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theory QuotMain
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imports QuotScript QuotProd Prove
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uses ("quotient_info.ML")
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("quotient.ML")
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("quotient_def.ML")
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begin
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locale QUOT_TYPE =
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fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
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and Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"
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and Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"
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assumes equivp: "equivp R"
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and rep_prop: "\<And>y. \<exists>x. Rep y = R x"
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and rep_inverse: "\<And>x. Abs (Rep x) = x"
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and abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"
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and rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
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begin
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definition
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ABS::"'a \<Rightarrow> 'b"
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where
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"ABS x \<equiv> Abs (R x)"
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definition
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REP::"'b \<Rightarrow> 'a"
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where
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"REP a = Eps (Rep a)"
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lemma lem9:
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shows "R (Eps (R x)) = R x"
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proof -
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have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)
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then have "R x (Eps (R x))" by (rule someI)
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then show "R (Eps (R x)) = R x"
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using equivp unfolding equivp_def by simp
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qed
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theorem thm10:
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shows "ABS (REP a) \<equiv> a"
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apply (rule eq_reflection)
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unfolding ABS_def REP_def
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proof -
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from rep_prop
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obtain x where eq: "Rep a = R x" by auto
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have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp
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also have "\<dots> = Abs (R x)" using lem9 by simp
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also have "\<dots> = Abs (Rep a)" using eq by simp
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also have "\<dots> = a" using rep_inverse by simp
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finally
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show "Abs (R (Eps (Rep a))) = a" by simp
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qed
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lemma REP_refl:
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shows "R (REP a) (REP a)"
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unfolding REP_def
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by (simp add: equivp[simplified equivp_def])
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lemma lem7:
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shows "(R x = R y) = (Abs (R x) = Abs (R y))"
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apply(rule iffI)
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apply(simp)
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apply(drule rep_inject[THEN iffD2])
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apply(simp add: abs_inverse)
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done
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theorem thm11:
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shows "R r r' = (ABS r = ABS r')"
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unfolding ABS_def
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by (simp only: equivp[simplified equivp_def] lem7)
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lemma REP_ABS_rsp:
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shows "R f (REP (ABS g)) = R f g"
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and "R (REP (ABS g)) f = R g f"
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by (simp_all add: thm10 thm11)
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lemma Quotient:
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"Quotient R ABS REP"
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apply(unfold Quotient_def)
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apply(simp add: thm10)
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apply(simp add: REP_refl)
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apply(subst thm11[symmetric])
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apply(simp add: equivp[simplified equivp_def])
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done
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lemma R_trans:
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assumes ab: "R a b"
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and bc: "R b c"
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shows "R a c"
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proof -
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have tr: "transp R" using equivp equivp_reflp_symp_transp[of R] by simp
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moreover have ab: "R a b" by fact
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moreover have bc: "R b c" by fact
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ultimately show "R a c" unfolding transp_def by blast
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qed
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lemma R_sym:
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assumes ab: "R a b"
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shows "R b a"
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proof -
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have re: "symp R" using equivp equivp_reflp_symp_transp[of R] by simp
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then show "R b a" using ab unfolding symp_def by blast
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qed
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lemma R_trans2:
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assumes ac: "R a c"
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and bd: "R b d"
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shows "R a b = R c d"
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using ac bd
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by (blast intro: R_trans R_sym)
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lemma REPS_same:
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shows "R (REP a) (REP b) \<equiv> (a = b)"
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proof -
