(* Title: HOLCF/Adm.thy ID: $Id: Adm.thy,v 1.12 2007/10/21 12:21:48 wenzelm Exp $ Author: Franz Regensburger *) header {* Admissibility and compactness *} theory Adm imports Cont begin defaultsort cpo subsection {* Definitions *} definition adm :: "('a::cpo => bool) => bool" where "adm P = (∀Y. chain Y --> (∀i. P (Y i)) --> P (\<Squnion>i. Y i))" definition compact :: "'a::cpo => bool" where "compact k = adm (λx. ¬ k \<sqsubseteq> x)" lemma admI: "(!!Y. [|chain Y; ∀i. P (Y i)|] ==> P (\<Squnion>i. Y i)) ==> adm P" by (unfold adm_def, fast) lemma triv_admI: "∀x. P x ==> adm P" by (rule admI, erule spec) lemma admD: "[|adm P; chain Y; ∀i. P (Y i)|] ==> P (\<Squnion>i. Y i)" by (unfold adm_def, fast) lemma compactI: "adm (λx. ¬ k \<sqsubseteq> x) ==> compact k" by (unfold compact_def) lemma compactD: "compact k ==> adm (λx. ¬ k \<sqsubseteq> x)" by (unfold compact_def) text {* improved admissibility introduction *} lemma admI2: "(!!Y. [|chain Y; ∀i. P (Y i); ∀i. ∃j>i. Y i ≠ Y j ∧ Y i \<sqsubseteq> Y j|] ==> P (\<Squnion>i. Y i)) ==> adm P" apply (rule admI) apply (erule (1) increasing_chain_adm_lemma) apply fast done subsection {* Admissibility on chain-finite types *} text {* for chain-finite (easy) types every formula is admissible *} lemma adm_max_in_chain: "∀Y. chain (Y::nat => 'a) --> (∃n. max_in_chain n Y) ==> adm (P::'a => bool)" by (auto simp add: adm_def maxinch_is_thelub) lemmas adm_chfin = chfin [THEN adm_max_in_chain, standard] lemma compact_chfin: "compact (x::'a::chfin)" by (rule compactI, rule adm_chfin) subsection {* Admissibility of special formulae and propagation *} lemma adm_not_free: "adm (λx. t)" by (rule admI, simp) lemma adm_conj: "[|adm P; adm Q|] ==> adm (λx. P x ∧ Q x)" by (fast elim: admD intro: admI) lemma adm_all: "∀y. adm (P y) ==> adm (λx. ∀y. P y x)" by (fast intro: admI elim: admD) lemma adm_ball: "∀y∈A. adm (P y) ==> adm (λx. ∀y∈A. P y x)" by (fast intro: admI elim: admD) lemmas adm_all2 = adm_all [rule_format] lemmas adm_ball2 = adm_ball [rule_format] text {* Admissibility for disjunction is hard to prove. It takes 5 Lemmas *} lemma adm_disj_lemma1: "[|chain (Y::nat => 'a::cpo); ∀i. ∃j≥i. P (Y j)|] ==> chain (λi. Y (LEAST j. i ≤ j ∧ P (Y j)))" apply (rule chainI) apply (erule chain_mono3) apply (rule Least_le) apply (rule LeastI2_ex) apply simp_all done lemmas adm_disj_lemma2 = LeastI_ex [of "λj. i ≤ j ∧ P (Y j)", standard] lemma adm_disj_lemma3: "[|chain (Y::nat => 'a::cpo); ∀i. ∃j≥i. P (Y j)|] ==> (\<Squnion>i. Y i) = (\<Squnion>i. Y (LEAST j. i ≤ j ∧ P (Y j)))" apply (frule (1) adm_disj_lemma1) apply (rule antisym_less) apply (rule lub_mono [rule_format], assumption+) apply (erule chain_mono3) apply (simp add: adm_disj_lemma2) apply (rule lub_range_mono, fast, assumption+) done lemma adm_disj_lemma4: "[|adm P; chain Y; ∀i. ∃j≥i. P (Y j)|] ==> P (\<Squnion>i. Y i)" apply (subst adm_disj_lemma3, assumption+) apply (erule admD) apply (simp add: adm_disj_lemma1) apply (simp add: adm_disj_lemma2) done lemma adm_disj_lemma5: "∀n::nat. P n ∨ Q n ==> (∀i. ∃j≥i. P j) ∨ (∀i. ∃j≥i. Q j)" apply (erule contrapos_pp) apply (clarsimp, rename_tac a b) apply (rule_tac x="max a b" in exI) apply (simp add: le_maxI1 le_maxI2) done lemma adm_disj: "[|adm P; adm Q|] ==> adm (λx. P x ∨ Q x)" apply (rule admI) apply (erule adm_disj_lemma5 [THEN disjE]) apply (erule (2) adm_disj_lemma4 [THEN disjI1]) apply (erule (2) adm_disj_lemma4 [THEN disjI2]) done lemma adm_imp: "[|adm (λx. ¬ P x); adm Q|] ==> adm (λx. P x --> Q x)" by (subst imp_conv_disj, rule adm_disj) lemma adm_iff: "[|adm (λx. P x --> Q x); adm (λx. Q x --> P x)|] ==> adm (λx. P x = Q x)" by (subst iff_conv_conj_imp, rule adm_conj) lemma adm_not_conj: "[|adm (λx. ¬ P x); adm (λx. ¬ Q x)|] ==> adm (λx. ¬ (P x ∧ Q x))" by (simp add: adm_imp) text {* admissibility and continuity *} lemma adm_less: "[|cont u; cont v|] ==> adm (λx. u x \<sqsubseteq> v x)" apply (rule admI) apply (simp add: cont2contlubE) apply (rule lub_mono) apply (erule (1) ch2ch_cont) apply (erule (1) ch2ch_cont) apply assumption done lemma adm_eq: "[|cont u; cont v|] ==> adm (λx. u x = v x)" by (simp add: po_eq_conv adm_conj adm_less) lemma adm_subst: "[|cont t; adm P|] ==> adm (λx. P (t x))" apply (rule admI) apply (simp add: cont2contlubE) apply (erule admD) apply (erule (1) ch2ch_cont) apply assumption done lemma adm_not_less: "cont t ==> adm (λx. ¬ t x \<sqsubseteq> u)" apply (rule admI) apply (drule_tac x=0 in spec) apply (erule contrapos_nn) apply (erule rev_trans_less) apply (erule cont2mono [THEN monofun_fun_arg]) apply (erule is_ub_thelub) done text {* admissibility and compactness *} lemma adm_compact_not_less: "[|compact k; cont t|] ==> adm (λx. ¬ k \<sqsubseteq> t x)" by (unfold compact_def, erule adm_subst) lemma adm_neq_compact: "[|compact k; cont t|] ==> adm (λx. t x ≠ k)" by (simp add: po_eq_conv adm_imp adm_not_less adm_compact_not_less) lemma adm_compact_neq: "[|compact k; cont t|] ==> adm (λx. k ≠ t x)" by (simp add: po_eq_conv adm_imp adm_not_less adm_compact_not_less) lemma compact_UU [simp, intro]: "compact ⊥" by (rule compactI, simp add: adm_not_free) lemma adm_not_UU: "cont t ==> adm (λx. t x ≠ ⊥)" by (simp add: adm_neq_compact) lemmas adm_lemmas [simp] = adm_not_free adm_conj adm_all2 adm_ball2 adm_disj adm_imp adm_iff adm_less adm_eq adm_not_less adm_compact_not_less adm_compact_neq adm_neq_compact adm_not_UU (* legacy ML bindings *) ML {* val adm_def = thm "adm_def"; val admI = thm "admI"; val triv_admI = thm "triv_admI"; val admD = thm "admD"; val adm_max_in_chain = thm "adm_max_in_chain"; val adm_chfin = thm "adm_chfin"; val admI2 = thm "admI2"; val adm_less = thm "adm_less"; val adm_conj = thm "adm_conj"; val adm_not_free = thm "adm_not_free"; val adm_not_less = thm "adm_not_less"; val adm_all = thm "adm_all"; val adm_all2 = thm "adm_all2"; val adm_ball = thm "adm_ball"; val adm_ball2 = thm "adm_ball2"; val adm_subst = thm "adm_subst"; val adm_not_UU = thm "adm_not_UU"; val adm_eq = thm "adm_eq"; val adm_disj_lemma1 = thm "adm_disj_lemma1"; val adm_disj_lemma2 = thm "adm_disj_lemma2"; val adm_disj_lemma3 = thm "adm_disj_lemma3"; val adm_disj_lemma4 = thm "adm_disj_lemma4"; val adm_disj_lemma5 = thm "adm_disj_lemma5"; val adm_disj = thm "adm_disj"; val adm_imp = thm "adm_imp"; val adm_iff = thm "adm_iff"; val adm_not_conj = thm "adm_not_conj"; val adm_lemmas = thms "adm_lemmas"; *} end
