src/HOL/Tools/Quotient/quotient_tacs.ML
author kuncar
Tue Apr 03 16:26:48 2012 +0200 (2012-04-03)
changeset 47308 9caab698dbe4
parent 46468 4db76d47b51a
child 51717 9e7d1c139569
permissions -rw-r--r--
new package Lifting - initial commit
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(*  Title:      HOL/Tools/Quotient/quotient_tacs.ML
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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 descend_procedure_tac: Proof.context -> thm list -> int -> tactic
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  val descend_tac: Proof.context -> thm list -> int -> tactic
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  val partiality_descend_procedure_tac: Proof.context -> thm list -> int -> tactic
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  val partiality_descend_tac: Proof.context -> thm list -> int -> tactic
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  val lift_procedure_tac: Proof.context -> thm list -> thm -> int -> tactic
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  val lift_tac: Proof.context -> thm list -> thm list -> int -> tactic
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  val lifted: Proof.context -> typ list -> thm list -> 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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(** 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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  |> Simplifier.set_subgoaler asm_simp_tac
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  |> Simplifier.set_mksimps (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? no! *)
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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 (Quotient_Info.equiv_rules ctxt))
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fun equiv_solver_tac ss = equiv_tac (Simplifier.the_context ss)
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val equiv_solver = 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_quotient3},
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     resolve_tac (Quotient_Info.quotient_rules ctxt)]))
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fun quotient_solver_tac ss = quotient_tac (Simplifier.the_context ss)
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val quotient_solver = 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 = Proof_Context.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_normalize 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) (Quotient_Info.equiv_rules 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) (Quotient_Info.equiv_rules ctxt)
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fun regularize_tac ctxt =
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  let
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    val thy = Proof_Context.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 =
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      Simplifier.simproc_global_i thy "" [ball_pat, bex_pat] (K (ball_bex_range_simproc))
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    val simpset =
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      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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    TRY o 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 = Proof_Context.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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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 = Proof_Context.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_rspQ3}
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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 = Proof_Context.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 $ (abs $ _))) =>
