--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Old_SMT/old_z3_proof_methods.ML Thu Aug 28 00:40:38 2014 +0200
@@ -0,0 +1,149 @@
+(* Title: HOL/Library/Old_SMT/old_z3_proof_methods.ML
+ Author: Sascha Boehme, TU Muenchen
+
+Proof methods for Z3 proof reconstruction.
+*)
+
+signature OLD_Z3_PROOF_METHODS =
+sig
+ val prove_injectivity: Proof.context -> cterm -> thm
+ val prove_ite: Proof.context -> cterm -> thm
+end
+
+structure Old_Z3_Proof_Methods: OLD_Z3_PROOF_METHODS =
+struct
+
+
+fun apply tac st =
+ (case Seq.pull (tac 1 st) of
+ NONE => raise THM ("tactic failed", 1, [st])
+ | SOME (st', _) => st')
+
+
+
+(* if-then-else *)
+
+val pull_ite = mk_meta_eq
+ @{lemma "f (if P then x else y) = (if P then f x else f y)" by simp}
+
+fun pull_ites_conv ct =
+ (Conv.rewr_conv pull_ite then_conv
+ Conv.binop_conv (Conv.try_conv pull_ites_conv)) ct
+
+fun prove_ite ctxt =
+ Old_Z3_Proof_Tools.by_tac ctxt (
+ CONVERSION (Conv.arg_conv (Conv.arg1_conv pull_ites_conv))
+ THEN' rtac @{thm refl})
+
+
+
+(* injectivity *)
+
+local
+
+val B = @{typ bool}
+fun mk_univ T = Const (@{const_name top}, HOLogic.mk_setT T)
+fun mk_inj_on T U =
+ Const (@{const_name inj_on}, (T --> U) --> HOLogic.mk_setT T --> B)
+fun mk_inv_into T U =
+ Const (@{const_name inv_into}, [HOLogic.mk_setT T, T --> U, U] ---> T)
+
+fun mk_inv_of ctxt ct =
+ let
+ val (dT, rT) = Term.dest_funT (Old_SMT_Utils.typ_of ct)
+ val inv = Old_SMT_Utils.certify ctxt (mk_inv_into dT rT)
+ val univ = Old_SMT_Utils.certify ctxt (mk_univ dT)
+ in Thm.mk_binop inv univ ct end
+
+fun mk_inj_prop ctxt ct =
+ let
+ val (dT, rT) = Term.dest_funT (Old_SMT_Utils.typ_of ct)
+ val inj = Old_SMT_Utils.certify ctxt (mk_inj_on dT rT)
+ val univ = Old_SMT_Utils.certify ctxt (mk_univ dT)
+ in Old_SMT_Utils.mk_cprop (Thm.mk_binop inj ct univ) end
+
+
+val disjE = @{lemma "~P | Q ==> P ==> Q" by fast}
+
+fun prove_inj_prop ctxt def lhs =
+ let
+ val (ct, ctxt') = Old_SMT_Utils.dest_all_cabs (Thm.rhs_of def) ctxt
+ val rule = disjE OF [Object_Logic.rulify ctxt' (Thm.assume lhs)]
+ in
+ Goal.init (mk_inj_prop ctxt' (Thm.dest_arg ct))
+ |> apply (rtac @{thm injI})
+ |> apply (Tactic.solve_tac [rule, rule RS @{thm sym}])
+ |> Goal.norm_result ctxt' o Goal.finish ctxt'
+ |> singleton (Variable.export ctxt' ctxt)
+ end
+
+fun prove_rhs ctxt def lhs =
+ Old_Z3_Proof_Tools.by_tac ctxt (
+ CONVERSION (Conv.top_sweep_conv (K (Conv.rewr_conv def)) ctxt)
+ THEN' REPEAT_ALL_NEW (match_tac @{thms allI})
+ THEN' rtac (@{thm inv_f_f} OF [prove_inj_prop ctxt def lhs]))
+
+
+fun expand thm ct =
+ let
+ val cpat = Thm.dest_arg (Thm.rhs_of thm)
+ val (cl, cr) = Thm.dest_binop (Thm.dest_arg (Thm.dest_arg1 ct))
+ val thm1 = Thm.instantiate (Thm.match (cpat, cl)) thm
+ val thm2 = Thm.instantiate (Thm.match (cpat, cr)) thm
+ in Conv.arg_conv (Conv.binop_conv (Conv.rewrs_conv [thm1, thm2])) ct end
+
+fun prove_lhs ctxt rhs =
+ let
+ val eq = Thm.symmetric (mk_meta_eq (Object_Logic.rulify ctxt (Thm.assume rhs)))
+ val conv = Old_SMT_Utils.binders_conv (K (expand eq)) ctxt
+ in
+ Old_Z3_Proof_Tools.by_tac ctxt (
+ CONVERSION (Old_SMT_Utils.prop_conv conv)
+ THEN' Simplifier.simp_tac (put_simpset HOL_ss ctxt))
+ end
+
+
+fun mk_inv_def ctxt rhs =
+ let
+ val (ct, ctxt') =
+ Old_SMT_Utils.dest_all_cbinders (Old_SMT_Utils.dest_cprop rhs) ctxt
+ val (cl, cv) = Thm.dest_binop ct
+ val (cg, (cargs, cf)) = Drule.strip_comb cl ||> split_last
+ val cu = fold_rev Thm.lambda cargs (mk_inv_of ctxt' (Thm.lambda cv cf))
+ in Thm.assume (Old_SMT_Utils.mk_cequals cg cu) end
+
+fun prove_inj_eq ctxt ct =
+ let
+ val (lhs, rhs) =
+ pairself Old_SMT_Utils.mk_cprop (Thm.dest_binop (Old_SMT_Utils.dest_cprop ct))
+ val lhs_thm = Thm.implies_intr rhs (prove_lhs ctxt rhs lhs)
+ val rhs_thm =
+ Thm.implies_intr lhs (prove_rhs ctxt (mk_inv_def ctxt rhs) lhs rhs)
+ in lhs_thm COMP (rhs_thm COMP @{thm iffI}) end
+
+
+val swap_eq_thm = mk_meta_eq @{thm eq_commute}
+val swap_disj_thm = mk_meta_eq @{thm disj_commute}
+
+fun swap_conv dest eq =
+ Old_SMT_Utils.if_true_conv ((op <) o pairself Term.size_of_term o dest)
+ (Conv.rewr_conv eq)
+
+val swap_eq_conv = swap_conv HOLogic.dest_eq swap_eq_thm
+val swap_disj_conv = swap_conv Old_SMT_Utils.dest_disj swap_disj_thm
+
+fun norm_conv ctxt =
+ swap_eq_conv then_conv
+ Conv.arg1_conv (Old_SMT_Utils.binders_conv (K swap_disj_conv) ctxt) then_conv
+ Conv.arg_conv (Old_SMT_Utils.binders_conv (K swap_eq_conv) ctxt)
+
+in
+
+fun prove_injectivity ctxt =
+ Old_Z3_Proof_Tools.by_tac ctxt (
+ CONVERSION (Old_SMT_Utils.prop_conv (norm_conv ctxt))
+ THEN' CSUBGOAL (uncurry (rtac o prove_inj_eq ctxt)))
+
+end
+
+end