author | berghofe |
Wed, 07 Feb 2007 18:04:44 +0100 | |
changeset 22280 | a20a203c8f41 |
parent 21646 | c07b5b0e8492 |
child 22662 | 3e492ba59355 |
permissions | -rw-r--r-- |
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(* Title: Pure/Proof/proof_rewrite_rules.ML |
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ID: $Id$ |
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Author: Stefan Berghofer, TU Muenchen |
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Simplification functions for proof terms involving meta level rules. |
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*) |
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signature PROOF_REWRITE_RULES = |
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sig |
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val rew : bool -> typ list -> Proofterm.proof -> Proofterm.proof option |
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val rprocs : bool -> (string * (typ list -> Proofterm.proof -> Proofterm.proof option)) list |
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val rewrite_terms : (term -> term) -> Proofterm.proof -> Proofterm.proof |
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val elim_defs : theory -> bool -> thm list -> Proofterm.proof -> Proofterm.proof |
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val elim_vars : (typ -> term) -> Proofterm.proof -> Proofterm.proof |
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val hhf_proof : term -> term -> Proofterm.proof -> Proofterm.proof |
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val un_hhf_proof : term -> term -> Proofterm.proof -> Proofterm.proof |
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end; |
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structure ProofRewriteRules : PROOF_REWRITE_RULES = |
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struct |
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open Proofterm; |
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fun rew b _ = |
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let |
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fun ?? x = if b then SOME x else NONE; |
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fun ax (prf as PAxm (s, prop, _)) Ts = |
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if b then PAxm (s, prop, SOME Ts) else prf; |
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fun ty T = if b then |
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let val Type (_, [Type (_, [U, _]), _]) = T |
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in SOME U end |
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else NONE; |
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val equal_intr_axm = ax equal_intr_axm []; |
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val equal_elim_axm = ax equal_elim_axm []; |
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val symmetric_axm = ax symmetric_axm [propT]; |
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fun rew' (PThm ("ProtoPure.protectD", _, _, _) % _ %% |
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(PThm ("ProtoPure.protectI", _, _, _) % _ %% prf)) = SOME prf |
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| rew' (PThm ("ProtoPure.conjunctionD1", _, _, _) % _ % _ %% |
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(PThm ("ProtoPure.conjunctionI", _, _, _) % _ % _ %% prf %% _)) = SOME prf |
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| rew' (PThm ("ProtoPure.conjunctionD2", _, _, _) % _ % _ %% |
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(PThm ("ProtoPure.conjunctionI", _, _, _) % _ % _ %% _ %% prf)) = SOME prf |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.equal_intr", _, _) % _ % _ %% prf %% _)) = SOME prf |
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| rew' (PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.equal_intr", _, _) % A % B %% prf1 %% prf2)) = |
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SOME (equal_intr_axm % B % A %% prf2 %% prf1) |
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|
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ A) % SOME (_ $ B) %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("prop", _)) % |
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_ % _ % _ %% (PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% |
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((tg as PThm ("ProtoPure.protectI", _, _, _)) % _ %% prf2)) = |
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SOME (tg %> B %% (equal_elim_axm %> A %> B %% prf1 %% prf2)) |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ A) % SOME (_ $ B) %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("prop", _)) % |
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_ % _ % _ %% (PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1)) %% |
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((tg as PThm ("ProtoPure.protectI", _, _, _)) % _ %% prf2)) = |
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SOME (tg %> B %% (equal_elim_axm %> A %> B %% |
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(symmetric_axm % ?? B % ?? A %% prf1) %% prf2)) |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME X % SOME Y %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==>", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2)) = |
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let |