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        (let
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          val thy = Proof_Context.theory_of ctxt;
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          val (ty_a, ty_b) = dest_funT (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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        handle TERM _ => no_tac)
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    | _ => no_tac))
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(* Injection means to prove that the regularized 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 using fun_relI
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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
   325
    - split applications of non-lifted type (cong_tac)
kaliszyk@35222
   326
    - apply extentionality
kaliszyk@35222
   327
    - assumption
kaliszyk@35222
   328
    - reflexivity of the relation
kaliszyk@35222
   329
*)
kaliszyk@35222
   330
fun injection_match_tac ctxt = SUBGOAL (fn (goal, i) =>
wenzelm@41444
   331
  (case bare_concl goal of
wenzelm@41444
   332
      (* (R1 ===> R2) (%x...) (%x...) ----> [|R1 x y|] ==> R2 (...x) (...y) *)
wenzelm@41444
   333
    (Const (@{const_name fun_rel}, _) $ _ $ _) $ (Abs _) $ (Abs _)
wenzelm@41444
   334
        => rtac @{thm fun_relI} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   335
wenzelm@41444
   336
      (* (op =) (Ball...) (Ball...) ----> (op =) (...) (...) *)
wenzelm@41444
   337
  | (Const (@{const_name HOL.eq},_) $
wenzelm@41444
   338
      (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
wenzelm@41444
   339
      (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
wenzelm@41444
   340
        => rtac @{thm ball_rsp} THEN' dtac @{thm QT_all}
kaliszyk@35222
   341
wenzelm@41444
   342
      (* (R1 ===> op =) (Ball...) (Ball...) ----> [|R1 x y|] ==> (Ball...x) = (Ball...y) *)
wenzelm@41444
   343
  | (Const (@{const_name fun_rel}, _) $ _ $ _) $
wenzelm@41444
   344
      (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
wenzelm@41444
   345
      (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
wenzelm@41444
   346
        => rtac @{thm fun_relI} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   347
wenzelm@41444
   348
      (* (op =) (Bex...) (Bex...) ----> (op =) (...) (...) *)
wenzelm@41444
   349
  | Const (@{const_name HOL.eq},_) $
wenzelm@41444
   350
      (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
wenzelm@41444
   351
      (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
wenzelm@41444
   352
        => rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex}
kaliszyk@35222
   353
wenzelm@41444
   354
      (* (R1 ===> op =) (Bex...) (Bex...) ----> [|R1 x y|] ==> (Bex...x) = (Bex...y) *)
wenzelm@41444
   355
  | (Const (@{const_name fun_rel}, _) $ _ $ _) $
wenzelm@41444
   356
      (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
wenzelm@41444
   357
      (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
wenzelm@41444
   358
        => rtac @{thm fun_relI} THEN' quot_true_tac ctxt unlam
kaliszyk@35222
   359
wenzelm@41444
   360
  | (Const (@{const_name fun_rel}, _) $ _ $ _) $
wenzelm@41444
   361
      (Const(@{const_name Bex1_rel},_) $ _) $ (Const(@{const_name Bex1_rel},_) $ _)
wenzelm@41444
   362
        => rtac @{thm bex1_rel_rsp} THEN' quotient_tac ctxt
kaliszyk@35222
   363
wenzelm@41444
   364
  | (_ $
wenzelm@41444
   365
      (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
wenzelm@41444
   366
      (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
wenzelm@46468
   367
        => rtac @{thm babs_rsp} THEN' quotient_tac ctxt
