theory QuotMainimports QuotScript QuotList 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 equiv: "EQUIV 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 equiv by (simp add: EQUIV_REFL_SYM_TRANS REFL_def) then have "R x (Eps (R x))" by (rule someI) then show "R (Eps (R x)) = R x" using equiv unfolding EQUIV_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: equiv[simplified EQUIV_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: equiv[simplified EQUIV_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: equiv[simplified EQUIV_def])donelemma R_trans: assumes ab: "R a b" and bc: "R b c" shows "R a c"proof - have tr: "TRANS R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp moreover have ab: "R a b" by fact moreover have bc: "R b c" by fact ultimately show "R a c" unfolding TRANS_def by blastqedlemma R_sym: assumes ab: "R a b" shows "R b a"proof - have re: "SYM R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp then show "R b a" using ab unfolding SYM_def by blastqedlemma R_trans2: assumes ac: "R a c" and bd: "R b d" shows "R a b = R c d"using ac bdby (blast intro: R_trans R_sym)lemma REPS_same: shows "R (REP a) (REP b) \<equiv> (a = b)"proof - have "R (REP a) (REP b) = (a = b)" proof assume as: "R (REP a) (REP b)" from rep_prop obtain x y where eqs: "Rep a = R x" "Rep b = R y" by blast from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp then have "R x (Eps (R y))" using lem9 by simp then have "R (Eps (R y)) x" using R_sym by blast then have "R y x" using lem9 by simp then have "R x y" using R_sym by blast then have "ABS x = ABS y" using thm11 by simp then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp then show "a = b" using rep_inverse by simp next assume ab: "a = b" have "REFL R" using equiv EQUIV_REFL_SYM_TRANS[of R] by simp then show "R (REP a) (REP b)" unfolding REFL_def using ab by auto qed then show "R (REP a) (REP b) \<equiv> (a = b)" by simpqedendlemma tst: "EQUIV bla"sorrylemma equiv_trans2: assumes e: "EQUIV R" and ac: "R a c" and bd: "R b d" shows "R a b = R c d"using ac bd eunfolding EQUIV_REFL_SYM_TRANS TRANS_def SYM_defby (blast)section {* type definition for the quotient type *}(* the auxiliary data for the quotient types *)use "quotient_info.ML"declare [[map list = (map, LIST_REL)]]declare [[map * = (prod_fun, prod_rel)]]declare [[map "fun" = (fun_map, FUN_REL)]]ML {* maps_lookup @{theory} "List.list" *}ML {* maps_lookup @{theory} "*" *}ML {* maps_lookup @{theory} "fun" *}(* definition of the quotient types *)(* FIXME: should be called quotient_typ.ML *)use "quotient.ML"(* lifting of constants *)use "quotient_def.ML"(* TODO: Consider defining it with an "if"; sth like: Babs p m = \<lambda>x. if x \<in> p then m x else undefined*)definition Babs :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"where "(x \<in> p) \<Longrightarrow> (Babs p m x = m x)"section {* ATOMIZE *}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 *}lemma prod_fun_id: "prod_fun id id \<equiv> id" by (rule eq_reflection) (simp add: prod_fun_def)lemma map_id: "map id \<equiv> id" apply (rule eq_reflection) apply (rule ext) apply (rule_tac list="x" in list.induct) apply (simp_all) donelemmas id_simps = FUN_MAP_I[THEN eq_reflection] id_apply[THEN eq_reflection] id_def[THEN eq_reflection,symmetric] prod_fun_id map_idML {*fun simp_ids thm = MetaSimplifier.rewrite_rule @{thms id_simps} thm*}section {* Debugging infrastructure