statement stringlengths 1 44.9k | proof stringlengths 0 114k | type stringclasses 26
values | symbolic_name stringlengths 1 82 | library stringclasses 153
values | filename stringlengths 13 83 | imports listlengths 0 0 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
HTT_coinduct: "\<lbrakk>t : R; R <= HTTgen(R)\<rbrakk> \<Longrightarrow> t : HTT" | apply (erule HTT_def [THEN def_coinduct])
apply assumption
done | lemma | HTT_coinduct | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTgen",
"apply",
"def_coinduct"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_coinduct3: "\<lbrakk>t : R; R <= HTTgen(lfp(\<lambda>x. HTTgen(x) Un R Un HTT))\<rbrakk> \<Longrightarrow> t : HTT" | apply (erule HTTgen_mono [THEN [3] HTT_def [THEN def_coinduct3]])
apply assumption
done | lemma | HTT_coinduct3 | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTgen",
"HTTgen_mono",
"Un",
"apply",
"def_coinduct3",
"lambda",
"lfp"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTTgenIs:
"true : HTTgen(R)"
"false : HTTgen(R)"
"\<lbrakk>a : R; b : R\<rbrakk> \<Longrightarrow> <a,b> : HTTgen(R)"
"\<And>b. (\<And>x. b(x) : R) \<Longrightarrow> lam x. b(x) : HTTgen(R)"
"one : HTTgen(R)"
"a : lfp(\<lambda>x. HTTgen(x) Un R Un HTT) \<Longrightarrow> inl(a) : HTTgen(lfp(\<lambda>x. HTTge... | unfolding data_defs by (genIs HTTgenXH HTTgen_mono)+ | lemma | HTTgenIs | CCL | src/CCL/Hered.thy | [] | [
"And",
"HTT",
"HTTgen",
"HTTgenXH",
"HTTgen_mono",
"Un",
"false",
"inl",
"inr",
"lam",
"lambda",
"lfp",
"one",
"succ",
"true",
"zero"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnitF: "Unit <= HTT" | by (simp add: subsetXH UnitXH HTT_rews) | lemma | UnitF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"Unit",
"add",
"subsetXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
BoolF: "Bool <= HTT" | by (fastforce simp: subsetXH BoolXH iff: HTT_rews) | lemma | BoolF | CCL | src/CCL/Hered.thy | [] | [
"Bool",
"HTT",
"iff",
"subsetXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
PlusF: "\<lbrakk>A <= HTT; B <= HTT\<rbrakk> \<Longrightarrow> A + B <= HTT" | by (fastforce simp: subsetXH PlusXH iff: HTT_rews) | lemma | PlusF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"iff",
"subsetXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
SigmaF: "\<lbrakk>A <= HTT; \<And>x. x:A \<Longrightarrow> B(x) <= HTT\<rbrakk> \<Longrightarrow> SUM x:A. B(x) <= HTT" | by (fastforce simp: subsetXH SgXH HTT_rews) | lemma | SigmaF | CCL | src/CCL/Hered.thy | [] | [
"And",
"HTT",
"subsetXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
"Nat <= HTT" | apply (simp add: subsetXH)
apply clarify
apply (erule Nat_ind)
apply (fastforce iff: HTT_rews)+
done | lemma | Nat | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"Nat",
"Nat_ind",
"add",
"apply",
"iff",
"subsetXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
NatF: "Nat <= HTT" | apply clarify
apply (erule HTT_coinduct3)
apply (fast intro: HTTgenIs elim!: HTTgen_mono [THEN ci3_RI] dest: NatXH [THEN iffD1])
done | lemma | NatF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTT_coinduct3",
"HTTgenIs",
"HTTgen_mono",
"Nat",
"NatXH",
"apply",
"ci3_RI",
"elim",
"iffD1"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ListF: "A <= HTT \<Longrightarrow> List(A) <= HTT" | apply clarify
apply (erule HTT_coinduct3)
apply (fast intro!: HTTgenIs elim!: HTTgen_mono [THEN ci3_RI]
subsetD [THEN HTTgen_mono [THEN ci3_AI]]
dest: ListXH [THEN iffD1])
done | lemma | ListF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTT_coinduct3",
"HTTgenIs",
"HTTgen_mono",
"List",
"apply",
