE('T) % x % y %% prf)) == \ 

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\ (sym % TYPE('T) % x % y %% (meta_eq_to_obj_eq % TYPE('T) % x % y %% prf))", 

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"(meta_eq_to_obj_eq % TYPE('T => 'U) % x1 % x2 %% \ 

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\ (abstract_rule % TYPE('T) % TYPE('U) % f % g %% prf)) == \ 
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\ (ext % TYPE('T) % TYPE('U) % f % g %% \ 
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\ (Lam (x::'T). meta_eq_to_obj_eq % TYPE('U) % f x % g x %% (prf % x)))", 
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"(meta_eq_to_obj_eq % TYPE('T) % x % y %% \ 

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\ (eq_reflection % TYPE('T) % x % y %% prf)) == prf", 

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"(meta_eq_to_obj_eq % TYPE('T1) % x1 % x2 %% (equal_elim % x3 % x4 %% \ 

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\ (combination % TYPE('T) % TYPE(prop) % x7 % x8 % C % D %% \ 
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\ (combination % TYPE('T) % TYPE('T3) % op == % op == % A % B %% \ 
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\ (axm.reflexive % TYPE('T4) % op ==) %% prf1) %% prf2) %% prf3)) == \ 
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\ (iffD1 % A = C % B = D %% \ 

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\ (cong % TYPE('T::type) % TYPE(bool) % op = A % op = B % C % D %% \ 
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\ (cong % TYPE('T) % TYPE('T=>bool) % \ 
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\ (op = :: 'T=>'T=>bool) % (op = :: 'T=>'T=>bool) % A % B %% \ 
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\ (HOL.refl % TYPE('T=>'T=>bool) % (op = :: 'T=>'T=>bool)) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % A % B %% prf1)) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % C % D %% prf2)) %% \ 

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\ (meta_eq_to_obj_eq % TYPE('T) % A % C %% prf3))", 
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"(meta_eq_to_obj_eq % TYPE('T1) % x1 % x2 %% (equal_elim % x3 % x4 %% \ 

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\ (axm.symmetric % TYPE('T2) % x5 % x6 %% \ 

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\ (combination % TYPE('T) % TYPE(prop) % x7 % x8 % C % D %% \ 
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\ (combination % TYPE('T) % TYPE('T3) % op == % op == % A % B %% \ 
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\ (axm.reflexive % TYPE('T4) % op ==) %% prf1) %% prf2)) %% prf3)) == \ 
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\ (iffD2 % A = C % B = D %% \ 

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\ (cong % TYPE('T::type) % TYPE(bool) % op = A % op = B % C % D %% \ 
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\ (cong % TYPE('T) % TYPE('T=>bool) % \ 
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\ (op = :: 'T=>'T=>bool) % (op = :: 'T=>'T=>bool) % A % B %% \ 
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\ (HOL.refl % TYPE('T=>'T=>bool) % (op = :: 'T=>'T=>bool)) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % A % B %% prf1)) %% \ 
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\ (meta_eq_to_obj_eq % TYPE('T) % C % D %% prf2)) %% \ 

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\ (meta_eq_to_obj_eq % TYPE('T) % B % D %% prf3))", 
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(** rewriting on bool: insert proper congruence rules for logical connectives **) 

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(* All *) 

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"(iffD1 % All P % All Q %% (cong % TYPE('T1) % TYPE('T2) % All % All % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % x1) %% (ext % TYPE('a) % TYPE(bool) % x2 % x3 %% prf)) %% prf') == \ 
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\ (allI % TYPE('a) % Q %% \ 
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\ (Lam x. \ 

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\ iffD1 % P x % Q x %% (prf % x) %% \ 

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\ (spec % TYPE('a) % P % x %% prf')))", 

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"(iffD2 % All P % All Q %% (cong % TYPE('T1) % TYPE('T2) % All % All % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % x1) %% (ext % TYPE('a) % TYPE(bool) % x2 % x3 %% prf)) %% prf') == \ 
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\ (allI % TYPE('a) % P %% \ 
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\ (Lam x. \ 

