theory QuotMainimports QuotScript Proveuses ("quotient_info.ML") ("quotient.ML") ("quotient_def.ML")beginlocale QUOT_TYPE = fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b" and Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)" assumes equivp: "equivp R" and rep_prop: "\<And>y. \<exists>x. Rep y = R x" and rep_inverse: "\<And>x. Abs (Rep x) = x" and abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)" and rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"begindefinition abs::"'a \<Rightarrow> 'b"where "abs x \<equiv> Abs (R x)"definition rep::"'b \<Rightarrow> 'a"where "rep a = Eps (Rep a)"lemma lem9: shows "R (Eps (R x)) = R x"proof - have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def) then have "R x (Eps (R x))" by (rule someI) then show "R (Eps (R x)) = R x" using equivp unfolding equivp_def by simpqedtheorem thm10: shows "abs (rep a) \<equiv> a" apply (rule eq_reflection) unfolding abs_def rep_defproof - from rep_prop obtain x where eq: "Rep a = R x" by auto have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp also have "\<dots> = Abs (R x)" using lem9 by simp also have "\<dots> = Abs (Rep a)" using eq by simp also have "\<dots> = a" using rep_inverse by simp finally show "Abs (R (Eps (Rep a))) = a" by simpqedlemma rep_refl: shows "R (rep a) (rep a)"unfolding rep_defby (simp add: equivp[simplified equivp_def])lemma lem7: shows "(R x = R y) = (Abs (R x) = Abs (R y))"apply(rule iffI)apply(simp)apply(drule rep_inject[THEN iffD2])apply(simp add: abs_inverse)donetheorem thm11: shows "R r r' = (abs r = abs r')"unfolding abs_defby (simp only: equivp[simplified equivp_def] lem7)lemma rep_abs_rsp: shows "R f (rep (abs g)) = R f g" and "R (rep (abs g)) f = R g f"by (simp_all add: thm10 thm11)lemma Quotient: "Quotient R abs rep"apply(unfold Quotient_def)apply(simp add: thm10)apply(simp add: rep_refl)apply(subst thm11[symmetric])apply(simp add: equivp[simplified equivp_def])doneendsection {* type definition for the quotient type *}(* the auxiliary data for the quotient types *)use "quotient_info.ML"ML {* print_mapsinfo @{context} *}declare [[map "fun" = (fun_map, fun_rel)]]lemmas [quot_thm] = fun_quotient lemmas [quot_respect] = quot_rel_rsp(* fun_map is not here since equivp is not true *)lemmas [quot_equiv] = identity_equivp(* definition of the quotient types *)(* FIXME: should be called quotient_typ.ML *)use "quotient.ML"(* lifting of constants *)use "quotient_def.ML"section {* Simset setup *}(* Since HOL_basic_ss is too "big" for us, *)(* we set up our own minimal simpset. *)ML {*fun mk_minimal_ss ctxt = Simplifier.context ctxt empty_ss setsubgoaler asm_simp_tac setmksimps (mksimps [])*}ML {*fun OF1 thm1 thm2 = thm2 RS thm1*}section {* Atomize Infrastructure *}lemma atomize_eqv[atomize]: shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)"proof assume "A \<equiv> B" then show "Trueprop A \<equiv> Trueprop B" by unfoldnext assume *: "Trueprop A \<equiv> Trueprop B" have "A = B" proof (cases A) case True have "A" by fact then show "A = B" using * by simp next case False have "\<not>A" by fact then show "A = B" using * by auto qed then show "A \<equiv> B" by (rule eq_reflection)qedML {*fun atomize_thm thm =let val thm' = Thm.freezeT (forall_intr_vars thm) val thm'' = ObjectLogic.atomize (cprop_of thm')in @{thm equal_elim_rule1} OF [thm'', thm']end*}section {* Infrastructure about id *}lemmas [id_simps] = fun_map_id[THEN eq_reflection] id_apply[THEN eq_reflection] id_def[THEN eq_reflection,symmetric]section {* Computation of the Regularize Goal *} (*Regularizing an rtrm means: - quantifiers over a type that needs lifting are replaced by bounded quantifiers, for example: \<forall>x. P \<Longrightarrow> \<forall>x \<in> (Respects R). P / All (Respects R) P the relation R is given by the rty and qty; - abstractions over a type that needs lifting are replaced by bounded abstractions: \<lambda>x. P \<Longrightarrow> Ball (Respects R) (\<lambda>x. P) - equalities over the type being lifted are replaced by corresponding relations: A = B \<Longrightarrow> A \<approx> B example with more complicated types of A, B: A = B \<Longrightarrow> (op = \<Longrightarrow> op \<approx>) A B*)ML {*(* builds the relation that is the argument of respects *)fun mk_resp_arg lthy (rty, qty) =let val thy = ProofContext.theory_of lthyin if rty = qty then HOLogic.eq_const rty else case (rty, qty) of (Type (s, tys), Type (s', tys')) => if s = s' then let val SOME map_info = maps_lookup thy s val args = map (mk_resp_arg lthy) (tys ~~ tys') in list_comb (Const (#relfun map_info, dummyT), args) end else let val SOME qinfo = quotdata_lookup_thy thy s' (* FIXME: check in this case that the rty and qty *) (* FIXME: correspond to each other *) val (s, _) = dest_Const (#rel qinfo) (* FIXME: the relation should only be the string *) (* FIXME: and the type needs to be calculated as below; *) (* FIXME: maybe one should actually have a term *) (* FIXME: and one needs to force it to have this type *) in Const (s, rty --> rty --> @{typ bool}) end | _ => HOLogic.eq_const dummyT (* FIXME: check that the types correspond to each other? *)end*}ML {*val mk_babs = Const (@{const_name Babs}, dummyT)val mk_ball = Const (@{const_name Ball}, dummyT)val mk_bex = Const (@{const_name Bex}, dummyT)val mk_resp = Const (@{const_name Respects}, dummyT)*}ML {*(* - applies f to the subterm of an abstraction, *)(* otherwise to the given term, *)(* - used by regularize, therefore abstracted *)(* variables do not have to be treated specially *)fun apply_subt f trm1 trm2 = case (trm1, trm2) of (Abs (x, T, t), Abs (x', T', t')) => Abs (x, T, f t t') | _ => f trm1 trm2(* the major type of All and Ex quantifiers *)fun qnt_typ ty = domain_type (domain_type ty) *}ML {*(* produces a regularized version of rtrm *)(* - the result is contains dummyT *)(* - does not need any special treatment of *)(* bound variables *)fun regularize_trm lthy rtrm qtrm = case (rtrm, qtrm) of (Abs (x, ty, t), Abs (x', ty', t')) => let val subtrm = Abs(x, ty, regularize_trm lthy t t') in if ty = ty' then subtrm else mk_babs $ (mk_resp $ mk_resp_arg lthy (ty, ty')) $ subtrm end | (Const (@{const_name "All"}, ty) $ t, Const (@{const_name "All"}, ty') $ t') => let val subtrm = apply_subt (regularize_trm lthy) t t' in if ty = ty' then Const (@{const_name "All"}, ty) $ subtrm else mk_ball $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm end | (Const (@{const_name "Ex"}, ty) $ t, Const (@{const_name "Ex"}, ty') $ t') => let val subtrm = apply_subt (regularize_trm lthy) t t' in if ty = ty' then Const (@{const_name "Ex"}, ty) $ subtrm else mk_bex $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm end | (* equalities need to be replaced by appropriate equivalence relations *) (Const (@{const_name "op ="}, ty), Const (@{const_name "op ="}, ty')) => if ty = ty' then rtrm else mk_resp_arg lthy (domain_type ty, domain_type ty') | (* in this case we check whether the given equivalence relation is correct *) (rel, Const (@{const_name "op ="}, ty')) => let val exc = LIFT_MATCH "regularise (relation mismatch)" val rel_ty = (fastype_of rel) handle TERM _ => raise exc val rel' = mk_resp_arg lthy (domain_type rel_ty, domain_type ty') in if rel' = rel then rtrm else raise exc end | (_, Const (s, Type(st, _))) => let fun same_name (Const (s, _)) (Const (s', _)) = (s = s') | same_name _ _ = false in (* TODO/FIXME: This test is not enough *) if same_name rtrm qtrm then rtrm else let val exc1 = LIFT_MATCH ("regularize (constant " ^ s ^ "(" ^ st ^ ") not found)") val exc2 = LIFT_MATCH ("regularize (constant " ^ s ^ "(" ^ st ^ ") mismatch)") val thy = ProofContext.theory_of lthy val rtrm' = (#rconst (qconsts_lookup thy qtrm)) handle NotFound => raise exc1 in if Pattern.matches thy (rtrm', rtrm) then rtrm else let val _ = tracing ("rtrm := " ^ Syntax.string_of_term @{context} rtrm); val _ = tracing ("rtrm':= " ^ Syntax.string_of_term @{context} rtrm'); in raise exc2 end end end | (t1 $ t2, t1' $ t2') => (regularize_trm lthy t1 t1') $ (regularize_trm lthy t2 t2') | (Free (x, ty), Free (x', ty')) => (* this case cannot arrise as we start with two fully atomized terms *) raise (LIFT_MATCH "regularize (frees)") | (Bound i, Bound i') => if i = i' then rtrm else raise (LIFT_MATCH "regularize (bounds mismatch)") | (rt, qt) => let val (rts, qts) = (Syntax.string_of_term lthy rt, Syntax.string_of_term lthy qt) in raise (LIFT_MATCH ("regularize failed (default: " ^ rts ^ "," ^ qts ^ ")")) end*}section {* Regularize Tactic *}ML {*fun equiv_tac ctxt = REPEAT_ALL_NEW (resolve_tac (equiv_rules_get ctxt))fun equiv_solver_tac ss = equiv_tac (Simplifier.the_context ss)val equiv_solver = Simplifier.mk_solver' "Equivalence goal solver" equiv_solver_tac*}ML {*fun prep_trm thy (x, (T, t)) = (cterm_of thy (Var (x, T)), cterm_of thy t)fun prep_ty thy (x, (S, ty)) = (ctyp_of thy (TVar (x, S)), ctyp_of thy ty)*}ML {*fun matching_prs thy pat trm =let val univ = Unify.matchers thy [(pat, trm)] val SOME (env, _) = Seq.pull univ val tenv = Vartab.dest (Envir.term_env env) val tyenv = Vartab.dest (Envir.type_env env)in (map (prep_ty thy) tyenv, map (prep_trm thy) tenv)end*}ML {*fun calculate_instance ctxt thm redex R1 R2 =let val thy = ProofContext.theory_of ctxt val goal = Const (@{const_name "equivp"}, dummyT) $ R2 |> Syntax.check_term ctxt |> HOLogic.mk_Trueprop val eqv_prem = Goal.prove ctxt [] [] goal (fn {context,...} => equiv_tac context 1) val thm = (@{thm eq_reflection} OF [thm OF [eqv_prem]]) val R1c = cterm_of thy R1 val thmi = Drule.instantiate' [] [SOME R1c] thm val inst = matching_prs thy (term_of (Thm.lhs_of thmi)) redex val thm2 = Drule.eta_contraction_rule (Drule.instantiate inst thmi)in SOME thm2endhandle _ => NONE(* FIXME/TODO: what is the place where the exception is raised: matching_prs? *)*}ML {*fun ball_bex_range_simproc ss redex =let val ctxt = Simplifier.the_context ssin case redex of (Const (@{const_name "Ball"}, _) $ (Const (@{const_name "Respects"}, _) $ (Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) => calculate_instance ctxt @{thm ball_reg_eqv_range} redex R1 R2 | (Const (@{const_name "Bex"}, _) $ (Const (@{const_name "Respects"}, _) $ (Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) => calculate_instance ctxt @{thm bex_reg_eqv_range} redex R1 R2 | _ => NONEend*}lemma eq_imp_rel: shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"by (simp add: equivp_reflp)(* Regularize Tactic *)(* 0. preliminary simplification step according to *)thm ball_reg_eqv bex_reg_eqv babs_reg_eqv (* the latter of no use *) ball_reg_eqv_range bex_reg_eqv_range(* 1. eliminating simple Ball/Bex instances*)thm ball_reg_right bex_reg_left(* 2. monos *)(* 3. commutation rules for ball and bex *)thm ball_all_comm bex_ex_comm(* 4. then rel-equality (which need to be instantiated to avoid loops *)thm eq_imp_rel(* 5. then simplification like 0 *)(* finally jump back to 1 *)ML {*fun quotient_tac ctxt = REPEAT_ALL_NEW (FIRST' [rtac @{thm identity_quotient}, resolve_tac (quotient_rules_get ctxt)])fun quotient_solver_tac ss = quotient_tac (Simplifier.the_context ss)val quotient_solver = Simplifier.mk_solver' "Quotient goal solver" quotient_solver_tac*}ML {*fun regularize_tac ctxt =let val thy = ProofContext.theory_of ctxt val pat_ball = @{term "Ball (Respects (R1 ===> R2)) P"} val pat_bex = @{term "Bex (Respects (R1 ===> R2)) P"} val simproc = Simplifier.simproc_i thy "" [pat_ball, pat_bex] (K (ball_bex_range_simproc)) val simpset = (mk_minimal_ss ctxt) addsimps @{thms ball_reg_eqv bex_reg_eqv babs_reg_eqv babs_simp} addsimprocs [simproc] addSolver equiv_solver addSolver quotient_solver (* TODO: Make sure that there are no list_rel, pair_rel etc involved *) (* can this cause loops in equiv_tac ? *) val eq_eqvs = map (OF1 @{thm