for testing tactics *}ML {*fun my_print_tac ctxt s thm =let val prems_str = prems_of thm |> map (Syntax.string_of_term ctxt) |> cat_lines val _ = tracing (s ^ "\n" ^ prems_str)in Seq.single thmendfun DT ctxt s tac = EVERY' [tac, K (my_print_tac ctxt ("after " ^ s))]*}section {* Infrastructure about definitions *}(* Does the same as 'subst' in a given theorem *)ML {*fun eqsubst_thm ctxt thms thm = let val goalstate = Goal.init (Thm.cprop_of thm) val a' = case (SINGLE (EqSubst.eqsubst_tac ctxt [0] thms 1) goalstate) of NONE => error "eqsubst_thm" | SOME th => cprem_of th 1 val tac = (EqSubst.eqsubst_tac ctxt [0] thms 1) THEN simp_tac HOL_ss 1 val goal = Logic.mk_equals (term_of (Thm.cprop_of thm), term_of a'); val cgoal = cterm_of (ProofContext.theory_of ctxt) goal val rt = Goal.prove_internal [] cgoal (fn _ => tac); in @{thm equal_elim_rule1} OF [rt, thm] end*}(* expects atomized definitions *)ML {*fun add_lower_defs_aux lthy thm = let val e1 = @{thm fun_cong} OF [thm]; val f = eqsubst_thm lthy @{thms fun_map.simps} e1; val g = simp_ids f in (simp_ids thm) :: (add_lower_defs_aux lthy g) end handle _ => [simp_ids thm]*}ML {*fun add_lower_defs lthy def = let val def_pre_sym = symmetric def val def_atom = atomize_thm def_pre_sym val defs_all = add_lower_defs_aux lthy def_atom in map Thm.varifyT defs_all end*}section {* Infrastructure for collecting theorems for starting the lifting *}ML {*fun lookup_quot_data lthy qty = let val qty_name = fst (dest_Type qty) val SOME quotdata = quotdata_lookup lthy qty_name (* cu: Changed the lookup\<dots>not sure whether this works *) (* TODO: Should no longer be needed *) val rty = Logic.unvarifyT (#rtyp quotdata) val rel = #rel quotdata val rel_eqv = #equiv_thm quotdata val rel_refl = @{thm EQUIV_REFL} OF [rel_eqv] in (rty, rel, rel_refl, rel_eqv) end*}ML {*fun lookup_quot_thms lthy qty_name = let val thy = ProofContext.theory_of lthy; val trans2 = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".R_trans2") val reps_same = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".REPS_same") val absrep = PureThy.get_thm thy ("QUOT_TYPE_I_" ^ qty_name ^ ".thm10") val quot = PureThy.get_thm thy ("QUOTIENT_" ^ qty_name) in (trans2, reps_same, absrep, quot) end*}ML {*fun lookup_quot_consts defs = let fun dest_term (a $ b) = (a, b); val def_terms = map (snd o Logic.dest_equals o concl_of) defs; in map (fst o dest_Const o snd o dest_term) def_terms end*}section {* Regularization *} (*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 rtm *)(* - the result is still not completely typed *)(* - 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 = regularize_trm lthy t t' in if ty = ty' then Abs (x, ty, 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 (* FIXME: Should = only be replaced, when fully applied? *) (* Then there must be a 2nd argument *) | (Const (@{const_name "op ="}, ty) $ t, Const (@{const_name "op ="}, ty') $ t') => let val subtrm = regularize_trm lthy t t' in if ty = ty' then Const (@{const_name "op ="}, ty) $ subtrm else mk_resp_arg lthy (domain_type ty, domain_type ty') $ subtrm end | (t1 $ t2, t1' $ t2') => (regularize_trm lthy t1 t1') $ (regularize_trm lthy t2 t2') | (Free (x, ty), Free (x', ty')) => if x = x' then rtrm (* FIXME: check whether types corresponds *) else raise (LIFT_MATCH "regularize (frees)") | (Bound i, Bound i') => if i = i' then rtrm else raise (LIFT_MATCH "regularize (bounds)") | (Const (s, ty), Const (s', ty')) => if s = s' andalso ty = ty' then rtrm else rtrm (* FIXME: check correspondence according to definitions *) | (rt, qt) => raise (LIFT_MATCH "regularize (default)")*}(*FIXME / TODO: needs to be adaptedTo prove that the raw theorem implies the regularised one, we try in order: - Reflexivity of the relation - Assumption - Elimnating quantifiers on both sides of toplevel implication - Simplifying implications on both sides of toplevel implication - Ball (Respects ?E) ?P = All ?P - (\<And>x. ?R x \<Longrightarrow> ?P x \<longrightarrow> ?Q x) \<Longrightarrow> All ?P \<longrightarrow> Ball ?R ?Q*)lemma universal_twice: assumes *: "\<And>x. (P x \<longrightarrow> Q x)" shows "(\<forall>x. P x) \<longrightarrow> (\<forall>x. Q x)"using * by autolemma implication_twice: assumes a: "c \<longrightarrow> a" assumes b: "b \<longrightarrow> d" shows "(a \<longrightarrow> b) \<longrightarrow> (c \<longrightarrow> d)"using a b by auto(* version of regularize_tac including debugging information *)ML {*fun regularize_tac ctxt rel_eqv rel_refl = (ObjectLogic.full_atomize_tac) THEN' REPEAT_ALL_NEW (FIRST' [(K (print_tac "start")) THEN' (K no_tac), DT ctxt "1" (FIRST' (map rtac rel_refl)), DT ctxt "2" atac, DT ctxt "3" (rtac @{thm universal_twice}), DT ctxt "4" (rtac @{thm impI} THEN' atac), DT ctxt "5" (rtac @{thm implication_twice}), DT ctxt "6" (EqSubst.eqsubst_tac ctxt [0] [(@{thm equiv_res_forall} OF [rel_eqv]), (@{thm equiv_res_exists} OF [rel_eqv])]), (* For a = b \<longrightarrow> a \<approx> b *) DT ctxt "7" (rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' (FIRST' (map rtac rel_refl))), DT ctxt "8" (rtac @{thm RIGHT_RES_FORALL_REGULAR}) ]);*}lemma move_forall: "(\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)) \<Longrightarrow> ((\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y))"by autolemma move_exists: "((\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)) \<Longrightarrow> ((\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y))"by autolemma [mono]: "(\<And>x. P x \<longrightarrow> Q x) \<Longrightarrow> (Ex P) \<longrightarrow> (Ex Q)"by blastlemma [mono]: "P \<longrightarrow> Q \<Longrightarrow> \<not>Q \<longrightarrow> \<not>P"by autolemma ball_respects_refl: fixes P Q::"'a \<Rightarrow> bool" and x::"'a" assumes a: "EQUIV R2" and b: "\<And>f. Q (f x) \<longrightarrow> P (f x)" shows "(Ball (Respects (R1 ===> R2)) (\<lambda>f. Q (f x)) \<longrightarrow> All (\<lambda>f. P (f x)))"apply(rule impI)apply(rule allI)apply(drule_tac x="\<lambda>y. f x" in bspec)apply(simp add: Respects_def IN_RESPECTS)apply(rule impI)using a EQUIV_REFL_SYM_TRANS[of "R2"]apply(simp add: REFL_def)using bapply(simp)donelemma bex_respects_refl: fixes P Q::"'a \<Rightarrow> bool" and x::"'a" assumes a: "EQUIV R2" and b: "\<And>f. P (f x) \<longrightarrow> Q (f x)" shows "(Ex (\<lambda>f. P (f x))) \<longrightarrow> (Bex (Respects (R1 ===> R2)) (\<lambda>f. Q (f x)))"apply(rule impI)apply(erule exE)thm bexIapply(rule_tac x="\<lambda>y. f x" in bexI)using bapply(simp)apply(simp add: Respects_def IN_RESPECTS)apply(rule impI)using a EQUIV_REFL_SYM_TRANS[of "R2"]apply(simp add: REFL_def)done(* FIXME: OPTION_EQUIV, PAIR_EQUIV, ... *)ML {*fun equiv_tac rel_eqvs = REPEAT_ALL_NEW(FIRST' [ FIRST' (map rtac rel_eqvs), rtac @{thm IDENTITY_EQUIV}, rtac @{thm LIST_EQUIV} ])*}ML {*fun SOLVES' tac = tac THEN_ALL_NEW (fn _ => no_tac)*}ML {*fun