
paulson@11451: (*  Title:      HOL/Hilbert_Choice.thy
paulson@14760:     ID: $Id$
paulson@11451:     Author:     Lawrence C Paulson
paulson@11451:     Copyright   2001  University of Cambridge
wenzelm@12023: *)
paulson@11451: 
paulson@14760: header {* Hilbert's Epsilon-Operator and the Axiom of Choice *}
paulson@11451: 
nipkow@15131: theory Hilbert_Choice
nipkow@15140: imports NatArith
haftmann@16417: uses ("Tools/meson.ML") ("Tools/specification_package.ML")
nipkow@15131: begin
wenzelm@12298: 
wenzelm@12298: subsection {* Hilbert's epsilon *}
wenzelm@12298: 
paulson@11451: consts
paulson@11451:   Eps           :: "('a => bool) => 'a"
paulson@11451: 
wenzelm@14872: syntax (epsilon)
wenzelm@14872:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3\<some>_./ _)" [0, 10] 10)
paulson@11451: syntax (HOL)
wenzelm@12298:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3@ _./ _)" [0, 10] 10)
paulson@11451: syntax
wenzelm@12298:   "_Eps"        :: "[pttrn, bool] => 'a"    ("(3SOME _./ _)" [0, 10] 10)
paulson@11451: translations
nipkow@13764:   "SOME x. P" == "Eps (%x. P)"
nipkow@13763: 
nipkow@13763: print_translation {*
nipkow@13763: (* to avoid eta-contraction of body *)
nipkow@13763: [("Eps", fn [Abs abs] =>
nipkow@13763:      let val (x,t) = atomic_abs_tr' abs
nipkow@13763:      in Syntax.const "_Eps" $ x $ t end)]
nipkow@13763: *}
paulson@11451: 
wenzelm@12298: axioms
wenzelm@12298:   someI: "P (x::'a) ==> P (SOME x. P x)"
paulson@11451: 
paulson@11451: 
wenzelm@12298: constdefs
wenzelm@12298:   inv :: "('a => 'b) => ('b => 'a)"
wenzelm@12298:   "inv(f :: 'a => 'b) == %y. SOME x. f x = y"
paulson@11454: 
wenzelm@12298:   Inv :: "'a set => ('a => 'b) => ('b => 'a)"
paulson@14760:   "Inv A f == %x. SOME y. y \<in> A & f y = x"
paulson@14760: 
paulson@14760: 
paulson@14760: subsection {*Hilbert's Epsilon-operator*}
paulson@14760: 
paulson@14760: text{*Easier to apply than @{text someI} if the witness comes from an
paulson@14760: existential formula*}
paulson@14760: lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
paulson@14760: apply (erule exE)
paulson@14760: apply (erule someI)
paulson@14760: done
paulson@14760: 
paulson@14760: text{*Easier to apply than @{text someI} because the conclusion has only one
paulson@14760: occurrence of @{term P}.*}
paulson@14760: lemma someI2: "[| P a;  !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760: by (blast intro: someI)
paulson@14760: 
paulson@14760: text{*Easier to apply than @{text someI2} if the witness comes from an
paulson@14760: existential formula*}
paulson@14760: lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
paulson@14760: by (blast intro: someI2)
paulson@14760: 
paulson@14760: lemma some_equality [intro]:
paulson@14760:      "[| P a;  !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
paulson@14760: by (blast intro: someI2)
paulson@14760: 
paulson@14760: lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
paulson@14760: by (blast intro: some_equality)
paulson@14760: 
paulson@14760: lemma some_eq_ex: "P (SOME x. P x) =  (\<exists>x. P x)"
paulson@14760: by (blast intro: someI)
paulson@14760: 
paulson@14760: lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
paulson@14760: apply (rule some_equality)
paulson@14760: apply (rule refl, assumption)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
paulson@14760: apply (rule some_equality)
paulson@14760: apply (rule refl)
paulson@14760: apply (erule sym)
paulson@14760: done
paulson@14760: 
paulson@14760: 
paulson@14760: subsection{*Axiom of Choice, Proved Using the Description Operator*}
paulson@14760: 
paulson@14760: text{*Used in @{text "Tools/meson.ML"}*}