"ci3_AI",
"ci3_RI",
"elim",
"iffD1",
"subsetD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ListsF: "A <= HTT \<Longrightarrow> Lists(A) <= HTT" | apply clarify
apply (erule HTT_coinduct3)
apply (fast intro!: HTTgenIs elim!: HTTgen_mono [THEN ci3_RI]
subsetD [THEN HTTgen_mono [THEN ci3_AI]] dest: ListsXH [THEN iffD1])
done | lemma | ListsF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTT_coinduct3",
"HTTgenIs",
"HTTgen_mono",
"Lists",
"apply",
"ci3_AI",
"ci3_RI",
"elim",
"iffD1",
"subsetD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
IListsF: "A <= HTT \<Longrightarrow> ILists(A) <= HTT" | apply clarify
apply (erule HTT_coinduct3)
apply (fast intro!: HTTgenIs elim!: HTTgen_mono [THEN ci3_RI]
subsetD [THEN HTTgen_mono [THEN ci3_AI]] dest: IListsXH [THEN iffD1])
done | lemma | IListsF | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTT_coinduct3",
"HTTgenIs",
"HTTgen_mono",
"ILists",
"apply",
"ci3_AI",
"ci3_RI",
"elim",
"iffD1",
"subsetD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp :: "['a set\<Rightarrow>'a set] \<Rightarrow> 'a set" | where \<comment> \<open>least fixed point\<close>
"lfp(f) == Inter({u. f(u) <= u})" | definition | lfp | CCL | src/CCL/Lfp.thy | [] | [
"Inter",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_lowerbound: "f(A) <= A \<Longrightarrow> lfp(f) <= A" | unfolding lfp_def by blast | lemma | lfp_lowerbound | CCL | src/CCL/Lfp.thy | [] | [
"lfp"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_greatest: "(\<And>u. f(u) <= u \<Longrightarrow> A<=u) \<Longrightarrow> A <= lfp(f)" | unfolding lfp_def by blast | lemma | lfp_greatest | CCL | src/CCL/Lfp.thy | [] | [
"And",
"lfp"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_lemma2: "mono(f) \<Longrightarrow> f(lfp(f)) <= lfp(f)" | by (rule lfp_greatest, rule subset_trans, drule monoD, rule lfp_lowerbound, assumption+) | lemma | lfp_lemma2 | CCL | src/CCL/Lfp.thy | [] | [
"lfp",
"lfp_greatest",
"lfp_lowerbound",
"mono",
"monoD",
"rule",
"subset_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_lemma3: "mono(f) \<Longrightarrow> lfp(f) <= f(lfp(f))" | by (rule lfp_lowerbound, frule monoD, drule lfp_lemma2, assumption+) | lemma | lfp_lemma3 | CCL | src/CCL/Lfp.thy | [] | [
"lfp",
"lfp_lemma2",
"lfp_lowerbound",
"mono",
"monoD",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_Tarski: "mono(f) \<Longrightarrow> lfp(f) = f(lfp(f))" | by (rule equalityI lfp_lemma2 lfp_lemma3 | assumption)+ | lemma | lfp_Tarski | CCL | src/CCL/Lfp.thy | [] | [
"equalityI",
"lfp",
"lfp_lemma2",
"lfp_lemma3",
"mono",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
induct:
assumes lfp: "a: lfp(f)"
and mono: "mono(f)"
and indhyp: "\<And>x. \<lbrakk>x: f(lfp(f) Int {x. P(x)})\<rbrakk> \<Longrightarrow> P(x)"
shows "P(a)" | apply (rule_tac a = a in Int_lower2 [THEN subsetD, THEN CollectD])
apply (rule lfp [THEN [2] lfp_lowerbound [THEN subsetD]])
apply (rule Int_greatest, rule subset_trans, rule Int_lower1 [THEN mono [THEN monoD]],
rule mono [THEN lfp_lemma2], rule CollectI [THEN subsetI], rule indhyp, assumption)
done | lemma | induct | CCL | src/CCL/Lfp.thy | [] | [
"And",
"CollectD",
"CollectI",
"Int",
"Int_greatest",
"Int_lower1",
"Int_lower2",
"and",
"apply",
"assumes",
"lfp",
"lfp_lemma2",
"lfp_lowerbound",
"mono",
"monoD",
"rule",
"shows",
"subsetD",
"subsetI",
"subset_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_lfp_Tarski: "\<lbrakk>h == lfp(f); mono(f)\<rbrakk> \<Longrightarrow> h = f(h)" | apply unfold
apply (drule lfp_Tarski)
apply assumption
done | lemma | def_lfp_Tarski | CCL | src/CCL/Lfp.thy | [] | [
