author  wenzelm 
Sun, 09 Nov 2014 17:04:14 +0100  
changeset 58957  c9e744ea8a38 
parent 58058  1a0b18176548 
child 59058  a78612c67ec0 
permissions  rwrr 
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(* Title: HOL/Library/Old_SMT/old_z3_proof_methods.ML 
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Author: Sascha Boehme, TU Muenchen 
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Proof methods for Z3 proof reconstruction. 

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*) 

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signature OLD_Z3_PROOF_METHODS = 
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sig 
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val prove_injectivity: Proof.context > cterm > thm 

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val prove_ite: Proof.context > cterm > thm 
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end 
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structure Old_Z3_Proof_Methods: OLD_Z3_PROOF_METHODS = 
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struct 
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fun apply tac st = 

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(case Seq.pull (tac 1 st) of 

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NONE => raise THM ("tactic failed", 1, [st]) 

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 SOME (st', _) => st') 

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(* ifthenelse *) 
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val pull_ite = mk_meta_eq 
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@{lemma "f (if P then x else y) = (if P then f x else f y)" by simp} 
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fun pull_ites_conv ct = 
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(Conv.rewr_conv pull_ite then_conv 
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Conv.binop_conv (Conv.try_conv pull_ites_conv)) ct 
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fun prove_ite ctxt = 
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Old_Z3_Proof_Tools.by_tac ctxt ( 
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CONVERSION (Conv.arg_conv (Conv.arg1_conv pull_ites_conv)) 
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THEN' rtac @{thm refl}) 
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(* injectivity *) 
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local 

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val B = @{typ bool} 

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fun mk_univ T = Const (@{const_name top}, HOLogic.mk_setT T) 
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fun mk_inj_on T U = 
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Const (@{const_name inj_on}, (T > U) > HOLogic.mk_setT T > B) 
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fun mk_inv_into T U = 
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Const (@{const_name inv_into}, [HOLogic.mk_setT T, T > U, U] > T) 
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fun mk_inv_of ctxt ct = 

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let 

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val (dT, rT) = Term.dest_funT (Old_SMT_Utils.typ_of ct) 
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val inv = Old_SMT_Utils.certify ctxt (mk_inv_into dT rT) 
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val univ = Old_SMT_Utils.certify ctxt (mk_univ dT) 
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in Thm.mk_binop inv univ ct end 
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fun mk_inj_prop ctxt ct = 

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let 

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val (dT, rT) = Term.dest_funT (Old_SMT_Utils.typ_of ct) 
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val inj = Old_SMT_Utils.certify ctxt (mk_inj_on dT rT) 
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val univ = Old_SMT_Utils.certify ctxt (mk_univ dT) 
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in Old_SMT_Utils.mk_cprop (Thm.mk_binop inj ct univ) end 
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val disjE = @{lemma "~P  Q ==> P ==> Q" by fast} 

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fun prove_inj_prop ctxt def lhs = 
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let 
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val (ct, ctxt') = Old_SMT_Utils.dest_all_cabs (Thm.rhs_of def) ctxt 
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val rule = disjE OF [Object_Logic.rulify ctxt' (Thm.assume lhs)] 
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in 
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Goal.init (mk_inj_prop ctxt' (Thm.dest_arg ct)) 

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> apply (rtac @{thm injI}) 
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> apply (Tactic.solve_tac [rule, rule RS @{thm sym}]) 
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> Goal.norm_result ctxt' o Goal.finish ctxt' 
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> singleton (Variable.export ctxt' ctxt) 
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end 

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fun prove_rhs ctxt def lhs = 
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Old_Z3_Proof_Tools.by_tac ctxt ( 
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CONVERSION (Conv.top_sweep_conv (K (Conv.rewr_conv def)) ctxt) 
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THEN' REPEAT_ALL_NEW (match_tac ctxt @{thms allI}) 
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THEN' rtac (@{thm inv_f_f} OF [prove_inj_prop ctxt def lhs])) 
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fun expand thm ct = 

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let 

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val cpat = Thm.dest_arg (Thm.rhs_of thm) 

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val (cl, cr) = Thm.dest_binop (Thm.dest_arg (Thm.dest_arg1 ct)) 

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val thm1 = Thm.instantiate (Thm.match (cpat, cl)) thm 

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val thm2 = Thm.instantiate (Thm.match (cpat, cr)) thm 

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in Conv.arg_conv (Conv.binop_conv (Conv.rewrs_conv [thm1, thm2])) ct end 

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fun prove_lhs ctxt rhs = 
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let 
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val eq = Thm.symmetric (mk_meta_eq (Object_Logic.rulify ctxt (Thm.assume rhs))) 
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val conv = Old_SMT_Utils.binders_conv (K (expand eq)) ctxt 
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in 
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Old_Z3_Proof_Tools.by_tac ctxt ( 
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CONVERSION (Old_SMT_Utils.prop_conv conv) 
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THEN' Simplifier.simp_tac (put_simpset HOL_ss ctxt)) 
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end 
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fun mk_inv_def ctxt rhs = 
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let 
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val (ct, ctxt') = 
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Old_SMT_Utils.dest_all_cbinders (Old_SMT_Utils.dest_cprop rhs) ctxt 
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val (cl, cv) = Thm.dest_binop ct 
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val (cg, (cargs, cf)) = Drule.strip_comb cl > split_last 

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val cu = fold_rev Thm.lambda cargs (mk_inv_of ctxt' (Thm.lambda cv cf)) 
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in Thm.assume (Old_SMT_Utils.mk_cequals cg cu) end 
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fun prove_inj_eq ctxt ct = 

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let 

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val (lhs, rhs) = 
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pairself Old_SMT_Utils.mk_cprop (Thm.dest_binop (Old_SMT_Utils.dest_cprop ct)) 
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val lhs_thm = Thm.implies_intr rhs (prove_lhs ctxt rhs lhs) 
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val rhs_thm = 
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Thm.implies_intr lhs (prove_rhs ctxt (mk_inv_def ctxt rhs) lhs rhs) 
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in lhs_thm COMP (rhs_thm COMP @{thm iffI}) end 
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val swap_eq_thm = mk_meta_eq @{thm eq_commute} 

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val swap_disj_thm = mk_meta_eq @{thm disj_commute} 

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fun swap_conv dest eq = 

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Old_SMT_Utils.if_true_conv ((op <) o pairself Term.size_of_term o dest) 
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(Conv.rewr_conv eq) 
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val swap_eq_conv = swap_conv HOLogic.dest_eq swap_eq_thm 

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val swap_disj_conv = swap_conv Old_SMT_Utils.dest_disj swap_disj_thm 
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fun norm_conv ctxt = 

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swap_eq_conv then_conv 

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Conv.arg1_conv (Old_SMT_Utils.binders_conv (K swap_disj_conv) ctxt) then_conv 
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Conv.arg_conv (Old_SMT_Utils.binders_conv (K swap_eq_conv) ctxt) 
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in 

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fun prove_injectivity ctxt = 
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Old_Z3_Proof_Tools.by_tac ctxt ( 
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CONVERSION (Old_SMT_Utils.prop_conv (norm_conv ctxt)) 
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THEN' CSUBGOAL (uncurry (rtac o prove_inj_eq ctxt))) 
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end 

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end 