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have "R (REP a) (REP b) = (a = b)"
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proof
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assume as: "R (REP a) (REP b)"
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from rep_prop
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obtain x y
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where eqs: "Rep a = R x" "Rep b = R y" by blast
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from eqs have "R (Eps (R x)) (Eps (R y))" using as unfolding REP_def by simp
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then have "R x (Eps (R y))" using lem9 by simp
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then have "R (Eps (R y)) x" using R_sym by blast
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then have "R y x" using lem9 by simp
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then have "R x y" using R_sym by blast
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then have "ABS x = ABS y" using thm11 by simp
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then have "Abs (Rep a) = Abs (Rep b)" using eqs unfolding ABS_def by simp
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then show "a = b" using rep_inverse by simp
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next
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assume ab: "a = b"
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have "reflp R" using equivp equivp_reflp_symp_transp[of R] by simp
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then show "R (REP a) (REP b)" unfolding reflp_def using ab by auto
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qed
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then show "R (REP a) (REP b) \<equiv> (a = b)" by simp
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qed
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end
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section {* type definition for the quotient type *}
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(* the auxiliary data for the quotient types *)
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use "quotient_info.ML"
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declare [[map * = (prod_fun, prod_rel)]]
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declare [[map "fun" = (fun_map, fun_rel)]]
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(* FIXME: This should throw an exception:
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declare [[map "option" = (bla, blu)]]
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*)
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(* identity quotient is not here as it has to be applied first *)
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lemmas [quotient_thm] =
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fun_quotient prod_quotient
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lemmas [quotient_rsp] =
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quot_rel_rsp pair_rsp
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(* fun_map is not here since equivp is not true *)
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(* TODO: option, ... *)
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lemmas [quotient_equiv] =
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identity_equivp prod_equivp
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ML {* maps_lookup @{theory} "*" *}
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ML {* maps_lookup @{theory} "fun" *}
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(* definition of the quotient types *)
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(* FIXME: should be called quotient_typ.ML *)
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use "quotient.ML"
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(* lifting of constants *)
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use "quotient_def.ML"
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section {* Simset setup *}
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(* since HOL_basic_ss is too "big", we need to set up *)
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(* our own minimal simpset *)
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ML {*
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fun mk_minimal_ss ctxt =
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Simplifier.context ctxt empty_ss
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setsubgoaler asm_simp_tac
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setmksimps (mksimps [])
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*}
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ML {*
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(* TODO/FIXME not needed anymore? *)
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fun SOLVES' tac = tac THEN_ALL_NEW (fn _ => no_tac)
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removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
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fun OF1 thm1 thm2 = thm2 RS thm1
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*}
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section {* atomize *}
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lemma atomize_eqv[atomize]:
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shows "(Trueprop A \<equiv> Trueprop B) \<equiv> (A \<equiv> B)"
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proof
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assume "A \<equiv> B"
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then show "Trueprop A \<equiv> Trueprop B" by unfold
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next
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assume *: "Trueprop A \<equiv> Trueprop B"
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have "A = B"
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proof (cases A)
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case True
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have "A" by fact
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then show "A = B" using * by simp
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next
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case False
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have "\<not>A" by fact
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then show "A = B" using * by auto
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qed
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then show "A \<equiv> B" by (rule eq_reflection)
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qed
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ML {*
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fun atomize_thm thm =
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let
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val thm' = Thm.freezeT (forall_intr_vars thm)
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val thm'' = ObjectLogic.atomize (cprop_of thm')
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in
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@{thm equal_elim_rule1} OF [thm'', thm']
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end
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*}
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section {* infrastructure about id *}
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(* TODO: think where should this be *)
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lemma prod_fun_id: "prod_fun id id \<equiv> id"
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by (rule eq_reflection) (simp add: prod_fun_def)
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(* FIXME: make it a list and add map_id to it *)
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lemmas id_simps =