paulson@14760: lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
paulson@14760: by (fast elim: someI)
paulson@14760: 
paulson@14760: lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
paulson@14760: by (fast elim: someI)
paulson@14760: 
paulson@14760: 
paulson@14760: subsection {*Function Inverse*}
paulson@14760: 
paulson@14760: lemma inv_id [simp]: "inv id = id"
paulson@14760: by (simp add: inv_def id_def)
paulson@14760: 
paulson@14760: text{*A one-to-one function has an inverse.*}
paulson@14760: lemma inv_f_f [simp]: "inj f ==> inv f (f x) = x"
paulson@14760: by (simp add: inv_def inj_eq)
paulson@14760: 
paulson@14760: lemma inv_f_eq: "[| inj f;  f x = y |] ==> inv f y = x"
paulson@14760: apply (erule subst)
paulson@14760: apply (erule inv_f_f)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma inj_imp_inv_eq: "[| inj f; \<forall>x. f(g x) = x |] ==> inv f = g"
paulson@14760: by (blast intro: ext inv_f_eq)
paulson@14760: 
paulson@14760: text{*But is it useful?*}
paulson@14760: lemma inj_transfer:
paulson@14760:   assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
paulson@14760:   shows "P x"
paulson@14760: proof -
paulson@14760:   have "f x \<in> range f" by auto
paulson@14760:   hence "P(inv f (f x))" by (rule minor)
paulson@14760:   thus "P x" by (simp add: inv_f_f [OF injf])
paulson@14760: qed
paulson@11451: 
paulson@11451: 
paulson@14760: lemma inj_iff: "(inj f) = (inv f o f = id)"
paulson@14760: apply (simp add: o_def expand_fun_eq)
paulson@14760: apply (blast intro: inj_on_inverseI inv_f_f)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
paulson@14760: by (blast intro: surjI inv_f_f)
paulson@14760: 
paulson@14760: lemma f_inv_f: "y \<in> range(f) ==> f(inv f y) = y"
paulson@14760: apply (simp add: inv_def)
paulson@14760: apply (fast intro: someI)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
paulson@14760: by (simp add: f_inv_f surj_range)
paulson@14760: 
paulson@14760: lemma inv_injective:
paulson@14760:   assumes eq: "inv f x = inv f y"
paulson@14760:       and x: "x: range f"
paulson@14760:       and y: "y: range f"
paulson@14760:   shows "x=y"
paulson@14760: proof -
paulson@14760:   have "f (inv f x) = f (inv f y)" using eq by simp
paulson@14760:   thus ?thesis by (simp add: f_inv_f x y) 
paulson@14760: qed
paulson@14760: 
paulson@14760: lemma inj_on_inv: "A <= range(f) ==> inj_on (inv f) A"
paulson@14760: by (fast intro: inj_onI elim: inv_injective injD)
paulson@14760: 
paulson@14760: lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
paulson@14760: by (simp add: inj_on_inv surj_range)
paulson@14760: 
paulson@14760: lemma surj_iff: "(surj f) = (f o inv f = id)"
paulson@14760: apply (simp add: o_def expand_fun_eq)
paulson@14760: apply (blast intro: surjI surj_f_inv_f)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
paulson@14760: apply (rule ext)
paulson@14760: apply (drule_tac x = "inv f x" in spec)
paulson@14760: apply (simp add: surj_f_inv_f)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
paulson@14760: by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
wenzelm@12372: 
paulson@14760: lemma inv_equality: "[| !!x. g (f x) = x;  !!y. f (g y) = y |] ==> inv f = g"
paulson@14760: apply (rule ext)
paulson@14760: apply (auto simp add: inv_def)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
paulson@14760: apply (rule inv_equality)
paulson@14760: apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760: done
paulson@14760: 
paulson@14760: (** bij(inv f) implies little about f.  Consider f::bool=>bool such that
paulson@14760:     f(True)=f(False)=True.  Then it's consistent with axiom someI that
paulson@14760:     inv f could be any function at all, including the identity function.