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val _ $ A $ C = Envir.beta_norm X; |
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val _ $ B $ D = Envir.beta_norm Y |
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in SOME (AbsP ("H1", ?? X, AbsP ("H2", ?? B, |
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equal_elim_axm %> C %> D %% incr_pboundvars 2 0 prf2 %% |
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(PBound 1 %% (equal_elim_axm %> B %> A %% |
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(symmetric_axm % ?? A % ?? B %% incr_pboundvars 2 0 prf1) %% PBound 0))))) |
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end |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME X % SOME Y %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==>", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2))) = |
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let |
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val _ $ A $ C = Envir.beta_norm Y; |
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val _ $ B $ D = Envir.beta_norm X |
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in SOME (AbsP ("H1", ?? X, AbsP ("H2", ?? A, |
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equal_elim_axm %> D %> C %% |
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(symmetric_axm % ?? C % ?? D %% incr_pboundvars 2 0 prf2) |
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%% (PBound 1 %% (equal_elim_axm %> A %> B %% incr_pboundvars 2 0 prf1 %% PBound 0))))) |
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end |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME X % SOME Y %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("all", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% |
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(PAxm ("ProtoPure.abstract_rule", _, _) % _ % _ %% prf))) = |
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let |
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val Const (_, T) $ P = Envir.beta_norm X; |
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val _ $ Q = Envir.beta_norm Y; |
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in SOME (AbsP ("H", ?? X, Abst ("x", ty T, |
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equal_elim_axm %> incr_boundvars 1 P $ Bound 0 %> incr_boundvars 1 Q $ Bound 0 %% |
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(incr_pboundvars 1 1 prf %> Bound 0) %% (PBound 0 %> Bound 0)))) |
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end |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME X % SOME Y %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("all", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% |
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(PAxm ("ProtoPure.abstract_rule", _, _) % _ % _ %% prf)))) = |
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let |
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val Const (_, T) $ P = Envir.beta_norm X; |
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val _ $ Q = Envir.beta_norm Y; |
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val t = incr_boundvars 1 P $ Bound 0; |
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val u = incr_boundvars 1 Q $ Bound 0 |
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in SOME (AbsP ("H", ?? X, Abst ("x", ty T, |
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equal_elim_axm %> t %> u %% |
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(symmetric_axm % ?? u % ?? t %% (incr_pboundvars 1 1 prf %> Bound 0)) |
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%% (PBound 0 %> Bound 0)))) |
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end |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME A % SOME C %% |
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(PAxm ("ProtoPure.transitive", _, _) % _ % SOME B % _ %% prf1 %% prf2) %% prf3) = |
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SOME (equal_elim_axm %> B %> C %% prf2 %% |
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(equal_elim_axm %> A %> B %% prf1 %% prf3)) |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % SOME A % SOME C %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.transitive", _, _) % _ % SOME B % _ %% prf1 %% prf2)) %% prf3) = |
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SOME (equal_elim_axm %> B %> C %% (symmetric_axm % ?? C % ?? B %% prf1) %% |
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(equal_elim_axm %> A %> B %% (symmetric_axm % ?? B % ?? A %% prf2) %% prf3)) |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf) = SOME prf |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _)) %% prf) = SOME prf |
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|
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| rew' (PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% prf)) = SOME prf |
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|
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ A $ C) % SOME (_ $ B $ D) %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2) %% prf3) %% prf4) = |
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SOME (equal_elim_axm %> C %> D %% prf2 %% |
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(equal_elim_axm %> A %> C %% prf3 %% |
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(equal_elim_axm %> B %> A %% (symmetric_axm % ?? A % ?? B %% prf1) %% prf4))) |