eq_imp_rel}) (equiv_rules_get ctxt)in simp_tac simpset THEN' REPEAT_ALL_NEW (CHANGED o FIRST' [ resolve_tac @{thms ball_reg_right bex_reg_left}, resolve_tac (Inductive.get_monos ctxt), resolve_tac @{thms ball_all_comm bex_ex_comm}, resolve_tac eq_eqvs, simp_tac simpset])end*}section {* Calculation of the Injected Goal *}(*Injecting repabs means: For abstractions: * If the type of the abstraction doesn't need lifting we recurse. * If it does we add RepAbs around the whole term and check if the variable needs lifting. * If it doesn't then we recurse * If it does we recurse and put 'RepAbs' around all occurences of the variable in the obtained subterm. This in combination with the RepAbs above will let us change the type of the abstraction with rewriting. For applications: * If the term is 'Respects' applied to anything we leave it unchanged * If the term needs lifting and the head is a constant that we know how to lift, we put a RepAbs and recurse * If the term needs lifting and the head is a free applied to subterms (if it is not applied we treated it in Abs branch) then we put RepAbs and recurse * Otherwise just recurse.*)ML {*fun mk_repabs lthy (T, T') trm = Quotient_Def.get_fun repF lthy (T, T') $ (Quotient_Def.get_fun absF lthy (T, T') $ trm)*}ML {*(* bound variables need to be treated properly, *)(* as the type of subterms need to be calculated *)(* in the abstraction case *)fun inj_repabs_trm lthy (rtrm, qtrm) = case (rtrm, qtrm) of (Const (@{const_name "Ball"}, T) $ r $ t, Const (@{const_name "All"}, _) $ t') => Const (@{const_name "Ball"}, T) $ r $ (inj_repabs_trm lthy (t, t')) | (Const (@{const_name "Bex"}, T) $ r $ t, Const (@{const_name "Ex"}, _) $ t') => Const (@{const_name "Bex"}, T) $ r $ (inj_repabs_trm lthy (t, t')) | (Const (@{const_name "Babs"}, T) $ r $ t, t' as (Abs _)) => let val rty = fastype_of rtrm val qty = fastype_of qtrm in mk_repabs lthy (rty, qty) (Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t'))) end | (Abs (x, T, t), Abs (x', T', t')) => let val rty = fastype_of rtrm val qty = fastype_of qtrm val (y, s) = Term.dest_abs (x, T, t) val (_, s') = Term.dest_abs (x', T', t') val yvar = Free (y, T) val result = Term.lambda_name (y, yvar) (inj_repabs_trm lthy (s, s')) in if rty = qty then result else mk_repabs lthy (rty, qty) result end | (t $ s, t' $ s') => (inj_repabs_trm lthy (t, t')) $ (inj_repabs_trm lthy (s, s')) | (Free (_, T), Free (_, T')) => if T = T' then rtrm else mk_repabs lthy (T, T') rtrm | (_, Const (@{const_name "op ="}, _)) => rtrm (* FIXME: check here that rtrm is the corresponding definition for the const *) (* Hasn't it already been checked in regularize? *) | (_, Const (_, T')) => let val rty = fastype_of rtrm in if rty = T' then rtrm else mk_repabs lthy (rty, T') rtrm end | _ => raise (LIFT_MATCH "injection")*}section {* Injection Tactic *}ML {*fun solve_quotient_assums ctxt thm =let val goal = hd (Drule.strip_imp_prems (cprop_of thm)) in thm OF [Goal.prove_internal [] goal (fn _ => quotient_tac ctxt 1)]endhandle _ => error "solve_quotient_assums failed. Maybe a quotient_thm is missing"*}definition "QUOT_TRUE x \<equiv> True"ML {*fun find_qt_asm asms = let fun find_fun trm = case trm of (Const(@{const_name Trueprop}, _) $ (Const (@{const_name QUOT_TRUE}, _) $ _)) => true | _ => false in case find_first find_fun asms of SOME (_ $ (_ $ (f $ a))) => (f, a) | SOME _ => error "find_qt_asm: no pair" | NONE => error "find_qt_asm: no assumption" end*}(*To prove that the regularised theorem implies the abs/rep injected, we try: 1) theorems 'trans2' from the appropriate QUOT_TYPE 2) remove lambdas from both sides: lambda_rsp_tac 3) remove Ball/Bex from the right hand side 4) use user-supplied RSP theorems 5) remove rep_abs from the right side 6) reflexivity of equality 7) split applications of lifted type (apply_rsp) 8) split applications of non-lifted type (cong_tac) 9) apply extentionality A) reflexivity of the relation