paulson@14760:     If inv f=id then inv f is a bijection, but inj f, surj(f) and
paulson@14760:     inv(inv f)=f all fail.
paulson@14760: **)
paulson@14760: 
paulson@14760: lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
paulson@14760: apply (rule inv_equality)
paulson@14760: apply (auto simp add: bij_def surj_f_inv_f)
paulson@14760: done
paulson@14760: 
paulson@14760: 
paulson@14760: lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
paulson@14760: by (simp add: image_eq_UN surj_f_inv_f)
paulson@14760: 
paulson@14760: lemma image_inv_f_f: "inj f ==> (inv f) ` (f ` A) = A"
paulson@14760: by (simp add: image_eq_UN)
paulson@14760: 
paulson@14760: lemma inv_image_comp: "inj f ==> inv f ` (f`X) = X"
paulson@14760: by (auto simp add: image_def)
paulson@14760: 
paulson@14760: lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
paulson@14760: apply auto
paulson@14760: apply (force simp add: bij_is_inj)
paulson@14760: apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
paulson@14760: done
paulson@14760: 
paulson@14760: lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A" 
paulson@14760: apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
paulson@14760: apply (blast intro: bij_is_inj [THEN inv_f_f, symmetric])
paulson@14760: done
paulson@14760: 
paulson@14760: 
paulson@14760: subsection {*Inverse of a PI-function (restricted domain)*}
paulson@14760: 
paulson@14760: lemma Inv_f_f: "[| inj_on f A;  x \<in> A |] ==> Inv A f (f x) = x"
paulson@14760: apply (simp add: Inv_def inj_on_def)
paulson@14760: apply (blast intro: someI2)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma f_Inv_f: "y \<in> f`A  ==> f (Inv A f y) = y"
paulson@14760: apply (simp add: Inv_def)
paulson@13585: apply (fast intro: someI2)
paulson@13585: done
paulson@11451: 
paulson@14760: lemma Inv_injective:
paulson@14760:   assumes eq: "Inv A f x = Inv A f y"
paulson@14760:       and x: "x: f`A"
paulson@14760:       and y: "y: f`A"
paulson@14760:   shows "x=y"
paulson@14760: proof -
paulson@14760:   have "f (Inv A f x) = f (Inv A f y)" using eq by simp
paulson@14760:   thus ?thesis by (simp add: f_Inv_f x y) 
paulson@14760: qed
paulson@14760: 
paulson@14760: lemma inj_on_Inv: "B <= f`A ==> inj_on (Inv A f) B"
paulson@14760: apply (rule inj_onI)
paulson@14760: apply (blast intro: inj_onI dest: Inv_injective injD)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma Inv_mem: "[| f ` A = B;  x \<in> B |] ==> Inv A f x \<in> A"
paulson@14760: apply (simp add: Inv_def)
paulson@14760: apply (fast intro: someI2)
paulson@14760: done
paulson@14760: 
paulson@14760: lemma Inv_f_eq: "[| inj_on f A; f x = y; x \<in> A |] ==> Inv A f y = x"
ballarin@14399:   apply (erule subst)
paulson@14760:   apply (erule Inv_f_f, assumption)
ballarin@14399:   done
ballarin@14399: 
ballarin@14399: lemma Inv_comp:
paulson@14760:   "[| inj_on f (g ` A); inj_on g A; x \<in> f ` g ` A |] ==>
ballarin@14399:   Inv A (f o g) x = (Inv A g o Inv (g ` A) f) x"
ballarin@14399:   apply simp
ballarin@14399:   apply (rule Inv_f_eq)
ballarin@14399:     apply (fast intro: comp_inj_on)
ballarin@14399:    apply (simp add: f_Inv_f Inv_mem)
ballarin@14399:   apply (simp add: Inv_mem)
ballarin@14399:   done
ballarin@14399: 
paulson@14760: 
paulson@14760: subsection {*Other Consequences of Hilbert's Epsilon*}
paulson@14760: 
paulson@14760: text {*Hilbert's Epsilon and the @{term split} Operator*}
paulson@14760: 
paulson@14760: text{*Looping simprule*}
