src/HOL/Tools/Quotient/quotient_tacs.ML
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Wed Jun 23 08:44:44 2010 +0200 (2010-06-23)
changeset 37493 2377d246a631
parent 36945 9bec62c10714
child 37560 1b5bbc4a14bc
permissions -rw-r--r--
Quotient package now uses Partial Equivalence instead place of equivalence
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(*  Title:      HOL/Tools/Quotient/quotient_tacs.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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Tactics for solving goal arising from lifting theorems to quotient
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types.
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*)
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signature QUOTIENT_TACS =
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sig
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  val regularize_tac: Proof.context -> int -> tactic
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  val injection_tac: Proof.context -> int -> tactic
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  val all_injection_tac: Proof.context -> int -> tactic
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  val clean_tac: Proof.context -> int -> tactic
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  val procedure_tac: Proof.context -> thm -> int -> tactic
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  val lift_tac: Proof.context -> thm list -> int -> tactic
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  val quotient_tac: Proof.context -> int -> tactic
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  val quot_true_tac: Proof.context -> (term -> term) -> int -> tactic
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  val lifted: typ list -> Proof.context -> thm -> thm
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  val lifted_attrib: attribute
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end;
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structure Quotient_Tacs: QUOTIENT_TACS =
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struct
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open Quotient_Info;
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open Quotient_Term;
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(** various helper fuctions **)
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(* Since HOL_basic_ss is too "big" for us, we *)
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(* need to set up our own minimal simpset.    *)
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fun mk_minimal_ss ctxt =
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  Simplifier.context ctxt empty_ss
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    setsubgoaler asm_simp_tac
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    setmksimps (mksimps [])
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(* composition of two theorems, used in maps *)
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fun OF1 thm1 thm2 = thm2 RS thm1
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fun atomize_thm thm =
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let
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  val thm' = Thm.legacy_freezeT (forall_intr_vars thm) (* FIXME/TODO: is this proper Isar-technology? *)
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  val thm'' = Object_Logic.atomize (cprop_of thm')
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in
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  @{thm equal_elim_rule1} OF [thm'', thm']
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end
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(*** Regularize Tactic ***)
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(** solvers for equivp and quotient assumptions **)
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fun equiv_tac ctxt =
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  REPEAT_ALL_NEW (resolve_tac (equiv_rules_get ctxt))
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fun equiv_solver_tac ss = equiv_tac (Simplifier.the_context ss)
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val equiv_solver = Simplifier.mk_solver' "Equivalence goal solver" equiv_solver_tac
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fun quotient_tac ctxt =
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  (REPEAT_ALL_NEW (FIRST'
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    [rtac @{thm identity_quotient},
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     resolve_tac (quotient_rules_get ctxt)]))
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fun quotient_solver_tac ss = quotient_tac (Simplifier.the_context ss)
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val quotient_solver =
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  Simplifier.mk_solver' "Quotient goal solver" quotient_solver_tac
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fun solve_quotient_assm ctxt thm =
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  case Seq.pull (quotient_tac ctxt 1 thm) of
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    SOME (t, _) => t
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  | _ => error "Solve_quotient_assm failed. Possibly a quotient theorem is missing."
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fun prep_trm thy (x, (T, t)) =
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  (cterm_of thy (Var (x, T)), cterm_of thy t)
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fun prep_ty thy (x, (S, ty)) =
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  (ctyp_of thy (TVar (x, S)), ctyp_of thy ty)
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fun get_match_inst thy pat trm =
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let
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  val univ = Unify.matchers thy [(pat, trm)]
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  val SOME (env, _) = Seq.pull univ           (* raises Bind, if no unifier *)  (* FIXME fragile *)
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  val tenv = Vartab.dest (Envir.term_env env)
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  val tyenv = Vartab.dest (Envir.type_env env)
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in
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  (map (prep_ty thy) tyenv, map (prep_trm thy) tenv)
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end
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(* Calculates the instantiations for the lemmas:
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      ball_reg_eqv_range and bex_reg_eqv_range
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   Since the left-hand-side contains a non-pattern '?P (f ?x)'
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   we rely on unification/instantiation to check whether the
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   theorem applies and return NONE if it doesn't.