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ A $ C) % SOME (_ $ B $ D) %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2) %% prf3)) %% prf4) = |
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SOME (equal_elim_axm %> A %> B %% prf1 %% |
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(equal_elim_axm %> C %> A %% (symmetric_axm % ?? A % ?? C %% prf3) %% |
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(equal_elim_axm %> D %> C %% (symmetric_axm % ?? C % ?? D %% prf2) %% prf4))) |
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|
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| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ B $ D) % SOME (_ $ A $ C) %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2)) %% prf3) %% prf4) = |
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SOME (equal_elim_axm %> D %> C %% (symmetric_axm % ?? C % ?? D %% prf2) %% |
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(equal_elim_axm %> B %> D %% prf3 %% |
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164 |
(equal_elim_axm %> A %> B %% prf1 %% prf4))) |
11522 | 165 |
|
19309 | 166 |
| rew' (PAxm ("ProtoPure.equal_elim", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
15531 | 168 |
(PAxm ("ProtoPure.equal_elim", _, _) % SOME (_ $ B $ D) % SOME (_ $ A $ C) %% |
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(PAxm ("ProtoPure.symmetric", _, _) % _ % _ %% |
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(PAxm ("ProtoPure.combination", _, _) % _ % _ % _ % _ %% |
15531 | 171 |
(PAxm ("ProtoPure.combination", _, _) % SOME (Const ("==", _)) % _ % _ % _ %% |
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(PAxm ("ProtoPure.reflexive", _, _) % _) %% prf1) %% prf2)) %% prf3)) %% prf4) = |
17137 | 173 |
SOME (equal_elim_axm %> B %> A %% (symmetric_axm % ?? A % ?? B %% prf1) %% |
174 |
(equal_elim_axm %> D %> B %% (symmetric_axm % ?? B % ?? D %% prf3) %% |
|
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(equal_elim_axm %> C %> D %% prf2 %% prf4))) |
11522 | 176 |
|
19309 | 177 |
| rew' ((prf as PAxm ("ProtoPure.combination", _, _) % |
15531 | 178 |
SOME ((eq as Const ("==", T)) $ t) % _ % _ % _) %% |
13257 | 179 |
(PAxm ("ProtoPure.reflexive", _, _) % _)) = |
180 |
let val (U, V) = (case T of |
|
181 |
Type (_, [U, V]) => (U, V) | _ => (dummyT, dummyT)) |
|
17137 | 182 |
in SOME (prf %% (ax combination_axm [V, U] %> eq % ?? eq % ?? t % ?? t %% |
183 |
(ax reflexive_axm [T] % ?? eq) %% (ax reflexive_axm [U] % ?? t))) |
|
13257 | 184 |
end |
185 |
||
19309 | 186 |
| rew' _ = NONE; |
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in rew' end; |
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188 |
|
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fun rprocs b = [("Pure/meta_equality", rew b)]; |
18708 | 190 |
val _ = Context.add_setup (Proofterm.add_prf_rprocs (rprocs false)); |
11522 | 191 |
|
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192 |
|
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(**** apply rewriting function to all terms in proof ****) |
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194 |
|
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195 |
fun rewrite_terms r = |
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let |
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197 |
fun rew_term Ts t = |
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let |
20076 | 199 |
val frees = map Free (Name.invent_list (add_term_names (t, [])) "xa" (length Ts) ~~ Ts); |
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val t' = r (subst_bounds (frees, t)); |
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fun strip [] t = t |
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| strip (_ :: xs) (Abs (_, _, t)) = strip xs t; |
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203 |
in |
19473 | 204 |
strip Ts (fold lambda frees t') |
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205 |
end; |
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206 |
|
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207 |
fun rew Ts (prf1 %% prf2) = rew Ts prf1 %% rew Ts prf2 |
15531 | 208 |
| rew Ts (prf % SOME t) = rew Ts prf % SOME (rew_term Ts t) |
209 |
| rew Ts (Abst (s, SOME T, prf)) = Abst (s, SOME T, rew (T :: Ts) prf) |
|
210 |
| rew Ts (AbsP (s, SOME t, prf)) = AbsP (s, SOME (rew_term Ts t), rew Ts prf) |
|
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211 |
| rew _ prf = prf |
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212 |
|
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in rew [] end; |
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214 |
|
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215 |
|
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(**** eliminate definitions in proof ****) |
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217 |
|
16861 | 218 |
fun vars_of t = rev (fold_aterms (fn v as Var _ => insert (op =) v | _ => I) t []); |
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219 |
|
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fun insert_refl defs Ts (prf1 %% prf2) = |
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221 |
insert_refl defs Ts prf1 %% insert_refl defs Ts prf2 |
15531 | 222 |
| insert_refl defs Ts (Abst (s, SOME T, prf)) = |
223 |
Abst (s, SOME T, insert_refl defs (T :: Ts) prf) |
|
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224 |
| insert_refl defs Ts (AbsP (s, t, prf)) = |
