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 |
|---|---|---|---|---|---|---|---|---|---|---|
SIM :: "[i,i,i set]\<Rightarrow>o" | where
"SIM(t,t',R) == (t=true \<and> t'=true) | (t=false \<and> t'=false) |
(\<exists>a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> <a,a'> : R \<and> <b,b'> : R) |
(\<exists>f f'. t=lam x. f(x) \<and> t'=lam x. f'(x) \<and> (ALL x.<f(x),f'(x)> : R))" | definition | SIM | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"a'",
"and",
"b'",
"exists",
"f'",
"false",
"lam",
"set",
"true",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
POgen :: "i set \<Rightarrow> i set" | where "POgen(R) == {p. \<exists>t t'. p=<t,t'> \<and> (t = bot | SIM(t,t',R))}" | definition | POgen | CCL | src/CCL/CCL.thy | [] | [
"SIM",
"and",
"bot",
"exists",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
EQgen :: "i set \<Rightarrow> i set" | where "EQgen(R) == {p. \<exists>t t'. p=<t,t'> \<and> (t = bot \<and> t' = bot | SIM(t,t',R))}" | definition | EQgen | CCL | src/CCL/CCL.thy | [] | [
"SIM",
"and",
"bot",
"exists",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
PO :: "i set" | where "PO == gfp(POgen)" | definition | PO | CCL | src/CCL/CCL.thy | [] | [
"POgen",
"gfp",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
EQ :: "i set" | where "EQ == gfp(EQgen)" | definition | EQ | CCL | src/CCL/CCL.thy | [] | [
"EQgen",
"gfp",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
where
po_refl: "a [= a" and
po_trans: "\<lbrakk>a [= b; b [= c\<rbrakk> \<Longrightarrow> a [= c" and
po_cong: "a [= b \<Longrightarrow> f(a) [= f(b)" and
(* Extend definition of [= to program fragments of higher type *)
po_abstractn: "(\<And>x. f(x) [= g(x)) \<Longrightarrow> (\<lambd... | axiomatization | where | CCL | src/CCL/CCL.thy | [] | [
"And",
"and",
"lambda",
"po_refl",
"po_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
where
eq_iff: "t = t' \<longleftrightarrow> t [= t' \<and> t' [= t" | axiomatization | where | CCL | src/CCL/CCL.thy | [] | [
"and",
"eq_iff"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
where
PO_iff: "t [= t' \<longleftrightarrow> <t,t'> : PO" and
EQ_iff: "t = t' \<longleftrightarrow> <t,t'> : EQ" | axiomatization | where | CCL | src/CCL/CCL.thy | [] | [
"EQ",
"PO",
"and"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
where
caseBtrue: "case(true,d,e,f,g) = d" and
caseBfalse: "case(false,d,e,f,g) = e" and
caseBpair: "case(<a,b>,d,e,f,g) = f(a,b)" and
caseBlam: "\<And>b. case(lam x. b(x),d,e,f,g) = g(b)" and
caseBbot: "case(bot,d,e,f,g) = bot" | axiomatization | where | CCL | src/CCL/CCL.thy | [] | [
"And",
"and",
"bot",
"case",
"false",
"lam",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
where
distinctness: "\<not> lam x. b(x) = bot" | axiomatization | where | CCL | src/CCL/CCL.thy | [] | [
"bot",
"lam",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
Dvg :: "i \<Rightarrow> o" | where "Dvg(t) == t = bot" | definition | Dvg | CCL | src/CCL/CCL.thy | [] | [
"bot",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Trm :: "i \<Rightarrow> o" | where "Trm(t) == \<not> Dvg(t)" | definition | Trm | CCL | src/CCL/CCL.thy | [] | [
"Dvg",
"not",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
fun_cong: "(f::'a\<Rightarrow>'b) = g \<Longrightarrow> f(x)=g(x)" | by simp | lemma | fun_cong | CCL | src/CCL/CCL.thy | [] | [] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
arg_cong: "x=y \<Longrightarrow> f(x)=f(y)" | by simp | lemma | arg_cong | CCL | src/CCL/CCL.thy | [] | [] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
abstractn: "(\<And>x. f(x) = g(x)) \<Longrightarrow> f = g" | apply (simp add: eq_iff)
apply (blast intro: po_abstractn)
done
lemmas caseBs = caseBtrue caseBfalse caseBpair caseBlam caseBbot | lemma | abstractn | CCL | src/CCL/CCL.thy | [] | [
"And",
"add",
"apply",
"eq_iff"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Trm_iff: "Trm(t) \<longleftrightarrow> \<not> t = bot" | by (simp add: Trm_def Dvg_def) | lemma | Trm_iff | CCL | src/CCL/CCL.thy | [] | [
"Trm",
"add",
"bot",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
Dvg_iff: "Dvg(t) \<longleftrightarrow> t = bot" | by (simp add: Trm_def Dvg_def) | lemma | Dvg_iff | CCL | src/CCL/CCL.thy | [] | [