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*)
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fun calculate_inst ctxt ball_bex_thm redex R1 R2 =
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let
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  val thy = ProofContext.theory_of ctxt
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  fun get_lhs thm = fst (Logic.dest_equals (Thm.concl_of thm))
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  val ty_inst = map (SOME o ctyp_of thy) [domain_type (fastype_of R2)]
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  val trm_inst = map (SOME o cterm_of thy) [R2, R1]
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in
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  case try (Drule.instantiate' ty_inst trm_inst) ball_bex_thm of
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    NONE => NONE
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  | SOME thm' =>
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      (case try (get_match_inst thy (get_lhs thm')) redex of
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        NONE => NONE
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      | SOME inst2 => try (Drule.instantiate inst2) thm')
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end
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fun ball_bex_range_simproc ss redex =
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let
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  val ctxt = Simplifier.the_context ss
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in
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  case redex of
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    (Const (@{const_name "Ball"}, _) $ (Const (@{const_name "Respects"}, _) $
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      (Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) =>
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        calculate_inst ctxt @{thm ball_reg_eqv_range[THEN eq_reflection]} redex R1 R2
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  | (Const (@{const_name "Bex"}, _) $ (Const (@{const_name "Respects"}, _) $
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      (Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) =>
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        calculate_inst ctxt @{thm bex_reg_eqv_range[THEN eq_reflection]} redex R1 R2
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  | _ => NONE
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end
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(* Regularize works as follows:
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  0. preliminary simplification step according to
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     ball_reg_eqv bex_reg_eqv babs_reg_eqv ball_reg_eqv_range bex_reg_eqv_range
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  1. eliminating simple Ball/Bex instances (ball_reg_right bex_reg_left)
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  2. monos
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  3. commutation rules for ball and bex (ball_all_comm bex_ex_comm)
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  4. then rel-equalities, which need to be instantiated with 'eq_imp_rel'
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     to avoid loops
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  5. then simplification like 0
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  finally jump back to 1
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*)
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fun reflp_get ctxt =
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  map_filter (fn th => if prems_of th = [] then SOME (OF1 @{thm equivp_reflp} th) else NONE
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    handle THM _ => NONE) (equiv_rules_get ctxt)
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val eq_imp_rel = @{lemma "equivp R ==> a = b --> R a b" by (simp add: equivp_reflp)}
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fun eq_imp_rel_get ctxt = map (OF1 eq_imp_rel) (equiv_rules_get ctxt)
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fun regularize_tac ctxt =
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let
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  val thy = ProofContext.theory_of ctxt
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  val ball_pat = @{term "Ball (Respects (R1 ===> R2)) P"}
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  val bex_pat  = @{term "Bex (Respects (R1 ===> R2)) P"}
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  val simproc = Simplifier.simproc_i thy "" [ball_pat, bex_pat] (K (ball_bex_range_simproc))
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  val simpset = (mk_minimal_ss ctxt)
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                       addsimps @{thms ball_reg_eqv bex_reg_eqv babs_reg_eqv babs_simp}
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                       addsimprocs [simproc]
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                       addSolver equiv_solver addSolver quotient_solver
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  val eq_eqvs = eq_imp_rel_get ctxt
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in
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  simp_tac simpset THEN'
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  REPEAT_ALL_NEW (CHANGED o FIRST'
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    [resolve_tac @{thms ball_reg_right bex_reg_left bex1_bexeq_reg},
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     resolve_tac (Inductive.get_monos ctxt),
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     resolve_tac @{thms ball_all_comm bex_ex_comm},
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     resolve_tac eq_eqvs,
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     simp_tac simpset])
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end
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(*** Injection Tactic ***)
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(* Looks for Quot_True assumptions, and in case its parameter
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   is an application, it returns the function and the argument.
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*)
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fun find_qt_asm asms =
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let
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  fun find_fun trm =
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    case trm of
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      (Const(@{const_name Trueprop}, _) $ (Const (@{const_name Quot_True}, _) $ _)) => true
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    | _ => false
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in
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 case find_first find_fun asms of
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   SOME (_ $ (_ $ (f $ a))) => SOME (f, a)
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 | _ => NONE
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end
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fun quot_true_simple_conv ctxt fnctn ctrm =
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  case (term_of ctrm) of
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    (Const (@{const_name Quot_True}, _) $ x) =>
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    let
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      val fx = fnctn x;
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      val thy = ProofContext.theory_of ctxt;
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      val cx = cterm_of thy x;
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      val cfx = cterm_of thy fx;
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      val cxt = ctyp_of thy (fastype_of x);
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      val cfxt = ctyp_of thy (fastype_of fx);
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      val thm = Drule.instantiate' [SOME cxt, SOME cfxt] [SOME cx, SOME cfx] @{thm QT_imp}
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    in
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      Conv.rewr_conv thm ctrm
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    end
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fun quot_true_conv ctxt fnctn ctrm =
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  case (term_of ctrm) of
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    (Const (@{const_name Quot_True}, _) $ _) =>
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      quot_true_simple_conv ctxt fnctn ctrm
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  | _ $ _ => Conv.comb_conv (quot_true_conv ctxt fnctn) ctrm
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  | Abs _ => Conv.abs_conv (fn (_, ctxt) => quot_true_conv ctxt fnctn) ctxt ctrm
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  | _ => Conv.all_conv ctrm
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fun quot_true_tac ctxt fnctn =
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   CONVERSION
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    ((Conv.params_conv ~1 (fn ctxt =>
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       (Conv.prems_conv ~1 (quot_true_conv ctxt fnctn)))) ctxt)
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fun dest_comb (f $ a) = (f, a)
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fun dest_bcomb ((_ $ l) $ r) = (l, r)
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fun unlam t =
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  case t of
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    (Abs a) => snd (Term.dest_abs a)
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  | _ => unlam (Abs("", domain_type (fastype_of t), (incr_boundvars 1 t) $ (Bound 0)))
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fun dest_fun_type (Type("fun", [T, S])) = (T, S)
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  | dest_fun_type _ = error "dest_fun_type"
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val bare_concl = HOLogic.dest_Trueprop o Logic.strip_assums_concl
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(* We apply apply_rsp only in case if the type needs lifting.