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fun_map_id[THEN eq_reflection]
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id_apply[THEN eq_reflection]
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id_def[THEN eq_reflection,symmetric]
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prod_fun_id
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ML {*
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fun simp_ids thm =
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MetaSimplifier.rewrite_rule @{thms id_simps} thm
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*}
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section {* Debugging infrastructure for testing tactics *}
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ML {*
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fun my_print_tac ctxt s i thm =
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let
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val prem_str = nth (prems_of thm) (i - 1)
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|> Syntax.string_of_term ctxt
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handle Subscript => "no subgoal"
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val _ = tracing (s ^ "\n" ^ prem_str)
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in
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Seq.single thm
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end *}
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ML {*
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fun DT ctxt s tac i thm =
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let
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val before_goal = nth (prems_of thm) (i - 1)
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|> Syntax.string_of_term ctxt
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val before_msg = ["before: " ^ s, before_goal, "after: " ^ s]
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|> cat_lines
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in
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EVERY [tac i, my_print_tac ctxt before_msg i] thm
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end
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fun NDT ctxt s tac thm = tac thm
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*}
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section {* Matching of terms and types *}
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ML {*
+ − 273
fun matches_typ (ty, ty') =
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case (ty, ty') of
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(_, TVar _) => true
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| (TFree x, TFree x') => x = x'
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| (Type (s, tys), Type (s', tys')) =>
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s = s' andalso
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if (length tys = length tys')
+ − 280
then (List.all matches_typ (tys ~~ tys'))
+ − 281
else false
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| _ => false
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*}
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ML {*
+ − 286
fun matches_term (trm, trm') =
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case (trm, trm') of
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(_, Var _) => true
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| (Const (s, ty), Const (s', ty')) => s = s' andalso matches_typ (ty, ty')
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| (Free (x, ty), Free (x', ty')) => x = x' andalso matches_typ (ty, ty')
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| (Bound i, Bound j) => i = j
+ − 292
| (Abs (_, T, t), Abs (_, T', t')) => matches_typ (T, T') andalso matches_term (t, t')
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| (t $ s, t' $ s') => matches_term (t, t') andalso matches_term (s, s')
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| _ => false
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*}
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section {* Regularization *}
+ − 298
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(*
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Regularizing an rtrm means:
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- quantifiers over a type that needs lifting are replaced by
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bounded quantifiers, for example:
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\<forall>x. P \<Longrightarrow> \<forall>x \<in> (Respects R). P / All (Respects R) P
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+ − 305
the relation R is given by the rty and qty;
+ − 306
+ − 307
- abstractions over a type that needs lifting are replaced
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by bounded abstractions:
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\<lambda>x. P \<Longrightarrow> Ball (Respects R) (\<lambda>x. P)
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+ − 311
- equalities over the type being lifted are replaced by
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corresponding relations:
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A = B \<Longrightarrow> A \<approx> B
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+ − 315
example with more complicated types of A, B:
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A = B \<Longrightarrow> (op = \<Longrightarrow> op \<approx>) A B
+ − 317
*)
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ML {*
+ − 320
(* builds the relation that is the argument of respects *)
+ − 321
fun mk_resp_arg lthy (rty, qty) =
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let
+ − 323
val thy = ProofContext.theory_of lthy
+ − 324
in
+ − 325
if rty = qty
+ − 326
then HOLogic.eq_const rty
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else
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case (rty, qty) of
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(Type (s, tys), Type (s', tys')) =>
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if s = s'
+ − 331
then let
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val SOME map_info = maps_lookup thy s
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val args = map (mk_resp_arg lthy) (tys ~~ tys')
+ − 334
in
+ − 335
list_comb (Const (#relfun map_info, dummyT), args)
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end
+ − 337
else let
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val SOME qinfo = quotdata_lookup_thy thy s'
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(* FIXME: check in this case that the rty and qty *)
+ − 340
(* FIXME: correspond to each other *)
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val (s, _) = dest_Const (#rel qinfo)
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(* FIXME: the relation should only be the string *)
+ − 343
(* FIXME: and the type needs to be calculated as below; *)
+ − 344
(* FIXME: maybe one should actually have a term *)
+ − 345
(* FIXME: and one needs to force it to have this type *)
+ − 346
in
+ − 347
Const (s, rty --> rty --> @{typ bool})
+ − 348
end
+ − 349
| _ => HOLogic.eq_const dummyT
+ − 350
(* FIXME: check that the types correspond to each other? *)
+ − 351
end
+ − 352
*}
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ML {*
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val mk_babs = Const (@{const_name Babs}, dummyT)
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val mk_ball = Const (@{const_name Ball}, dummyT)
+ − 357
val mk_bex = Const (@{const_name Bex}, dummyT)
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val mk_resp = Const (@{const_name Respects}, dummyT)
+ − 359
*}
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ML {*