"Dvg",
"add",
"bot"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
eq_lemma: "\<lbrakk>x=a; y=b; x=y\<rbrakk> \<Longrightarrow> a=b" | by simp | lemma | eq_lemma | CCL | src/CCL/CCL.thy | [] | [] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ccl_injs:
"<a,b> = <a',b'> \<longleftrightarrow> (a=a' \<and> b=b')"
"\<And>b b'. (lam x. b(x) = lam x. b'(x)) \<longleftrightarrow> ((ALL z. b(z)=b'(z)))" | by (inj_rl caseBs) | lemma | ccl_injs | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"And",
"a'",
"and",
"b'",
"lam"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
pair_inject: "<a,b> = <a',b'> \<Longrightarrow> (a = a' \<Longrightarrow> b = b' \<Longrightarrow> R) \<Longrightarrow> R" | by (simp add: ccl_injs) | lemma | pair_inject | CCL | src/CCL/CCL.thy | [] | [
"a'",
"add",
"b'",
"ccl_injs"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lem: "t=t' \<Longrightarrow> case(t,b,c,d,e) = case(t',b,c,d,e)" | by simp | lemma | lem | CCL | src/CCL/CCL.thy | [] | [
"case"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
XHlemma1: "\<lbrakk>A=B; a:B \<longleftrightarrow> P\<rbrakk> \<Longrightarrow> a:A \<longleftrightarrow> P" | by simp | lemma | XHlemma1 | CCL | src/CCL/CCL.thy | [] | [] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
XHlemma2: "(P(t,t') \<longleftrightarrow> Q) \<Longrightarrow> (<t,t'> : {p. \<exists>t t'. p=<t,t'> \<and> P(t,t')} \<longleftrightarrow> Q)" | by blast | lemma | XHlemma2 | CCL | src/CCL/CCL.thy | [] | [
"and",
"exists"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
POgen_mono: "mono(\<lambda>X. POgen(X))" | apply (unfold POgen_def SIM_def)
apply (rule monoI)
apply blast
done | lemma | POgen_mono | CCL | src/CCL/CCL.thy | [] | [
"POgen",
"apply",
"lambda",
"mono",
"monoI",
"rule",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
POgenXH:
"<t,t'> : POgen(R) \<longleftrightarrow> t= bot | (t=true \<and> t'=true) | (t=false \<and> t'=false) |
(EX a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> <a,a'> : R \<and> <b,b'> : R) |
(EX f f'. t=lam x. f(x) \<and> t'=lam x. f'(x) \<and> (ALL x. <f(x),f'(x)> : R))" | apply (unfold POgen_def SIM_def)
apply (rule iff_refl [THEN XHlemma2])
done | lemma | POgenXH | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"EX",
"POgen",
"XHlemma2",
"a'",
"and",
"apply",
"b'",
"bot",
"f'",
"false",
"iff_refl",
"lam",
"rule",
"true",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
poXH:
"t [= t' \<longleftrightarrow> t=bot | (t=true \<and> t'=true) | (t=false \<and> t'=false) |
(EX a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> a [= a' \<and> b [= b') |
(EX f f'. t=lam x. f(x) \<and> t'=lam x. f'(x) \<and> (ALL x. f(x) [= f'(x)))" | apply (simp add: PO_iff del: ex_simps)
apply (rule POgen_mono
[THEN PO_def [THEN def_gfp_Tarski], THEN XHlemma1, unfolded POgen_def SIM_def])
apply (rule iff_refl [THEN XHlemma2])
done | lemma | poXH | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"EX",
"POgen_mono",
"XHlemma1",
"XHlemma2",
"a'",
"add",
"and",
"apply",
"b'",
"bot",
"def_gfp_Tarski",
"del",
"ex_simps",
"f'",
"false",
"iff_refl",
"lam",
"rule",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_bot: "bot [= b" | apply (rule poXH [THEN iffD2])
apply simp
done | lemma | po_bot | CCL | src/CCL/CCL.thy | [] | [
"apply",
"bot",
"iffD2",
"poXH",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
bot_poleast: "a [= bot \<Longrightarrow> a=bot" | apply (drule poXH [THEN iffD1])
apply simp
done | lemma | bot_poleast | CCL | src/CCL/CCL.thy | [] | [
"apply",
"bot",
"iffD1",
"poXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_pair: "<a,b> [= <a',b'> \<longleftrightarrow> a [= a' \<and> b [= b'" | apply (rule poXH [THEN iff_trans])
apply simp
done | lemma | po_pair | CCL | src/CCL/CCL.thy | [] | [
"a'",
"and",
"apply",
"b'",
"iff_trans",
"poXH",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_lam: "lam x. f(x) [= lam x. f'(x) \<longleftrightarrow> (ALL x. f(x) [= f'(x))" | apply (rule poXH [THEN iff_trans])
apply fastforce
done
lemmas ccl_porews = po_bot po_pair po_lam | lemma | po_lam | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"apply",
"f'",
"iff_trans",
"lam",
"poXH",
"po_bot",
"po_pair",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