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   This is the case if the type of the data in the Quot_True
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   assumption is different from the corresponding type in the goal.
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*)
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val apply_rsp_tac =
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  Subgoal.FOCUS (fn {concl, asms, context,...} =>
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  let
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    val bare_concl = HOLogic.dest_Trueprop (term_of concl)
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    val qt_asm = find_qt_asm (map term_of asms)
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  in
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    case (bare_concl, qt_asm) of
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      (R2 $ (f $ x) $ (g $ y), SOME (qt_fun, qt_arg)) =>
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         if fastype_of qt_fun = fastype_of f
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         then no_tac
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         else
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           let
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             val ty_x = fastype_of x
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             val ty_b = fastype_of qt_arg
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             val ty_f = range_type (fastype_of f)
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             val thy = ProofContext.theory_of context
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             val ty_inst = map (SOME o (ctyp_of thy)) [ty_x, ty_b, ty_f]
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             val t_inst = map (SOME o (cterm_of thy)) [R2, f, g, x, y];
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             val inst_thm = Drule.instantiate' ty_inst
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               ([NONE, NONE, NONE] @ t_inst) @{thm apply_rsp}
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           in
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             (rtac inst_thm THEN' SOLVED' (quotient_tac context)) 1
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           end
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    | _ => no_tac
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  end)
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(* Instantiates and applies 'equals_rsp'. Since the theorem is
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   complex we rely on instantiation to tell us if it applies
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*)
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fun equals_rsp_tac R ctxt =
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let
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  val thy = ProofContext.theory_of ctxt
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in
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  case try (cterm_of thy) R of (* There can be loose bounds in R *)
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    SOME ctm =>
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      let
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        val ty = domain_type (fastype_of R)
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      in
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        case try (Drule.instantiate' [SOME (ctyp_of thy ty)]
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          [SOME (cterm_of thy R)]) @{thm equals_rsp} of
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          SOME thm => rtac thm THEN' quotient_tac ctxt
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        | NONE => K no_tac
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      end
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  | _ => K no_tac
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end
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fun rep_abs_rsp_tac ctxt =
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  SUBGOAL (fn (goal, i) =>
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    case (try bare_concl goal) of
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      SOME (rel $ _ $ (rep $ (Bound _ $ _))) => no_tac
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    | SOME (rel $ _ $ (rep $ (abs $ _))) =>
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        let
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          val thy = ProofContext.theory_of ctxt;
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          val (ty_a, ty_b) = dest_fun_type (fastype_of abs);
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          val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b];
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        in
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          case try (map (SOME o (cterm_of thy))) [rel, abs, rep] of
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            SOME t_inst =>
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              (case try (Drule.instantiate' ty_inst t_inst) @{thm rep_abs_rsp} of
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                SOME inst_thm => (rtac inst_thm THEN' quotient_tac ctxt) i
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              | NONE => no_tac)
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          | NONE => no_tac
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        end
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    | _ => no_tac)
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(* Injection means to prove that the regularised theorem implies
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   the abs/rep injected one.