"apply",
"lfp",
"lfp_Tarski",
"mono",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_induct: "\<lbrakk>A == lfp(f); a:A; mono(f); \<And>x. x: f(A Int {x. P(x)}) \<Longrightarrow> P(x)\<rbrakk> \<Longrightarrow> P(a)" | apply (rule induct [of concl: P a])
apply simp
apply assumption
apply blast
done | lemma | def_induct | CCL | src/CCL/Lfp.thy | [] | [
"And",
"Int",
"apply",
"induct",
"lfp",
"mono",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lfp_mono: "\<lbrakk>mono(g); \<And>Z. f(Z) <= g(Z)\<rbrakk> \<Longrightarrow> lfp(f) <= lfp(g)" | apply (rule lfp_lowerbound)
apply (rule subset_trans)
apply (erule meta_spec)
apply (erule lfp_lemma2)
done | lemma | lfp_mono | CCL | src/CCL/Lfp.thy | [] | [
"And",
"apply",
"lfp",
"lfp_lemma2",
"lfp_lowerbound",
"meta_spec",
"mono",
"rule",
"subset_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
set :: ("term") "term" .. | instance | set | CCL | src/CCL/Set.thy | [] | [
"set",
"term"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
Collect :: "['a \<Rightarrow> o] \<Rightarrow> 'a set"
and mem :: "['a, 'a set] \<Rightarrow> o" (infixl \<open>:\<close> 50)
where mem_Collect_iff: "(a : Collect(P)) \<longleftrightarrow> P(a)"
and set_extension: "A = B \<longleftrightarrow> (ALL x. x:A \<longleftrightarrow> x:B)"
syntax
"_Coll" :: "[idt, o] \... | axiomatization | Collect | CCL | src/CCL/Set.thy | [] | [
"ALL",
"and",
"close",
"infixl",
"lambda",
"mem",
"open",
"set",
"syntax",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
CollectI: "P(a) \<Longrightarrow> a : {x. P(x)}" | apply (rule mem_Collect_iff [THEN iffD2])
apply assumption
done | lemma | CollectI | CCL | src/CCL/Set.thy | [] | [
"apply",
"iffD2",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
CollectD: "a : {x. P(x)} \<Longrightarrow> P(a)" | apply (erule mem_Collect_iff [THEN iffD1])
done
lemmas CollectE = CollectD [elim_format] | lemma | CollectD | CCL | src/CCL/Set.thy | [] | [
"CollectE",
"apply",
"iffD1"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
set_ext: "(\<And>x. x:A \<longleftrightarrow> x:B) \<Longrightarrow> A = B" | apply (rule set_extension [THEN iffD2])
apply simp
done | lemma | set_ext | CCL | src/CCL/Set.thy | [] | [
"And",
"apply",
"iffD2",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Ball :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o" | where "Ball(A, P) == ALL x. x:A \<longrightarrow> P(x)" | definition | Ball | CCL | src/CCL/Set.thy | [] | [
"ALL",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Bex :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o" | where "Bex(A, P) == EX x. x:A \<and> P(x)"
syntax
"_Ball" :: "[idt, 'a set, o] \<Rightarrow> o" (\<open>(\<open>notation=\<open>binder ALL:\<close>\<close>ALL _:_./ _)\<close> [0, 0, 0] 10)
"_Bex" :: "[idt, 'a set, o] \<Rightarrow> o" (\<open>(\<open>notation=\<open>binder EX:\<close>\<close>EX _:_./ _)\<close> ... | definition | Bex | CCL | src/CCL/Set.thy | [] | [
"ALL",
"Ball",
"EX",
"and",
"close",
"lambda",
"open",
"set",
"syntax",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ballI: "(\<And>x. x:A \<Longrightarrow> P(x)) \<Longrightarrow> ALL x:A. P(x)" | by (simp add: Ball_def) | lemma | ballI | CCL | src/CCL/Set.thy | [] | [
"ALL",
"And",
"add"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bspec: "\<lbrakk>ALL x:A. P(x); x:A\<rbrakk> \<Longrightarrow> P(x)" | by (simp add: Ball_def) | lemma | bspec | CCL | src/CCL/Set.thy | [] | [
"ALL",