case_pocong:
assumes 1: "t [= t'"
and 2: "a [= a'"
and 3: "b [= b'"
and 4: "\<And>x y. c(x,y) [= c'(x,y)"
and 5: "\<And>u. d(u) [= d'(u)"
shows "case(t,a,b,c,d) [= case(t',a',b',c',d')" | apply (rule 1 [THEN po_cong, THEN po_trans])
apply (rule 2 [THEN po_cong, THEN po_trans])
apply (rule 3 [THEN po_cong, THEN po_trans])
apply (rule 4 [THEN po_abstractn, THEN po_abstractn, THEN po_cong, THEN po_trans])
apply (rule_tac f1 = "\<lambda>d. case (t',a',b',c',d)" in
5 [THEN po_abstractn, THEN po_c... | lemma | case_pocong | CCL | src/CCL/CCL.thy | [] | [
"And",
"a'",
"and",
"apply",
"assumes",
"b'",
"c'",
"case",
"f1",
"lambda",
"po_refl",
"po_trans",
"rule",
"shows"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
apply_pocong: "\<lbrakk>f [= f'; a [= a'\<rbrakk> \<Longrightarrow> f ` a [= f' ` a'" | unfolding ccl_data_defs
apply (rule case_pocong, (rule po_refl | assumption)+)
apply (erule po_cong)
done | lemma | apply_pocong | CCL | src/CCL/CCL.thy | [] | [
"a'",
"apply",
"case_pocong",
"f'",
"po_refl",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
npo_lam_bot: "\<not> lam x. b(x) [= bot" | apply (rule notI)
apply (drule bot_poleast)
apply (erule distinctness [THEN notE])
done | lemma | npo_lam_bot | CCL | src/CCL/CCL.thy | [] | [
"apply",
"bot",
"bot_poleast",
"lam",
"not",
"notE",
"notI",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_lemma: "\<lbrakk>x=a; y=b; x[=y\<rbrakk> \<Longrightarrow> a[=b" | by simp | lemma | po_lemma | CCL | src/CCL/CCL.thy | [] | [] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
npo_pair_lam: "\<not> <a,b> [= lam x. f(x)" | apply (rule notI)
apply (rule npo_lam_bot [THEN notE])
apply (erule case_pocong [THEN caseBlam [THEN caseBpair [THEN po_lemma]]])
apply (rule po_refl npo_lam_bot)+
done | lemma | npo_pair_lam | CCL | src/CCL/CCL.thy | [] | [
"apply",
"case_pocong",
"lam",
"not",
"notE",
"notI",
"npo_lam_bot",
"po_lemma",
"po_refl",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
npo_lam_pair: "\<not> lam x. f(x) [= <a,b>" | apply (rule notI)
apply (rule npo_lam_bot [THEN notE])
apply (erule case_pocong [THEN caseBpair [THEN caseBlam [THEN po_lemma]]])
apply (rule po_refl npo_lam_bot)+
done | lemma | npo_lam_pair | CCL | src/CCL/CCL.thy | [] | [
"apply",
"case_pocong",
"lam",
"not",
"notE",
"notI",
"npo_lam_bot",
"po_lemma",
"po_refl",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
npo_rls1:
"\<not> true [= false"
"\<not> false [= true"
"\<not> true [= <a,b>"
"\<not> <a,b> [= true"
"\<not> true [= lam x. f(x)"
"\<not> lam x. f(x) [= true"
"\<not> false [= <a,b>"
"\<not> <a,b> [= false"
"\<not> false [= lam x. f(x)"
"\<not> lam x. f(x) [= false" | by (rule notI, drule case_pocong, erule_tac [5] rev_mp, simp_all,
(rule po_refl npo_lam_bot)+)+
lemmas npo_rls = npo_pair_lam npo_lam_pair npo_rls1 | lemma | npo_rls1 | CCL | src/CCL/CCL.thy | [] | [
"case_pocong",
"false",
"lam",
"not",
"notI",
"npo_lam_bot",
"npo_lam_pair",
"npo_pair_lam",
"po_refl",
"rev_mp",
"rule",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_coinduct: "\<lbrakk><t,u> : R; R <= POgen(R)\<rbrakk> \<Longrightarrow> t [= u" | apply (rule PO_def [THEN def_coinduct, THEN PO_iff [THEN iffD2]])
apply assumption+
done | lemma | po_coinduct | CCL | src/CCL/CCL.thy | [] | [
"POgen",
"apply",
"def_coinduct",
"iffD2",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
EQgen_mono: "mono(\<lambda>X. EQgen(X))" | apply (unfold EQgen_def SIM_def)
apply (rule monoI)
apply blast
done | lemma | EQgen_mono | CCL | src/CCL/CCL.thy | [] | [
"EQgen",
"apply",
"lambda",
"mono",
"monoI",
"rule",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
EQgenXH:
"<t,t'> : EQgen(R) \<longleftrightarrow> (t=bot \<and> t'=bot) | (t=true \<and> t'=true) |
(t=false \<and> t'=false) |
(EX a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> <a,a'> : R \<and> <b,b'> : R) |
(EX f f'. t=lam x. f(x) \<an... | apply (unfold EQgen_def SIM_def)
apply (rule iff_refl [THEN XHlemma2])
done | lemma | EQgenXH | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"EQgen",
"EX",
"XHlemma2",
"a'",
"and",
"apply",
"b'",
"bot",
"f'",
"false",
"iff_refl",
"lam",
"rule",
"true",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
eqXH:
"t=t' \<longleftrightarrow> (t=bot \<and> t'=bot) | (t=true \<and> t'=true) | (t=false \<and> t'=false) |
(EX a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> a=a' \<and> b=b') |
(EX f f'. t=lam x. f(x) \<and> t'=lam x. f'(x) \<and> (ALL x. f(x)=f'(x)))" | apply (subgoal_tac "<t,t'> : EQ \<longleftrightarrow>
(t=bot \<and> t'=bot) | (t=true \<and> t'=true) | (t=false \<and> t'=false) |
(EX a a' b b'. t=<a,b> \<and> t'=<a',b'> \<and> <a,a'> : EQ \<and> <b,b'> : EQ) |
(EX f f'. t=lam x. f (x) \<and> t'=lam x. f' (x) \<and> (ALL x. <f (x) ,f' (x) > : EQ))")
ap... | lemma | eqXH | CCL | src/CCL/CCL.thy | [] | [
"ALL",
"EQ",
"EQgen_mono",
"EX",
"XHlemma1",
"XHlemma2",
"a'",
"add",
"and",
"apply",
"b'",
"bot",
"def_gfp_Tarski",
"f'",
"false",
"iff_refl",
"iff_sym",
"lam",
"rev_mp",
"rule",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
eq_coinduct: "\<lbrakk><t,u> : R; R <= EQgen(R)\<rbrakk> \<Longrightarrow> t = u" | apply (rule EQ_def [THEN def_coinduct, THEN EQ_iff [THEN iffD2]])
apply assumption+
done | lemma | eq_coinduct | CCL | src/CCL/CCL.thy | [] | [
"EQgen",
"apply",
"def_coinduct",
"iffD2",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
eq_coinduct3:
"\<lbrakk><t,u> : R; R <= EQgen(lfp(\<lambda>x. EQgen(x) Un R Un EQ))\<rbrakk> \<Longrightarrow> t = u" | apply (rule EQ_def [THEN def_coinduct3, THEN EQ_iff [THEN iffD2]])
apply (rule EQgen_mono | assumption)+
done | lemma | eq_coinduct3 | CCL | src/CCL/CCL.thy | [] | [
"EQ",
"EQgen",
"EQgen_mono",
"Un",
"apply",
"def_coinduct3",
"iffD2",
"lambda",
"lfp",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
cond_eta: "(EX f. t=lam x. f(x)) \<Longrightarrow> t = lam x.(t ` x)" | by (auto simp: apply_def) | lemma | cond_eta | CCL | src/CCL/CCL.thy | [] | [
"EX",
"auto",
"lam"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
exhaustion: "(t=bot) | (t=true) | (t=false) | (EX a b. t=<a,b>) | (EX f. t=lam x. f(x))" | apply (cut_tac refl [THEN eqXH [THEN iffD1]])
apply blast
done | lemma | exhaustion | CCL | src/CCL/CCL.thy | [] | [
"EX",
"apply",
"bot",
"eqXH",
"false",
"iffD1",
"lam",
"refl",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
term_case:
"\<lbrakk>P(bot); P(true); P(false); \<And>x y. P(<x,y>); \<And>b. P(lam x. b(x))\<rbrakk> \<Longrightarrow> P(t)" | using exhaustion [of t] by blast | lemma | term_case | CCL | src/CCL/CCL.thy | [] | [
"And",
"bot",
"exhaustion",
"false",
"lam",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
idgen :: "i \<Rightarrow> i" | where "idgen(f) == lam t. case(t,true,false, \<lambda>x y.<f`x, f`y>, \<lambda>u. lam x. f ` u(x))" | definition | idgen | CCL | src/CCL/Fix.thy | [] | [
"case",
"false",
"lam",
"lambda",
"true",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
INCL :: "[i\<Rightarrow>o]\<Rightarrow>o" where
INCL_def: "INCL(\<lambda>x. P(x)) == (ALL f.(ALL n:Nat. P(f^n`bot)) \<longrightarrow> P(fix(f)))" and
po_INCL: "INCL(\<lambda>x. a(x) [= b(x))" and
INCL_subst: "INCL(P) \<Longrightarrow> INCL(\<lambda>x. P((g::i\<Rightarrow>i)(x)))" | axiomatization | INCL | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"and",
"bot",
"fix",
"lambda",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | ||
fix_ind:
assumes base: "P(bot)"
and step: "\<And>x. P(x) \<Longrightarrow> P(f(x))"
and incl: "INCL(P)"
shows "P(fix(f))" | apply (rule incl [unfolded INCL_def, rule_format])
apply (rule Nat_ind [THEN ballI], assumption)
apply simp_all
apply (rule base)
apply (erule step)
done | lemma | fix_ind | CCL | src/CCL/Fix.thy | [] | [
"And",
"INCL",
"Nat_ind",
"and",
"apply",
"assumes",
"ballI",
"bot",
"fix",
"rule",
"shows",
"step"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
inclXH: "INCL(P) \<longleftrightarrow> (ALL f. (ALL n:Nat. P(f ^ n ` bot)) \<longrightarrow> P(fix(f)))" | by (simp add: INCL_def) | lemma | inclXH | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"INCL",
"add",
"bot",
"fix"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
inclI: "\<lbrakk>\<And>f. ALL n:Nat. P(f^n`bot) \<Longrightarrow> P(fix(f))\<rbrakk> \<Longrightarrow> INCL(\<lambda>x. P(x))" | unfolding inclXH by blast | lemma | inclI | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"And",
"INCL",
"bot",
"fix",
"inclXH",
"lambda"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