+ − 362
(* - applies f to the subterm of an abstraction, *)
+ − 363
(* otherwise to the given term, *)
+ − 364
(* - used by regularize, therefore abstracted *)
+ − 365
(* variables do not have to be treated specially *)
+ − 366
+ − 367
fun apply_subt f trm1 trm2 =
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case (trm1, trm2) of
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(Abs (x, T, t), Abs (x', T', t')) => Abs (x, T, f t t')
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| _ => f trm1 trm2
+ − 371
+ − 372
(* the major type of All and Ex quantifiers *)
+ − 373
fun qnt_typ ty = domain_type (domain_type ty)
+ − 374
*}
+ − 375
+ − 376
ML {*
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(* produces a regularized version of rtm *)
+ − 378
(* - the result is still not completely typed *)
+ − 379
(* - does not need any special treatment of *)
+ − 380
(* bound variables *)
+ − 381
+ − 382
fun regularize_trm lthy rtrm qtrm =
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case (rtrm, qtrm) of
+ − 384
(Abs (x, ty, t), Abs (x', ty', t')) =>
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let
+ − 386
val subtrm = Abs(x, ty, regularize_trm lthy t t')
+ − 387
in
+ − 388
if ty = ty'
+ − 389
then subtrm
+ − 390
else mk_babs $ (mk_resp $ mk_resp_arg lthy (ty, ty')) $ subtrm
+ − 391
end
+ − 392
+ − 393
| (Const (@{const_name "All"}, ty) $ t, Const (@{const_name "All"}, ty') $ t') =>
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let
+ − 395
val subtrm = apply_subt (regularize_trm lthy) t t'
+ − 396
in
+ − 397
if ty = ty'
+ − 398
then Const (@{const_name "All"}, ty) $ subtrm
+ − 399
else mk_ball $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm
+ − 400
end
+ − 401
+ − 402
| (Const (@{const_name "Ex"}, ty) $ t, Const (@{const_name "Ex"}, ty') $ t') =>
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let
+ − 404
val subtrm = apply_subt (regularize_trm lthy) t t'
+ − 405
in
+ − 406
if ty = ty'
+ − 407
then Const (@{const_name "Ex"}, ty) $ subtrm
+ − 408
else mk_bex $ (mk_resp $ mk_resp_arg lthy (qnt_typ ty, qnt_typ ty')) $ subtrm
+ − 409
end
+ − 410
+ − 411
| (* equalities need to be replaced by appropriate equivalence relations *)
+ − 412
(Const (@{const_name "op ="}, ty), Const (@{const_name "op ="}, ty')) =>
+ − 413
if ty = ty'
+ − 414
then rtrm
+ − 415
else mk_resp_arg lthy (domain_type ty, domain_type ty')
+ − 416
+ − 417
| (* in this case we check whether the given equivalence relation is correct *)
+ − 418
(rel, Const (@{const_name "op ="}, ty')) =>
+ − 419
let
+ − 420
val exc = LIFT_MATCH "regularise (relation mismatch)"
+ − 421
val rel_ty = (fastype_of rel) handle TERM _ => raise exc
+ − 422
val rel' = mk_resp_arg lthy (domain_type rel_ty, domain_type ty')
+ − 423
in
+ − 424
if rel' = rel
+ − 425
then rtrm
+ − 426
else raise exc
+ − 427
end
+ − 428
| (_, Const (s, _)) =>
+ − 429
let
+ − 430
fun same_name (Const (s, _)) (Const (s', _)) = (s = s')
+ − 431
| same_name _ _ = false
+ − 432
in
+ − 433
if same_name rtrm qtrm
+ − 434
then rtrm
+ − 435
else
+ − 436
let
+ − 437
fun exc1 s = LIFT_MATCH ("regularize (constant " ^ s ^ " not found)")
+ − 438
val exc2 = LIFT_MATCH ("regularize (constant mismatch)")
+ − 439
val thy = ProofContext.theory_of lthy
+ − 440
val rtrm' = (#rconst (qconsts_lookup thy s)) handle NotFound => raise (exc1 s)
+ − 441
in
+ − 442
if matches_term (rtrm, rtrm')
+ − 443
then rtrm
+ − 444
else raise exc2
+ − 445
end
+ − 446
end
+ − 447
+ − 448
| (t1 $ t2, t1' $ t2') =>
+ − 449
(regularize_trm lthy t1 t1') $ (regularize_trm lthy t2 t2')
+ − 450
+ − 451
| (Free (x, ty), Free (x', ty')) =>
+ − 452
(* this case cannot arrise as we start with two fully atomized terms *)
+ − 453
raise (LIFT_MATCH "regularize (frees)")
+ − 454
+ − 455
| (Bound i, Bound i') =>
+ − 456
if i = i'
+ − 457
then rtrm
+ − 458
else raise (LIFT_MATCH "regularize (bounds mismatch)")
+ − 459
+ − 460
| (rt, qt) =>
+ − 461
raise (LIFT_MATCH "regularize (default)")
+ − 462
*}
+ − 463
+ − 464
ML {*
+ − 465
fun equiv_tac ctxt =
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+ − 466
(K (print_tac "equiv tac")) THEN'
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first attempt to deal with Babs in regularise and cleaning (not yet working)
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+ − 467
REPEAT_ALL_NEW (resolve_tac (equiv_rules_get ctxt))
597
+ − 468
*}
+ − 469
+ − 470
ML {*
+ − 471
fun equiv_solver_tac ss = equiv_tac (Simplifier.the_context ss)
+ − 472
val equiv_solver = Simplifier.mk_solver' "Equivalence goal solver" equiv_solver_tac
+ − 473
*}
+ − 474
+ − 475
ML {*
+ − 476
fun prep_trm thy (x, (T, t)) =
+ − 477
(cterm_of thy (Var (x, T)), cterm_of thy t)
+ − 478
+ − 479
fun prep_ty thy (x, (S, ty)) =
+ − 480
(ctyp_of thy (TVar (x, S)), ctyp_of thy ty)
+ − 481
*}
+ − 482
+ − 483
ML {*
+ − 484
fun matching_prs thy pat trm =
+ − 485
let
+ − 486
val univ = Unify.matchers thy [(pat, trm)]
+ − 487
val SOME (env, _) = Seq.pull univ
+ − 488
val tenv = Vartab.dest (Envir.term_env env)
+ − 489
val tyenv = Vartab.dest (Envir.type_env env)
+ − 490
in
+ − 491
(map (prep_ty thy) tyenv, map (prep_trm thy) tenv)
+ − 492
end
+ − 493
*}
+ − 494
+ − 495
ML {*
+ − 496
fun calculate_instance ctxt thm redex R1 R2 =
+ − 497
let
+ − 498
val thy = ProofContext.theory_of ctxt
+ − 499
val goal = Const (@{const_name "equivp"}, dummyT) $ R2
+ − 500
|> Syntax.check_term ctxt
+ − 501
|> HOLogic.mk_Trueprop
+ − 502
val eqv_prem = Goal.prove ctxt [] [] goal (fn {context,...} => equiv_tac context 1)
+ − 503
val thm = (@{thm eq_reflection} OF [thm OF [eqv_prem]])
+ − 504
val R1c = cterm_of thy R1
+ − 505
val thmi = Drule.instantiate' [] [SOME R1c] thm
+ − 506
val inst = matching_prs thy (term_of (Thm.lhs_of thmi)) redex
+ − 507
val thm2 = Drule.eta_contraction_rule (Drule.instantiate inst thmi)
+ − 508
in
+ − 509
SOME thm2
+ − 510
end
+ − 511
handle _ => NONE
+ − 512
(* FIXME/TODO: what is the place where the exception can be raised: matching_prs? *)
+ − 513
*}
+ − 514
+ − 515
ML {*
+ − 516
fun ball_bex_range_simproc ss redex =
+ − 517
let
+ − 518
val ctxt = Simplifier.the_context ss
+ − 519
in
+ − 520
case redex of
+ − 521
(Const (@{const_name "Ball"}, _) $ (Const (@{const_name "Respects"}, _) $
+ − 522
(Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) =>
+ − 523
calculate_instance ctxt @{thm ball_reg_eqv_range} redex R1 R2
+ − 524
| (Const (@{const_name "Bex"}, _) $ (Const (@{const_name "Respects"}, _) $
+ − 525
(Const (@{const_name "fun_rel"}, _) $ R1 $ R2)) $ _) =>
+ − 526
calculate_instance ctxt @{thm bex_reg_eqv_range} redex R1 R2
+ − 527
| _ => NONE
+ − 528
end
+ − 529
*}
+ − 530
+ − 531
lemma eq_imp_rel:
+ − 532
shows "equivp R \<Longrightarrow> a = b \<longrightarrow> R a b"
+ − 533
by (simp add: equivp_reflp)
+ − 534
+ − 535
(* FIXME/TODO: How does regularizing work? *)
+ − 536
(* FIXME/TODO: needs to be adapted
+ − 537
+ − 538
To prove that the raw theorem implies the regularised one,
+ − 539
we try in order:
+ − 540
+ − 541
- Reflexivity of the relation
+ − 542
- Assumption
+ − 543
- Elimnating quantifiers on both sides of toplevel implication
+ − 544
- Simplifying implications on both sides of toplevel implication
+ − 545
- Ball (Respects ?E) ?P = All ?P
+ − 546
- (\<And>x. ?R x \<Longrightarrow> ?P x \<longrightarrow> ?Q x) \<Longrightarrow> All ?P \<longrightarrow> Ball ?R ?Q
+ − 547
+ − 548
*)
+ − 549
ML {*
+ − 550
fun regularize_tac ctxt =
+ − 551
let
+ − 552
val thy = ProofContext.theory_of ctxt
+ − 553
val pat_ball = @{term "Ball (Respects (R1 ===> R2)) P"}
+ − 554
val pat_bex = @{term "Bex (Respects (R1 ===> R2)) P"}
+ − 555
val simproc = Simplifier.simproc_i thy "" [pat_ball, pat_bex] (K (ball_bex_range_simproc))
+ − 556
val simpset = (mk_minimal_ss ctxt)
605
120e479ed367
first attempt to deal with Babs in regularise and cleaning (not yet working)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 557
addsimps @{thms ball_reg_eqv bex_reg_eqv babs_reg_eqv}
