fact stringlengths 5 2k | type stringclasses 22
values | library stringclasses 1
value | imports listlengths 0 15.1k | filename stringlengths 13 83 | symbolic_name stringlengths 1 82 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
psp : "\<lbrakk>F \<in> A \<longmapsto> A'; F \<in> B co B'; st_set(B')\<rbrakk>\<Longrightarrow> F:(A \<inter> B') \<longmapsto> ((A' \<inter> B) \<union> (B' - B))" apply (subgoal_tac "F \<in> program \<and> st_set (A) \<and> st_set (A') \<and> st_set (B) ") prefer 2 apply (blast dest!: constrainsD2 leadsToD2) apply ... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | psp | |
psp2 : "\<lbrakk>F \<in> A \<longmapsto> A'; F \<in> B co B'; st_set(B')\<rbrakk> \<Longrightarrow> F \<in> (B' \<inter> A) \<longmapsto> ((B \<inter> A') \<union> (B' - B))" by (simp (no_asm_simp) add: psp Int_ac) | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | psp2 | |
psp_unless : "\<lbrakk>F \<in> A \<longmapsto> A'; F \<in> B unless B'; st_set(B); st_set(B')\<rbrakk> \<Longrightarrow> F \<in> (A \<inter> B) \<longmapsto> ((A' \<inter> B) \<union> B')" unfolding unless_def apply (subgoal_tac "st_set (A) \<and>st_set (A') ") prefer 2 apply (blast dest: leadsToD2) apply (drule psp, a... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | psp_unless | |
leadsTo_wf_induct_aux : "\<lbrakk>wf(r); m \<in> I; field(r)<=I; F \<in> program; st_set(B); \<forall>m \<in> I. F \<in> (A \<inter> f-``{m}) \<longmapsto> ((A \<inter> f-``(converse(r)``{m})) \<union> B)\<rbrakk> \<Longrightarrow> F \<in> (A \<inter> f-``{m}) \<longmapsto> B" apply (erule_tac a = m in wf_induct2, simp... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_wf_induct_aux | |
leadsTo_wf_induct : "\<lbrakk>wf(r); field(r)<=I; A<=f-``I; F \<in> program; st_set(A); st_set(B); \<forall>m \<in> I. F \<in> (A \<inter> f-``{m}) \<longmapsto> ((A \<inter> f-``(converse(r)``{m})) \<union> B)\<rbrakk> \<Longrightarrow> F \<in> A \<longmapsto> B" apply (rule_tac b = A in subst) defer 1 apply (rule_tac... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_wf_induct | |
nat_measure_field : "field(measure(nat, \<lambda>x. x)) = nat" unfolding field_def apply (simp add: measure_def) apply (rule equalityI, force, clarify) apply (erule_tac V = "x\<notin>range (y)" for y in thin_rl) apply (erule nat_induct) apply (rule_tac [2] b = "succ (succ (xa))" in domainI) apply (rule_tac b = "succ (0... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | nat_measure_field | |
Image_inverse_lessThan : "k<A \<Longrightarrow> measure(A, \<lambda>x. x) -`` {k} = k" apply (rule equalityI) apply (auto simp add: measure_def) apply (blast intro: ltD) apply (rule vimageI) prefer 2 apply blast apply (simp add: lt_Ord lt_Ord2 Ord_mem_iff_lt) apply (blast intro: lt_trans) done (*Alternative proof is vi... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | Image_inverse_lessThan | |
lessThan_induct : "\<lbrakk>A<=f-``nat; F \<in> program; st_set(A); st_set(B); \<forall>m \<in> nat. F:(A \<inter> f-``{m}) \<longmapsto> ((A \<inter> f -`` m) \<union> B)\<rbrakk> \<Longrightarrow> F \<in> A \<longmapsto> B" apply (rule_tac A1 = nat and f1 = "\<lambda>x. x" in wf_measure [THEN leadsTo_wf_induct]) appl... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | lessThan_induct | |
wlt_type : "wlt(F,B) <=state" by (unfold wlt_def, auto) | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | wlt_type | |
wlt_st_set : "st_set(wlt(F, B))" unfolding st_set_def apply (rule wlt_type) done declare wlt_st_set [iff] | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | wlt_st_set | |
wlt_leadsTo_iff : "F \<in> wlt(F, B) \<longmapsto> B \<longleftrightarrow> (F \<in> program \<and> st_set(B))" unfolding wlt_def apply (blast dest: leadsToD2 intro!: leadsTo_Union) done (* \<lbrakk>F \<in> program; st_set(B)\<rbrakk> \<Longrightarrow> F \<in> wlt(F, B) \<longmapsto> B *) lemmas wlt_leadsTo = conjI [THE... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | wlt_leadsTo_iff | |