kaliszyk@35222
   368
wenzelm@41444
   369
  | Const (@{const_name HOL.eq},_) $ (R $ _ $ _) $ (_ $ _ $ _) =>
wenzelm@41444
   370
     (rtac @{thm refl} ORELSE'
wenzelm@41444
   371
      (equals_rsp_tac R ctxt THEN' RANGE [
wenzelm@41444
   372
         quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)]))
kaliszyk@35222
   373
wenzelm@41444
   374
      (* reflexivity of operators arising from Cong_tac *)
wenzelm@41444
   375
  | Const (@{const_name HOL.eq},_) $ _ $ _ => rtac @{thm refl}
kaliszyk@35222
   376
wenzelm@41444
   377
     (* respectfulness of constants; in particular of a simple relation *)
wenzelm@41444
   378
  | _ $ (Const _) $ (Const _)  (* fun_rel, list_rel, etc but not equality *)
wenzelm@45279
   379
      => resolve_tac (Quotient_Info.rsp_rules ctxt) THEN_ALL_NEW quotient_tac ctxt
kaliszyk@35222
   380
wenzelm@41444
   381
      (* R (...) (Rep (Abs ...)) ----> R (...) (...) *)
wenzelm@41444
   382
      (* observe map_fun *)
wenzelm@41444
   383
  | _ $ _ $ _
wenzelm@41444
   384
      => (rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt)
wenzelm@41444
   385
         ORELSE' rep_abs_rsp_tac ctxt
kaliszyk@35222
   386
wenzelm@41444
   387
  | _ => K no_tac) i)
kaliszyk@35222
   388
kaliszyk@35222
   389
fun injection_step_tac ctxt rel_refl =
wenzelm@41444
   390
  FIRST' [
kaliszyk@35222
   391
    injection_match_tac ctxt,
kaliszyk@35222
   392
kaliszyk@35222
   393
    (* R (t $ ...) (t' $ ...) ----> apply_rsp   provided type of t needs lifting *)
kaliszyk@35222
   394
    apply_rsp_tac ctxt THEN'
kaliszyk@35222
   395
                 RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)],
kaliszyk@35222
   396
kaliszyk@35222
   397
    (* (op =) (t $ ...) (t' $ ...) ----> Cong   provided type of t does not need lifting *)
kaliszyk@35222
   398
    (* merge with previous tactic *)
kaliszyk@35222
   399
    Cong_Tac.cong_tac @{thm cong} THEN'
kaliszyk@35222
   400
                 RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)],
kaliszyk@35222
   401
kaliszyk@44285
   402
    (* resolving with R x y assumptions *)
kaliszyk@44285
   403
    atac,
kaliszyk@44285
   404
kaliszyk@35222
   405
    (* (op =) (%x...) (%y...) ----> (op =) (...) (...) *)
kaliszyk@35222
   406
    rtac @{thm ext} THEN' quot_true_tac ctxt unlam,
kaliszyk@35222
   407
kaliszyk@35222
   408
    (* reflexivity of the basic relations *)
kaliszyk@35222
   409
    (* R ... ... *)
kaliszyk@35222
   410
    resolve_tac rel_refl]
kaliszyk@35222
   411
kaliszyk@35222
   412
fun injection_tac ctxt =
wenzelm@41444
   413
  let
wenzelm@41444
   414
    val rel_refl = reflp_get ctxt
wenzelm@41444
   415
  in
wenzelm@41444
   416
    injection_step_tac ctxt rel_refl
wenzelm@41444
   417
  end
kaliszyk@35222
   418
kaliszyk@35222
   419
fun all_injection_tac ctxt =
kaliszyk@35222
   420
  REPEAT_ALL_NEW (injection_tac ctxt)
kaliszyk@35222
   421
kaliszyk@35222
   422
kaliszyk@35222
   423
kaliszyk@35222
   424
(*** Cleaning of the Theorem ***)
kaliszyk@35222
   425
haftmann@40602
   426
(* expands all map_funs, except in front of the (bound) variables listed in xs *)
haftmann@40602
   427
fun map_fun_simple_conv xs ctrm =
wenzelm@41444
   428
  (case term_of ctrm of
haftmann@40602
   429
    ((Const (@{const_name "map_fun"}, _) $ _ $ _) $ h $ _) =>
kaliszyk@35222
   430
        if member (op=) xs h
kaliszyk@35222
   431
        then Conv.all_conv ctrm
haftmann@40602
   432
        else Conv.rewr_conv @{thm map_fun_apply [THEN eq_reflection]} ctrm
wenzelm@41444
   433
  | _ => Conv.all_conv ctrm)
kaliszyk@35222
   434
haftmann@40602
   435
fun map_fun_conv xs ctxt ctrm =
wenzelm@41444
   436
  (case term_of ctrm of
wenzelm@41444
   437
    _ $ _ =>
wenzelm@41444
   438