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\ iffD2 % P x % Q x %% (prf % x) %% \ 

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\ (spec % TYPE('a) % Q % x %% prf')))", 
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(* Ex *) 

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"(iffD1 % Ex P % Ex Q %% (cong % TYPE('T1) % TYPE('T2) % Ex % Ex % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % x1) %% (ext % TYPE('a) % TYPE(bool) % x2 % x3 %% prf)) %% prf') == \ 
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\ (exE % TYPE('a) % P % EX x. Q x %% prf' %% \ 
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\ (Lam x H : P x. \ 

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\ exI % TYPE('a) % Q % x %% \ 

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\ (iffD1 % P x % Q x %% (prf % x) %% H)))", 

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"(iffD2 % Ex P % Ex Q %% (cong % TYPE('T1) % TYPE('T2) % Ex % Ex % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % x1) %% (ext % TYPE('a) % TYPE(bool) % x2 % x3 %% prf)) %% prf') == \ 
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\ (exE % TYPE('a) % Q % EX x. P x %% prf' %% \ 
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\ (Lam x H : Q x. \ 

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\ exI % TYPE('a) % P % x %% \ 

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\ (iffD2 % P x % Q x %% (prf % x) %% H)))", 

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(* & *) 

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"(iffD1 % A & C % B & D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op & % op & % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op &) %% prf1) %% prf2) %% prf3) == \ 
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\ (conjI % B % D %% \ 
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\ (iffD1 % A % B %% prf1 %% (conjunct1 % A % C %% prf3)) %% \ 

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\ (iffD1 % C % D %% prf2 %% (conjunct2 % A % C %% prf3)))", 

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"(iffD2 % A & C % B & D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op & % op & % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op &) %% prf1) %% prf2) %% prf3) == \ 
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\ (conjI % A % C %% \ 
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\ (iffD2 % A % B %% prf1 %% (conjunct1 % B % D %% prf3)) %% \ 

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\ (iffD2 % C % D %% prf2 %% (conjunct2 % B % D %% prf3)))", 

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"(cong % TYPE(bool) % TYPE(bool) % op & A % op & A % B % C %% \ 
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\ (HOL.refl % TYPE(bool=>bool) % op & A)) == \ 
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\ (cong % TYPE(bool) % TYPE(bool) % op & A % op & A % B % C %% \ 
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\ (cong % TYPE(bool) % TYPE(bool=>bool) % \ 
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\ (op & :: bool=>bool=>bool) % (op & :: bool=>bool=>bool) % A % A %% \ 
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\ (HOL.refl % TYPE(bool=>bool=>bool) % (op & :: bool=>bool=>bool)) %% \ 
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\ (HOL.refl % TYPE(bool) % A)))", 
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(*  *) 
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"(iffD1 % A  C % B  D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op  % op  % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op  ) %% prf1) %% prf2) %% prf3) == \ 
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\ (disjE % A % C % B  D %% prf3 %% \ 
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\ (Lam H : A. disjI1 % B % D %% (iffD1 % A % B %% prf1 %% H)) %% \ 

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\ (Lam H : C. disjI2 % D % B %% (iffD1 % C % D %% prf2 %% H)))", 

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"(iffD2 % A  C % B  D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op  % op  % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op  ) %% prf1) %% prf2) %% prf3) == \ 
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\ (disjE % B % D % A  C %% prf3 %% \ 
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\ (Lam H : B. disjI1 % A % C %% (iffD2 % A % B %% prf1 %% H)) %% \ 

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\ (Lam H : D. disjI2 % C % A %% (iffD2 % C % D %% prf2 %% H)))", 