lemma admI:
(!!Y. [| chain Y; ∀i. P (Y i) |] ==> P (LUB i. Y i)) ==> adm P
lemma triv_admI:
∀x. P x ==> adm P
lemma admD:
[| adm P; chain Y; ∀i. P (Y i) |] ==> P (LUB i. Y i)
lemma compactI:
adm (λx. ¬ k << x) ==> compact k
lemma compactD:
compact k ==> adm (λx. ¬ k << x)
lemma admI2:
(!!Y. [| chain Y; ∀i. P (Y i); ∀i. ∃j>i. Y i ≠ Y j ∧ Y i << Y j |]
==> P (LUB i. Y i))
==> adm P
lemma adm_max_in_chain:
∀Y. chain Y --> (∃n. max_in_chain n Y) ==> adm P
lemma adm_chfin:
adm P
lemma compact_chfin:
compact x
lemma adm_not_free:
adm (λx. t)
lemma adm_conj:
[| adm P; adm Q |] ==> adm (λx. P x ∧ Q x)
lemma adm_all:
∀y. adm (P y) ==> adm (λx. ∀y. P y x)
lemma adm_ball:
∀y∈A. adm (P y) ==> adm (λx. ∀y∈A. P y x)
lemma adm_all2:
(!!y. adm (P y)) ==> adm (λx. ∀y. P y x)
lemma adm_ball2:
(!!y. y ∈ A ==> adm (P y)) ==> adm (λx. ∀y∈A. P y x)
lemma adm_disj_lemma1:
[| chain Y; ∀i. ∃j≥i. P (Y j) |] ==> chain (λi. Y (LEAST j. i ≤ j ∧ P (Y j)))
lemma adm_disj_lemma2:
∃x≥i. P (Y x)
==> i ≤ (LEAST j. i ≤ j ∧ P (Y j)) ∧ P (Y (LEAST j. i ≤ j ∧ P (Y j)))
lemma adm_disj_lemma3:
[| chain Y; ∀i. ∃j≥i. P (Y j) |]
==> (LUB i. Y i) = (LUB i. Y (LEAST j. i ≤ j ∧ P (Y j)))
lemma adm_disj_lemma4:
[| adm P; chain Y; ∀i. ∃j≥i. P (Y j) |] ==> P (LUB i. Y i)
lemma adm_disj_lemma5:
∀n. P n ∨ Q n ==> (∀i. ∃j≥i. P j) ∨ (∀i. ∃j≥i. Q j)
lemma adm_disj:
[| adm P; adm Q |] ==> adm (λx. P x ∨ Q x)
lemma adm_imp:
[| adm (λx. ¬ P x); adm Q |] ==> adm (λx. P x --> Q x)
lemma adm_iff:
[| adm (λx. P x --> Q x); adm (λx. Q x --> P x) |] ==> adm (λx. P x = Q x)
lemma adm_not_conj:
[| adm (λx. ¬ P x); adm (λx. ¬ Q x) |] ==> adm (λx. ¬ (P x ∧ Q x))
lemma adm_less:
[| cont u; cont v |] ==> adm (λx. u x << v x)
lemma adm_eq:
[| cont u; cont v |] ==> adm (λx. u x = v x)
lemma adm_subst:
[| cont t; adm P |] ==> adm (λx. P (t x))
lemma adm_not_less:
cont t ==> adm (λx. ¬ t x << u)
lemma adm_compact_not_less:
[| compact k; cont t |] ==> adm (λx. ¬ k << t x)
lemma adm_neq_compact:
[| compact k; cont t |] ==> adm (λx. t x ≠ k)
lemma adm_compact_neq:
[| compact k; cont t |] ==> adm (λx. k ≠ t x)
lemma compact_UU:
compact UU
lemma adm_not_UU:
cont t ==> adm (λx. t x ≠ UU)
lemma adm_lemmas:
adm (λx. t)
[| adm P; adm Q |] ==> adm (λx. P x ∧ Q x)
(!!y. adm (P y)) ==> adm (λx. ∀y. P y x)
(!!y. y ∈ A ==> adm (P y)) ==> adm (λx. ∀y∈A. P y x)
[| adm P; adm Q |] ==> adm (λx. P x ∨ Q x)
[| adm (λx. ¬ P x); adm Q |] ==> adm (λx. P x --> Q x)
[| adm (λx. P x --> Q x); adm (λx. Q x --> P x) |] ==> adm (λx. P x = Q x)
[| cont u; cont v |] ==> adm (λx. u x << v x)
[| cont u; cont v |] ==> adm (λx. u x = v x)
cont t ==> adm (λx. ¬ t x << u)
[| compact k; cont t |] ==> adm (λx. ¬ k << t x)
[| compact k; cont t |] ==> adm (λx. k ≠ t x)
[| compact k; cont t |] ==> adm (λx. t x ≠ k)
cont t ==> adm (λx. t x ≠ UU)