paulson@14760: lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
paulson@14760: by (simp add: split_Pair_apply)
paulson@14760: 
paulson@14760: lemma Eps_split: "Eps (split P) = (SOME xy. P (fst xy) (snd xy))"
paulson@14760: by (simp add: split_def)
paulson@14760: 
paulson@14760: lemma Eps_split_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: 
paulson@14760: text{*A relation is wellfounded iff it has no infinite descending chain*}
paulson@14760: lemma wf_iff_no_infinite_down_chain:
paulson@14760:   "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
paulson@14760: apply (simp only: wf_eq_minimal)
paulson@14760: apply (rule iffI)
paulson@14760:  apply (rule notI)
paulson@14760:  apply (erule exE)
paulson@14760:  apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
paulson@14760: apply (erule contrapos_np, simp, clarify)
paulson@14760: apply (subgoal_tac "\<forall>n. nat_rec x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
paulson@14760:  apply (rule_tac x = "nat_rec x (%i y. @z. z:Q & (z,y) :r)" in exI)
paulson@14760:  apply (rule allI, simp)
paulson@14760:  apply (rule someI2_ex, blast, blast)
paulson@14760: apply (rule allI)
paulson@14760: apply (induct_tac "n", simp_all)
paulson@14760: apply (rule someI2_ex, blast+)
paulson@14760: done
paulson@14760: 
paulson@14760: text{*A dynamically-scoped fact for TFL *}
wenzelm@12298: lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
wenzelm@12298:   by (blast intro: someI)
paulson@11451: 
wenzelm@12298: 
wenzelm@12298: subsection {* Least value operator *}
paulson@11451: 
paulson@11451: constdefs
wenzelm@12298:   LeastM :: "['a => 'b::ord, 'a => bool] => 'a"
paulson@14760:   "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
paulson@11451: 
paulson@11451: syntax
wenzelm@12298:   "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"    ("LEAST _ WRT _. _" [0, 4, 10] 10)
paulson@11451: translations
wenzelm@12298:   "LEAST x WRT m. P" == "LeastM m (%x. P)"
paulson@11451: 
paulson@11451: lemma LeastMI2:
wenzelm@12298:   "P x ==> (!!y. P y ==> m x <= m y)
wenzelm@12298:     ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
wenzelm@12298:     ==> Q (LeastM m P)"
paulson@14760:   apply (simp add: LeastM_def)
paulson@14208:   apply (rule someI2_ex, blast, blast)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma LeastM_equality:
wenzelm@12298:   "P k ==> (!!x. P x ==> m k <= m x)
wenzelm@12298:     ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
paulson@14208:   apply (rule LeastMI2, assumption, blast)
wenzelm@12298:   apply (blast intro!: order_antisym)
wenzelm@12298:   done
paulson@11451: 
paulson@11454: lemma wf_linord_ex_has_least:
paulson@14760:   "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
paulson@14760:     ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
wenzelm@12298:   apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
paulson@14208:   apply (drule_tac x = "m`Collect P" in spec, force)
wenzelm@12298:   done
paulson@11454: 
paulson@11454: lemma ex_has_least_nat:
paulson@14760:     "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
wenzelm@12298:   apply (simp only: pred_nat_trancl_eq_le [symmetric])
wenzelm@12298:   apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
paulson@16796:    apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
wenzelm@12298:   done
paulson@11454: 
wenzelm@12298: lemma LeastM_nat_lemma:
paulson@14760:     "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