"add"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ballE: "\<lbrakk>ALL x:A. P(x); P(x) \<Longrightarrow> Q; \<not> x:A \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" | unfolding Ball_def by blast | lemma | ballE | CCL | src/CCL/Set.thy | [] | [
"ALL",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bexI: "\<lbrakk>P(x); x:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)" | unfolding Bex_def by blast | lemma | bexI | CCL | src/CCL/Set.thy | [] | [
"EX"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bexCI: "\<lbrakk>EX x:A. \<not>P(x) \<Longrightarrow> P(a); a:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)" | unfolding Bex_def by blast | lemma | bexCI | CCL | src/CCL/Set.thy | [] | [
"EX",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bexE: "\<lbrakk>EX x:A. P(x); \<And>x. \<lbrakk>x:A; P(x)\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" | unfolding Bex_def by blast | lemma | bexE | CCL | src/CCL/Set.thy | [] | [
"And",
"EX"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ball_rew: "(ALL x:A. True) \<longleftrightarrow> True" | by (blast intro: ballI) | lemma | ball_rew | CCL | src/CCL/Set.thy | [] | [
"ALL",
"True",
"ballI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ball_cong:
"\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
(ALL x:A. P(x)) \<longleftrightarrow> (ALL x:A'. P'(x))" | by (blast intro: ballI elim: ballE) | lemma | ball_cong | CCL | src/CCL/Set.thy | [] | [
"ALL",
"And",
"ballE",
"ballI",
"elim"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bex_cong:
"\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
(EX x:A. P(x)) \<longleftrightarrow> (EX x:A'. P'(x))" | by (blast intro: bexI elim: bexE) | lemma | bex_cong | CCL | src/CCL/Set.thy | [] | [
"And",
"EX",
"bexE",
"bexI",
"elim"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subset :: "['a set, 'a set] \<Rightarrow> o" (infixl \<open><=\<close> 50) | where "A <= B == ALL x:A. x:B" | definition | subset | CCL | src/CCL/Set.thy | [] | [
"ALL",
"close",
"infixl",
"open",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
mono :: "['a set \<Rightarrow> 'b set] \<Rightarrow> o" | where "mono(f) == (ALL A B. A <= B \<longrightarrow> f(A) <= f(B))" | definition | mono | CCL | src/CCL/Set.thy | [] | [
"ALL",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
singleton :: "'a \<Rightarrow> 'a set" (\<open>(\<open>open_block notation=\<open>mixfix singleton\<close>\<close>{_})\<close>) | where "{a} == {x. x=a}" | definition | singleton | CCL | src/CCL/Set.thy | [] | [
"close",
"open",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
empty :: "'a set" (\<open>{}\<close>) | where "{} == {x. False}" | definition | empty | CCL | src/CCL/Set.thy | [] | [
"False",
"close",
"open",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Un :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl \<open>Un\<close> 65) | where "A Un B == {x. x:A | x:B}" | definition | Un | CCL | src/CCL/Set.thy | [] | [
"close",
"infixl",
"open",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Int :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl \<open>Int\<close> 70) | where "A Int B == {x. x:A \<and> x:B}" | definition | Int | CCL | src/CCL/Set.thy | [] | [
"and",
"close",
"infixl",
"open",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Compl :: "('a set) \<Rightarrow> 'a set" | where "Compl(A) == {x. \<not>x:A}" | definition | Compl | CCL | src/CCL/Set.thy | [] | [
"not",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INTER :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set" | where "INTER(A, B) == {y. ALL x:A. y: B(x)}" | definition | INTER | CCL | src/CCL/Set.thy | [] | [
