--- a/QuotMain.thy Mon Dec 07 14:00:36 2009 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1191 +0,0 @@
-theory QuotMain
-imports QuotScript QuotList QuotProd Prove
-uses ("quotient_info.ML")
- ("quotient.ML")
- ("quotient_def.ML")
-begin
-
-
-locale 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)"
-begin
-
-definition
- 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 simp
-qed
-
-theorem thm10:
- shows "ABS (REP a) \<equiv> a"
- apply (rule eq_reflection)
- unfolding ABS_def REP_def
-proof -
- 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 simp
-qed
-
-lemma REP_refl:
- shows "R (REP a) (REP a)"
-unfolding REP_def
-by (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)
-done
-
-theorem thm11:
- shows "R r r' = (ABS r = ABS r')"
-unfolding ABS_def
-by (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])
-done
-
-lemma R_trans:
- assumes ab: "R a b"
- and bc: "R b c"
- shows "R a c"
-proof -
- have tr: "transp R" using equivp equivp_reflp_symp_transp[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 transp_def by blast
-qed
-
-lemma R_sym:
- assumes ab: "R a b"
- shows "R b a"
-proof -
- have re: "symp R" using equivp equivp_reflp_symp_transp[of R] by simp
- then show "R b a" using ab unfolding symp_def by blast
-qed
-
-lemma R_trans2:
- assumes ac: "R a c"
- and bd: "R b d"
- shows "R a b = R c d"
-using ac bd
-by (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 "reflp R" using equivp equivp_reflp_symp_transp[of R] by simp
- then show "R (REP a) (REP b)" unfolding reflp_def using ab by auto
- qed
- then show "R (REP a) (REP b) \<equiv> (a = b)" by simp
-qed
-
-end
-
-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)]]
-
-(* identity quotient is not here as it has to be applied first *)
-lemmas [quotient_thm] =
- fun_quotient list_quotient prod_quotient
-
-lemmas [quotient_rsp] =
- quot_rel_rsp nil_rsp cons_rsp foldl_rsp pair_rsp
-
-(* fun_map is not here since equivp is not true *)
-(* TODO: option, ... *)
-lemmas [quotient_equiv] =
- identity_equivp list_equivp prod_equivp
-
-
-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"
-
-section {* Simset setup *}
-
-(* since HOL_basic_ss is too "big", we need to set up *)
-(* our own minimal simpset *)
-ML {*
-fun mk_minimal_ss ctxt =
- Simplifier.context ctxt empty_ss
- setsubgoaler asm_simp_tac
- setmksimps (mksimps [])
-*}
-
-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 unfold
-next
- 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)
-qed
-
-ML {*
-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)
- done
-
-lemmas id_simps =
- fun_map_id[THEN eq_reflection]
- id_apply[THEN eq_reflection]
- id_def[THEN eq_reflection,symmetric]
- prod_fun_id map_id
-
-ML {*
-fun simp_ids thm =
- MetaSimplifier.rewrite_rule @{thms id_simps} thm
-*}
-
-section {* Debugging infrastructure for testing tactics *}
-
-ML {*
-fun my_print_tac ctxt s i thm =
-let
- val prem_str = nth (prems_of thm) (i - 1)
- |> Syntax.string_of_term ctxt
- handle Subscript => "no subgoal"
- val _ = tracing (s ^ "\n" ^ prem_str)
-in
- Seq.single thm
-end *}
-
-ML {*
-fun DT ctxt s tac i thm =
-let
- val before_goal = nth (prems_of thm) (i - 1)
- |> Syntax.string_of_term ctxt
- val before_msg = ["before: " ^ s, before_goal, "after: " ^ s]
- |> cat_lines
-in
- EVERY [tac i, my_print_tac ctxt before_msg i] thm
-end
-
-fun NDT ctxt s tac thm = tac thm
-*}
-
-section {* Matching of terms and types *}
-
-ML {*
-fun matches_typ (ty, ty') =
- case (ty, ty') of
- (_, TVar _) => true
- | (TFree x, TFree x') => x = x'
- | (Type (s, tys), Type (s', tys')) =>
- s = s' andalso
- if (length tys = length tys')
- then (List.all matches_typ (tys ~~ tys'))
- else false
- | _ => false
-*}
-
-ML {*
-fun matches_term (trm, trm') =
- case (trm, trm') of
- (_, Var _) => true
- | (Const (s, ty), Const (s', ty')) => s = s' andalso matches_typ (ty, ty')
- | (Free (x, ty), Free (x', ty')) => x = x' andalso matches_typ (ty, ty')
- | (Bound i, Bound j) => i = j
- | (Abs (_, T, t), Abs (_, T', t')) => matches_typ (T, T') andalso matches_term (t, t')
- | (t $ s, t' $ s') => matches_term (t, t') andalso matches_term (s, s')
- | _ => false
-*}
-
-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
- (* 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 equivp_reflp} 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
-*}
-
-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 lthy
-in