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   The deterministic part:
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    - remove lambdas from both sides
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    - prove Ball/Bex/Babs equalities using ball_rsp, bex_rsp, babs_rsp
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    - prove Ball/Bex relations unfolding fun_rel_id
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    - reflexivity of equality
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    - prove equality of relations using equals_rsp
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    - use user-supplied RSP theorems
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    - solve 'relation of relations' goals using quot_rel_rsp
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    - remove rep_abs from the right side
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      (Lambdas under respects may have left us some assumptions)
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   Then in order:
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    - split applications of lifted type (apply_rsp)
kaliszyk@35222
   326
    - split applications of non-lifted type (cong_tac)
kaliszyk@35222
   327
    - apply extentionality
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   328
    - assumption
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   329
    - reflexivity of the relation
kaliszyk@35222
   330
*)
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   331
fun injection_match_tac ctxt = SUBGOAL (fn (goal, i) =>
kaliszyk@35222
   332
(case (bare_concl goal) of
kaliszyk@35222
   333
    (* (R1 ===> R2) (%x...) (%x...) ----> [|R1 x y|] ==> R2 (...x) (...y) *)
kaliszyk@35222
   334
  (Const (@{const_name fun_rel}, _) $ _ $ _) $ (Abs _) $ (Abs _)
kaliszyk@35222
   335
      => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   336
kaliszyk@35222
   337
    (* (op =) (Ball...) (Ball...) ----> (op =) (...) (...) *)
kaliszyk@35222
   338
| (Const (@{const_name "op ="},_) $
kaliszyk@35222
   339
    (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
kaliszyk@35222
   340
    (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
kaliszyk@35222
   341
      => rtac @{thm ball_rsp} THEN' dtac @{thm QT_all}
kaliszyk@35222
   342
kaliszyk@35222
   343
    (* (R1 ===> op =) (Ball...) (Ball...) ----> [|R1 x y|] ==> (Ball...x) = (Ball...y) *)
kaliszyk@35222
   344
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
kaliszyk@35222
   345
    (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
kaliszyk@35222
   346
    (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
kaliszyk@35222
   347
      => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   348
kaliszyk@35222
   349
    (* (op =) (Bex...) (Bex...) ----> (op =) (...) (...) *)
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   350
| Const (@{const_name "op ="},_) $
kaliszyk@35222
   351
    (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
kaliszyk@35222
   352
    (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
kaliszyk@35222
   353
      => rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex}
kaliszyk@35222
   354
kaliszyk@35222
   355
    (* (R1 ===> op =) (Bex...) (Bex...) ----> [|R1 x y|] ==> (Bex...x) = (Bex...y) *)
kaliszyk@35222
   356
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
kaliszyk@35222
   357
    (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
kaliszyk@35222
   358
    (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
kaliszyk@35222
   359
      => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   360
kaliszyk@35222
   361
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
kaliszyk@35222
   362
    (Const(@{const_name Bex1_rel},_) $ _) $ (Const(@{const_name Bex1_rel},_) $ _)
kaliszyk@35222
   363
      => rtac @{thm bex1_rel_rsp} THEN' quotient_tac ctxt
kaliszyk@35222
   364
kaliszyk@35222
   365
| (_ $
kaliszyk@35222
   366
    (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
kaliszyk@35222
   367
    (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
kaliszyk@35222
   368
      => rtac @{thm babs_rsp} THEN' RANGE [quotient_tac ctxt]
kaliszyk@35222
   369
kaliszyk@35222
   370
| Const (@{const_name "op ="},_) $ (R $ _ $ _) $ (_ $ _ $ _) =>
kaliszyk@35222
   371
   (rtac @{thm refl} ORELSE'
kaliszyk@35222
   372
    (equals_rsp_tac R ctxt THEN' RANGE [
kaliszyk@35222
   373
       quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)]))
kaliszyk@35222
   374
kaliszyk@35222
   375
    (* reflexivity of operators arising from Cong_tac *)
kaliszyk@35222
   376
| Const (@{const_name "op ="},_) $ _ $ _ => rtac @{thm refl}
kaliszyk@35222
   377
kaliszyk@35222
   378
   (* respectfulness of constants; in particular of a simple relation *)