597
+ − 558
addsimprocs [simproc] addSolver equiv_solver
+ − 559
(* TODO: Make sure that there are no list_rel, pair_rel etc involved *)
605
120e479ed367
first attempt to deal with Babs in regularise and cleaning (not yet working)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 560
(* can this cause loops in equiv_tac ? *)
597
+ − 561
val eq_eqvs = map (fn x => @{thm eq_imp_rel} OF [x]) (equiv_rules_get ctxt)
+ − 562
in
+ − 563
simp_tac simpset THEN'
605
120e479ed367
first attempt to deal with Babs in regularise and cleaning (not yet working)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 564
REPEAT_ALL_NEW (CHANGED o FIRST' [
597
+ − 565
rtac @{thm ball_reg_right},
+ − 566
rtac @{thm bex_reg_left},
+ − 567
resolve_tac (Inductive.get_monos ctxt),
+ − 568
rtac @{thm ball_all_comm},
+ − 569
rtac @{thm bex_ex_comm},
605
120e479ed367
first attempt to deal with Babs in regularise and cleaning (not yet working)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 570
resolve_tac eq_eqvs,
606
+ − 571
(* should be equivalent to the above, but causes loops: *)
605
120e479ed367
first attempt to deal with Babs in regularise and cleaning (not yet working)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 572
(* rtac @{thm eq_imp_rel} THEN' SOLVES' (equiv_tac ctxt), *)
606
+ − 573
(* the culprit is aslread rtac @{thm eq_imp_rel} *)
597
+ − 574
simp_tac simpset])
+ − 575
end
+ − 576
*}
+ − 577
+ − 578
section {* Injections of rep and abses *}
+ − 579
+ − 580
(*
+ − 581
Injecting repabs means:
+ − 582
+ − 583
For abstractions:
+ − 584
* If the type of the abstraction doesn't need lifting we recurse.
+ − 585
* If it does we add RepAbs around the whole term and check if the
+ − 586
variable needs lifting.
+ − 587
* If it doesn't then we recurse
+ − 588
* If it does we recurse and put 'RepAbs' around all occurences
+ − 589
of the variable in the obtained subterm. This in combination
+ − 590
with the RepAbs above will let us change the type of the
+ − 591
abstraction with rewriting.
+ − 592
For applications:
+ − 593
* If the term is 'Respects' applied to anything we leave it unchanged
+ − 594
* If the term needs lifting and the head is a constant that we know
+ − 595
how to lift, we put a RepAbs and recurse
+ − 596
* If the term needs lifting and the head is a free applied to subterms
+ − 597
(if it is not applied we treated it in Abs branch) then we
+ − 598
put RepAbs and recurse
+ − 599
* Otherwise just recurse.
+ − 600
*)
+ − 601
+ − 602
ML {*
+ − 603
fun mk_repabs lthy (T, T') trm =
+ − 604
Quotient_Def.get_fun repF lthy (T, T')
+ − 605
$ (Quotient_Def.get_fun absF lthy (T, T') $ trm)
+ − 606
*}
+ − 607
+ − 608
ML {*
+ − 609
(* bound variables need to be treated properly, *)
+ − 610
(* as the type of subterms need to be calculated *)
+ − 611
(* in the abstraction case *)
+ − 612
+ − 613
fun inj_repabs_trm lthy (rtrm, qtrm) =
+ − 614
case (rtrm, qtrm) of
+ − 615
(Const (@{const_name "Ball"}, T) $ r $ t, Const (@{const_name "All"}, _) $ t') =>
+ − 616
Const (@{const_name "Ball"}, T) $ r $ (inj_repabs_trm lthy (t, t'))
+ − 617
+ − 618
| (Const (@{const_name "Bex"}, T) $ r $ t, Const (@{const_name "Ex"}, _) $ t') =>
+ − 619
Const (@{const_name "Bex"}, T) $ r $ (inj_repabs_trm lthy (t, t'))
+ − 620
+ − 621
| (Const (@{const_name "Babs"}, T) $ r $ t, t' as (Abs _)) =>
+ − 622
Const (@{const_name "Babs"}, T) $ r $ (inj_repabs_trm lthy (t, t'))
+ − 623
+ − 624
| (Abs (x, T, t), Abs (x', T', t')) =>
+ − 625
let
+ − 626
val rty = fastype_of rtrm
+ − 627
val qty = fastype_of qtrm
+ − 628
val (y, s) = Term.dest_abs (x, T, t)
+ − 629
val (_, s') = Term.dest_abs (x', T', t')
+ − 630
val yvar = Free (y, T)
+ − 631
val result = Term.lambda_name (y, yvar) (inj_repabs_trm lthy (s, s'))
+ − 632
in
+ − 633
if rty = qty
+ − 634
then result
+ − 635
else mk_repabs lthy (rty, qty) result
+ − 636
end
+ − 637
+ − 638
| (t $ s, t' $ s') =>
+ − 639
(inj_repabs_trm lthy (t, t')) $ (inj_repabs_trm lthy (s, s'))
+ − 640
+ − 641
| (Free (_, T), Free (_, T')) =>
+ − 642
if T = T'
+ − 643
then rtrm
+ − 644
else mk_repabs lthy (T, T') rtrm
+ − 645
+ − 646
| (_, Const (@{const_name "op ="}, _)) => rtrm
+ − 647
+ − 648
(* FIXME: check here that rtrm is the corresponding definition for the const *)
+ − 649
| (_, Const (_, T')) =>
+ − 650
let
+ − 651
val rty = fastype_of rtrm
+ − 652
in
+ − 653
if rty = T'
+ − 654
then rtrm
+ − 655
else mk_repabs lthy (rty, T') rtrm
+ − 656
end
+ − 657
+ − 658
| _ => raise (LIFT_MATCH "injection")
+ − 659
*}
+ − 660
+ − 661
section {* RepAbs Injection Tactic *}
+ − 662
+ − 663
ML {*
+ − 664
fun quotient_tac ctxt =
+ − 665
REPEAT_ALL_NEW (FIRST'
+ − 666
[rtac @{thm identity_quotient},
+ − 667
resolve_tac (quotient_rules_get ctxt)])
+ − 668
*}
+ − 669
+ − 670
(* solver for the simplifier *)
+ − 671
ML {*
+ − 672
fun quotient_solver_tac ss = quotient_tac (Simplifier.the_context ss)
+ − 673
val quotient_solver = Simplifier.mk_solver' "Quotient goal solver" quotient_solver_tac
+ − 674
*}
+ − 675
+ − 676
ML {*
+ − 677
fun solve_quotient_assums ctxt thm =
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 678
let
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 679
val goal = hd (Drule.strip_imp_prems (cprop_of thm))
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 680
in
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 681
thm OF [Goal.prove_internal [] goal (fn _ => quotient_tac ctxt 1)]
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 682
end
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 683
handle _ => error "solve_quotient_assums failed. Maybe a quotient_thm is missing"
597
+ − 684
*}
+ − 685
+ − 686
(* Not used *)
+ − 687
(* It proves the Quotient assumptions by calling quotient_tac *)
+ − 688
ML {*
+ − 689
fun solve_quotient_assum i ctxt thm =
+ − 690
let
+ − 691
val tac =
+ − 692
(compose_tac (false, thm, i)) THEN_ALL_NEW
+ − 693
(quotient_tac ctxt);
+ − 694
val gc = Drule.strip_imp_concl (cprop_of thm);
+ − 695
in
+ − 696
Goal.prove_internal [] gc (fn _ => tac 1)
+ − 697
end
+ − 698
handle _ => error "solve_quotient_assum"
+ − 699
*}
+ − 700
+ − 701
definition
+ − 702
"QUOT_TRUE x \<equiv> True"
+ − 703
+ − 704
ML {*
+ − 705
fun find_qt_asm asms =
+ − 706
let
+ − 707
fun find_fun trm =
+ − 708
case trm of
+ − 709
(Const(@{const_name Trueprop}, _) $ (Const (@{const_name QUOT_TRUE}, _) $ _)) => true
+ − 710
| _ => false
+ − 711
in
+ − 712
case find_first find_fun asms of
+ − 713
SOME (_ $ (_ $ (f $ a))) => (f, a)
+ − 714
| SOME _ => error "find_qt_asm: no pair"
+ − 715
| NONE => error "find_qt_asm: no assumption"
+ − 716
end
+ − 717
*}
+ − 718
+ − 719
(*
+ − 720
To prove that the regularised theorem implies the abs/rep injected,
+ − 721
we try:
+ − 722
+ − 723
1) theorems 'trans2' from the appropriate QUOT_TYPE
+ − 724
2) remove lambdas from both sides: lambda_rsp_tac
+ − 725
3) remove Ball/Bex from the right hand side
+ − 726
4) use user-supplied RSP theorems
+ − 727
5) remove rep_abs from the right side
+ − 728
6) reflexivity of equality
+ − 729
7) split applications of lifted type (apply_rsp)
+ − 730
8) split applications of non-lifted type (cong_tac)
+ − 731
9) apply extentionality
+ − 732
A) reflexivity of the relation
+ − 733
B) assumption
+ − 734
(Lambdas under respects may have left us some assumptions)
+ − 735
C) proving obvious higher order equalities by simplifying fun_rel
+ − 736
(not sure if it is still needed?)
+ − 737
D) unfolding lambda on one side
+ − 738
E) simplifying (= ===> =) for simpler respectfulness
+ − 739
+ − 740
*)
+ − 741
+ − 742
lemma quot_true_dests:
+ − 743
shows QT_all: "QUOT_TRUE (All P) \<Longrightarrow> QUOT_TRUE P"
+ − 744
and QT_ex: "QUOT_TRUE (Ex P) \<Longrightarrow> QUOT_TRUE P"
+ − 745
and QT_lam: "QUOT_TRUE (\<lambda>x. P x) \<Longrightarrow> (\<And>x. QUOT_TRUE (P x))"
+ − 746
and QT_ext: "(\<And>x. QUOT_TRUE (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (QUOT_TRUE a \<Longrightarrow> f = g)"
+ − 747