      (Conv.comb_conv (map_fun_conv xs ctxt) then_conv
wenzelm@41444
   439
        map_fun_simple_conv xs) ctrm
wenzelm@41444
   440
  | Abs _ => Conv.abs_conv (fn (x, ctxt) => map_fun_conv ((term_of x)::xs) ctxt) ctxt ctrm
wenzelm@41444
   441
  | _ => Conv.all_conv ctrm)
kaliszyk@35222
   442
haftmann@40602
   443
fun map_fun_tac ctxt = CONVERSION (map_fun_conv [] ctxt)
kaliszyk@35222
   444
kaliszyk@35222
   445
(* custom matching functions *)
kaliszyk@35222
   446
fun mk_abs u i t =
wenzelm@41444
   447
  if incr_boundvars i u aconv t then Bound i
wenzelm@41444
   448
  else
wenzelm@41444
   449
    case t of
wenzelm@41444
   450
      t1 $ t2 => mk_abs u i t1 $ mk_abs u i t2
wenzelm@41444
   451
    | Abs (s, T, t') => Abs (s, T, mk_abs u (i + 1) t')
wenzelm@41444
   452
    | Bound j => if i = j then error "make_inst" else t
wenzelm@41444
   453
    | _ => t
kaliszyk@35222
   454
kaliszyk@35222
   455
fun make_inst lhs t =
wenzelm@41444
   456
  let
wenzelm@41444
   457
    val _ $ (Abs (_, _, (_ $ ((f as Var (_, Type ("fun", [T, _]))) $ u)))) = lhs;
wenzelm@41444
   458
    val _ $ (Abs (_, _, (_ $ g))) = t;
wenzelm@41444
   459
  in
wenzelm@41444
   460
    (f, Abs ("x", T, mk_abs u 0 g))
wenzelm@41444
   461
  end
kaliszyk@35222
   462
kaliszyk@35222
   463
fun make_inst_id lhs t =
wenzelm@41444
   464
  let
wenzelm@41444
   465
    val _ $ (Abs (_, _, (f as Var (_, Type ("fun", [T, _]))) $ u)) = lhs;
wenzelm@41444
   466
    val _ $ (Abs (_, _, g)) = t;
wenzelm@41444
   467
  in
wenzelm@41444
   468
    (f, Abs ("x", T, mk_abs u 0 g))
wenzelm@41444
   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 =
wenzelm@41444
   479
  (case term_of ctrm of
haftmann@40602
   480
    (Const (@{const_name map_fun}, _) $ r1 $ a2) $ (Abs _) =>
kaliszyk@35222
   481
      let
wenzelm@42361
   482
        val thy = Proof_Context.theory_of ctxt
wenzelm@40840
   483
        val (ty_b, ty_a) = dest_funT (fastype_of r1)
wenzelm@40840
   484
        val (ty_c, ty_d) = dest_funT (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)
wenzelm@41228
   489
        val thm3 = Raw_Simplifier.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)
wenzelm@43333
   494
        val thm4 = Drule.instantiate_normalize ([], [(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
wenzelm@41444
   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 =
wenzelm@41444
   526
  let
urbanc@45350
   527
    val thy =  Proof_Context.theory_of lthy
urbanc@45350
   528
    val defs = map (Thm.symmetric o #def) (Quotient_Info.dest_quotconsts_global thy)
wenzelm@45279
   529
    val prs = Quotient_Info.prs_rules lthy
wenzelm@45279
   530
    val ids = Quotient_Info.id_simps lthy
wenzelm@41444
   531
    val thms =
kuncar@47308
   532
      @{thms Quotient3_abs_rep Quotient3_rel_rep babs_prs all_prs ex_prs ex1_prs} @ ids @ prs @ defs
kaliszyk@35222
   533
wenzelm@41444
   534
    val ss = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver
wenzelm@41444
   535
  in
wenzelm@41451
   536
    EVERY' [
wenzelm@41451
   537
      map_fun_tac lthy,
wenzelm@41451
   538
      lambda_prs_tac lthy,
wenzelm@41451
   539
      simp_tac ss,
wenzelm@41451
   540
      TRY o rtac refl]
wenzelm@41444
   541
  end
kaliszyk@35222
   542
kaliszyk@35222
   543
urbanc@38718
   544
(* Tactic for Generalising Free Variables in a Goal *)
kaliszyk@35222
   545
kaliszyk@35222
   546
fun inst_spec ctrm =
wenzelm@41444
   547
  Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}
kaliszyk@35222
   548
kaliszyk@35222
   549
fun inst_spec_tac ctrms =
kaliszyk@35222
   550
  EVERY' (map (dtac o inst_spec) ctrms)
kaliszyk@35222
   551
kaliszyk@35222
   552
fun all_list xs trm =
kaliszyk@35222
   553
  fold (fn (x, T) => fn t' => HOLogic.mk_all (x, T, t')) xs trm