B) assumption (Lambdas under respects may have left us some assumptions) C) proving obvious higher order equalities by simplifying fun_rel (not sure if it is still needed?) D) unfolding lambda on one side E) simplifying (= ===> =) for simpler respectfulness*)lemma quot_true_dests: shows QT_all: "QUOT_TRUE (All P) \<Longrightarrow> QUOT_TRUE P" and QT_ex: "QUOT_TRUE (Ex P) \<Longrightarrow> QUOT_TRUE P" and QT_lam: "QUOT_TRUE (\<lambda>x. P x) \<Longrightarrow> (\<And>x. QUOT_TRUE (P x))" and QT_ext: "(\<And>x. QUOT_TRUE (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (QUOT_TRUE a \<Longrightarrow> f = g)"by (simp_all add: QUOT_TRUE_def ext)lemma QUOT_TRUE_imp: "QUOT_TRUE a \<equiv> QUOT_TRUE b"by (simp add: QUOT_TRUE_def)lemma regularize_to_injection: "(QUOT_TRUE l \<Longrightarrow> y) \<Longrightarrow> (l = r) \<longrightarrow> y" by(auto simp add: QUOT_TRUE_def)ML {*fun quot_true_conv1 ctxt fnctn ctrm = case (term_of ctrm) of (Const (@{const_name QUOT_TRUE}, _) $ x) => let val fx = fnctn x; val thy = ProofContext.theory_of ctxt; val cx = cterm_of thy x; val cfx = cterm_of thy fx; val cxt = ctyp_of thy (fastype_of x); val cfxt = ctyp_of thy (fastype_of fx); val thm = Drule.instantiate' [SOME cxt, SOME cfxt] [SOME cx, SOME cfx] @{thm QUOT_TRUE_imp} in Conv.rewr_conv thm ctrm end*}ML {*fun quot_true_conv ctxt fnctn ctrm = case (term_of ctrm) of (Const (@{const_name QUOT_TRUE}, _) $ _) => quot_true_conv1 ctxt fnctn ctrm | _ $ _ => Conv.comb_conv (quot_true_conv ctxt fnctn) ctrm | Abs _ => Conv.abs_conv (fn (_, ctxt) => quot_true_conv ctxt fnctn) ctxt ctrm | _ => Conv.all_conv ctrm*}ML {*fun quot_true_tac ctxt fnctn = CONVERSION ((Conv.params_conv ~1 (fn ctxt => (Conv.prems_conv ~1 (quot_true_conv ctxt fnctn)))) ctxt)*}ML {* fun dest_comb (f $ a) = (f, a) *}ML {* fun dest_bcomb ((_ $ l) $ r) = (l, r) *}(* TODO: Can this be done easier? *)ML {*fun unlam t = case t of (Abs a) => snd (Term.dest_abs a) | _ => unlam (Abs("", domain_type (fastype_of t), (incr_boundvars 1 t) $ (Bound 0)))*}ML {*fun dest_fun_type (Type("fun", [T, S])) = (T, S) | dest_fun_type _ = error "dest_fun_type"*}ML {*val bare_concl = HOLogic.dest_Trueprop o Logic.strip_assums_concl*}ML {*val apply_rsp_tac = Subgoal.FOCUS (fn {concl, asms, context,...} => case ((HOLogic.dest_Trueprop (term_of concl))) of ((R2 $ (f $ x) $ (g $ y))) => (let val (asmf, asma) = find_qt_asm (map term_of asms); in if (fastype_of asmf) = (fastype_of f) then no_tac else let val ty_a = fastype_of x; val ty_b = fastype_of asma; val ty_c = range_type (type_of f); val thy = ProofContext.theory_of context; val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c]; val thm = Drule.instantiate' ty_inst [] @{thm apply_rsp} val te = solve_quotient_assums context thm val t_inst = map (SOME o (cterm_of thy)) [R2, f, g, x, y]; val thm = Drule.instantiate' [] t_inst te in compose_tac (false, thm, 2) 1 end end handle ERROR "find_qt_asm: no pair" => no_tac) | _ => no_tac)*}ML {*fun equals_rsp_tac R ctxt = let val ty = domain_type (fastype_of R); val thy = ProofContext.theory_of ctxt val thm = Drule.instantiate' [SOME (ctyp_of thy ty)] [SOME (cterm_of thy R)] @{thm equals_rsp} in rtac thm THEN' quotient_tac ctxt end handle THM _ => K no_tac | TYPE _ => K no_tac | TERM _ => K no_tac*}ML {*fun rep_abs_rsp_tac ctxt = SUBGOAL (fn (goal, i) => case (bare_concl goal) of (rel $ _ $ (rep $ (abs $ _))) => (let val thy = ProofContext.theory_of ctxt; val (ty_a, ty_b) = dest_fun_type (fastype_of abs); val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b]; val t_inst = map (SOME o (cterm_of thy)) [rel, abs, rep]; val thm = Drule.instantiate' ty_inst t_inst @{thm rep_abs_rsp} val te = solve_quotient_assums ctxt thm in rtac te i end handle _ => no_tac) | _ => no_tac)*}ML {*fun inj_repabs_tac_match ctxt = SUBGOAL (fn (goal, i) =>(case (bare_concl goal) of (* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *) ((Const (@{const_name