regularize_tac ctxt rel_eqvs rel_refl = let val subs1 = map (fn x => @{thm equiv_res_forall} OF [x]) rel_eqvs val subs2 = map (fn x => @{thm equiv_res_exists} OF [x]) rel_eqvs val subs = map (fn x => @{thm eq_reflection} OF [x]) (subs1 @ subs2) in (ObjectLogic.full_atomize_tac) THEN' (simp_tac ((Simplifier.context ctxt empty_ss) addsimps subs)) THEN' REPEAT_ALL_NEW (FIRST' [ (rtac @{thm RIGHT_RES_FORALL_REGULAR}), (rtac @{thm LEFT_RES_EXISTS_REGULAR}), (resolve_tac (Inductive.get_monos ctxt)), (rtac @{thm ball_respects_refl} THEN' (RANGE [SOLVES' (equiv_tac rel_eqvs)])), (rtac @{thm bex_respects_refl} THEN' (RANGE [SOLVES' (equiv_tac rel_eqvs)])), rtac @{thm move_forall}, rtac @{thm move_exists}, (rtac @{thm impI} THEN' (asm_full_simp_tac HOL_ss) THEN' FIRST' (map rtac rel_refl)) ]) end*}section {* Injections of REP and ABSes *}(*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 *)fun inj_repabs_trm lthy (rtrm, qtrm) =let val rty = fastype_of rtrm val qty = fastype_of qtrmin 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') => Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t')) | (Abs (x, T, t), Abs (x', T', t')) => let val (y, s) = Term.dest_abs (x, T, t) val (_, s') = Term.dest_abs (x', T', t') val yvar = Free (y, T) val result = lambda yvar (inj_repabs_trm lthy (s, s')) in if rty = qty then result else mk_repabs lthy (rty, qty) result end | _ => (* FIXME / TODO: this is a case that needs to be looked at *) (* - variables get a rep-abs insde and outside an application *) (* - constants only get a rep-abs on the outside of the application *) (* - applications get a rep-abs insde and outside an application *) let val (rhead, rargs) = strip_comb rtrm val (qhead, qargs) = strip_comb qtrm val rargs' = map (inj_repabs_trm lthy) (rargs ~~ qargs) in if rty = qty then case (rhead, qhead) of (Free (_, T), Free (_, T')) => if T = T' then list_comb (rhead, rargs') else list_comb (mk_repabs lthy (T, T') rhead, rargs') | _ => list_comb (rhead, rargs') else case (rhead, qhead, length rargs') of (Const _, Const _, 0) => mk_repabs lthy (rty, qty) rhead | (Free (_, T), Free (_, T'), 0) => mk_repabs lthy (T, T') rhead | (Const _, Const _, _) => mk_repabs lthy (rty, qty) (list_comb (rhead, rargs')) | (Free (x, T), Free (x', T'), _) => mk_repabs lthy (rty, qty) (list_comb (mk_repabs lthy (T, T') rhead, rargs')) | (Abs _, Abs _, _ ) => mk_repabs lthy (rty, qty) (list_comb (inj_repabs_trm lthy (rhead, qhead), rargs')) | _ => raise (LIFT_MATCH "injection") endend*}section {* RepAbs Injection Tactic *}(*To prove that the regularised theorem implies the abs/rep injected, we firstatomize it and then try: 1) theorems 'trans2' from the appropriate QUOT_TYPE 2) remove lambdas from both sides (LAMBDA_RES_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 extentionality10) reflexivity of the relation11) assumption (Lambdas under respects may have left us some assumptions)12) proving obvious higher order equalities by simplifying fun_rel (not sure if it is still needed?)13) unfolding lambda on one side14) simplifying (= ===> =) for simpler respectfullness*)ML {*fun instantiate_tac thm = Subgoal.FOCUS (fn {concl, ...