- 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 = 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, _)) =>
- let
- fun same_name (Const (s, _)) (Const (s', _)) = (s = s')
- | same_name _ _ = false
- in
- if same_name rtrm qtrm
- then rtrm
- else
- let
- fun exc1 s = LIFT_MATCH ("regularize (constant " ^ s ^ " not found)")
- val exc2 = LIFT_MATCH ("regularize (constant mismatch)")
- val thy = ProofContext.theory_of lthy
- val rtrm' = (#rconst (qconsts_lookup thy s)) handle NotFound => raise (exc1 s)
- in
- if matches_term (rtrm, rtrm')
- then rtrm
- else raise exc2
- 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) =>
- raise (LIFT_MATCH "regularize (default)")
-*}
-
-ML {*
-fun equiv_tac ctxt =
- REPEAT_ALL_NEW (FIRST'
- [resolve_tac (equiv_rules_get ctxt)])
-*}
-
-ML {*
-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 thm2
-end
-handle _ => NONE
-(* FIXME/TODO: what is the place where the exception can be raised: matching_prs? *)
-*}
-
-ML {*
-fun ball_bex_range_simproc ss redex =
-let
- val ctxt = Simplifier.the_context ss
-in
- 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
- | _ => NONE
-end
-*}
-
-lemma eq_imp_rel:
- shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"
-by (simp add: equivp_reflp)
-
-(* FIXME/TODO: How does regularizing work? *)
-(* FIXME/TODO: needs to be adapted
-
-To 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
-
-*)
-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}
- addsimprocs [simproc] addSolver equiv_solver
- (* TODO: Make sure that there are no list_rel, pair_rel etc involved *)
- val eq_eqvs = map (fn x => @{thm eq_imp_rel} OF [x]) (equiv_rules_get ctxt)
-in
- ObjectLogic.full_atomize_tac THEN'
- simp_tac simpset THEN'
- REPEAT_ALL_NEW (FIRST' [
- rtac @{thm ball_reg_right},
- rtac @{thm bex_reg_left},
- resolve_tac (Inductive.get_monos ctxt),
- rtac @{thm ball_all_comm},
- rtac @{thm bex_ex_comm},
- resolve_tac eq_eqvs,
- simp_tac simpset])
-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 *)
-(* 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 _)) =>
- Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t'))
-
- | (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 *)
- | (_, 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 {* RepAbs Injection Tactic *}
-
-ML {*
-fun quotient_tac ctxt =
- REPEAT_ALL_NEW (FIRST'
- [rtac @{thm identity_quotient},
- resolve_tac (quotient_rules_get ctxt)])
-*}
-
-(* solver for the simplifier *)
-ML {*
-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 solve_quotient_assums ctxt thm =
- let val gl = hd (Drule.strip_imp_prems (cprop_of thm)) in
- thm OF [Goal.prove_internal [] gl (fn _ => quotient_tac ctxt 1)]
- end
- handle _ => error "solve_quotient_assums failed. Maybe a quotient_thm is missing"
-*}
-
-(* Not used *)
-(* It proves the Quotient assumptions by calling quotient_tac *)
-ML {*
-fun solve_quotient_assum i ctxt thm =
- let
- val tac =
- (compose_tac (false, thm, i)) THEN_ALL_NEW
- (quotient_tac ctxt);
- val gc = Drule.strip_imp_concl (cprop_of thm);
- in
- Goal.prove_internal [] gc (fn _ => tac 1)
- end
- handle _ => error "solve_quotient_assum"
-*}
-
-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)"
-apply(simp_all add: QUOT_TRUE_def ext)
-done
-
-lemma QUOT_TRUE_i: "(QUOT_TRUE (a :: bool) \<Longrightarrow> P) \<Longrightarrow> P"
-by (simp add: QUOT_TRUE_def)
-
-lemma QUOT_TRUE_imp: "QUOT_TRUE a \<equiv> QUOT_TRUE b"
-by (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 SOLVES' tac = tac THEN_ALL_NEW (fn _ => 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 trans2 = 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]
-
- (* reflexivity of operators arising from Cong_tac *)
-| Const (@{const_name "op ="},_) $ _ $ _
- => rtac @{thm refl} ORELSE'
- (resolve_tac trans2 THEN' RANGE [
- quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)])
-
-(* TODO: These patterns should should be somehow combined and generalized... *)
-| (Const (@{const_name fun_rel}, _) $ _ $ _) $
- (Const (@{const_name fun_rel}, _) $ _ $ _) $
- (Const (@{const_name fun_rel}, _) $ _ $ _)
- => rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt
-
-| (Const (@{const_name fun_rel}, _) $ _ $ _) $
- (Const (@{const_name prod_rel}, _) $ _ $ _) $
- (Const (@{const_name prod_rel}, _) $ _ $ _)
- => rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt
-
- (* 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 ---> *)
-| _ $ _ $ _
- => rep_abs_rsp_tac ctxt
-
-| _ => error "inj_repabs_tac not a relation"