kaliszyk@35222
   379
| _ $ (Const _) $ (Const _)  (* fun_rel, list_rel, etc but not equality *)
kaliszyk@35222
   380
    => resolve_tac (rsp_rules_get ctxt) THEN_ALL_NEW quotient_tac ctxt
kaliszyk@35222
   381
kaliszyk@35222
   382
    (* R (...) (Rep (Abs ...)) ----> R (...) (...) *)
kaliszyk@35222
   383
    (* observe fun_map *)
kaliszyk@35222
   384
| _ $ _ $ _
kaliszyk@35222
   385
    => (rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt)
kaliszyk@35222
   386
       ORELSE' rep_abs_rsp_tac ctxt
kaliszyk@35222
   387
kaliszyk@35222
   388
| _ => K no_tac
kaliszyk@35222
   389
) i)
kaliszyk@35222
   390
kaliszyk@35222
   391
fun injection_step_tac ctxt rel_refl =
kaliszyk@35222
   392
 FIRST' [
kaliszyk@35222
   393
    injection_match_tac ctxt,
kaliszyk@35222
   394
kaliszyk@35222
   395
    (* R (t $ ...) (t' $ ...) ----> apply_rsp   provided type of t needs lifting *)
kaliszyk@35222
   396
    apply_rsp_tac ctxt THEN'
kaliszyk@35222
   397
                 RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)],
kaliszyk@35222
   398
kaliszyk@35222
   399
    (* (op =) (t $ ...) (t' $ ...) ----> Cong   provided type of t does not need lifting *)
kaliszyk@35222
   400
    (* merge with previous tactic *)
kaliszyk@35222
   401
    Cong_Tac.cong_tac @{thm cong} THEN'
kaliszyk@35222
   402
                 RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)],
kaliszyk@35222
   403
kaliszyk@35222
   404
    (* (op =) (%x...) (%y...) ----> (op =) (...) (...) *)
kaliszyk@35222
   405
    rtac @{thm ext} THEN' quot_true_tac ctxt unlam,
kaliszyk@35222
   406
kaliszyk@35222
   407
    (* resolving with R x y assumptions *)
kaliszyk@35222
   408
    atac,
kaliszyk@35222
   409
kaliszyk@35222
   410
    (* reflexivity of the basic relations *)
kaliszyk@35222
   411
    (* R ... ... *)
kaliszyk@35222
   412
    resolve_tac rel_refl]
kaliszyk@35222
   413
kaliszyk@35222
   414
fun injection_tac ctxt =
kaliszyk@35222
   415
let
kaliszyk@37493
   416
  val rel_refl = reflp_get ctxt
kaliszyk@35222
   417
in
kaliszyk@35222
   418
  injection_step_tac ctxt rel_refl
kaliszyk@35222
   419
end
kaliszyk@35222
   420
kaliszyk@35222
   421
fun all_injection_tac ctxt =
kaliszyk@35222
   422
  REPEAT_ALL_NEW (injection_tac ctxt)
kaliszyk@35222
   423
kaliszyk@35222
   424
kaliszyk@35222
   425
kaliszyk@35222
   426
(*** Cleaning of the Theorem ***)
kaliszyk@35222
   427
kaliszyk@35222
   428
(* expands all fun_maps, except in front of the (bound) variables listed in xs *)
kaliszyk@35222
   429
fun fun_map_simple_conv xs ctrm =
kaliszyk@35222
   430
  case (term_of ctrm) of
kaliszyk@35222
   431
    ((Const (@{const_name "fun_map"}, _) $ _ $ _) $ h $ _) =>
kaliszyk@35222
   432
        if member (op=) xs h
kaliszyk@35222
   433
        then Conv.all_conv ctrm
kaliszyk@35222
   434
        else Conv.rewr_conv @{thm fun_map_def[THEN eq_reflection]} ctrm
kaliszyk@35222
   435
  | _ => Conv.all_conv ctrm
kaliszyk@35222
   436
kaliszyk@35222
   437
fun fun_map_conv xs ctxt ctrm =
kaliszyk@35222
   438
  case (term_of ctrm) of
kaliszyk@35222
   439
      _ $ _ => (Conv.comb_conv (fun_map_conv xs ctxt) then_conv
kaliszyk@35222
   440
                fun_map_simple_conv xs) ctrm
kaliszyk@35222
   441
    | Abs _ => Conv.abs_conv (fn (x, ctxt) => fun_map_conv ((term_of x)::xs) ctxt) ctxt ctrm
kaliszyk@35222
   442
    | _ => Conv.all_conv ctrm
kaliszyk@35222
   443
kaliszyk@35222
   444
fun fun_map_tac ctxt = CONVERSION (fun_map_conv [] ctxt)
kaliszyk@35222
   445
kaliszyk@35222
   446
(* custom matching functions *)
kaliszyk@35222
   447
fun mk_abs u i t =
kaliszyk@35222
   448
  if incr_boundvars i u aconv t then Bound i else
kaliszyk@35222
   449
  case t of
kaliszyk@35222
   450
    t1 $ t2 => mk_abs u i t1 $ mk_abs u i t2
kaliszyk@35222
   451
  | Abs (s, T, t') => Abs (s, T, mk_abs u (i + 1) t')
kaliszyk@35222
   452
  | Bound j => if i = j then error "make_inst" else t
kaliszyk@35222
   453
  | _ => t
kaliszyk@35222
   454
kaliszyk@35222
   455
fun make_inst lhs t =
kaliszyk@35222
   456
let
kaliszyk@35222
   457
  val _ $ (Abs (_, _, (_ $ ((f as Var (_, Type ("fun", [T, _]))) $ u)))) = lhs;
kaliszyk@35222
   458
  val _ $ (Abs (_, _, (_ $ g))) = t;
kaliszyk@35222
   459
in
kaliszyk@35222
   460
  (f, Abs ("x", T, mk_abs u 0 g))
kaliszyk@35222
   461
end
kaliszyk@35222
   462
kaliszyk@35222
   463
fun make_inst_id lhs t =
kaliszyk@35222
   464
let
kaliszyk@35222
   465
  val _ $ (Abs (_, _, (f as Var (_, Type ("fun", [T, _]))) $ u)) = lhs;
kaliszyk@35222
   466
  val _ $ (Abs (_, _, g)) = t;
kaliszyk@35222
   467
in
kaliszyk@35222
   468
  (f, Abs ("x", T, mk_abs u 0 g))
kaliszyk@35222
   469
end
kaliszyk@35222
   470
kaliszyk@35222
   471
(* Simplifies a redex using the 'lambda_prs' theorem.