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"(cong % TYPE(bool) % TYPE(bool) % op  A % op  A % B % C %% \ 
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\ (HOL.refl % TYPE(bool=>bool) % op  A)) == \ 
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\ (cong % TYPE(bool) % TYPE(bool) % op  A % op  A % B % C %% \ 
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\ (cong % TYPE(bool) % TYPE(bool=>bool) % \ 
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\ (op  :: bool=>bool=>bool) % (op  :: bool=>bool=>bool) % A % A %% \ 
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\ (HOL.refl % TYPE(bool=>bool=>bool) % (op  :: bool=>bool=>bool)) %% \ 
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\ (HOL.refl % TYPE(bool) % A)))", 
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(* > *) 
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"(iffD1 % A > C % B > D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op > % op > % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op > ) %% prf1) %% prf2) %% prf3) == \ 
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\ (impI % B % D %% (Lam H: B. iffD1 % C % D %% prf2 %% \ 
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\ (mp % A % C %% prf3 %% (iffD2 % A % B %% prf1 %% H))))", 

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"(iffD2 % A > C % B > D %% (cong % TYPE('T1) % TYPE('T2) % x1 % x2 % C % D %% \ 

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\ (cong % TYPE('T3) % TYPE('T4) % op > % op > % A % B %% \ 

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\ (HOL.refl % TYPE('T5) % op > ) %% prf1) %% prf2) %% prf3) == \ 
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\ (impI % A % C %% (Lam H: A. iffD2 % C % D %% prf2 %% \ 
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\ (mp % B % D %% prf3 %% (iffD1 % A % B %% prf1 %% H))))", 

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"(cong % TYPE(bool) % TYPE(bool) % op > A % op > A % B % C %% \ 
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\ (HOL.refl % TYPE(bool=>bool) % op > A)) == \ 
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\ (cong % TYPE(bool) % TYPE(bool) % op > A % op > A % B % C %% \ 
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\ (cong % TYPE(bool) % TYPE(bool=>bool) % \ 
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\ (op > :: bool=>bool=>bool) % (op > :: bool=>bool=>bool) % A % A %% \ 
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\ (HOL.refl % TYPE(bool=>bool=>bool) % (op > :: bool=>bool=>bool)) %% \ 
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\ (HOL.refl % TYPE(bool) % A)))", 
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(* ~ *) 
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"(iffD1 % ~ P % ~ Q %% (cong % TYPE('T1) % TYPE('T2) % Not % Not % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % Not) %% prf1) %% prf2) == \ 
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\ (notI % Q %% (Lam H: Q. \ 
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\ notE % P % False %% prf2 %% (iffD2 % P % Q %% prf1 %% H)))", 

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"(iffD2 % ~ P % ~ Q %% (cong % TYPE('T1) % TYPE('T2) % Not % Not % P % Q %% \ 

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\ (HOL.refl % TYPE('T3) % Not) %% prf1) %% prf2) == \ 
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\ (notI % P %% (Lam H: P. \ 
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\ notE % Q % False %% prf2 %% (iffD1 % P % Q %% prf1 %% H)))", 

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(* = *) 

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"(iffD1 % B % D %% \ 

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\ (iffD1 % A = C % B = D %% (cong % TYPE(bool) % TYPE('T1) % x1 % x2 % C % D %% \ 
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\ (cong % TYPE(bool) % TYPE('T2) % op = % op = % A % B %% \ 
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\ (HOL.refl % TYPE('T3) % op =) %% prf1) %% prf2) %% prf3) %% prf4) == \ 
13404  211 
\ (iffD1 % C % D %% prf2 %% \ 
212 
\ (iffD1 % A % C %% prf3 %% (iffD2 % A % B %% prf1 %% prf4)))", 

213 

214 
"(iffD2 % B % D %% \ 

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\ (iffD1 % A = C % B = D %% (cong % TYPE(bool) % TYPE('T1) % x1 % x2 % C % D %% \ 
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\ (cong % TYPE(bool) % TYPE('T2) % op = % op = % A % B %% \ 
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\ (HOL.refl % TYPE('T3) % op =) %% prf1) %% prf2) %% prf3) %% prf4) == \ 
13404  218 
\ (iffD1 % A % B %% prf1 %% \ 
219 
\ (iffD2 % A % C %% prf3 %% (iffD2 % C % D %% prf2 %% prf4)))", 