inclD: "\<lbrakk>INCL(P); \<And>n. n:Nat \<Longrightarrow> P(f^n`bot)\<rbrakk> \<Longrightarrow> P(fix(f))" | unfolding inclXH by blast | lemma | inclD | CCL | src/CCL/Fix.thy | [] | [
"And",
"INCL",
"Nat",
"bot",
"fix",
"inclXH"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
inclE: "\<lbrakk>INCL(P); (ALL n:Nat. P(f^n`bot)) \<longrightarrow> P(fix(f)) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" | by (blast dest: inclD) | lemma | inclE | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"INCL",
"bot",
"fix",
"inclD"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
npo_INCL: "INCL(\<lambda>x. \<not> a(x) [= t)" | apply (rule inclI)
apply (drule bspec)
apply (rule zeroT)
apply (erule contrapos)
apply (rule po_trans)
prefer 2
apply assumption
apply (subst napplyBzero)
apply (rule po_cong, rule po_bot)
done | lemma | npo_INCL | CCL | src/CCL/Fix.thy | [] | [
"INCL",
"apply",
"bspec",
"contrapos",
"inclI",
"lambda",
"napplyBzero",
"not",
"po_bot",
"po_trans",
"rule",
"subst",
"zeroT"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
conj_INCL: "\<lbrakk>INCL(P); INCL(Q)\<rbrakk> \<Longrightarrow> INCL(\<lambda>x. P(x) \<and> Q(x))" | by (blast intro!: inclI dest!: inclD) | lemma | conj_INCL | CCL | src/CCL/Fix.thy | [] | [
"INCL",
"and",
"inclD",
"inclI",
"lambda"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
all_INCL: "(\<And>a. INCL(P(a))) \<Longrightarrow> INCL(\<lambda>x. ALL a. P(a,x))" | by (blast intro!: inclI dest!: inclD) | lemma | all_INCL | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"And",
"INCL",
"inclD",
"inclI",
"lambda"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
ball_INCL: "(\<And>a. a:A \<Longrightarrow> INCL(P(a))) \<Longrightarrow> INCL(\<lambda>x. ALL a:A. P(a,x))" | by (blast intro!: inclI dest!: inclD) | lemma | ball_INCL | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"And",
"INCL",
"inclD",
"inclI",
"lambda"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
eq_INCL: "INCL(\<lambda>x. a(x) = (b(x)::'a::prog))" | apply (simp add: eq_iff)
apply (rule conj_INCL po_INCL)+
done | lemma | eq_INCL | CCL | src/CCL/Fix.thy | [] | [
"INCL",
"add",
"apply",
"conj_INCL",
"eq_iff",
"lambda",
"prog",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
fix_idgenfp: "idgen(fix(idgen)) = fix(idgen)" | apply (rule fixB [symmetric])
done | lemma | fix_idgenfp | CCL | src/CCL/Fix.thy | [] | [
"apply",
"fix",
"fixB",
"idgen",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
id_idgenfp: "idgen(lam x. x) = lam x. x" | apply (simp add: idgen_def)
apply (rule term_case [THEN allI])
apply simp_all
done | lemma | id_idgenfp | CCL | src/CCL/Fix.thy | [] | [
"add",
"allI",
"apply",
"idgen",
"lam",
"rule",
"term_case"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
idgenfp_lam: "idgen(d) = d \<Longrightarrow> d = lam x. ?f(x)" | apply (unfold idgen_def)
apply (erule ssubst)
apply (rule refl)
done | schematic_goal | idgenfp_lam | CCL | src/CCL/Fix.thy | [] | [
"apply",
"idgen",
"lam",
"refl",
"rule",
"ssubst",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
l_lemma: "\<lbrakk>a = b; a ` t = u\<rbrakk> \<Longrightarrow> b ` t = u" | by (simp add: idgen_def) | lemma | l_lemma | CCL | src/CCL/Fix.thy | [] | [
"add"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
idgen_lemmas:
"idgen(d) = d \<Longrightarrow> d ` bot = bot"
"idgen(d) = d \<Longrightarrow> d ` true = true"
"idgen(d) = d \<Longrightarrow> d ` false = false"
"idgen(d) = d \<Longrightarrow> d ` <a,b> = <d ` a,d ` b>"
"idgen(d) = d \<Longrightarrow> d ` (lam x. f(x)) = lam x. d ` f(x)" | by (erule l_lemma, simp add: idgen_def)+ | lemma | idgen_lemmas | CCL | src/CCL/Fix.thy | [] | [
"add",
"bot",
"false",
"idgen",
"l_lemma",
"lam",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_eta:
"\<lbrakk>ALL x. t ` x [= u ` x; EX f. t=lam x. f(x); EX f. u=lam x. f(x)\<rbrakk> \<Longrightarrow> t [= u" | apply (drule cond_eta)+
apply (erule ssubst)
apply (erule ssubst)
apply (rule po_lam [THEN iffD2])
apply simp
done | lemma | po_eta | CCL | src/CCL/Fix.thy | [] | [
"ALL",
"EX",
"apply",
"cond_eta",
"iffD2",
"lam",
"po_lam",
"rule",
"ssubst"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
po_eta_lemma: "idgen(d) = d \<Longrightarrow> d = lam x. ?f(x)" | apply (unfold idgen_def)