paulson@14760:   apply (simp add: LeastM_def)
wenzelm@12298:   apply (rule someI_ex)
wenzelm@12298:   apply (erule ex_has_least_nat)
wenzelm@12298:   done
paulson@11454: 
paulson@11454: lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1, standard]
paulson@11454: 
paulson@11454: lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
paulson@14208: by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
paulson@11454: 
paulson@11451: 
wenzelm@12298: subsection {* Greatest value operator *}
paulson@11451: 
paulson@11451: constdefs
wenzelm@12298:   GreatestM :: "['a => 'b::ord, 'a => bool] => 'a"
paulson@14760:   "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
wenzelm@12298: 
wenzelm@12298:   Greatest :: "('a::ord => bool) => 'a"    (binder "GREATEST " 10)
wenzelm@12298:   "Greatest == GreatestM (%x. x)"
paulson@11451: 
paulson@11451: syntax
wenzelm@12298:   "_GreatestM" :: "[pttrn, 'a=>'b::ord, bool] => 'a"
wenzelm@12298:       ("GREATEST _ WRT _. _" [0, 4, 10] 10)
paulson@11451: 
paulson@11451: translations
wenzelm@12298:   "GREATEST x WRT m. P" == "GreatestM m (%x. P)"
paulson@11451: 
paulson@11451: lemma GreatestMI2:
wenzelm@12298:   "P x ==> (!!y. P y ==> m y <= m x)
wenzelm@12298:     ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
wenzelm@12298:     ==> Q (GreatestM m P)"
paulson@14760:   apply (simp add: GreatestM_def)
paulson@14208:   apply (rule someI2_ex, blast, blast)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma GreatestM_equality:
wenzelm@12298:  "P k ==> (!!x. P x ==> m x <= m k)
wenzelm@12298:     ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
paulson@14208:   apply (rule_tac m = m in GreatestMI2, assumption, blast)
wenzelm@12298:   apply (blast intro!: order_antisym)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma Greatest_equality:
wenzelm@12298:   "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
paulson@14760:   apply (simp add: Greatest_def)
paulson@14208:   apply (erule GreatestM_equality, blast)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemma ex_has_greatest_nat_lemma:
paulson@14760:   "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
paulson@14760:     ==> \<exists>y. P y & ~ (m y < m k + n)"
paulson@15251:   apply (induct n, force)
wenzelm@12298:   apply (force simp add: le_Suc_eq)
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma ex_has_greatest_nat:
paulson@14760:   "P k ==> \<forall>y. P y --> m y < b
paulson@14760:     ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
wenzelm@12298:   apply (rule ccontr)
wenzelm@12298:   apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
paulson@14208:     apply (subgoal_tac [3] "m k <= b", auto)
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma GreatestM_nat_lemma:
paulson@14760:   "P k ==> \<forall>y. P y --> m y < b
paulson@14760:     ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
paulson@14760:   apply (simp add: GreatestM_def)
wenzelm@12298:   apply (rule someI_ex)
paulson@14208:   apply (erule ex_has_greatest_nat, assumption)
wenzelm@12298:   done
paulson@11451: 
paulson@11451: lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1, standard]
paulson@11451: 
wenzelm@12298: lemma GreatestM_nat_le:
paulson@14760:   "P x ==> \<forall>y. P y --> m y < b
wenzelm@12298:     ==> (m x::nat) <= m (GreatestM m P)"
wenzelm@12298:   apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec])
wenzelm@12298:   done