kaliszyk@35222
   472
   First instantiates the types and known subterms.
kaliszyk@35222
   473
   Then solves the quotient assumptions to get Rep2 and Abs1
kaliszyk@35222
   474
   Finally instantiates the function f using make_inst
kaliszyk@35222
   475
   If Rep2 is an identity then the pattern is simpler and
kaliszyk@35222
   476
   make_inst_id is used
kaliszyk@35222
   477
*)
kaliszyk@35222
   478
fun lambda_prs_simple_conv ctxt ctrm =
kaliszyk@35222
   479
  case (term_of ctrm) of
kaliszyk@35222
   480
    (Const (@{const_name fun_map}, _) $ r1 $ a2) $ (Abs _) =>
kaliszyk@35222
   481
      let
kaliszyk@35222
   482
        val thy = ProofContext.theory_of ctxt
kaliszyk@35222
   483
        val (ty_b, ty_a) = dest_fun_type (fastype_of r1)
kaliszyk@35222
   484
        val (ty_c, ty_d) = dest_fun_type (fastype_of a2)
kaliszyk@35222
   485
        val tyinst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c, ty_d]
kaliszyk@35222
   486
        val tinst = [NONE, NONE, SOME (cterm_of thy r1), NONE, SOME (cterm_of thy a2)]
kaliszyk@35222
   487
        val thm1 = Drule.instantiate' tyinst tinst @{thm lambda_prs[THEN eq_reflection]}
kaliszyk@35222
   488
        val thm2 = solve_quotient_assm ctxt (solve_quotient_assm ctxt thm1)
kaliszyk@35222
   489
        val thm3 = MetaSimplifier.rewrite_rule @{thms id_apply[THEN eq_reflection]} thm2
kaliszyk@35222
   490
        val (insp, inst) =
kaliszyk@35222
   491
          if ty_c = ty_d
kaliszyk@35222
   492
          then make_inst_id (term_of (Thm.lhs_of thm3)) (term_of ctrm)
kaliszyk@35222
   493
          else make_inst (term_of (Thm.lhs_of thm3)) (term_of ctrm)
kaliszyk@35222
   494
        val thm4 = Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) thm3
kaliszyk@35222
   495
      in
kaliszyk@35222
   496
        Conv.rewr_conv thm4 ctrm
kaliszyk@35222
   497
      end
kaliszyk@35222
   498
  | _ => Conv.all_conv ctrm
kaliszyk@35222
   499
wenzelm@36936
   500
fun lambda_prs_conv ctxt = Conv.top_conv lambda_prs_simple_conv ctxt
kaliszyk@35222
   501
fun lambda_prs_tac ctxt = CONVERSION (lambda_prs_conv ctxt)
kaliszyk@35222
   502
kaliszyk@35222
   503
kaliszyk@35222
   504
(* Cleaning consists of:
kaliszyk@35222
   505
kaliszyk@35222
   506
  1. unfolding of ---> in front of everything, except
kaliszyk@35222
   507
     bound variables (this prevents lambda_prs from
kaliszyk@35222
   508
     becoming stuck)
kaliszyk@35222
   509
kaliszyk@35222
   510
  2. simplification with lambda_prs
kaliszyk@35222
   511
kaliszyk@35222
   512
  3. simplification with:
kaliszyk@35222
   513
kaliszyk@35222
   514
      - Quotient_abs_rep Quotient_rel_rep
kaliszyk@35222
   515
        babs_prs all_prs ex_prs ex1_prs
kaliszyk@35222
   516
kaliszyk@35222
   517
      - id_simps and preservation lemmas and
kaliszyk@35222
   518
kaliszyk@35222
   519
      - symmetric versions of the definitions
kaliszyk@35222
   520
        (that is definitions of quotient constants
kaliszyk@35222
   521
         are folded)
kaliszyk@35222
   522
kaliszyk@35222
   523
  4. test for refl
kaliszyk@35222
   524
*)
kaliszyk@35222
   525
fun clean_tac lthy =
kaliszyk@35222
   526
let
wenzelm@36945
   527
  val defs = map (Thm.symmetric o #def) (qconsts_dest lthy)
kaliszyk@35222
   528
  val prs = prs_rules_get lthy
kaliszyk@35222
   529