apply (erule sym)
done | schematic_goal | po_eta_lemma | CCL | src/CCL/Fix.thy | [] | [
"apply",
"idgen",
"lam",
"sym",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lemma1:
"idgen(d) = d \<Longrightarrow>
{p. EX a b. p=<a,b> \<and> (EX t. a=fix(idgen) ` t \<and> b = d ` t)} <=
POgen({p. EX a b. p=<a,b> \<and> (EX t. a=fix(idgen) ` t \<and> b = d ` t)})" | apply clarify
apply (rule_tac t = t in term_case)
apply (simp_all add: POgenXH idgen_lemmas idgen_lemmas [OF fix_idgenfp])
apply blast
apply fast
done | lemma | lemma1 | CCL | src/CCL/Fix.thy | [] | [
"EX",
"POgen",
"POgenXH",
"add",
"and",
"apply",
"fix",
"fix_idgenfp",
"idgen",
"idgen_lemmas",
"term_case"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
fix_least_idgen: "idgen(d) = d \<Longrightarrow> fix(idgen) [= d" | apply (rule allI [THEN po_eta])
apply (rule lemma1 [THEN [2] po_coinduct])
apply (blast intro: po_eta_lemma fix_idgenfp)+
done | lemma | fix_least_idgen | CCL | src/CCL/Fix.thy | [] | [
"allI",
"apply",
"fix",
"fix_idgenfp",
"idgen",
"lemma1",
"po_coinduct",
"po_eta",
"po_eta_lemma",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
lemma2:
"idgen(d) = d \<Longrightarrow>
{p. EX a b. p=<a,b> \<and> b = d ` a} <= POgen({p. EX a b. p=<a,b> \<and> b = d ` a})" | apply clarify
apply (rule_tac t = a in term_case)
apply (simp_all add: POgenXH idgen_lemmas)
apply fast
done | lemma | lemma2 | CCL | src/CCL/Fix.thy | [] | [
"EX",
"POgen",
"POgenXH",
"add",
"and",
"apply",
"idgen",
"idgen_lemmas",
"term_case"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
id_least_idgen: "idgen(d) = d \<Longrightarrow> lam x. x [= d" | apply (rule allI [THEN po_eta])
apply (rule lemma2 [THEN [2] po_coinduct])
apply simp
apply (fast intro: po_eta_lemma fix_idgenfp)+
done | lemma | id_least_idgen | CCL | src/CCL/Fix.thy | [] | [
"allI",
"apply",
"fix_idgenfp",
"idgen",
"lam",
"lemma2",
"po_coinduct",
"po_eta",
"po_eta_lemma",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
reachability: "fix(idgen) = lam x. x" | apply (fast intro: eq_iff [THEN iffD2]
id_idgenfp [THEN fix_least_idgen] fix_idgenfp [THEN id_least_idgen])
done | lemma | reachability | CCL | src/CCL/Fix.thy | [] | [
"apply",
"eq_iff",
"fix",
"fix_idgenfp",
"fix_least_idgen",
"id_idgenfp",
"id_least_idgen",
"idgen",
"iffD2",
"lam"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
id_apply: "f = lam x. x \<Longrightarrow> f`t = t" | apply (erule ssubst)
apply (rule applyB)
done | lemma | id_apply | CCL | src/CCL/Fix.thy | [] | [
"apply",
"applyB",
"lam",
"rule",
"ssubst"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
term_ind:
assumes 1: "P(bot)" and 2: "P(true)" and 3: "P(false)"
and 4: "\<And>x y. \<lbrakk>P(x); P(y)\<rbrakk> \<Longrightarrow> P(<x,y>)"
and 5: "\<And>u.(\<And>x. P(u(x))) \<Longrightarrow> P(lam x. u(x))"
and 6: "INCL(P)"
shows "P(t)" | apply (rule reachability [THEN id_apply, THEN subst])
apply (rule_tac x = t in spec)
apply (rule fix_ind)
apply (unfold idgen_def)
apply (rule allI)
apply (subst applyBbot)
apply (rule 1)
apply (rule allI)
apply (rule applyB [THEN ssubst])
apply (rule_tac t = "xa" in term_case)
appl... | lemma | term_ind | CCL | src/CCL/Fix.thy | [] | [
"And",
"INCL",
"allI",
"all_INCL",
"and",
"apply",
"applyB",
"applyBbot",
"assumes",
"bot",
"false",
"fix_ind",
"id_apply",
"lam",
"reachability",
"rule",
"shows",
"spec",
"ssubst",
"subst",
"term_case",
"true",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp :: "['a set\<Rightarrow>'a set] \<Rightarrow> 'a set" | where \<comment> \<open>greatest fixed point\<close>
"gfp(f) == Union({u. u <= f(u)})" | definition | gfp | CCL | src/CCL/Gfp.thy | [] | [
"Union",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_upperbound: "A <= f(A) \<Longrightarrow> A <= gfp(f)" | unfolding gfp_def by blast | lemma | gfp_upperbound | CCL | src/CCL/Gfp.thy | [] | [
"gfp"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_least: "(\<And>u. u <= f(u) \<Longrightarrow> u <= A) \<Longrightarrow> gfp(f) <= A" | unfolding gfp_def by blast | lemma | gfp_least | CCL | src/CCL/Gfp.thy | [] | [
"And",
"gfp"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_lemma2: "mono(f) \<Longrightarrow> gfp(f) <= f(gfp(f))" | by (rule gfp_least, rule subset_trans, assumption, erule monoD,