apply(simp_all add: QUOT_TRUE_def ext)
+ − 748
done
+ − 749
+ − 750
lemma QUOT_TRUE_i: "(QUOT_TRUE (a :: bool) \<Longrightarrow> P) \<Longrightarrow> P"
+ − 751
by (simp add: QUOT_TRUE_def)
+ − 752
+ − 753
lemma QUOT_TRUE_imp: "QUOT_TRUE a \<equiv> QUOT_TRUE b"
+ − 754
by (simp add: QUOT_TRUE_def)
+ − 755
+ − 756
ML {*
+ − 757
fun quot_true_conv1 ctxt fnctn ctrm =
+ − 758
case (term_of ctrm) of
+ − 759
(Const (@{const_name QUOT_TRUE}, _) $ x) =>
+ − 760
let
+ − 761
val fx = fnctn x;
+ − 762
val thy = ProofContext.theory_of ctxt;
+ − 763
val cx = cterm_of thy x;
+ − 764
val cfx = cterm_of thy fx;
+ − 765
val cxt = ctyp_of thy (fastype_of x);
+ − 766
val cfxt = ctyp_of thy (fastype_of fx);
+ − 767
val thm = Drule.instantiate' [SOME cxt, SOME cfxt] [SOME cx, SOME cfx] @{thm QUOT_TRUE_imp}
+ − 768
in
+ − 769
Conv.rewr_conv thm ctrm
+ − 770
end
+ − 771
*}
+ − 772
+ − 773
ML {*
+ − 774
fun quot_true_conv ctxt fnctn ctrm =
+ − 775
case (term_of ctrm) of
+ − 776
(Const (@{const_name QUOT_TRUE}, _) $ _) =>
+ − 777
quot_true_conv1 ctxt fnctn ctrm
+ − 778
| _ $ _ => Conv.comb_conv (quot_true_conv ctxt fnctn) ctrm
+ − 779
| Abs _ => Conv.abs_conv (fn (_, ctxt) => quot_true_conv ctxt fnctn) ctxt ctrm
+ − 780
| _ => Conv.all_conv ctrm
+ − 781
*}
+ − 782
+ − 783
ML {*
+ − 784
fun quot_true_tac ctxt fnctn = CONVERSION
+ − 785
((Conv.params_conv ~1 (fn ctxt =>
+ − 786
(Conv.prems_conv ~1 (quot_true_conv ctxt fnctn)))) ctxt)
+ − 787
*}
+ − 788
+ − 789
ML {* fun dest_comb (f $ a) = (f, a) *}
+ − 790
ML {* fun dest_bcomb ((_ $ l) $ r) = (l, r) *}
+ − 791
(* TODO: Can this be done easier? *)
+ − 792
ML {*
+ − 793
fun unlam t =
+ − 794
case t of
+ − 795
(Abs a) => snd (Term.dest_abs a)
+ − 796
| _ => unlam (Abs("", domain_type (fastype_of t), (incr_boundvars 1 t) $ (Bound 0)))
+ − 797
*}
+ − 798
+ − 799
ML {*
+ − 800
fun dest_fun_type (Type("fun", [T, S])) = (T, S)
+ − 801
| dest_fun_type _ = error "dest_fun_type"
+ − 802
*}
+ − 803
+ − 804
ML {*
+ − 805
val bare_concl = HOLogic.dest_Trueprop o Logic.strip_assums_concl
+ − 806
*}
+ − 807
+ − 808
ML {*
+ − 809
val apply_rsp_tac =
+ − 810
Subgoal.FOCUS (fn {concl, asms, context,...} =>
+ − 811
case ((HOLogic.dest_Trueprop (term_of concl))) of
+ − 812
((R2 $ (f $ x) $ (g $ y))) =>
+ − 813
(let
+ − 814
val (asmf, asma) = find_qt_asm (map term_of asms);
+ − 815
in
+ − 816
if (fastype_of asmf) = (fastype_of f) then no_tac else let
+ − 817
val ty_a = fastype_of x;
+ − 818
val ty_b = fastype_of asma;
+ − 819
val ty_c = range_type (type_of f);
+ − 820
val thy = ProofContext.theory_of context;
+ − 821
val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c];
+ − 822
val thm = Drule.instantiate' ty_inst [] @{thm apply_rsp}
+ − 823
val te = solve_quotient_assums context thm
+ − 824
val t_inst = map (SOME o (cterm_of thy)) [R2, f, g, x, y];
+ − 825
val thm = Drule.instantiate' [] t_inst te
+ − 826
in
+ − 827
compose_tac (false, thm, 2) 1
+ − 828
end
+ − 829
end
+ − 830
handle ERROR "find_qt_asm: no pair" => no_tac)
+ − 831
| _ => no_tac)
+ − 832
*}
+ − 833
+ − 834
ML {*
+ − 835
fun rep_abs_rsp_tac ctxt =
+ − 836
SUBGOAL (fn (goal, i) =>
+ − 837
case (bare_concl goal) of
+ − 838
(rel $ _ $ (rep $ (abs $ _))) =>
+ − 839
(let
+ − 840
val thy = ProofContext.theory_of ctxt;
+ − 841
val (ty_a, ty_b) = dest_fun_type (fastype_of abs);
+ − 842
val ty_inst = map (SOME o (ctyp_of thy)) [ty_a, ty_b];
+ − 843
val t_inst = map (SOME o (cterm_of thy)) [rel, abs, rep];
+ − 844
val thm = Drule.instantiate' ty_inst t_inst @{thm rep_abs_rsp}
+ − 845
val te = solve_quotient_assums ctxt thm
+ − 846
in
+ − 847
rtac te i
+ − 848
end
+ − 849
handle _ => no_tac)
+ − 850
| _ => no_tac)
+ − 851
*}
+ − 852
+ − 853
ML {*
+ − 854
fun inj_repabs_tac_match ctxt trans2 = SUBGOAL (fn (goal, i) =>
+ − 855
(case (bare_concl goal) of
+ − 856
(* (R1 ===> R2) (\<lambda>x\<dots>) (\<lambda>y\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> R2 (\<dots>x) (\<dots>y) *)
+ − 857
((Const (@{const_name fun_rel}, _) $ _ $ _) $ (Abs _) $ (Abs _))
+ − 858
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
+ − 859
+ − 860
(* (op =) (Ball\<dots>) (Ball\<dots>) ----> (op =) (\<dots>) (\<dots>) *)
+ − 861
| (Const (@{const_name "op ="},_) $
+ − 862
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
+ − 863
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
+ − 864
=> rtac @{thm ball_rsp} THEN' dtac @{thm QT_all}
+ − 865
+ − 866
(* (R1 ===> op =) (Ball\<dots>) (Ball\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Ball\<dots>x) = (Ball\<dots>y) *)
+ − 867
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
+ − 868
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
+ − 869
(Const(@{const_name Ball},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
+ − 870
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
+ − 871
+ − 872
(* (op =) (Bex\<dots>) (Bex\<dots>) ----> (op =) (\<dots>) (\<dots>) *)
+ − 873
| Const (@{const_name "op ="},_) $
+ − 874
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
+ − 875
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
+ − 876
=> rtac @{thm bex_rsp} THEN' dtac @{thm QT_ex}
+ − 877
+ − 878
(* (R1 ===> op =) (Bex\<dots>) (Bex\<dots>) ----> \<lbrakk>R1 x y\<rbrakk> \<Longrightarrow> (Bex\<dots>x) = (Bex\<dots>y) *)
+ − 879
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
+ − 880
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
+ − 881
(Const(@{const_name Bex},_) $ (Const (@{const_name Respects}, _) $ _) $ _)
+ − 882
=> rtac @{thm fun_rel_id} THEN' quot_true_tac ctxt unlam
+ − 883
+ − 884
| (_ $
+ − 885
(Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _) $
+ − 886
(Const(@{const_name Babs},_) $ (Const (@{const_name Respects}, _) $ _) $ _))
+ − 887
=> rtac @{thm babs_rsp} THEN' RANGE [quotient_tac ctxt]
+ − 888
+ − 889
(* reflexivity of operators arising from Cong_tac *)
+ − 890
| Const (@{const_name "op ="},_) $ _ $ _
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 891
=> rtac @{thm refl} (*ORELSE'
597
+ − 892
(resolve_tac trans2 THEN' RANGE [
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 893
quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)])*)
597
+ − 894
+ − 895
(* TODO: These patterns should should be somehow combined and generalized... *)
+ − 896
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
+ − 897
(Const (@{const_name fun_rel}, _) $ _ $ _) $
+ − 898
(Const (@{const_name fun_rel}, _) $ _ $ _)
+ − 899
=> rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt
+ − 900
+ − 901
| (Const (@{const_name fun_rel}, _) $ _ $ _) $
+ − 902
(Const (@{const_name prod_rel}, _) $ _ $ _) $
+ − 903
(Const (@{const_name prod_rel}, _) $ _ $ _)
+ − 904
=> rtac @{thm quot_rel_rsp} THEN_ALL_NEW quotient_tac ctxt
+ − 905
+ − 906
(* respectfulness of constants; in particular of a simple relation *)
+ − 907
| _ $ (Const _) $ (Const _) (* fun_rel, list_rel, etc but not equality *)
+ − 908
=> resolve_tac (rsp_rules_get ctxt) THEN_ALL_NEW quotient_tac ctxt
+ − 909
+ − 910
(* R (\<dots>) (Rep (Abs \<dots>)) ----> R (\<dots>) (\<dots>) *)
+ − 911
(* observe ---> *)
+ − 912
| _ $ _ $ _
+ − 913
=> rep_abs_rsp_tac ctxt
+ − 914
+ − 915
| _ => error "inj_repabs_tac not a relation"
+ − 916
) i)
+ − 917
*}
+ − 918
+ − 919
ML {*
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 920
fun inj_repabs_step_tac ctxt rel_refl trans2 =
597
+ − 921
(FIRST' [
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 922
NDT ctxt "0" (inj_repabs_tac_match ctxt trans2),
597
+ − 923
(* R (t $ \<dots>) (t' $ \<dots>) ----> apply_rsp provided type of t needs lifting *)
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 924
597
+ − 925
NDT ctxt "A" (apply_rsp_tac ctxt THEN'
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 926
RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)]),
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 927
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 928
(* TODO/FIXME: I had to move this after the apply_rsp_tac - is this right *)