220 

221 
"(iffD1 % A % C %% \ 

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\ (iffD2 % A = C % B = D %% (cong % TYPE(bool) % TYPE('T1) % x1 % x2 % C % D %% \ 
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\ (cong % TYPE(bool) % TYPE('T2) % op = % op = % A % B %% \ 
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\ (HOL.refl % TYPE('T3) % op =) %% prf1) %% prf2) %% prf3) %% prf4)== \ 
13404  225 
\ (iffD2 % C % D %% prf2 %% \ 
226 
\ (iffD1 % B % D %% prf3 %% (iffD1 % A % B %% prf1 %% prf4)))", 

227 

228 
"(iffD2 % A % C %% \ 

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\ (iffD2 % A = C % B = D %% (cong % TYPE(bool) % TYPE('T1) % x1 % x2 % C % D %% \ 
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\ (cong % TYPE(bool) % TYPE('T2) % op = % op = % A % B %% \ 
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\ (HOL.refl % TYPE('T3) % op =) %% prf1) %% prf2) %% prf3) %% prf4) == \ 
13404  232 
\ (iffD2 % A % B %% prf1 %% \ 
233 
\ (iffD2 % B % D %% prf3 %% (iffD1 % C % D %% prf2 %% prf4)))", 

234 

235 
"(cong % TYPE(bool) % TYPE(bool) % op = A % op = A % B % C %% \ 

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\ (HOL.refl % TYPE(bool=>bool) % op = A)) == \ 
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\ (cong % TYPE(bool) % TYPE(bool) % op = A % op = A % B % C %% \ 
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\ (cong % TYPE(bool) % TYPE(bool=>bool) % \ 
13404  239 
\ (op = :: bool=>bool=>bool) % (op = :: bool=>bool=>bool) % A % A %% \ 
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\ (HOL.refl % TYPE(bool=>bool=>bool) % (op = :: bool=>bool=>bool)) %% \ 
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\ (HOL.refl % TYPE(bool) % A)))", 
13404  242 

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(** transitivity, reflexivity, and symmetry **) 
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244 

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"(iffD1 % A % C %% (HOL.trans % TYPE(bool) % A % B % C %% prf1 %% prf2) %% prf3) == \ 
13404  246 
\ (iffD1 % B % C %% prf2 %% (iffD1 % A % B %% prf1 %% prf3))", 
247 

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"(iffD2 % A % C %% (HOL.trans % TYPE(bool) % A % B % C %% prf1 %% prf2) %% prf3) == \ 
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\ (iffD2 % A % B %% prf1 %% (iffD2 % B % C %% prf2 %% prf3))", 
250 

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"(iffD1 % A % A %% (HOL.refl % TYPE(bool) % A) %% prf) == prf", 
13404  252 

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"(iffD2 % A % A %% (HOL.refl % TYPE(bool) % A) %% prf) == prf", 
13404  254 

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"(iffD1 % A % B %% (sym % TYPE(bool) % B % A %% prf)) == (iffD2 % B % A %% prf)", 
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"(iffD2 % A % B %% (sym % TYPE(bool) % B % A %% prf)) == (iffD1 % B % A %% prf)", 
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13404  259 
(** normalization of HOL proofs **) 
260 

261 
"(mp % A % B %% (impI % A % B %% prf)) == prf", 

262 

263 
"(impI % A % B %% (mp % A % B %% prf)) == prf", 

264 

265 
"(spec % TYPE('a) % P % x %% (allI % TYPE('a) % P %% prf)) == prf % x", 

266 

267 
"(allI % TYPE('a) % P %% (Lam x::'a. spec % TYPE('a) % P % x %% prf)) == prf", 

268 

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"(exE % TYPE('a) % P % Q %% (exI % TYPE('a) % P % x %% prf1) %% prf2) == (prf2 % x %% prf1)", 
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"(exE % TYPE('a) % P % Q %% prf %% (exI % TYPE('a) % P)) == prf", 
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13404  273 
"(disjE % P % Q % R %% (disjI1 % P % Q %% prf1) %% prf2 %% prf3) == (prf2 %% prf1)", 
274 