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: text {* \medskip Specialization to @{text GREATEST}. *}
wenzelm@12298: 
paulson@14760: lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
paulson@14760:   apply (simp add: Greatest_def)
paulson@14208:   apply (rule GreatestM_natI, auto)
wenzelm@12298:   done
paulson@11451: 
wenzelm@12298: lemma Greatest_le:
paulson@14760:     "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
paulson@14760:   apply (simp add: Greatest_def)
paulson@14208:   apply (rule GreatestM_nat_le, auto)
wenzelm@12298:   done
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: subsection {* The Meson proof procedure *}
paulson@11451: 
wenzelm@12298: subsubsection {* Negation Normal Form *}
wenzelm@12298: 
wenzelm@12298: text {* de Morgan laws *}
wenzelm@12298: 
wenzelm@12298: lemma meson_not_conjD: "~(P&Q) ==> ~P | ~Q"
wenzelm@12298:   and meson_not_disjD: "~(P|Q) ==> ~P & ~Q"
wenzelm@12298:   and meson_not_notD: "~~P ==> P"
paulson@14760:   and meson_not_allD: "!!P. ~(\<forall>x. P(x)) ==> \<exists>x. ~P(x)"
paulson@14760:   and meson_not_exD: "!!P. ~(\<exists>x. P(x)) ==> \<forall>x. ~P(x)"
wenzelm@12298:   by fast+
paulson@11451: 
wenzelm@12298: text {* Removal of @{text "-->"} and @{text "<->"} (positive and
wenzelm@12298: negative occurrences) *}
wenzelm@12298: 
wenzelm@12298: lemma meson_imp_to_disjD: "P-->Q ==> ~P | Q"
wenzelm@12298:   and meson_not_impD: "~(P-->Q) ==> P & ~Q"
wenzelm@12298:   and meson_iff_to_disjD: "P=Q ==> (~P | Q) & (~Q | P)"
wenzelm@12298:   and meson_not_iffD: "~(P=Q) ==> (P | Q) & (~P | ~Q)"
wenzelm@12298:     -- {* Much more efficient than @{prop "(P & ~Q) | (Q & ~P)"} for computing CNF *}
wenzelm@12298:   by fast+
wenzelm@12298: 
wenzelm@12298: 
wenzelm@12298: subsubsection {* Pulling out the existential quantifiers *}
wenzelm@12298: 
wenzelm@12298: text {* Conjunction *}
wenzelm@12298: 
paulson@14760: lemma meson_conj_exD1: "!!P Q. (\<exists>x. P(x)) & Q ==> \<exists>x. P(x) & Q"
paulson@14760:   and meson_conj_exD2: "!!P Q. P & (\<exists>x. Q(x)) ==> \<exists>x. P & Q(x)"
wenzelm@12298:   by fast+
wenzelm@12298: 
paulson@11451: 
wenzelm@12298: text {* Disjunction *}
wenzelm@12298: 
paulson@14760: lemma meson_disj_exD: "!!P Q. (\<exists>x. P(x)) | (\<exists>x. Q(x)) ==> \<exists>x. P(x) | Q(x)"
wenzelm@12298:   -- {* DO NOT USE with forall-Skolemization: makes fewer schematic variables!! *}
wenzelm@12298:   -- {* With ex-Skolemization, makes fewer Skolem constants *}
paulson@14760:   and meson_disj_exD1: "!!P Q. (\<exists>x. P(x)) | Q ==> \<exists>x. P(x) | Q"
paulson@14760:   and meson_disj_exD2: "!!P Q. P | (\<exists>x. Q(x)) ==> \<exists>x. P | Q(x)"
wenzelm@12298:   by fast+
wenzelm@12298: 
paulson@11451: 
wenzelm@12298: subsubsection {* Generating clauses for the Meson Proof Procedure *}
wenzelm@12298: 
wenzelm@12298: text {* Disjunctions *}
wenzelm@12298: 
wenzelm@12298: lemma meson_disj_assoc: "(P|Q)|R ==> P|(Q|R)"
wenzelm@12298:   and meson_disj_comm: "P|Q ==> Q|P"
wenzelm@12298:   and meson_disj_FalseD1: "False|P ==> P"
wenzelm@12298:   and meson_disj_FalseD2: "P|False ==> P"
wenzelm@12298:   by fast+
paulson@11451: 
paulson@14760: 
paulson@14760: subsection{*Lemmas for Meson, the Model Elimination Procedure*}
paulson@14760: 
paulson@14760: 
paulson@14760: text{* Generation of contrapositives *}
paulson@14760: 
paulson@14760: text{*Inserts negated disjunct after removing the negation; P is a literal.