} => let val pat = Drule.strip_imp_concl (cprop_of thm) val insts = Thm.match (pat, concl) in rtac (Drule.instantiate insts thm) 1 end handle _ => no_tac)*}ML {*fun quotient_tac quot_thms = REPEAT_ALL_NEW (FIRST' [ rtac @{thm FUN_QUOTIENT}, FIRST' (map rtac quot_thms), rtac @{thm IDENTITY_QUOTIENT}, (* For functional identity quotients, (op = ---> op =) *) (* TODO: think about the other way around, if we need to shorten the relation *) CHANGED o (simp_tac (HOL_ss addsimps @{thms id_simps})) ])*}ML {*fun LAMBDA_RES_TAC ctxt i st = (case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of (_ $ (_ $ (Abs(_, _, _)) $ (Abs(_, _, _)))) => (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI}) | _ => fn _ => no_tac) i st*}ML {*fun WEAK_LAMBDA_RES_TAC ctxt i st = (case (term_of o #concl o fst) (Subgoal.focus ctxt i st) of (_ $ (_ $ _ $ (Abs(_, _, _)))) => (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI}) | (_ $ (_ $ (Abs(_, _, _)) $ _)) => (EqSubst.eqsubst_tac ctxt [0] @{thms FUN_REL.simps}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm allI}) THEN' (rtac @{thm impI}) | _ => fn _ => no_tac) i st*}ML {* (* Legacy *)fun needs_lift (rty as Type (rty_s, _)) ty = case ty of Type (s, tys) => (s = rty_s) orelse (exists (needs_lift rty) tys) | _ => false*}ML {*fun APPLY_RSP_TAC rty = Subgoal.FOCUS (fn {concl, ...} => let val (_ $ (R $ (f $ _) $ (_ $ _))) = term_of concl; val pat = Drule.strip_imp_concl (cprop_of @{thm APPLY_RSP}); val insts = Thm.match (pat, concl) in if needs_lift rty (type_of f) then rtac (Drule.instantiate insts @{thm APPLY_RSP}) 1 else no_tac end handle _ => no_tac)*}ML {*val ball_rsp_tac = Subgoal.FOCUS (fn {concl, context = ctxt, ...} => let val _ $ (_ $ (Const (@{const_name Ball}, _) $ _) $ (Const (@{const_name Ball}, _) $ _)) = term_of concl in ((simp_tac (HOL_ss addsimps @{thms FUN_REL.simps})) THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI} THEN' instantiate_tac @{thm RES_FORALL_RSP} ctxt THEN' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))) 1 end handle _ => no_tac)*}ML {*val bex_rsp_tac = Subgoal.FOCUS (fn {concl, context = ctxt, ...} => let val _ $ (_ $ (Const (@{const_name Bex}, _) $ _) $ (Const (@{const_name Bex}, _) $ _)) = term_of concl in ((simp_tac (HOL_ss addsimps @{thms FUN_REL.simps})) THEN' rtac @{thm allI} THEN' rtac @{thm allI} THEN' rtac @{thm impI} THEN' instantiate_tac @{thm RES_EXISTS_RSP} ctxt THEN' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))) 1 end handle _ => no_tac)*}ML {*fun r_mk_comb_tac ctxt rty quot_thms rel_refl trans2 rsp_thms = (FIRST' [ FIRST' (map rtac trans2), LAMBDA_RES_TAC ctxt, rtac @{thm RES_FORALL_RSP}, ball_rsp_tac ctxt, rtac @{thm RES_EXISTS_RSP}, bex_rsp_tac ctxt, FIRST' (map rtac rsp_thms), rtac refl, (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [SOLVES' (quotient_tac quot_thms)])), (APPLY_RSP_TAC rty ctxt THEN' (RANGE [SOLVES' (quotient_tac quot_thms), SOLVES' (quotient_tac quot_thms)])), Cong_Tac.cong_tac @{thm cong}, rtac @{thm ext}, FIRST' (map rtac rel_refl), atac, SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps})), WEAK_LAMBDA_RES_TAC ctxt, CHANGED o (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ})) ])*}(*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_RES_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 extentionality10) reflexivity of the relation11) assumption (Lambdas under respects may have left us some assumptions)12) proving obvious higher order equalities by simplifying fun_rel (not sure if it is still needed?)13) unfolding lambda on one side14) simplifying (= ===> =) for simpler respectfulness*)ML {*fun r_mk_comb_tac' ctxt rty quot_thms reflex_thm trans_thm