-) i)
-*}
-
-ML {*
-fun inj_repabs_tac ctxt rel_refl trans2 =
- (FIRST' [
- inj_repabs_tac_match ctxt trans2,
- (* R (t $ \<dots>) (t' $ \<dots>) ----> apply_rsp provided type of t needs lifting *)
- NDT ctxt "A" (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 *)
- NDT ctxt "B" (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>) *)
- NDT ctxt "C" (rtac @{thm ext} THEN' quot_true_tac ctxt unlam),
- (* resolving with R x y assumptions *)
- NDT ctxt "E" (atac),
- (* reflexivity of the basic relations *)
- (* R \<dots> \<dots> *)
- NDT ctxt "D" (resolve_tac rel_refl)
- ])
-*}
-
-ML {*
-fun all_inj_repabs_tac ctxt rel_refl trans2 =
- REPEAT_ALL_NEW (inj_repabs_tac ctxt rel_refl trans2)
-*}
-
-section {* Cleaning of the theorem *}
-
-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 lambda_prs_simple_conv ctxt ctrm =
- case (term_of ctrm) of
- ((Const (@{const_name fun_map}, _) $ r1 $ (a2 as (Const (s,_)))) $ (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 ts = MetaSimplifier.rewrite_rule @{thms id_simps} te
- val _ = tracing ("te rule:\n" ^ (Syntax.string_of_term ctxt (prop_of te)));
- val tl = Thm.lhs_of ts
- val (insp, inst) = make_inst (term_of tl) (term_of ctrm)
- val ti = Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) ts
- val _ = if not (s = @{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 ("ts rule:\n" ^ (Syntax.string_of_term ctxt (prop_of ts)));
- tracing ("instantiated rule:\n" ^ (Syntax.string_of_term ctxt (prop_of ti))))
- else ()
- in
- Conv.rewr_conv ti ctrm
- end
- | _ => Conv.all_conv ctrm
-*}
-
-ML {*
-val lambda_prs_conv =
- More_Conv.top_conv lambda_prs_simple_conv
-
-fun lambda_prs_tac ctxt = CONVERSION (lambda_prs_conv ctxt)
-*}
-
-(*
- Cleaning the theorem consists of three rewriting steps.
- The first two need to be done before fun_map is unfolded
-
- 1) lambda_prs:
- (Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) ----> f
-
- Implemented as conversion since it is not a pattern.
-
- 2) all_prs (the same for exists):
- Ball (Respects R) ((abs ---> id) f) ----> All f
-
- Rewriting with definitions from the argument defs
- (rep ---> abs) oldConst ----> newconst
-
- 3) Quotient_rel_rep:
- Rel (Rep x) (Rep y) ----> x = y
-
- Quotient_abs_rep:
- Abs (Rep x) ----> x
-
- id_simps; fun_map.simps
-*)
-
-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: shouldn't the definitions already be varified? *)
- val thms1 = @{thms all_prs ex_prs} @ defs
- val thms2 = @{thms eq_reflection[OF fun_map.simps]}
- @ @{thms id_simps Quotient_abs_rep Quotient_rel_rep}
- fun simps thms = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver
- in
- EVERY' [lambda_prs_tac lthy,
- simp_tac (simps thms1),
- simp_tac (simps thms2),
- TRY o rtac refl]
- 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 trm
-
-fun apply_under_Trueprop f =
- HOLogic.dest_Trueprop #> f #> HOLogic.mk_Trueprop
-
-fun 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 simp
-
-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 = [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 _ = warning "Regularization done."
- val inj_goal =
- Syntax.check_term ctxt (inj_repabs_trm ctxt (reg_goal, qtrm'))
- handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm
- val _ = warning "RepAbs Injection done."
-in
- 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 ctxt rthm =
- ObjectLogic.full_atomize_tac
- THEN' gen_frees_tac ctxt
- THEN' CSUBGOAL (fn (gl, i) =>
- let
- val rthm' = atomize_thm rthm
- val rule = procedure_inst ctxt (prop_of rthm') (term_of gl)
- val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb gl))] @{thm QUOT_TRUE_i}
- in
- (rtac rule THEN' RANGE [rtac rthm', (fn _ => all_tac), rtac thm]) i
- end)
-*}
-
-ML {*
-(* FIXME/TODO should only get as arguments the rthm like procedure_tac *)
-
-fun lift_tac ctxt rthm =
- ObjectLogic.full_atomize_tac
- THEN' gen_frees_tac ctxt
- THEN' CSUBGOAL (fn (gl, i) =>
- let
- val rthm' = atomize_thm rthm
- val rule = procedure_inst ctxt (prop_of rthm') (term_of gl)
- val rel_refl = map (fn x => @{thm equivp_reflp} OF [x]) (equiv_rules_get ctxt)
- val quotients = quotient_rules_get ctxt
- val trans2 = map (fn x => @{thm equals_rsp} OF [x]) quotients
- val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb gl))] @{thm QUOT_TRUE_i}
- in
- (rtac rule THEN'
- RANGE [rtac rthm',
- regularize_tac ctxt,
- rtac thm THEN' all_inj_repabs_tac ctxt rel_refl trans2,
- clean_tac ctxt]) i
- end)
-*}
-
-end
-