275 
"(disjE % P % Q % R %% (disjI2 % Q % P %% prf1) %% prf2 %% prf3) == (prf3 %% prf1)", 

276 

277 
"(conjunct1 % P % Q %% (conjI % P % Q %% prf1 %% prf2)) == prf1", 

278 

279 
"(conjunct2 % P % Q %% (conjI % P % Q %% prf1 %% prf2)) == prf2", 

280 

281 
"(iffD1 % A % B %% (iffI % A % B %% prf1 %% prf2)) == prf1", 

282 

283 
"(iffD2 % A % B %% (iffI % A % B %% prf1 %% prf2)) == prf2"]; 

284 

285 

286 
(** Replace congruence rules by substitution rules **) 

287 

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fun strip_cong ps (PThm (_, (("HOL.cong", _, _), _)) % _ % _ % SOME x % SOME y %% 
13404  289 
prf1 %% prf2) = strip_cong (((x, y), prf2) :: ps) prf1 
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 strip_cong ps (PThm (_, (("HOL.refl", _, _), _)) % SOME f) = SOME (f, ps) 
15531  291 
 strip_cong _ _ = NONE; 
13404  292 

28814  293 
val subst_prf = fst (strip_combt (Thm.proof_of subst)); 
294 
val sym_prf = fst (strip_combt (Thm.proof_of sym)); 

13404  295 

296 
fun make_subst Ts prf xs (_, []) = prf 

297 
 make_subst Ts prf xs (f, ((x, y), prf') :: ps) = 

298 
let val T = fastype_of1 (Ts, x) 

299 
in if x aconv y then make_subst Ts prf (xs @ [x]) (f, ps) 

15531  300 
else change_type (SOME [T]) subst_prf %> x %> y %> 
13404  301 
Abs ("z", T, list_comb (incr_boundvars 1 f, 
302 
map (incr_boundvars 1) xs @ Bound 0 :: 

303 
map (incr_boundvars 1 o snd o fst) ps)) %% prf' %% 

304 
make_subst Ts prf (xs @ [x]) (f, ps) 

305 
end; 

306 

307 
fun make_sym Ts ((x, y), prf) = 

15531  308 
((y, x), change_type (SOME [fastype_of1 (Ts, x)]) sym_prf %> x %> y %% prf); 
13404  309 

22277  310 
fun mk_AbsP P t = AbsP ("H", Option.map HOLogic.mk_Trueprop P, t); 
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fun elim_cong_aux Ts (PThm (_, (("HOL.iffD1", _, _), _)) % _ % _ %% prf1 %% prf2) = 
15570  313 
Option.map (make_subst Ts prf2 []) (strip_cong [] prf1) 
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 elim_cong_aux Ts (PThm (_, (("HOL.iffD1", _, _), _)) % P % _ %% prf) = 
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Option.map (mk_AbsP P o make_subst Ts (PBound 0) []) 
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(strip_cong [] (incr_pboundvars 1 0 prf)) 
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 elim_cong_aux Ts (PThm (_, (("HOL.iffD2", _, _), _)) % _ % _ %% prf1 %% prf2) = 
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Option.map (make_subst Ts prf2 [] o 
13404  319 
apsnd (map (make_sym Ts))) (strip_cong [] prf1) 
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 elim_cong_aux Ts (PThm (_, (("HOL.iffD2", _, _), _)) % _ % P %% prf) = 
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Option.map (mk_AbsP P o make_subst Ts (PBound 0) [] o 
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apsnd (map (make_sym Ts))) (strip_cong [] (incr_pboundvars 1 0 prf)) 
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 elim_cong_aux _ _ = NONE; 
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324 

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fun elim_cong Ts prf = Option.map (rpair no_skel) (elim_cong_aux Ts prf); 
13404  326 

327 
end; 