"ALL",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UNION :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set" | where "UNION(A, B) == {y. EX x:A. y: B(x)}"
syntax
"_INTER" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" (\<open>(\<open>notation=\<open>binder INT:\<close>\<close>INT _:_./ _)\<close> [0, 0, 0] 10)
"_UNION" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" (\<open>(\<open>notation=\<open>binder UN:\<close>\<cl... | definition | UNION | CCL | src/CCL/Set.thy | [] | [
"EX",
"INTER",
"and",
"close",
"lambda",
"open",
"set",
"syntax",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Inter :: "(('a set)set) \<Rightarrow> 'a set" | where "Inter(S) == (INT x:S. x)" | definition | Inter | CCL | src/CCL/Set.thy | [] | [
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Union :: "(('a set)set) \<Rightarrow> 'a set" | where "Union(S) == (UN x:S. x)" | definition | Union | CCL | src/CCL/Set.thy | [] | [
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subsetI: "(\<And>x. x:A \<Longrightarrow> x:B) \<Longrightarrow> A <= B" | unfolding subset_def by (blast intro: ballI) | lemma | subsetI | CCL | src/CCL/Set.thy | [] | [
"And",
"ballI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subsetD: "\<lbrakk>A <= B; c:A\<rbrakk> \<Longrightarrow> c:B" | unfolding subset_def by (blast elim: ballE) | lemma | subsetD | CCL | src/CCL/Set.thy | [] | [
"ballE",
"elim"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subsetCE: "\<lbrakk>A <= B; \<not>(c:A) \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" | by (blast dest: subsetD) | lemma | subsetCE | CCL | src/CCL/Set.thy | [] | [
"not",
"subsetD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subset_refl: "A <= A" | by (blast intro: subsetI) | lemma | subset_refl | CCL | src/CCL/Set.thy | [] | [
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subset_trans: "\<lbrakk>A <= B; B <= C\<rbrakk> \<Longrightarrow> A <= C" | by (blast intro: subsetI dest: subsetD) | lemma | subset_trans | CCL | src/CCL/Set.thy | [] | [
"subsetD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
subset_antisym: "\<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> A = B" | by (blast intro: set_ext dest: subsetD)
lemmas equalityI = subset_antisym | lemma | subset_antisym | CCL | src/CCL/Set.thy | [] | [
"equalityI",
"set_ext",
"subsetD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
equalityD1: "A = B \<Longrightarrow> A<=B"
and equalityD2: "A = B \<Longrightarrow> B<=A" | by (simp_all add: subset_refl) | lemma | equalityD1 | CCL | src/CCL/Set.thy | [] | [
"add",
"and",
"equalityD2",
"subset_refl"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
equalityE: "\<lbrakk>A = B; \<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" | by (simp add: subset_refl) | lemma | equalityE | CCL | src/CCL/Set.thy | [] | [
"add",
"subset_refl"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
equalityCE: "\<lbrakk>A = B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P; \<lbrakk>\<not> c:A; \<not> c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" | by (blast elim: equalityE subsetCE) | lemma | equalityCE | CCL | src/CCL/Set.thy | [] | [
"elim",
"equalityE",
"not",
"subsetCE"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
trivial_set: "{x. x:A} = A" | by (blast intro: equalityI subsetI CollectI dest: CollectD) | lemma | trivial_set | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"CollectI",
"equalityI",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnI1: "c:A \<Longrightarrow> c : A Un B"