fun_rel}, _) $ _ $ _) $ (Abs _) $ (Abs _)) => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam (* (op =) (Ball\<dots>) (Ball\<dots>) ----> (op =) (\<dots>) (\<dots>) *)| (Const (@{const_name "op ="},_) $ (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $ (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _)) => rtac @{thm ball_rsp} THEN' dtac @{thm QT_all} (* (R1 ===> op =) (Ball\<dots>) (Ball\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Ball\<dots>x) = (Ball\<dots>y) *)| (Const (@{const_name fun_rel}, _) $ _ $ _) $ (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $ (Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam (* (op =) (Bex\<dots>) (Bex\<dots>) ----> (op =) (\<dots>) (\<dots>) *)| Const (@{const_name "op ="},_) $ (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $ (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) => rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex} (* (R1 ===> op =) (Bex\<dots>) (Bex\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Bex\<dots>x) = (Bex\<dots>y) *)| (Const (@{const_name fun_rel}, _) $ _ $ _) $ (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $ (Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) => rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam| (_ $ (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $ (Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _)) => rtac @{thm babs_rsp} THEN' RANGE [quotient_tac ctxt]| Const (@{const_name "op ="},_) $ (R $ _ $ _) $ (_ $ _ $ _) => (rtac @{thm refl} ORELSE' (equals_rsp_tac R ctxt THEN' RANGE [ quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)])) (* reflexivity of operators arising from Cong_tac *)| Const (@{const_name "op ="},_) $ _ $ _ => rtac @{thm refl} (* respectfulness of constants; in particular of a simple relation *)| _ $ (Const _) $ (Const _) (* fun_rel, list_rel, etc but not equality *) => resolve_tac (rsp_rules_get ctxt) THEN_ALL_NEW quotient_tac ctxt (* R (\<dots>) (Rep (Abs \<dots>)) ----> R (\<dots>) (\<dots>) *) (* observe ---> *)| _ $ _ $ _ => (rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt) ORELSE' rep_abs_rsp_tac ctxt| _ => error "inj_repabs_tac not a relation") i)*}ML {*fun inj_repabs_step_tac ctxt rel_refl = (FIRST' [ inj_repabs_tac_match ctxt, (* R (t $ \<dots>) (t' $ \<dots>) ----> apply_rsp provided type of t needs lifting *) apply_rsp_tac ctxt THEN' RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)], (* (op =) (t $ \<dots>) (t' $ \<dots>) ----> Cong provided type of t does not need lifting *) (* merge with previous tactic *) Cong_Tac.cong_tac @{thm cong} THEN' RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)], (* (op =) (\<lambda>x\<dots>) (\<lambda>x\<dots>) ----> (op =) (\<dots>) (\<dots>) *) rtac @{thm ext} THEN' quot_true_tac ctxt unlam, (* resolving with R x y assumptions *) atac, (* reflexivity of the basic relations *) (* R \<dots> \<dots> *) resolve_tac rel_refl])*}ML {*fun inj_repabs_tac ctxt =let val rel_refl = map (OF1 @{thm equivp_reflp}) (equiv_rules_get ctxt)in inj_repabs_step_tac ctxt rel_reflendfun all_inj_repabs_tac ctxt = REPEAT_ALL_NEW (inj_repabs_tac ctxt)*}section {* Cleaning of the Theorem *}ML {*fun fun_map_simple_conv xs ctxt ctrm = case (term_of ctrm) of ((Const (@{const_name "fun_map"}, _) $ _ $ _) $ h $ _) => if (member (op=) xs h) then Conv.all_conv ctrm else Conv.rewr_conv @{thm fun_map.simps[THEN eq_reflection]} ctrm | _ => Conv.all_conv ctrmfun fun_map_conv xs ctxt ctrm = case (term_of ctrm) of _ $ _ => (Conv.comb_conv (fun_map_conv xs ctxt) then_conv fun_map_simple_conv xs ctxt) ctrm | Abs _ => Conv.abs_conv (fn (x, ctxt) => fun_map_conv ((term_of x)::xs) ctxt) ctxt ctrm | _ => Conv.all_conv ctrmfun fun_map_tac ctxt = CONVERSION (fun_map_conv [] ctxt)*}(* Since the patterns for the lhs are different; there are 2 different make-insts *)(* 1: does ? \<rightarrow> id *)(* 2: does ? \<rightarrow> non-id *)ML {*fun