paulson@14760:   Model elimination requires assuming the negation of every attempted subgoal,
paulson@14760:   hence the negated disjuncts.*}
paulson@14760: lemma make_neg_rule: "~P|Q ==> ((~P==>P) ==> Q)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: text{*Version for Plaisted's "Postive refinement" of the Meson procedure*}
paulson@14760: lemma make_refined_neg_rule: "~P|Q ==> (P ==> Q)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: text{*@{term P} should be a literal*}
paulson@14760: lemma make_pos_rule: "P|Q ==> ((P==>~P) ==> Q)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: text{*Versions of @{text make_neg_rule} and @{text make_pos_rule} that don't
paulson@14760: insert new assumptions, for ordinary resolution.*}
paulson@14760: 
paulson@14760: lemmas make_neg_rule' = make_refined_neg_rule
paulson@14760: 
paulson@14760: lemma make_pos_rule': "[|P|Q; ~P|] ==> Q"
paulson@14760: by blast
paulson@14760: 
paulson@14760: text{* Generation of a goal clause -- put away the final literal *}
paulson@14760: 
paulson@14760: lemma make_neg_goal: "~P ==> ((~P==>P) ==> False)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: lemma make_pos_goal: "P ==> ((P==>~P) ==> False)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: 
paulson@14760: subsubsection{* Lemmas for Forward Proof*}
paulson@14760: 
paulson@14760: text{*There is a similarity to congruence rules*}
paulson@14760: 
paulson@14760: (*NOTE: could handle conjunctions (faster?) by
paulson@14760:     nf(th RS conjunct2) RS (nf(th RS conjunct1) RS conjI) *)
paulson@14760: lemma conj_forward: "[| P'&Q';  P' ==> P;  Q' ==> Q |] ==> P&Q"
paulson@14760: by blast
paulson@14760: 
paulson@14760: lemma disj_forward: "[| P'|Q';  P' ==> P;  Q' ==> Q |] ==> P|Q"
paulson@14760: by blast
paulson@14760: 
paulson@14760: (*Version of @{text disj_forward} for removal of duplicate literals*)
paulson@14760: lemma disj_forward2:
paulson@14760:     "[| P'|Q';  P' ==> P;  [| Q'; P==>False |] ==> Q |] ==> P|Q"
paulson@14760: apply blast 
paulson@14760: done
paulson@14760: 
paulson@14760: lemma all_forward: "[| \<forall>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<forall>x. P(x)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: lemma ex_forward: "[| \<exists>x. P'(x);  !!x. P'(x) ==> P(x) |] ==> \<exists>x. P(x)"
paulson@14760: by blast
paulson@14760: 
paulson@14760: ML
paulson@14760: {*
paulson@14760: val inv_def = thm "inv_def";
paulson@14760: val Inv_def = thm "Inv_def";
paulson@14760: 
paulson@14760: val someI = thm "someI";
paulson@14760: val someI_ex = thm "someI_ex";
paulson@14760: val someI2 = thm "someI2";
paulson@14760: val someI2_ex = thm "someI2_ex";
paulson@14760: val some_equality = thm "some_equality";
paulson@14760: val some1_equality = thm "some1_equality";
paulson@14760: val some_eq_ex = thm "some_eq_ex";
paulson@14760: val some_eq_trivial = thm "some_eq_trivial";
paulson@14760: val some_sym_eq_trivial = thm "some_sym_eq_trivial";