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 929
NDT ctxt "B" (resolve_tac trans2 THEN'
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 930
RANGE [quot_true_tac ctxt (fst o dest_bcomb), quot_true_tac ctxt (snd o dest_bcomb)]),
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 931
597
+ − 932
(* (op =) (t $ \<dots>) (t' $ \<dots>) ----> Cong provided type of t does not need lifting *)
+ − 933
(* merge with previous tactic *)
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 934
NDT ctxt "C" (Cong_Tac.cong_tac @{thm cong} THEN'
597
+ − 935
(RANGE [quot_true_tac ctxt (fst o dest_comb), quot_true_tac ctxt (snd o dest_comb)])),
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 936
597
+ − 937
(* (op =) (\<lambda>x\<dots>) (\<lambda>x\<dots>) ----> (op =) (\<dots>) (\<dots>) *)
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 938
NDT ctxt "D" (rtac @{thm ext} THEN' quot_true_tac ctxt unlam),
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 939
597
+ − 940
(* resolving with R x y assumptions *)
+ − 941
NDT ctxt "E" (atac),
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 942
597
+ − 943
(* reflexivity of the basic relations *)
+ − 944
(* R \<dots> \<dots> *)
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 945
NDT ctxt "F" (resolve_tac rel_refl)
597
+ − 946
])
+ − 947
*}
+ − 948
+ − 949
ML {*
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 950
fun inj_repabs_tac ctxt =
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 951
let
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 952
val rel_refl = map (fn x => @{thm equivp_reflp} OF [x]) (equiv_rules_get ctxt)
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 953
val trans2 = map (fn x => @{thm equals_rsp} OF [x]) (quotient_rules_get ctxt)
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 954
in
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 955
inj_repabs_step_tac ctxt rel_refl trans2
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 956
end
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 957
*}
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 958
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 959
ML {*
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 960
fun all_inj_repabs_tac ctxt =
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 961
REPEAT_ALL_NEW (inj_repabs_tac ctxt)
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 962
(* if this is too slow we can inline the code above *)
597
+ − 963
*}
+ − 964
+ − 965
section {* Cleaning of the theorem *}
+ − 966
+ − 967
ML {*
+ − 968
fun make_inst lhs t =
+ − 969
let
+ − 970
val _ $ (Abs (_, _, (f as Var (_, Type ("fun", [T, _]))) $ u)) = lhs;
+ − 971
val _ $ (Abs (_, _, g)) = t;
+ − 972
fun mk_abs i t =
+ − 973
if incr_boundvars i u aconv t then Bound i
+ − 974
else (case t of
+ − 975
t1 $ t2 => mk_abs i t1 $ mk_abs i t2
+ − 976
| Abs (s, T, t') => Abs (s, T, mk_abs (i + 1) t')
+ − 977
| Bound j => if i = j then error "make_inst" else t
+ − 978
| _ => t);
+ − 979
in (f, Abs ("x", T, mk_abs 0 g)) end;
+ − 980
*}
+ − 981
602
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 982
(* Since the patterns for the lhs are different; there are 2 different make-insts *)
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 983
ML {*
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 984
fun make_inst2 lhs t =
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 985
let
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 986
val _ $ (Abs (_, _, (_ $ ((f as Var (_, Type ("fun", [T, _]))) $ u)))) = lhs;
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 987
val _ = tracing "a";
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 988
val _ $ (Abs (_, _, (_ $ g))) = t;
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 989
fun mk_abs i t =
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 990
if incr_boundvars i u aconv t then Bound i
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 991
else (case t of
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 992
t1 $ t2 => mk_abs i t1 $ mk_abs i t2
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 993
| Abs (s, T, t') => Abs (s, T, mk_abs (i + 1) t')
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 994
| Bound j => if i = j then error "make_inst" else t
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 995
| _ => t);
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 996
in (f, Abs ("x", T, mk_abs 0 g)) end;
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 997
*}
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 998
597
+ − 999
ML {*
+ − 1000
fun lambda_prs_simple_conv ctxt ctrm =
+ − 1001
case (term_of ctrm) of
602
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1002
((Const (@{const_name fun_map}, _) $ r1 $ a2) $ (Abs _)) =>
608
+ − 1003
(let
597
+ − 1004
val thy = ProofContext.theory_of ctxt
+ − 1005
val (ty_b, ty_a) = dest_fun_type (fastype_of r1)
+ − 1006
val (ty_c, ty_d) = dest_fun_type (fastype_of a2)
+ − 1007
val tyinst = map (SOME o (ctyp_of thy)) [ty_a, ty_b, ty_c, ty_d]
+ − 1008
val tinst = [NONE, NONE, SOME (cterm_of thy r1), NONE, SOME (cterm_of thy a2)]
+ − 1009
val lpi = Drule.instantiate' tyinst tinst @{thm lambda_prs}
+ − 1010
val te = @{thm eq_reflection} OF [solve_quotient_assums ctxt (solve_quotient_assums ctxt lpi)]
+ − 1011
val _ = tracing ("te rule:\n" ^ (Syntax.string_of_term ctxt (prop_of te)));
602
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1012
val ti =
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1013
(let
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1014
val ts = MetaSimplifier.rewrite_rule @{thms id_simps} te
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1015
val (insp, inst) = make_inst (term_of (Thm.lhs_of ts)) (term_of ctrm)
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1016
in
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1017
Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) ts
608
+ − 1018
end handle _ => (* TODO handle only Bind | Error "make_inst" *)
602
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1019
let
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1020
val (insp, inst) = make_inst2 (term_of (Thm.lhs_of te)) (term_of ctrm)
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1021
val td = Drule.instantiate ([], [(cterm_of thy insp, cterm_of thy inst)]) te
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1022
in
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1023
MetaSimplifier.rewrite_rule @{thms id_simps} td
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1024
end);
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1025
val _ = if not (Term.is_Const a2 andalso fst (dest_Const a2) = @{const_name "id"}) then
597
+ − 1026
(tracing "lambda_prs";
+ − 1027
tracing ("redex:\n" ^ (Syntax.string_of_term ctxt (term_of ctrm)));
+ − 1028
tracing ("lpi rule:\n" ^ (Syntax.string_of_term ctxt (prop_of lpi)));
+ − 1029
tracing ("te rule:\n" ^ (Syntax.string_of_term ctxt (prop_of te)));
602
e56eeb9fedb3
make_inst for lambda_prs where the second quotient is not identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
diff
changeset
+ − 1030
tracing ("ti rule:\n" ^ (Syntax.string_of_term ctxt (prop_of ti))))
597
+ − 1031
else ()
+ − 1032
in
+ − 1033
Conv.rewr_conv ti ctrm
+ − 1034
end
608
+ − 1035
handle _ => Conv.all_conv ctrm)
597
+ − 1036
| _ => Conv.all_conv ctrm
+ − 1037
*}
+ − 1038
+ − 1039
ML {*
+ − 1040
val lambda_prs_conv =
+ − 1041
More_Conv.top_conv lambda_prs_simple_conv
+ − 1042
+ − 1043
fun lambda_prs_tac ctxt = CONVERSION (lambda_prs_conv ctxt)
+ − 1044
*}
+ − 1045
+ − 1046
(*
+ − 1047
Cleaning the theorem consists of three rewriting steps.
+ − 1048
The first two need to be done before fun_map is unfolded
+ − 1049
+ − 1050
1) lambda_prs:
+ − 1051
(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) ----> f
+ − 1052
+ − 1053
Implemented as conversion since it is not a pattern.