  val ids = id_simps_get lthy
kaliszyk@35222
   530
  val thms = @{thms Quotient_abs_rep Quotient_rel_rep babs_prs all_prs ex_prs ex1_prs} @ ids @ prs @ defs
kaliszyk@35222
   531
kaliszyk@35222
   532
  val ss = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver
kaliszyk@35222
   533
in
kaliszyk@35222
   534
  EVERY' [fun_map_tac lthy,
kaliszyk@35222
   535
          lambda_prs_tac lthy,
kaliszyk@35222
   536
          simp_tac ss,
kaliszyk@35222
   537
          TRY o rtac refl]
kaliszyk@35222
   538
end
kaliszyk@35222
   539
kaliszyk@35222
   540
kaliszyk@35222
   541
kaliszyk@35222
   542
(** Tactic for Generalising Free Variables in a Goal **)
kaliszyk@35222
   543
kaliszyk@35222
   544
fun inst_spec ctrm =
kaliszyk@35222
   545
   Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}
kaliszyk@35222
   546
kaliszyk@35222
   547
fun inst_spec_tac ctrms =
kaliszyk@35222
   548
  EVERY' (map (dtac o inst_spec) ctrms)
kaliszyk@35222
   549
kaliszyk@35222
   550
fun all_list xs trm =
kaliszyk@35222
   551
  fold (fn (x, T) => fn t' => HOLogic.mk_all (x, T, t')) xs trm
kaliszyk@35222
   552
kaliszyk@35222
   553
fun apply_under_Trueprop f =
kaliszyk@35222
   554
  HOLogic.dest_Trueprop #> f #> HOLogic.mk_Trueprop
kaliszyk@35222
   555
kaliszyk@35222
   556
fun gen_frees_tac ctxt =
kaliszyk@35222
   557
  SUBGOAL (fn (concl, i) =>
kaliszyk@35222
   558
    let
kaliszyk@35222
   559
      val thy = ProofContext.theory_of ctxt
kaliszyk@35222
   560
      val vrs = Term.add_frees concl []
kaliszyk@35222
   561
      val cvrs = map (cterm_of thy o Free) vrs
kaliszyk@35222
   562
      val concl' = apply_under_Trueprop (all_list vrs) concl
kaliszyk@35222
   563
      val goal = Logic.mk_implies (concl', concl)
kaliszyk@35222
   564
      val rule = Goal.prove ctxt [] [] goal
kaliszyk@35222
   565
        (K (EVERY1 [inst_spec_tac (rev cvrs), atac]))
kaliszyk@35222
   566
    in
kaliszyk@35222
   567
      rtac rule i
kaliszyk@35222
   568
    end)
kaliszyk@35222
   569
kaliszyk@35222
   570
kaliszyk@35222
   571
(** The General Shape of the Lifting Procedure **)
kaliszyk@35222
   572
kaliszyk@35222
   573
(* - A is the original raw theorem
kaliszyk@35222
   574
   - B is the regularized theorem
kaliszyk@35222
   575
   - C is the rep/abs injected version of B
kaliszyk@35222
   576
   - D is the lifted theorem
kaliszyk@35222
   577
kaliszyk@35222
   578
   - 1st prem is the regularization step
kaliszyk@35222
   579
   - 2nd prem is the rep/abs injection step
kaliszyk@35222
   580
   - 3rd prem is the cleaning part
kaliszyk@35222
   581
kaliszyk@35222
   582
   the Quot_True premise in 2nd records the lifted theorem
kaliszyk@35222
   583
*)
kaliszyk@35222
   584
val lifting_procedure_thm =
kaliszyk@35222
   585
  @{lemma  "[|A;
kaliszyk@35222
   586
              A --> B;
kaliszyk@35222
   587
              Quot_True D ==> B = C;
kaliszyk@35222
   588
              C = D|] ==> D"
kaliszyk@35222
   589
      by (simp add: Quot_True_def)}
kaliszyk@35222
   590
kaliszyk@35222
   591
fun lift_match_error ctxt msg rtrm qtrm =
kaliszyk@35222
   592
let
kaliszyk@35222
   593
  val rtrm_str = Syntax.string_of_term ctxt rtrm
kaliszyk@35222
   594
  val qtrm_str = Syntax.string_of_term ctxt qtrm
kaliszyk@35222
   595
  val msg = cat_lines [enclose "[" "]" msg, "The quotient theorem", qtrm_str,