leadsTo_subset : "F \<in> A \<longmapsto> B \<Longrightarrow> A \<subseteq> wlt(F, B)" unfolding wlt_def apply (frule leadsToD2) apply (auto simp add: st_set_def) done (*Misra's property W2*) | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_subset | |
leadsTo_eq_subset_wlt : "F \<in> A \<longmapsto> B \<longleftrightarrow> (A \<subseteq> wlt(F,B) \<and> F \<in> program \<and> st_set(B))" apply auto apply (blast dest: leadsToD2 leadsTo_subset intro: leadsTo_weaken_L wlt_leadsTo)+ done (*Misra's property W4*) | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_eq_subset_wlt | |
wlt_increasing : "\<lbrakk>F \<in> program; st_set(B)\<rbrakk> \<Longrightarrow> B \<subseteq> wlt(F,B)" apply (rule leadsTo_subset) apply (simp (no_asm_simp) add: leadsTo_eq_subset_wlt [THEN iff_sym] subset_imp_leadsTo) done (*Used in the Trans case below*) | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | wlt_increasing | |
leadsTo_123_aux : "\<lbrakk>B \<subseteq> A2; F \<in> (A1 - B) co (A1 \<union> B); F \<in> (A2 - C) co (A2 \<union> C)\<rbrakk> \<Longrightarrow> F \<in> (A1 \<union> A2 - C) co (A1 \<union> A2 \<union> C)" apply (unfold constrains_def st_set_def, blast) done (*Lemma (1,2,3) of Misra's draft book, Chapter 4, "Progress"... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_123_aux | |
leadsTo_123 : "F \<in> A \<longmapsto> A' \<Longrightarrow> \<exists>B \<in> Pow(state). A<=B \<and> F \<in> B \<longmapsto> A' \<and> F \<in> (B-A') co (B \<union> A')" apply (frule leadsToD2) apply (erule leadsTo_induct) txt\<open>Basis\<close> apply (blast dest: ensuresD constrainsD2 st_setD) txt\<open>Trans\<close>... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | leadsTo_123 | |
wlt_constrains_wlt : "\<lbrakk>F \<in> program; st_set(B)\<rbrakk> \<Longrightarrow>F \<in> (wlt(F, B) - B) co (wlt(F,B))" apply (cut_tac F = F in wlt_leadsTo [THEN leadsTo_123], assumption, blast) apply clarify apply (subgoal_tac "Ba = wlt (F,B) ") prefer 2 apply (blast dest: leadsTo_eq_subset_wlt [THEN iffD1], clarif... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | wlt_constrains_wlt | |
completion_aux : "\<lbrakk>W = wlt(F, (B' \<union> C)); F \<in> A \<longmapsto> (A' \<union> C); F \<in> A' co (A' \<union> C); F \<in> B \<longmapsto> (B' \<union> C); F \<in> B' co (B' \<union> C)\<rbrakk> \<Longrightarrow> F \<in> (A \<inter> B) \<longmapsto> ((A' \<inter> B') \<union> C)" apply (subgoal_tac "st_set... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | completion_aux | |
finite_completion_aux : "\<lbrakk>I \<in> Fin(X); F \<in> program; st_set(C)\<rbrakk> \<Longrightarrow> (\<forall>i \<in> I. F \<in> (A(i)) \<longmapsto> (A'(i) \<union> C)) \<longrightarrow> (\<forall>i \<in> I. F \<in> (A'(i)) co (A'(i) \<union> C)) \<longrightarrow> F \<in> (\<Inter>i \<in> I. A(i)) \<longmapsto> ((... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | finite_completion_aux | |
finite_completion : "\<lbrakk>I \<in> Fin(X); \<And>i. i \<in> I \<Longrightarrow> F \<in> A(i) \<longmapsto> (A'(i) \<union> C); \<And>i. i \<in> I \<Longrightarrow> F \<in> A'(i) co (A'(i) \<union> C); F \<in> program; st_set(C)\<rbrakk> \<Longrightarrow> F \<in> (\<Inter>i \<in> I. A(i)) \<longmapsto> ((\<Inter>i \<... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | finite_completion | |
stable_completion : "\<lbrakk>F \<in> A \<longmapsto> A'; F \<in> stable(A'); F \<in> B \<longmapsto> B'; F \<in> stable(B')\<rbrakk> \<Longrightarrow> F \<in> (A \<inter> B) \<longmapsto> (A' \<inter> B')" unfolding stable_def apply (rule_tac C1 = 0 in completion [THEN leadsTo_weaken_R], simp+) apply (blast dest: lead... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | stable_completion | |
finite_stable_completion : "\<lbrakk>I \<in> Fin(X); (\<And>i. i \<in> I \<Longrightarrow> F \<in> A(i) \<longmapsto> A'(i)); (\<And>i. i \<in> I \<Longrightarrow> F \<in> stable(A'(i))); F \<in> program\<rbrakk> \<Longrightarrow> F \<in> (\<Inter>i \<in> I. A(i)) \<longmapsto> (\<Inter>i \<in> I. A'(i))" unfolding sta... | lemma | src | [
"UNITY",
"ZFC"
] | src/ZF/UNITY/WFair.thy | finite_stable_completion |
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