rsp_thms = REPEAT_ALL_NEW (FIRST' [ (K (print_tac "start")) THEN' (K no_tac), DT ctxt "1" (rtac trans_thm), DT ctxt "2" (LAMBDA_RES_TAC ctxt), DT ctxt "3" (rtac @{thm RES_FORALL_RSP}), DT ctxt "4" (ball_rsp_tac ctxt), DT ctxt "5" (rtac @{thm RES_EXISTS_RSP}), DT ctxt "6" (bex_rsp_tac ctxt), DT ctxt "7" (FIRST' (map rtac rsp_thms)), DT ctxt "8" (rtac refl), DT ctxt "9" ((instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [SOLVES' (quotient_tac quot_thms)]))), DT ctxt "A" ((APPLY_RSP_TAC rty ctxt THEN' (RANGE [SOLVES' (quotient_tac quot_thms), SOLVES' (quotient_tac quot_thms)]))), DT ctxt "B" (Cong_Tac.cong_tac @{thm cong}), DT ctxt "C" (rtac @{thm ext}), DT ctxt "D" (rtac reflex_thm), DT ctxt "E" (atac), DT ctxt "F" (SOLVES' (simp_tac (HOL_ss addsimps @{thms FUN_REL.simps}))), DT ctxt "G" (WEAK_LAMBDA_RES_TAC ctxt), DT ctxt "H" (CHANGED o (asm_full_simp_tac (HOL_ss addsimps @{thms FUN_REL_EQ}))) ])*}section {* Cleaning of the theorem *}ML {*fun applic_prs lthy absrep (rty, qty) = let fun mk_rep (T, T') tm = (Quotient_Def.get_fun repF lthy (T, T')) $ tm; fun mk_abs (T, T') tm = (Quotient_Def.get_fun absF lthy (T, T')) $ tm; val (raty, rgty) = Term.strip_type rty; val (qaty, qgty) = Term.strip_type qty; val vs = map (fn _ => "x") qaty; val ((fname :: vfs), lthy') = Variable.variant_fixes ("f" :: vs) lthy; val f = Free (fname, qaty ---> qgty); val args = map Free (vfs ~~ qaty); val rhs = list_comb(f, args); val largs = map2 mk_rep (raty ~~ qaty) args; val lhs = mk_abs (rgty, qgty) (list_comb((mk_rep (raty ---> rgty, qaty ---> qgty) f), largs)); val llhs = Syntax.check_term lthy lhs; val eq = Logic.mk_equals (llhs, rhs); val ceq = cterm_of (ProofContext.theory_of lthy') eq; val sctxt = HOL_ss addsimps (@{thms fun_map.simps id_simps} @ absrep); val t = Goal.prove_internal [] ceq (fn _ => simp_tac sctxt 1) val t_id = MetaSimplifier.rewrite_rule @{thms id_simps} t; in singleton (ProofContext.export lthy' lthy) t_id end*}ML {*fun find_aps_all rtm qtm = case (rtm, qtm) of (Abs(_, T1, s1), Abs(_, T2, s2)) => find_aps_all (subst_bound ((Free ("x", T1)), s1)) (subst_bound ((Free ("x", T2)), s2)) | (((f1 as (Free (_, T1))) $ a1), ((f2 as (Free (_, T2))) $ a2)) => let val sub = (find_aps_all f1 f2) @ (find_aps_all a1 a2) in if T1 = T2 then sub else (T1, T2) :: sub end | ((f1 $ a1), (f2 $ a2)) => (find_aps_all f1 f2) @ (find_aps_all a1 a2) | _ => [];fun find_aps rtm qtm = distinct (op =) (find_aps_all rtm qtm)*}ML {*fun allex_prs_tac lthy quot = (EqSubst.eqsubst_tac lthy [0] @{thms FORALL_PRS[symmetric] EXISTS_PRS[symmetric]}) THEN' (quotient_tac quot);*}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 lambda_prs_conv1 ctxt quot_thms ctrm = case (term_of ctrm) of ((Const (@{const_name "fun_map"}, _) $ r1 $ a2) $ (Abs _)) => let val (_, [ty_b, ty_a]) = dest_Type (fastype_of r1); val (_, [ty_c, ty_d]) = dest_Type (fastype_of a2); val thy = ProofContext.theory_of ctxt; val [cty_a, cty_b, cty_c, cty_d] = map (ctyp_of thy) [ty_a, ty_b, ty_c, ty_d] val tyinst = [SOME cty_a, SOME cty_b, SOME cty_c, SOME cty_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 tac = (compose_tac (false, lpi, 2)) THEN_ALL_NEW (quotient_tac quot_thms); val gc = Drule.strip_imp_concl (cprop_of lpi); val t = Goal.prove_internal [] gc (fn _ => tac 1) val te = @{thm eq_reflection} OF [t] val ts = MetaSimplifier.rewrite_rule @{thms id_simps} te val tl = Thm.lhs_of ts(* val _ = tracing (Syntax.string_of_term @{context} (term_of ctrm));*)(* val _ = tracing (Syntax.string_of_term @{context} (term_of tl));*) val insts = matching_prs (ProofContext.theory_of ctxt) (term_of tl) (term_of ctrm); val ti = Drule.eta_contraction_rule (Drule.instantiate insts ts);(* val _ = tracing (Syntax.string_of_term @{context} (term_of (cprop_of ti)));*) in Conv.rewr_conv ti ctrm end(* TODO: We can add a proper error message... *) handle Bind => Conv.all_conv ctrm*}(* quot stands for the QUOTIENT theorems: *) (* could be potentially all of them *)ML {*fun lambda_prs_conv ctxt quot ctrm = case (term_of ctrm) of (Const (@{const_name "fun_map"}, _) $ _ $ _) $ (Abs _) => (Conv.arg_conv (Conv.abs_conv (fn (_, ctxt) => lambda_prs_conv ctxt quot) ctxt) then_conv (lambda_prs_conv1 ctxt quot)) ctrm | _ $ _ => Conv.comb_conv (lambda_prs_conv ctxt quot) ctrm | Abs _ => Conv.abs_conv (fn (_, ctxt) => lambda_prs_conv ctxt quot) ctxt ctrm | _ => Conv.all_conv ctrm*}ML {*fun lambda_prs_tac ctxt quot = CSUBGOAL (fn (goal, i) => CONVERSION (Conv.params_conv ~1 (fn ctxt => (Conv.prems_conv ~1 (lambda_prs_conv ctxt quot) then_conv Conv.concl_conv ~1 (lambda_prs_conv ctxt quot))) ctxt) i)*}ML {*fun clean_tac lthy quot defs aps = let val lower = flat (map (add_lower_defs lthy) defs) val absrep = map (fn x => @{thm QUOTIENT_ABS_REP} OF [x]) quot val reps_same = map (fn x => @{thm QUOTIENT_REL_REP} OF [x]) quot val aps_thms = map (applic_prs lthy absrep) aps in EVERY' [TRY o REPEAT_ALL_NEW (allex_prs_tac lthy quot), lambda_prs_tac lthy quot, TRY o REPEAT_ALL_NEW (EqSubst.eqsubst_tac lthy [0] aps_thms), TRY o REPEAT_ALL_NEW (EqSubst.eqsubst_tac lthy [0] lower), simp_tac (HOL_ss addsimps reps_same)] end*}section {* 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 {* General outline 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 *)(* *)(* - b is the regularization step *)(* - c is the rep/abs injection step *)(* - d is the cleaning part *)lemma lifting_procedure: assumes a: "A" and b: "A \<Longrightarrow> B" and c: "B = C" and d: "C = D" shows "D" using a b c d by simpML {*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 = [enclose "[" "]" fun_str, "The quotient theorem\n", qtrm_str, "and the lifted theorem\n", rtrm_str, "do not match"]in error (space_implode " " msg)end*}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), SOME (cterm_of thy inj_goal)] @{thm lifting_procedure}end*}(* Left for debugging *)ML {*fun procedure_tac lthy rthm = ObjectLogic.full_atomize_tac THEN' gen_frees_tac lthy THEN' Subgoal.FOCUS (fn {context, concl, ...} => let val rthm' = atomize_thm rthm val rule = procedure_inst context (prop_of rthm') (term_of concl) in EVERY1 [rtac rule, rtac rthm'] end) lthy*}ML {*(* FIXME/TODO should only get as arguments the rthm like procedure_tac *)fun lift_tac lthy rthm rel_eqv rty quot rsp_thms defs = ObjectLogic.full_atomize_tac THEN' gen_frees_tac lthy THEN' Subgoal.FOCUS (fn {context, concl, ...} => let val rthm' = atomize_thm rthm val rule = procedure_inst context (prop_of rthm') (term_of concl) val aps = find_aps (prop_of rthm') (term_of concl) val rel_refl = map (fn x => @{thm EQUIV_REFL} OF [x]) rel_eqv val trans2 = map (fn x => @{thm equiv_trans2} OF [x]) rel_eqv in EVERY1 [rtac rule, RANGE [rtac rthm', regularize_tac lthy rel_eqv rel_refl, REPEAT_ALL_NEW (r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms), clean_tac lthy quot defs aps]] end) lthy*}end