+ − 1054
+ − 1055
2) all_prs (the same for exists):
+ − 1056
Ball (Respects R) ((abs ---> id) f) ----> All f
+ − 1057
+ − 1058
Rewriting with definitions from the argument defs
+ − 1059
(rep ---> abs) oldConst ----> newconst
+ − 1060
+ − 1061
3) Quotient_rel_rep:
+ − 1062
Rel (Rep x) (Rep y) ----> x = y
+ − 1063
+ − 1064
Quotient_abs_rep:
+ − 1065
Abs (Rep x) ----> x
+ − 1066
+ − 1067
id_simps; fun_map.simps
+ − 1068
*)
+ − 1069
+ − 1070
ML {*
+ − 1071
fun clean_tac lthy =
+ − 1072
let
+ − 1073
val thy = ProofContext.theory_of lthy;
+ − 1074
val defs = map (Thm.varifyT o symmetric o #def) (qconsts_dest thy)
+ − 1075
(* FIXME: shouldn't the definitions already be varified? *)
+ − 1076
val thms1 = @{thms all_prs ex_prs} @ defs
+ − 1077
val thms2 = @{thms eq_reflection[OF fun_map.simps]}
+ − 1078
@ @{thms id_simps Quotient_abs_rep Quotient_rel_rep}
+ − 1079
fun simps thms = (mk_minimal_ss lthy) addsimps thms addSolver quotient_solver
+ − 1080
in
+ − 1081
EVERY' [lambda_prs_tac lthy,
+ − 1082
simp_tac (simps thms1),
+ − 1083
simp_tac (simps thms2),
+ − 1084
TRY o rtac refl]
+ − 1085
end
+ − 1086
*}
+ − 1087
+ − 1088
section {* Genralisation of free variables in a goal *}
+ − 1089
+ − 1090
ML {*
+ − 1091
fun inst_spec ctrm =
+ − 1092
Drule.instantiate' [SOME (ctyp_of_term ctrm)] [NONE, SOME ctrm] @{thm spec}
+ − 1093
+ − 1094
fun inst_spec_tac ctrms =
+ − 1095
EVERY' (map (dtac o inst_spec) ctrms)
+ − 1096
+ − 1097
fun all_list xs trm =
+ − 1098
fold (fn (x, T) => fn t' => HOLogic.mk_all (x, T, t')) xs trm
+ − 1099
+ − 1100
fun apply_under_Trueprop f =
+ − 1101
HOLogic.dest_Trueprop #> f #> HOLogic.mk_Trueprop
+ − 1102
+ − 1103
fun gen_frees_tac ctxt =
+ − 1104
SUBGOAL (fn (concl, i) =>
+ − 1105
let
+ − 1106
val thy = ProofContext.theory_of ctxt
+ − 1107
val vrs = Term.add_frees concl []
+ − 1108
val cvrs = map (cterm_of thy o Free) vrs
+ − 1109
val concl' = apply_under_Trueprop (all_list vrs) concl
+ − 1110
val goal = Logic.mk_implies (concl', concl)
+ − 1111
val rule = Goal.prove ctxt [] [] goal
+ − 1112
(K (EVERY1 [inst_spec_tac (rev cvrs), atac]))
+ − 1113
in
+ − 1114
rtac rule i
+ − 1115
end)
+ − 1116
*}
+ − 1117
+ − 1118
section {* General outline of the lifting procedure *}
+ − 1119
+ − 1120
(* - A is the original raw theorem *)
+ − 1121
(* - B is the regularized theorem *)
+ − 1122
(* - C is the rep/abs injected version of B *)
+ − 1123
(* - D is the lifted theorem *)
+ − 1124
(* *)
+ − 1125
(* - b is the regularization step *)
+ − 1126
(* - c is the rep/abs injection step *)
+ − 1127
(* - d is the cleaning part *)
+ − 1128
+ − 1129
lemma lifting_procedure:
+ − 1130
assumes a: "A"
606
+ − 1131
and b: "A \<longrightarrow> B"
597
+ − 1132
and c: "B = C"
+ − 1133
and d: "C = D"
+ − 1134
shows "D"
+ − 1135
using a b c d
+ − 1136
by simp
+ − 1137
+ − 1138
ML {*
+ − 1139
fun lift_match_error ctxt fun_str rtrm qtrm =
+ − 1140
let
+ − 1141
val rtrm_str = Syntax.string_of_term ctxt rtrm
+ − 1142
val qtrm_str = Syntax.string_of_term ctxt qtrm
+ − 1143
val msg = [enclose "[" "]" fun_str, "The quotient theorem\n", qtrm_str,
+ − 1144
"and the lifted theorem\n", rtrm_str, "do not match"]
+ − 1145
in
+ − 1146
error (space_implode " " msg)
+ − 1147
end
+ − 1148
*}
+ − 1149
+ − 1150
ML {*
+ − 1151
fun procedure_inst ctxt rtrm qtrm =
+ − 1152
let
+ − 1153
val thy = ProofContext.theory_of ctxt
+ − 1154
val rtrm' = HOLogic.dest_Trueprop rtrm
+ − 1155
val qtrm' = HOLogic.dest_Trueprop qtrm
+ − 1156
val reg_goal =
+ − 1157
Syntax.check_term ctxt (regularize_trm ctxt rtrm' qtrm')
+ − 1158
handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm
+ − 1159
val _ = warning "Regularization done."
+ − 1160
val inj_goal =
+ − 1161
Syntax.check_term ctxt (inj_repabs_trm ctxt (reg_goal, qtrm'))
+ − 1162
handle (LIFT_MATCH s) => lift_match_error ctxt s rtrm qtrm
+ − 1163
val _ = warning "RepAbs Injection done."
+ − 1164
in
+ − 1165
Drule.instantiate' []
+ − 1166
[SOME (cterm_of thy rtrm'),
+ − 1167
SOME (cterm_of thy reg_goal),
+ − 1168
SOME (cterm_of thy inj_goal)] @{thm lifting_procedure}
+ − 1169
end
+ − 1170
*}
+ − 1171
+ − 1172
(* Left for debugging *)
+ − 1173
ML {*
+ − 1174
fun procedure_tac ctxt rthm =
+ − 1175
ObjectLogic.full_atomize_tac
+ − 1176
THEN' gen_frees_tac ctxt
+ − 1177
THEN' CSUBGOAL (fn (gl, i) =>
+ − 1178
let
+ − 1179
val rthm' = atomize_thm rthm
+ − 1180
val rule = procedure_inst ctxt (prop_of rthm') (term_of gl)
+ − 1181
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb gl))] @{thm QUOT_TRUE_i}
+ − 1182
in
+ − 1183
(rtac rule THEN' RANGE [rtac rthm', (fn _ => all_tac), rtac thm]) i
+ − 1184
end)
+ − 1185
*}
+ − 1186
+ − 1187
ML {*
+ − 1188
fun lift_tac ctxt rthm =
+ − 1189
ObjectLogic.full_atomize_tac
+ − 1190
THEN' gen_frees_tac ctxt
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1191
THEN' CSUBGOAL (fn (goal, i) =>
597
+ − 1192
let
+ − 1193
val rthm' = atomize_thm rthm
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1194
val rule = procedure_inst ctxt (prop_of rthm') (term_of goal)
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1195
val thm = Drule.instantiate' [] [SOME (snd (Thm.dest_comb goal))] @{thm QUOT_TRUE_i}
597
+ − 1196
in
+ − 1197
(rtac rule THEN'
+ − 1198
RANGE [rtac rthm',
+ − 1199
regularize_tac ctxt,
610
2bee5ca44ef5
removed "global" data and lookup functions; had to move a tactic out from the inj_repabs_match tactic since apply_rsp interferes with a trans2 rule for ===>
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1200
rtac thm THEN' all_inj_repabs_tac ctxt,
597
+ − 1201
clean_tac ctxt]) i
+ − 1202
end)
+ − 1203
*}
+ − 1204
+ − 1205
end
+ − 1206