and UnI2: "c:B \<Longrightarrow> c : A Un B" | unfolding Un_def by (blast intro: CollectI)+ | lemma | UnI1 | CCL | src/CCL/Set.thy | [] | [
"CollectI",
"Un",
"UnI2",
"Un_def",
"and"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnCI: "(\<not>c:B \<Longrightarrow> c:A) \<Longrightarrow> c : A Un B" | by (blast intro: UnI1 UnI2) | lemma | UnCI | CCL | src/CCL/Set.thy | [] | [
"Un",
"UnI1",
"UnI2",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnE: "\<lbrakk>c : A Un B; c:A \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" | unfolding Un_def by (blast dest: CollectD) | lemma | UnE | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"Un",
"Un_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
IntI: "\<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> c : A Int B" | unfolding Int_def by (blast intro: CollectI) | lemma | IntI | CCL | src/CCL/Set.thy | [] | [
"CollectI",
"Int",
"Int_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
IntD1: "c : A Int B \<Longrightarrow> c:A"
and IntD2: "c : A Int B \<Longrightarrow> c:B" | unfolding Int_def by (blast dest: CollectD)+ | lemma | IntD1 | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"Int",
"IntD2",
"Int_def",
"and"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
IntE: "\<lbrakk>c : A Int B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" | by (blast dest: IntD1 IntD2) | lemma | IntE | CCL | src/CCL/Set.thy | [] | [
"Int",
"IntD1",
"IntD2"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ComplI: "(c:A \<Longrightarrow> False) \<Longrightarrow> c : Compl(A)" | unfolding Compl_def by (blast intro: CollectI) | lemma | ComplI | CCL | src/CCL/Set.thy | [] | [
"CollectI",
"Compl",
"False"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ComplD: "c : Compl(A) \<Longrightarrow> \<not>c:A" | unfolding Compl_def by (blast dest: CollectD)
lemmas ComplE = ComplD [elim_format] | lemma | ComplD | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"Compl",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
empty_eq: "{x. False} = {}" | by (simp add: empty_def) | lemma | empty_eq | CCL | src/CCL/Set.thy | [] | [
"False",
"add",
"empty_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
emptyD: "a : {} \<Longrightarrow> P" | unfolding empty_def by (blast dest: CollectD)
lemmas emptyE = emptyD [elim_format] | lemma | emptyD | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"emptyE",
"empty_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
not_emptyD:
assumes "\<not> A={}"
shows "EX x. x:A" | proof -
have "\<not> (EX x. x:A) \<Longrightarrow> A = {}"
by (rule equalityI) (blast intro!: subsetI elim!: emptyD)+
with assms show ?thesis by blast
qed | lemma | not_emptyD | CCL | src/CCL/Set.thy | [] | [
"EX",
"assumes",
"elim",
"emptyD",
"equalityI",
"not",
"rule",
"shows",
"subsetI",
"with"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
singletonI: "a : {a}" | unfolding singleton_def by (blast intro: CollectI) | lemma | singletonI | CCL | src/CCL/Set.thy | [] | [
"CollectI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
singletonD: "b : {a} \<Longrightarrow> b=a" | unfolding singleton_def by (blast dest: CollectD)
lemmas singletonE = singletonD [elim_format] | lemma | singletonD | CCL | src/CCL/Set.thy | [] | [
"CollectD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UN_I: "\<lbrakk>a:A; b: B(a)\<rbrakk> \<Longrightarrow> b: (UN x:A. B(x))" | unfolding UNION_def by (blast intro: bexI CollectI) | lemma | UN_I | CCL | src/CCL/Set.thy | [] | [
"CollectI",