make_inst lhs t = let val _ $ (Abs (_, _, (f as Var (_, Type ("fun", [T, _]))) $ u)) = lhs; val _ $ (Abs (_, _, g)) = t; fun mk_abs i t = if incr_boundvars i u aconv t then Bound i else (case t of t1 $ t2 => mk_abs i t1 $ mk_abs i t2 | Abs (s, T, t') => Abs (s, T, mk_abs (i + 1) t') | Bound j => if i = j then error "make_inst" else t | _ => t); in (f, Abs ("x", T, mk_abs 0 g)) end;*}ML {*fun make_inst2 lhs t = let val _ $ (Abs (_, _, (_ $ ((f as Var (_, Type ("fun", [T, _]))) $ u)))) = lhs; val _ $ (Abs (_, _, (_ $ g))) = t; fun mk_abs i t = if incr_boundvars i u aconv t then Bound i else (case t of t1 $ t2 => mk_abs i t1 $ mk_abs i t2 | Abs (s, T, t') => Abs (s, T, mk_abs (i + 1) t') | Bound j => if i = j then error "make_inst" else t | _ => t); in (f, Abs ("x", T, mk_abs 0 g)) end;*}ML {*fun lambda_prs_simple_conv ctxt ctrm = case (term_of ctrm) of ((Const (@{const_name fun_map}, _) $ r1 $ a2) $ (Abs _)) => (let val thy = ProofContext.theory_of ctxt val (ty_b, ty_a) = dest_fun_type (fastype_of r1) val (ty_c, ty_d) = dest_fun_type (fastype_of a2) val tyinst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c, ty_d] val tinst = [NONE, NONE, SOME (cterm_of thy r1), NONE, SOME (cterm_of thy a2)] val lpi = Drule.instantiate' tyinst tinst @{thm lambda_prs} val te = @{thm eq_reflection} OF [solve_quotient_assums ctxt (solve_quotient_assums ctxt lpi)] val ti = (let val ts = MetaSimplifier.rewrite_rule (id_simps_get ctxt) te val (insp, inst) = make_inst (term_of (Thm.lhs_of ts)) (term_of ctrm) in Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) ts end handle _ => (* TODO handle only Bind | Error "make_inst" *) let val ts = MetaSimplifier.rewrite_rule (id_simps_get ctxt) te val _ = tracing ("ts rule:\n" ^ (Syntax.string_of_term ctxt (prop_of ts))); val _ = tracing ("redex:\n" ^ (Syntax.string_of_term ctxt (term_of ctrm))); val (insp, inst) = make_inst2 (term_of (Thm.lhs_of ts)) (term_of ctrm) in Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) ts end handle _ => (* TODO handle only Bind | Error "make_inst" *) let val (insp, inst) = make_inst2 (term_of (Thm.lhs_of te)) (term_of ctrm) val td = Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) te in MetaSimplifier.rewrite_rule (id_simps_get ctxt) td end); val _ = if not (Term.is_Const a2 andalso fst (dest_Const a2) = @{const_name "id"}) then (tracing "lambda_prs"; tracing ("redex:\n" ^ (Syntax.string_of_term ctxt (term_of ctrm))); tracing ("lpi rule:\n" ^ (Syntax.string_of_term ctxt (prop_of lpi))); tracing ("te rule:\n" ^ (Syntax.string_of_term ctxt (prop_of te))); tracing ("ti rule:\n" ^ (Syntax.string_of_term ctxt (prop_of ti)))) else () in Conv.rewr_conv ti ctrm end handle _ => Conv.all_conv ctrm) | _ => Conv.all_conv ctrm*}ML {*val lambda_prs_conv = More_Conv.top_conv lambda_prs_simple_convfun lambda_prs_tac ctxt = CONVERSION (lambda_prs_conv ctxt)*}(* 1. folding of definitions and preservation lemmas; *)(* and simplification with *)thm babs_prs all_prs ex_prs (* 2. unfolding of ---> in front of everything, except *)(* bound variables *)thm fun_map.simps(* 3. simplification with *)thm lambda_prs(* 4. simplification with *)thm Quotient_abs_rep Quotient_rel_rep id_simps (* 5. Test for refl *)ML {*fun clean_tac lthy = let val thy = ProofContext.theory_of lthy; val defs = map (Thm.varifyT o symmetric o #def) (qconsts_dest thy) (* FIXME: why is the Thm.varifyT needed: example where it fails is LamEx *) val thms1 = defs @ (prs_rules_get lthy) @ @{thms babs_prs all_prs ex_prs} val thms2 = @{thms Quotient_abs_rep Quotient_rel_rep} @ (id_simps_get lthy) fun simps thms = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver in EVERY' [simp_tac (simps thms1), fun_map_tac lthy, lambda_prs_tac lthy, simp_tac (simps thms2), TRY o rtac refl] end*}section {* Tactic for Genralisation of Free Variables in a Goal *}ML {*fun inst_spec ctrm = Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}fun inst_spec_tac