rule gfp_upperbound, assumption) | lemma | gfp_lemma2 | CCL | src/CCL/Gfp.thy | [] | [
"gfp",
"gfp_least",
"gfp_upperbound",
"mono",
"monoD",
"rule",
"subset_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_lemma3: "mono(f) \<Longrightarrow> f(gfp(f)) <= gfp(f)" | by (rule gfp_upperbound, frule monoD, rule gfp_lemma2, assumption+) | lemma | gfp_lemma3 | CCL | src/CCL/Gfp.thy | [] | [
"gfp",
"gfp_lemma2",
"gfp_upperbound",
"mono",
"monoD",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_Tarski: "mono(f) \<Longrightarrow> gfp(f) = f(gfp(f))" | by (rule equalityI gfp_lemma2 gfp_lemma3 | assumption)+ | lemma | gfp_Tarski | CCL | src/CCL/Gfp.thy | [] | [
"equalityI",
"gfp",
"gfp_lemma2",
"gfp_lemma3",
"mono",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct: "\<lbrakk>a: A; A <= f(A)\<rbrakk> \<Longrightarrow> a : gfp(f)" | by (blast dest: gfp_upperbound) | lemma | coinduct | CCL | src/CCL/Gfp.thy | [] | [
"gfp",
"gfp_upperbound"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct2_lemma: "\<lbrakk>A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> A Un gfp(f) <= f(A Un gfp(f))" | apply (rule subset_trans)
prefer 2
apply (erule mono_Un)
apply (rule subst, erule gfp_Tarski)
apply (erule Un_least)
apply (rule Un_upper2)
done | lemma | coinduct2_lemma | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"Un_least",
"Un_upper2",
"apply",
"gfp",
"gfp_Tarski",
"mono",
"mono_Un",
"rule",
"subset_trans",
"subst"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct2: "\<lbrakk>a: A; A <= f(A) Un gfp(f); mono(f)\<rbrakk> \<Longrightarrow> a : gfp(f)" | apply (rule coinduct)
prefer 2
apply (erule coinduct2_lemma, assumption)
apply blast
done | lemma | coinduct2 | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"apply",
"coinduct",
"coinduct2_lemma",
"gfp",
"mono",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct3_mono_lemma: "mono(f) \<Longrightarrow> mono(\<lambda>x. f(x) Un A Un B)" | by (rule monoI) (blast dest: monoD) | lemma | coinduct3_mono_lemma | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"lambda",
"mono",
"monoD",
"monoI",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct3_lemma:
assumes prem: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
and mono: "mono(f)"
shows "lfp(\<lambda>x. f(x) Un A Un gfp(f)) <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))" | apply (rule subset_trans)
apply (rule mono [THEN coinduct3_mono_lemma, THEN lfp_lemma3])
apply (rule Un_least [THEN Un_least])
apply (rule subset_refl)
apply (rule prem)
apply (rule mono [THEN gfp_Tarski, THEN equalityD1, THEN subset_trans])
apply (rule mono [THEN monoD])
apply (subst mono [THEN coind... | lemma | coinduct3_lemma | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"Un_least",
"Un_upper2",
"and",
"apply",
"assumes",
"coinduct3_mono_lemma",
"equalityD1",
"gfp",
"gfp_Tarski",
"lambda",
"lfp",
"lfp_Tarski",
"lfp_lemma3",
"mono",
"monoD",
"rule",
"shows",
"subset_refl",
"subset_trans",
"subst"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
coinduct3:
assumes 1: "a:A"
and 2: "A <= f(lfp(\<lambda>x. f(x) Un A Un gfp(f)))"
and 3: "mono(f)"
shows "a : gfp(f)" | apply (rule coinduct)
prefer 2
apply (rule coinduct3_lemma [OF 2 3])
apply (subst lfp_Tarski [OF coinduct3_mono_lemma, OF 3])
using 1 apply blast
done | lemma | coinduct3 | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"and",
"apply",
"assumes",
"coinduct",
"coinduct3_lemma",
"coinduct3_mono_lemma",
"gfp",
"lambda",
"lfp",
"lfp_Tarski",
"mono",
"rule",
"shows",
"subst"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_gfp_Tarski: "\<lbrakk>h == gfp(f); mono(f)\<rbrakk> \<Longrightarrow> h = f(h)" | apply unfold
apply (erule gfp_Tarski)
done | lemma | def_gfp_Tarski | CCL | src/CCL/Gfp.thy | [] | [
"apply",
"gfp",
"gfp_Tarski",
"mono",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_coinduct: "\<lbrakk>h == gfp(f); a:A; A <= f(A)\<rbrakk> \<Longrightarrow> a: h" | apply unfold
apply (erule coinduct)
apply assumption
done | lemma | def_coinduct | CCL | src/CCL/Gfp.thy | [] | [
"apply",
"coinduct",
"gfp",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_coinduct2: "\<lbrakk>h == gfp(f); a:A; A <= f(A) Un h; mono(f)\<rbrakk> \<Longrightarrow> a: h" | apply unfold