"bexI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UN_E: "\<lbrakk>b : (UN x:A. B(x)); \<And>x. \<lbrakk>x:A; b: B(x)\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" | unfolding UNION_def by (blast dest: CollectD elim: bexE) | lemma | UN_E | CCL | src/CCL/Set.thy | [] | [
"And",
"CollectD",
"bexE",
"elim"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UN_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (UN x:A. C(x)) = (UN x:B. D(x))" | by (simp add: UNION_def cong: bex_cong) | lemma | UN_cong | CCL | src/CCL/Set.thy | [] | [
"And",
"add",
"bex_cong",
"cong"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INT_I: "(\<And>x. x:A \<Longrightarrow> b: B(x)) \<Longrightarrow> b : (INT x:A. B(x))" | unfolding INTER_def by (blast intro: CollectI ballI) | lemma | INT_I | CCL | src/CCL/Set.thy | [] | [
"And",
"CollectI",
"ballI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INT_D: "\<lbrakk>b : (INT x:A. B(x)); a:A\<rbrakk> \<Longrightarrow> b: B(a)" | unfolding INTER_def by (blast dest: CollectD bspec) | lemma | INT_D | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"bspec"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INT_E: "\<lbrakk>b : (INT x:A. B(x)); b: B(a) \<Longrightarrow> R; \<not> a:A \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" | unfolding INTER_def by (blast dest: CollectD bspec) | lemma | INT_E | CCL | src/CCL/Set.thy | [] | [
"CollectD",
"bspec",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INT_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (INT x:A. C(x)) = (INT x:B. D(x))" | by (simp add: INTER_def cong: ball_cong) | lemma | INT_cong | CCL | src/CCL/Set.thy | [] | [
"And",
"add",
"ball_cong",
"cong"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnionI: "\<lbrakk>X:C; A:X\<rbrakk> \<Longrightarrow> A : Union(C)" | unfolding Union_def by (blast intro: UN_I) | lemma | UnionI | CCL | src/CCL/Set.thy | [] | [
"UN_I",
"Union"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
UnionE: "\<lbrakk>A : Union(C); \<And>X. \<lbrakk> A:X; X:C\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" | unfolding Union_def by (blast elim: UN_E) | lemma | UnionE | CCL | src/CCL/Set.thy | [] | [
"And",
"UN_E",
"Union",
"elim"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
InterI: "(\<And>X. X:C \<Longrightarrow> A:X) \<Longrightarrow> A : Inter(C)" | unfolding Inter_def by (blast intro: INT_I) | lemma | InterI | CCL | src/CCL/Set.thy | [] | [
"And",
"INT_I",
"Inter"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
InterD: "\<lbrakk>A : Inter(C); X:C\<rbrakk> \<Longrightarrow> A:X" | unfolding Inter_def by (blast dest: INT_D) | lemma | InterD | CCL | src/CCL/Set.thy | [] | [
"INT_D",
"Inter"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
InterE: "\<lbrakk>A : Inter(C); A:X \<Longrightarrow> R; \<not> X:C \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" | unfolding Inter_def by (blast elim: INT_E) | lemma | InterE | CCL | src/CCL/Set.thy | [] | [
"INT_E",
"Inter",
"elim",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Union_upper: "B:A \<Longrightarrow> B <= Union(A)" | by (blast intro: subsetI UnionI) | lemma | Union_upper | CCL | src/CCL/Set.thy | [] | [
"Union",
"UnionI",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Union_least: "(\<And>X. X:A \<Longrightarrow> X<=C) \<Longrightarrow> Union(A) <= C" | by (blast intro: subsetI dest: subsetD elim: UnionE) | lemma | Union_least | CCL | src/CCL/Set.thy | [] | [
"And",
"Union",
"UnionE",
"elim",
"subsetD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Inter_lower: "B:A \<Longrightarrow> Inter(A) <= B" | by (blast intro: subsetI dest: InterD) | lemma | Inter_lower | CCL | src/CCL/Set.thy | [] | [