kaliszyk@35222
   554
kaliszyk@35222
   555
fun apply_under_Trueprop f =
kaliszyk@35222
   556
  HOLogic.dest_Trueprop #> f #> HOLogic.mk_Trueprop
kaliszyk@35222
   557
kaliszyk@35222
   558
fun gen_frees_tac ctxt =
kaliszyk@35222
   559
  SUBGOAL (fn (concl, i) =>
kaliszyk@35222
   560
    let
wenzelm@42361
   561
      val thy = Proof_Context.theory_of ctxt
kaliszyk@35222
   562
      val vrs = Term.add_frees concl []
kaliszyk@35222
   563
      val cvrs = map (cterm_of thy o Free) vrs
kaliszyk@35222
   564
      val concl' = apply_under_Trueprop (all_list vrs) concl
kaliszyk@35222
   565
      val goal = Logic.mk_implies (concl', concl)
kaliszyk@35222
   566
      val rule = Goal.prove ctxt [] [] goal
kaliszyk@35222
   567
        (K (EVERY1 [inst_spec_tac (rev cvrs), atac]))
kaliszyk@35222
   568
    in
kaliszyk@35222
   569
      rtac rule i
kaliszyk@35222
   570
    end)
kaliszyk@35222
   571
kaliszyk@35222
   572
kaliszyk@35222
   573
(** The General Shape of the Lifting Procedure **)
kaliszyk@35222
   574
kaliszyk@35222
   575
(* - A is the original raw theorem
kaliszyk@35222
   576
   - B is the regularized theorem
kaliszyk@35222
   577
   - C is the rep/abs injected version of B
kaliszyk@35222
   578
   - D is the lifted theorem
kaliszyk@35222
   579
kaliszyk@35222
   580
   - 1st prem is the regularization step
kaliszyk@35222
   581
   - 2nd prem is the rep/abs injection step
kaliszyk@35222
   582
   - 3rd prem is the cleaning part
kaliszyk@35222
   583
kaliszyk@35222
   584
   the Quot_True premise in 2nd records the lifted theorem
kaliszyk@35222
   585
*)
urbanc@45782
   586
val procedure_thm =
kaliszyk@35222
   587
  @{lemma  "[|A;
kaliszyk@35222
   588
              A --> B;
kaliszyk@35222
   589
              Quot_True D ==> B = C;
kaliszyk@35222
   590
              C = D|] ==> D"
kaliszyk@35222
   591
      by (simp add: Quot_True_def)}
kaliszyk@35222
   592
urbanc@45782
   593
(* in case of partial equivalence relations, this form of the 
urbanc@45782
   594
   procedure theorem results in solvable proof obligations 
urbanc@45782
   595
*)
urbanc@45782
   596
val partiality_procedure_thm =
urbanc@45782
   597
  @{lemma  "[|B; 
urbanc@45782
   598
              Quot_True D ==> B = C;
urbanc@45782
   599
              C = D|] ==> D"
urbanc@45782
   600
      by (simp add: Quot_True_def)}
urbanc@45782
   601
kaliszyk@35222
   602
fun lift_match_error ctxt msg rtrm qtrm =
wenzelm@41444
   603
  let
wenzelm@41444
   604
    val rtrm_str = Syntax.string_of_term ctxt rtrm
wenzelm@41444
   605
    val qtrm_str = Syntax.string_of_term ctxt qtrm
wenzelm@41444
   606
    val msg = cat_lines [enclose "[" "]" msg, "The quotient theorem", qtrm_str,
wenzelm@41444
   607
      "", "does not match with original theorem", rtrm_str]
wenzelm@41444
   608
  in
wenzelm@41444
   609
    error msg
wenzelm@41444
   610
  end
kaliszyk@35222
   611
kaliszyk@35222
   612
fun procedure_inst ctxt rtrm qtrm =
wenzelm@41444
   613
  let
wenzelm@42361
   614
    val thy = Proof_Context.theory_of ctxt
wenzelm@41444
   615
    val rtrm' = HOLogic.dest_Trueprop rtrm
wenzelm@41444
   616
    val qtrm' = HOLogic.dest_Trueprop qtrm
wenzelm@41451
   617
    val reg_goal = Quotient_Term.regularize_trm_chk ctxt (rtrm', qtrm')
wenzelm@41451
   618
      handle Quotient_Term.LIFT_MATCH msg => lift_match_error ctxt msg rtrm qtrm
wenzelm@41451
   619
    val inj_goal = Quotient_Term.inj_repabs_trm_chk ctxt (reg_goal, qtrm')
wenzelm@41451
   620
      handle Quotient_Term.LIFT_MATCH msg => lift_match_error ctxt msg rtrm qtrm
wenzelm@41444
   621
  in
wenzelm@41444
   622
    Drule.instantiate' []
wenzelm@41444
   623
      [SOME (cterm_of thy rtrm'),
wenzelm@41444
   624
       SOME (cterm_of thy reg_goal),