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225 |
AbsP (s, t, insert_refl defs Ts prf) |
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226 |
| insert_refl defs Ts prf = (case strip_combt prf of |
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227 |
(PThm (s, _, prop, SOME Ts), ts) => |
20664 | 228 |
if member (op =) defs s then |
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229 |
let |
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230 |
val vs = vars_of prop; |
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231 |
val tvars = term_tvars prop; |
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232 |
val (_, rhs) = Logic.dest_equals prop; |
18185 | 233 |
val rhs' = Term.betapplys (subst_TVars (map fst tvars ~~ Ts) |
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234 |
(foldr (fn p => Abs ("", dummyT, abstract_over p)) rhs vs), |
19466 | 235 |
map the ts); |
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|
236 |
in |
15531 | 237 |
change_type (SOME [fastype_of1 (Ts, rhs')]) reflexive_axm %> rhs' |
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|
238 |
end |
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|
239 |
else prf |
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|
240 |
| (_, []) => prf |
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|
241 |
| (prf', ts) => proof_combt' (insert_refl defs Ts prf', ts)); |
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242 |
|
17203 | 243 |
fun elim_defs thy r defs prf = |
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244 |
let |
13341 | 245 |
val defs' = map (Logic.dest_equals o prop_of o Drule.abs_def) defs |
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|
246 |
val defnames = map Thm.get_name defs; |
13341 | 247 |
val f = if not r then I else |
248 |
let |
|
249 |
val cnames = map (fst o dest_Const o fst) defs'; |
|
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|
250 |
val thms = maps (fn (s, ps) => |
20664 | 251 |
if member (op =) defnames s then [] |
15531 | 252 |
else map (pair s o SOME o fst) (filter_out (fn (p, _) => |
13646 | 253 |
null (term_consts p inter cnames)) ps)) |
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|
254 |
(Symtab.dest (thms_of_proof prf Symtab.empty)) |
17203 | 255 |
in Reconstruct.expand_proof thy thms end |
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|
256 |
in |
17203 | 257 |
rewrite_terms (Pattern.rewrite_term thy defs' []) |
13341 | 258 |
(insert_refl defnames [] (f prf)) |
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|
259 |
end; |
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|
260 |
|
13608 | 261 |
|
262 |
(**** eliminate all variables that don't occur in the proposition ****) |
|
263 |
||
264 |
fun elim_vars mk_default prf = |
|
265 |
let |
|
266 |
val prop = Reconstruct.prop_of prf; |
|
19309 | 267 |
val tv = Term.add_vars prop []; |
268 |
val tf = Term.add_frees prop []; |
|
269 |
||
270 |
fun hidden_variable (Var v) = not (member (op =) tv v) |
|
271 |
| hidden_variable (Free f) = not (member (op =) tf f) |
|
272 |
| hidden_variable _ = false; |
|
13917 | 273 |
|
274 |
fun mk_default' T = list_abs |
|
275 |
(apfst (map (pair "x")) (apsnd mk_default (strip_type T))); |
|
276 |
||
277 |
fun elim_varst (t $ u) = elim_varst t $ elim_varst u |
|
278 |
| elim_varst (Abs (s, T, t)) = Abs (s, T, elim_varst t) |
|
19309 | 279 |
| elim_varst (t as Free (x, T)) = if member (op =) tf (x, T) then t else mk_default' T |
280 |
| elim_varst (t as Var (xi, T)) = if member (op =) tv (xi, T) then t else mk_default' T |
|
281 |
| elim_varst t = t; |
|
13608 | 282 |
in |
19309 | 283 |
map_proof_terms (fn t => |
284 |
if Term.exists_subterm hidden_variable t then Envir.beta_norm (elim_varst t) else t) I prf |
|
13608 | 285 |
end; |
286 |
||
22280 | 287 |
|
288 |
(**** convert between hhf and non-hhf form ****) |
|
289 |
||
290 |
fun hhf_proof P Q prf = |
|
291 |
let |
|
292 |
val params = Logic.strip_params Q; |
|
293 |
val Hs = Logic.strip_assums_hyp P; |
|
294 |
val Hs' = Logic.strip_assums_hyp Q; |
|
295 |
val k = length Hs; |
|
296 |
val l = length params; |
|
297 |
fun mk_prf i j Hs Hs' (Const ("all", _) $ Abs (_, _, P)) prf = |
|
298 |
mk_prf i (j - 1) Hs Hs' P (prf %> Bound j) |
|
299 |
| mk_prf i j (H :: Hs) (H' :: Hs') (Const ("==>", _) $ _ $ P) prf = |
|
300 |
mk_prf (i - 1) j Hs Hs' P (prf %% un_hhf_proof H' H (PBound i)) |
|
301 |
| mk_prf _ _ _ _ _ prf = prf |
|
302 |
in |
|
303 |
prf |> Proofterm.incr_pboundvars k l |> mk_prf (k - 1) (l - 1) Hs Hs' P |> |
|
304 |
fold_rev (fn P => fn prf => AbsP ("H", SOME P, prf)) Hs' |> |
|
305 |
fold_rev (fn (s, T) => fn prf => Abst (s, SOME T, prf)) params |
|
306 |
end |
|
307 |
and un_hhf_proof P Q prf = |
|
308 |
let |
|
309 |
val params = Logic.strip_params Q; |
|
310 |
val Hs = Logic.strip_assums_hyp P; |
|
311 |
val Hs' = Logic.strip_assums_hyp Q; |
|
312 |
val k = length Hs; |
|
313 |
val l = length params; |
|
314 |
fun mk_prf (Const ("all", _) $ Abs (s, T, P)) prf = |
|
315 |
Abst (s, SOME T, mk_prf P prf) |
|
316 |
| mk_prf (Const ("==>", _) $ P $ Q) prf = |
|
317 |
AbsP ("H", SOME P, mk_prf Q prf) |
|
318 |
| mk_prf _ prf = prf |
|
319 |
in |
|
320 |
prf |> Proofterm.incr_pboundvars k l |> |
|
321 |
fold (fn i => fn prf => prf %> Bound i) (l - 1 downto 0) |> |
|
322 |
fold (fn ((H, H'), i) => fn prf => prf %% hhf_proof H' H (PBound i)) |
|
323 |
(Hs ~~ Hs' ~~ (k - 1 downto 0)) |> |
|
324 |
mk_prf Q |
|
325 |
end; |
|
326 |
||
11522 | 327 |
end; |