kaliszyk@35222
   596
    "", "does not match with original theorem", rtrm_str]
kaliszyk@35222
   597
in
kaliszyk@35222
   598
  error msg
kaliszyk@35222
   599
end
kaliszyk@35222
   600
kaliszyk@35222
   601
fun procedure_inst ctxt rtrm qtrm =
kaliszyk@35222
   602
let
kaliszyk@35222
   603
  val thy = ProofContext.theory_of ctxt
kaliszyk@35222
   604
  val rtrm' = HOLogic.dest_Trueprop rtrm
kaliszyk@35222
   605
  val qtrm' = HOLogic.dest_Trueprop qtrm
kaliszyk@35222
   606
  val reg_goal = regularize_trm_chk ctxt (rtrm', qtrm')
kaliszyk@35222
   607
    handle (LIFT_MATCH msg) => lift_match_error ctxt msg rtrm qtrm
kaliszyk@35222
   608
  val inj_goal = inj_repabs_trm_chk ctxt (reg_goal, qtrm')
kaliszyk@35222
   609
    handle (LIFT_MATCH msg) => lift_match_error ctxt msg rtrm qtrm
kaliszyk@35222
   610
in
kaliszyk@35222
   611
  Drule.instantiate' []
kaliszyk@35222
   612
    [SOME (cterm_of thy rtrm'),
kaliszyk@35222
   613
     SOME (cterm_of thy reg_goal),
kaliszyk@35222
   614
     NONE,
kaliszyk@35222
   615
     SOME (cterm_of thy inj_goal)] lifting_procedure_thm
kaliszyk@35222
   616
end
kaliszyk@35222
   617
kaliszyk@35222
   618
(* the tactic leaves three subgoals to be proved *)
kaliszyk@35222
   619
fun procedure_tac ctxt rthm =
wenzelm@35625
   620
  Object_Logic.full_atomize_tac
kaliszyk@35222
   621
  THEN' gen_frees_tac ctxt
kaliszyk@35222
   622
  THEN' SUBGOAL (fn (goal, i) =>
kaliszyk@35222
   623
    let
kaliszyk@35222
   624
      val rthm' = atomize_thm rthm
kaliszyk@35222
   625
      val rule = procedure_inst ctxt (prop_of rthm') goal
kaliszyk@35222
   626
    in
kaliszyk@35222
   627
      (rtac rule THEN' rtac rthm') i
kaliszyk@35222
   628
    end)
kaliszyk@35222
   629
kaliszyk@35222
   630
kaliszyk@35222
   631
(* Automatic Proofs *)
kaliszyk@35222
   632
kaliszyk@35222
   633
fun lift_tac ctxt rthms =
kaliszyk@35222
   634
let
kaliszyk@35222
   635
  fun mk_tac rthm =
kaliszyk@35222
   636
    procedure_tac ctxt rthm
kaliszyk@36850
   637
    THEN' RANGE
kaliszyk@36850
   638
      [regularize_tac ctxt,
kaliszyk@36850
   639
       all_injection_tac ctxt,
kaliszyk@36850
   640
       clean_tac ctxt]
kaliszyk@35222
   641
in
kaliszyk@35222
   642
  simp_tac (mk_minimal_ss ctxt) (* unfolding multiple &&& *)
kaliszyk@35222
   643
  THEN' RANGE (map mk_tac rthms)
kaliszyk@35222
   644
end
kaliszyk@35222
   645
kaliszyk@35990
   646
fun lifted qtys ctxt thm =
kaliszyk@35222
   647
let
kaliszyk@36214
   648
  (* When the theorem is atomized, eta redexes are contracted,
kaliszyk@36214
   649
     so we do it both in the original theorem *)
kaliszyk@36214
   650
  val thm' = Drule.eta_contraction_rule thm
kaliszyk@36214
   651
  val ((_, [thm'']), ctxt') = Variable.import false [thm'] ctxt
kaliszyk@36214
   652
  val goal = (quotient_lift_all qtys ctxt' o prop_of) thm''
kaliszyk@35222
   653
in
kaliszyk@36214
   654
  Goal.prove ctxt' [] [] goal (K (lift_tac ctxt' [thm'] 1))
kaliszyk@35222
   655
  |> singleton (ProofContext.export ctxt' ctxt)
kaliszyk@35222
   656
end;
kaliszyk@35222
   657
kaliszyk@35990
   658
(* An Attribute which automatically constructs the qthm *)
kaliszyk@35990
   659
val lifted_attrib = Thm.rule_attribute (fn ctxt => lifted [] (Context.proof_of ctxt))
kaliszyk@35222
   660
kaliszyk@35222
   661
end; (* structure *)