wenzelm@41444
   625
       NONE,
urbanc@45782
   626
       SOME (cterm_of thy inj_goal)] procedure_thm
wenzelm@41444
   627
  end
kaliszyk@35222
   628
urbanc@37593
   629
urbanc@38860
   630
(* Since we use Ball and Bex during the lifting and descending,
kaliszyk@38862
   631
   we cannot deal with lemmas containing them, unless we unfold
kaliszyk@38862
   632
   them by default. *)
urbanc@38860
   633
kaliszyk@38862
   634
val default_unfolds = @{thms Ball_def Bex_def}
urbanc@38860
   635
urbanc@38860
   636
urbanc@37593
   637
(** descending as tactic **)
urbanc@37593
   638
urbanc@38859
   639
fun descend_procedure_tac ctxt simps =
wenzelm@41444
   640
  let
wenzelm@41444
   641
    val ss = (mk_minimal_ss ctxt) addsimps (simps @ default_unfolds)
wenzelm@41444
   642
  in
wenzelm@41444
   643
    full_simp_tac ss
wenzelm@41444
   644
    THEN' Object_Logic.full_atomize_tac
wenzelm@41444
   645
    THEN' gen_frees_tac ctxt
wenzelm@41444
   646
    THEN' SUBGOAL (fn (goal, i) =>
wenzelm@41444
   647
      let
wenzelm@45279
   648
        val qtys = map #qtyp (Quotient_Info.dest_quotients ctxt)
wenzelm@41451
   649
        val rtrm = Quotient_Term.derive_rtrm ctxt qtys goal
wenzelm@41444
   650
        val rule = procedure_inst ctxt rtrm  goal
wenzelm@41444
   651
      in
wenzelm@41444
   652
        rtac rule i
wenzelm@41444
   653
      end)
wenzelm@41444
   654
  end
urbanc@37593
   655
urbanc@38859
   656
fun descend_tac ctxt simps =
wenzelm@41444
   657
  let
wenzelm@41444
   658
    val mk_tac_raw =
wenzelm@41444
   659
      descend_procedure_tac ctxt simps
wenzelm@41444
   660
      THEN' RANGE
wenzelm@41444
   661
        [Object_Logic.rulify_tac THEN' (K all_tac),
wenzelm@41444
   662
         regularize_tac ctxt,
wenzelm@41444
   663
         all_injection_tac ctxt,
wenzelm@41444
   664
         clean_tac ctxt]
wenzelm@41444
   665
  in
wenzelm@41444
   666
    Goal.conjunction_tac THEN_ALL_NEW mk_tac_raw
wenzelm@41444
   667
  end
kaliszyk@35222
   668
kaliszyk@35222
   669
urbanc@45782
   670
(** descending for partial equivalence relations **)
urbanc@45782
   671
urbanc@45782
   672
fun partiality_procedure_inst ctxt rtrm qtrm =
urbanc@45782
   673
  let
urbanc@45782
   674
    val thy = Proof_Context.theory_of ctxt
urbanc@45782
   675
    val rtrm' = HOLogic.dest_Trueprop rtrm
urbanc@45782
   676
    val qtrm' = HOLogic.dest_Trueprop qtrm
urbanc@45782
   677
    val reg_goal = Quotient_Term.regularize_trm_chk ctxt (rtrm', qtrm')
urbanc@45782
   678
      handle Quotient_Term.LIFT_MATCH msg => lift_match_error ctxt msg rtrm qtrm
urbanc@45782
   679
    val inj_goal = Quotient_Term.inj_repabs_trm_chk ctxt (reg_goal, qtrm')
urbanc@45782
   680
      handle Quotient_Term.LIFT_MATCH msg => lift_match_error ctxt msg rtrm qtrm
urbanc@45782
   681
  in
urbanc@45782
   682
    Drule.instantiate' []
urbanc@45782
   683
      [SOME (cterm_of thy reg_goal),
urbanc@45782
   684
       NONE,
urbanc@45782
   685
       SOME (cterm_of thy inj_goal)] partiality_procedure_thm
urbanc@45782
   686
  end
urbanc@45782
   687
urbanc@45782
   688
urbanc@45782
   689
fun partiality_descend_procedure_tac ctxt simps =
urbanc@45782
   690
  let
urbanc@45782
   691
    val ss = (mk_minimal_ss ctxt) addsimps (simps @ default_unfolds)
urbanc@45782
   692
  in
urbanc@45782
   693
    full_simp_tac ss
urbanc@45782
   694
    THEN' Object_Logic.full_atomize_tac
urbanc@45782
   695
    THEN' gen_frees_tac ctxt
urbanc@45782
   696
    THEN' SUBGOAL (fn (goal, i) =>
urbanc@45782
   697
      let
urbanc@45782
   698
        val qtys = map #qtyp (Quotient_Info.dest_quotients ctxt)
urbanc@45782
   699
        val rtrm = Quotient_Term.derive_rtrm ctxt qtys goal
urbanc@45782
   700