ctrms = EVERY' (map (dtac o inst_spec) ctrms)fun all_list xs trm = fold (fn (x, T) => fn t' => HOLogic.mk_all (x, T, t')) xs trmfun apply_under_Trueprop f = HOLogic.dest_Trueprop #> f #> HOLogic.mk_Truepropfun gen_frees_tac ctxt = SUBGOAL (fn (concl, i) => let val thy = ProofContext.theory_of ctxt val vrs = Term.add_frees concl [] val cvrs = map (cterm_of thy o Free) vrs val concl' = apply_under_Trueprop (all_list vrs) concl val goal = Logic.mk_implies (concl', concl) val rule = Goal.prove ctxt [] [] goal (K (EVERY1 [inst_spec_tac (rev cvrs), atac])) in rtac rule i end) *}section {* The General Shape of the Lifting Procedure *}(* - A is the original raw theorem *)(* - B is the regularized theorem *)(* - C is the rep/abs injected version of B *)(* - D is the lifted theorem *)(* *)(* - 1st prem is the regularization step *)(* - 2nd prem is the rep/abs injection step *)(* - 3rd prem is the cleaning part *)(* *)(* the QUOT_TRUE premise in 2 records the lifted theorem *)ML {* val lifting_procedure = @{lemma "\<lbrakk>A; A \<longrightarrow> B; QUOT_TRUE D \<Longrightarrow> B = C; C = D\<rbrakk> \<Longrightarrow> D" by (simp add: QUOT_TRUE_def)}*}ML {*fun lift_match_error ctxt fun_str rtrm qtrm =let val rtrm_str = Syntax.string_of_term ctxt rtrm val qtrm_str = Syntax.string_of_term ctxt qtrm val msg = cat_lines [enclose "[" "]" fun_str, "The quotient theorem", qtrm_str, "", "does not match with original theorem", rtrm_str]in error msgend*}ML {* fun procedure_inst ctxt rtrm qtrm =let val thy = ProofContext.theory_of ctxt val rtrm' = HOLogic.dest_Trueprop rtrm val qtrm' = HOLogic.dest_Trueprop qtrm val reg_goal = Syntax.check_term ctxt (regularize_trm ctxt rtrm' qtrm') handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm val inj_goal = Syntax.check_term ctxt (inj_repabs_trm ctxt (reg_goal, qtrm')) handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrmin Drule.instantiate' [] [SOME (cterm_of thy rtrm'), SOME (cterm_of thy reg_goal), NONE, SOME (cterm_of thy inj_goal)] lifting_procedureend*}ML {*(* the tactic leaves three subgoals to be proved *)fun procedure_tac ctxt rthm = ObjectLogic.full_atomize_tac THEN' gen_frees_tac ctxt THEN' CSUBGOAL (fn (goal, i) => let val rthm' = atomize_thm rthm val rule = procedure_inst ctxt (prop_of rthm') (term_of goal) in (rtac rule THEN' rtac rthm') i end)*}section {* Automatic Proofs *}ML {*fun SOLVES' tac = tac THEN_ALL_NEW (K no_tac)(* prints warning, if goal is unsolved *)fun WARN (tac, msg) i st = case Seq.pull ((SOLVES' tac) i st) of NONE => (warning msg; Seq.single st) | seqcell => Seq.make (fn () => seqcell)fun RANGE_WARN xs = RANGE (map WARN xs)*}ML {*localval msg1 = "Regularize proof failed."val msg2 = cat_lines ["Injection proof failed.", "This is probably due to missing respects lemmas.", "Try invoking the injection method manually to see", "which lemmas are missing."]val msg3 = "Cleaning proof failed."infun lift_tac ctxt rthm = procedure_tac ctxt rthm THEN' RANGE_WARN [(regularize_tac ctxt, msg1), (all_inj_repabs_tac ctxt, msg2), (clean_tac ctxt, msg3)]end*}section {* Methods / Interface *}ML {*fun mk_method1 tac thm ctxt = SIMPLE_METHOD (HEADGOAL (tac ctxt thm)) fun mk_method2 tac ctxt = SIMPLE_METHOD (HEADGOAL (tac ctxt)) *}method_setup lifting = {* Attrib.thm >> (mk_method1 lift_tac) *} {* Lifting of theorems to quotient types. *}method_setup lifting_setup = {* Attrib.thm >> (mk_method1 procedure_tac) *} {* Sets up the three goals for the lifting procedure. *}method_setup regularize = {* Scan.succeed (mk_method2 regularize_tac) *} {* Proves automatically the regularization goals from the lifting procedure. *}method_setup injection = {* Scan.succeed (mk_method2 all_inj_repabs_tac) *} {* Proves automatically the rep/abs injection goals from the lifting procedure. *}method_setup cleaning = {* Scan.succeed (mk_method2 clean_tac) *} {* Proves automatically the cleaning goals from the lifting procedure. *}end