"Inter",
"InterD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Inter_greatest: "(\<And>X. X:A \<Longrightarrow> C<=X) \<Longrightarrow> C <= Inter(A)" | by (blast intro: subsetI InterI dest: subsetD) | lemma | Inter_greatest | CCL | src/CCL/Set.thy | [] | [
"And",
"Inter",
"InterI",
"subsetD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Un_upper1: "A <= A Un B" | by (blast intro: subsetI UnI1) | lemma | Un_upper1 | CCL | src/CCL/Set.thy | [] | [
"Un",
"UnI1",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Un_upper2: "B <= A Un B" | by (blast intro: subsetI UnI2) | lemma | Un_upper2 | CCL | src/CCL/Set.thy | [] | [
"Un",
"UnI2",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Un_least: "\<lbrakk>A<=C; B<=C\<rbrakk> \<Longrightarrow> A Un B <= C" | by (blast intro: subsetI elim: UnE dest: subsetD) | lemma | Un_least | CCL | src/CCL/Set.thy | [] | [
"Un",
"UnE",
"elim",
"subsetD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Int_lower1: "A Int B <= A" | by (blast intro: subsetI elim: IntE) | lemma | Int_lower1 | CCL | src/CCL/Set.thy | [] | [
"Int",
"IntE",
"elim",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Int_lower2: "A Int B <= B" | by (blast intro: subsetI elim: IntE) | lemma | Int_lower2 | CCL | src/CCL/Set.thy | [] | [
"Int",
"IntE",
"elim",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Int_greatest: "\<lbrakk>C<=A; C<=B\<rbrakk> \<Longrightarrow> C <= A Int B" | by (blast intro: subsetI IntI dest: subsetD) | lemma | Int_greatest | CCL | src/CCL/Set.thy | [] | [
"Int",
"IntI",
"subsetD",
"subsetI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
monoI: "(\<And>A B. A <= B \<Longrightarrow> f(A) <= f(B)) \<Longrightarrow> mono(f)" | unfolding mono_def by blast | lemma | monoI | CCL | src/CCL/Set.thy | [] | [
"And",
"mono",
"mono_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
monoD: "\<lbrakk>mono(f); A <= B\<rbrakk> \<Longrightarrow> f(A) <= f(B)" | unfolding mono_def by blast | lemma | monoD | CCL | src/CCL/Set.thy | [] | [
"mono",
"mono_def"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
mono_Un: "mono(f) \<Longrightarrow> f(A) Un f(B) <= f(A Un B)" | by (blast intro: Un_least dest: monoD intro: Un_upper1 Un_upper2) | lemma | mono_Un | CCL | src/CCL/Set.thy | [] | [
"Un",
"Un_least",
"Un_upper1",
"Un_upper2",
"mono",
"monoD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
mono_Int: "mono(f) \<Longrightarrow> f(A Int B) <= f(A) Int f(B)" | by (blast intro: Int_greatest dest: monoD intro: Int_lower1 Int_lower2) | lemma | mono_Int | CCL | src/CCL/Set.thy | [] | [
"Int",
"Int_greatest",
"Int_lower1",
"Int_lower2",
"mono",
"monoD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
mem_rews:
"(a : A Un B) \<longleftrightarrow> (a:A | a:B)"
"(a : A Int B) \<longleftrightarrow> (a:A \<and> a:B)"
"(a : Compl(B)) \<longleftrightarrow> (\<not>a:B)"
"(a : {b}) \<longleftrightarrow> (a=b)"
"(a : {}) \<longleftrightarrow> False"
"(a : {x. P(x)}) \<longleftrightarrow> P(a)... | by blast+
lemmas [simp] = trivial_set empty_eq mem_rews
and [cong] = ball_cong bex_cong INT_cong UN_cong | lemma | mem_rews | CCL | src/CCL/Set.thy | [] | [
"Compl",
"False",
"INT_cong",
"Int",
"UN_cong",
"Un",
"and",
"ball_cong",
"bex_cong",
"cong",
"empty_eq",
"not",
"trivial_set"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Int_absorb: "A Int A = A" | by (blast intro: equalityI) | lemma | Int_absorb | CCL | src/CCL/Set.thy | [] | [
"Int",
"equalityI"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 |
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