        val rule = partiality_procedure_inst ctxt rtrm  goal
urbanc@45782
   701
      in
urbanc@45782
   702
        rtac rule i
urbanc@45782
   703
      end)
urbanc@45782
   704
  end
urbanc@45782
   705
urbanc@45782
   706
fun partiality_descend_tac ctxt simps =
urbanc@45782
   707
  let
urbanc@45782
   708
    val mk_tac_raw =
urbanc@45782
   709
      partiality_descend_procedure_tac ctxt simps
urbanc@45782
   710
      THEN' RANGE
urbanc@45782
   711
        [Object_Logic.rulify_tac THEN' (K all_tac),
urbanc@45782
   712
         all_injection_tac ctxt,
urbanc@45782
   713
         clean_tac ctxt]
urbanc@45782
   714
  in
urbanc@45782
   715
    Goal.conjunction_tac THEN_ALL_NEW mk_tac_raw
urbanc@45782
   716
  end
urbanc@45782
   717
urbanc@45782
   718
urbanc@45782
   719
urbanc@38625
   720
(** lifting as a tactic **)
urbanc@37593
   721
urbanc@38718
   722
urbanc@37593
   723
(* the tactic leaves three subgoals to be proved *)
urbanc@38859
   724
fun lift_procedure_tac ctxt simps rthm =
wenzelm@41444
   725
  let
wenzelm@41444
   726
    val ss = (mk_minimal_ss ctxt) addsimps (simps @ default_unfolds)
wenzelm@41444
   727
  in
wenzelm@41444
   728
    full_simp_tac ss
wenzelm@41444
   729
    THEN' Object_Logic.full_atomize_tac
wenzelm@41444
   730
    THEN' gen_frees_tac ctxt
wenzelm@41444
   731
    THEN' SUBGOAL (fn (goal, i) =>
wenzelm@41444
   732
      let
wenzelm@41444
   733
        (* full_atomize_tac contracts eta redexes,
wenzelm@41444
   734
           so we do it also in the original theorem *)
wenzelm@41444
   735
        val rthm' =
wenzelm@41444
   736
          rthm |> full_simplify ss
wenzelm@41444
   737
               |> Drule.eta_contraction_rule
wenzelm@41444
   738
               |> Thm.forall_intr_frees
wenzelm@41444
   739
               |> atomize_thm
urbanc@38717
   740
wenzelm@41444
   741
        val rule = procedure_inst ctxt (prop_of rthm') goal
wenzelm@41444
   742
      in
wenzelm@41444
   743
        (rtac rule THEN' rtac rthm') i
wenzelm@41444
   744
      end)
wenzelm@41444
   745
  end
urbanc@38625
   746
wenzelm@41444
   747
fun lift_single_tac ctxt simps rthm =
urbanc@38859
   748
  lift_procedure_tac ctxt simps rthm
urbanc@38625
   749
  THEN' RANGE
urbanc@38625
   750
    [ regularize_tac ctxt,
urbanc@38625
   751
      all_injection_tac ctxt,
urbanc@38625
   752
      clean_tac ctxt ]
urbanc@38625
   753
urbanc@38859
   754
fun lift_tac ctxt simps rthms =
wenzelm@41444
   755
  Goal.conjunction_tac
urbanc@38859
   756
  THEN' RANGE (map (lift_single_tac ctxt simps) rthms)
urbanc@38625
   757
urbanc@38625
   758
urbanc@38625
   759
(* automated lifting with pre-simplification of the theorems;
urbanc@38625
   760
   for internal usage *)
urbanc@38625
   761
fun lifted ctxt qtys simps rthm =
wenzelm@41444
   762
  let
wenzelm@41444
   763
    val ((_, [rthm']), ctxt') = Variable.import true [rthm] ctxt
wenzelm@41451
   764
    val goal = Quotient_Term.derive_qtrm ctxt' qtys (prop_of rthm')
wenzelm@41444
   765
  in
wenzelm@41444
   766
    Goal.prove ctxt' [] [] goal
wenzelm@41444
   767
      (K (HEADGOAL (lift_single_tac ctxt' simps rthm')))
wenzelm@42361
   768
    |> singleton (Proof_Context.export ctxt' ctxt)
wenzelm@41444
   769
  end
kaliszyk@35222
   770
urbanc@37593
   771
urbanc@38625
   772
(* lifting as an attribute *)
kaliszyk@35222
   773
wenzelm@41444
   774
val lifted_attrib = Thm.rule_attribute (fn context =>
urbanc@37593
   775
  let
urbanc@37593
   776
    val ctxt = Context.proof_of context
wenzelm@45279
   777
    val qtys = map #qtyp (Quotient_Info.dest_quotients ctxt)
urbanc@37593
   778
  in
urbanc@38625
   779
    lifted ctxt qtys []
urbanc@37593
   780
  end)
kaliszyk@35222
   781
kaliszyk@35222
   782
end; (* structure *)