paulson@14760: val choice = thm "choice";
paulson@14760: val bchoice = thm "bchoice";
paulson@14760: val inv_id = thm "inv_id";
paulson@14760: val inv_f_f = thm "inv_f_f";
paulson@14760: val inv_f_eq = thm "inv_f_eq";
paulson@14760: val inj_imp_inv_eq = thm "inj_imp_inv_eq";
paulson@14760: val inj_transfer = thm "inj_transfer";
paulson@14760: val inj_iff = thm "inj_iff";
paulson@14760: val inj_imp_surj_inv = thm "inj_imp_surj_inv";
paulson@14760: val f_inv_f = thm "f_inv_f";
paulson@14760: val surj_f_inv_f = thm "surj_f_inv_f";
paulson@14760: val inv_injective = thm "inv_injective";
paulson@14760: val inj_on_inv = thm "inj_on_inv";
paulson@14760: val surj_imp_inj_inv = thm "surj_imp_inj_inv";
paulson@14760: val surj_iff = thm "surj_iff";
paulson@14760: val surj_imp_inv_eq = thm "surj_imp_inv_eq";
paulson@14760: val bij_imp_bij_inv = thm "bij_imp_bij_inv";
paulson@14760: val inv_equality = thm "inv_equality";
paulson@14760: val inv_inv_eq = thm "inv_inv_eq";
paulson@14760: val o_inv_distrib = thm "o_inv_distrib";
paulson@14760: val image_surj_f_inv_f = thm "image_surj_f_inv_f";
paulson@14760: val image_inv_f_f = thm "image_inv_f_f";
paulson@14760: val inv_image_comp = thm "inv_image_comp";
paulson@14760: val bij_image_Collect_eq = thm "bij_image_Collect_eq";
paulson@14760: val bij_vimage_eq_inv_image = thm "bij_vimage_eq_inv_image";
paulson@14760: val Inv_f_f = thm "Inv_f_f";
paulson@14760: val f_Inv_f = thm "f_Inv_f";
paulson@14760: val Inv_injective = thm "Inv_injective";
paulson@14760: val inj_on_Inv = thm "inj_on_Inv";
paulson@14760: val split_paired_Eps = thm "split_paired_Eps";
paulson@14760: val Eps_split = thm "Eps_split";
paulson@14760: val Eps_split_eq = thm "Eps_split_eq";
paulson@14760: val wf_iff_no_infinite_down_chain = thm "wf_iff_no_infinite_down_chain";
paulson@14760: val Inv_mem = thm "Inv_mem";
paulson@14760: val Inv_f_eq = thm "Inv_f_eq";
paulson@14760: val Inv_comp = thm "Inv_comp";
paulson@14760: val tfl_some = thm "tfl_some";
paulson@14760: val make_neg_rule = thm "make_neg_rule";
paulson@14760: val make_refined_neg_rule = thm "make_refined_neg_rule";
paulson@14760: val make_pos_rule = thm "make_pos_rule";
paulson@14760: val make_neg_rule' = thm "make_neg_rule'";
paulson@14760: val make_pos_rule' = thm "make_pos_rule'";
paulson@14760: val make_neg_goal = thm "make_neg_goal";
paulson@14760: val make_pos_goal = thm "make_pos_goal";
paulson@14760: val conj_forward = thm "conj_forward";
paulson@14760: val disj_forward = thm "disj_forward";
paulson@14760: val disj_forward2 = thm "disj_forward2";
paulson@14760: val all_forward = thm "all_forward";
paulson@14760: val ex_forward = thm "ex_forward";
paulson@14760: *}
paulson@14760: 
paulson@14760: 
paulson@11451: use "Tools/meson.ML"
paulson@16563: setup Meson.skolemize_setup
paulson@11451: 
skalberg@14115: use "Tools/specification_package.ML"
skalberg@14115: 
paulson@11451: end