apply (erule coinduct2)
apply assumption
apply assumption
done | lemma | def_coinduct2 | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"apply",
"coinduct2",
"gfp",
"mono",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
def_coinduct3: "\<lbrakk>h == gfp(f); a:A; A <= f(lfp(\<lambda>x. f(x) Un A Un h)); mono(f)\<rbrakk> \<Longrightarrow> a: h" | apply unfold
apply (erule coinduct3)
apply assumption
apply assumption
done | lemma | def_coinduct3 | CCL | src/CCL/Gfp.thy | [] | [
"Un",
"apply",
"coinduct3",
"gfp",
"lambda",
"lfp",
"mono",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
gfp_mono: "\<lbrakk>mono(f); \<And>Z. f(Z) <= g(Z)\<rbrakk> \<Longrightarrow> gfp(f) <= gfp(g)" | apply (rule gfp_upperbound)
apply (rule subset_trans)
apply (rule gfp_lemma2)
apply assumption
apply (erule meta_spec)
done | lemma | gfp_mono | CCL | src/CCL/Gfp.thy | [] | [
"And",
"apply",
"gfp",
"gfp_lemma2",
"gfp_upperbound",
"meta_spec",
"mono",
"rule",
"subset_trans"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTTgen :: "i set \<Rightarrow> i set" | where
"HTTgen(R) ==
{t. t=true | t=false | (EX a b. t= <a, b> \<and> a : R \<and> b : R) |
(EX f. t = lam x. f(x) \<and> (ALL x. f(x) : R))}" | definition | HTTgen | CCL | src/CCL/Hered.thy | [] | [
"ALL",
"EX",
"and",
"false",
"lam",
"set",
"true",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT :: "i set" | where "HTT == gfp(HTTgen)" | definition | HTT | CCL | src/CCL/Hered.thy | [] | [
"HTTgen",
"gfp",
"set",
"where"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTTgen_mono: "mono(\<lambda>X. HTTgen(X))" | apply (unfold HTTgen_def)
apply (rule monoI)
apply blast
done | lemma | HTTgen_mono | CCL | src/CCL/Hered.thy | [] | [
"HTTgen",
"apply",
"lambda",
"mono",
"monoI",
"rule",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTTgenXH:
"t : HTTgen(A) \<longleftrightarrow> t=true | t=false | (EX a b. t=<a,b> \<and> a : A \<and> b : A) |
(EX f. t=lam x. f(x) \<and> (ALL x. f(x) : A))" | apply (unfold HTTgen_def)
apply blast
done | lemma | HTTgenXH | CCL | src/CCL/Hered.thy | [] | [
"ALL",
"EX",
"HTTgen",
"and",
"apply",
"false",
"lam",
"true",
"unfold"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTTXH:
"t : HTT \<longleftrightarrow> t=true | t=false | (EX a b. t=<a,b> \<and> a : HTT \<and> b : HTT) |
(EX f. t=lam x. f(x) \<and> (ALL x. f(x) : HTT))" | apply (rule HTTgen_mono [THEN HTT_def [THEN def_gfp_Tarski], THEN XHlemma1, unfolded HTTgen_def])
apply blast
done | lemma | HTTXH | CCL | src/CCL/Hered.thy | [] | [
"ALL",
"EX",
"HTT",
"HTTgen_mono",
"XHlemma1",
"and",
"apply",
"def_gfp_Tarski",
"false",
"lam",
"rule",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_bot: "\<not> bot : HTT" | by (blast dest: HTTXH [THEN iffD1]) | lemma | HTT_bot | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTXH",
"bot",
"iffD1",
"not"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_true: "true : HTT" | by (blast intro: HTTXH [THEN iffD2]) | lemma | HTT_true | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTXH",
"iffD2",
"true"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_false: "false : HTT" | by (blast intro: HTTXH [THEN iffD2]) | lemma | HTT_false | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTXH",
"false",
"iffD2"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_pair: "<a,b> : HTT \<longleftrightarrow> a : HTT \<and> b : HTT" | apply (rule HTTXH [THEN iff_trans])
apply blast
done | lemma | HTT_pair | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"HTTXH",
"and",
"apply",
"iff_trans",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_lam: "lam x. f(x) : HTT \<longleftrightarrow> (ALL x. f(x) : HTT)" | apply (rule HTTXH [THEN iff_trans])
apply auto
done
lemmas HTT_rews1 = HTT_bot HTT_true HTT_false HTT_pair HTT_lam | lemma | HTT_lam | CCL | src/CCL/Hered.thy | [] | [
"ALL",
"HTT",
"HTTXH",
"HTT_bot",
"HTT_false",
"HTT_pair",
"HTT_true",
"apply",
"auto",
"iff_trans",
"lam",
"rule"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 | |
HTT_rews2:
"one : HTT"
"inl(a) : HTT \<longleftrightarrow> a : HTT"
"inr(b) : HTT \<longleftrightarrow> b : HTT"
"zero : HTT"
"succ(n) : HTT \<longleftrightarrow> n : HTT"
"[] : HTT"
"x$xs : HTT \<longleftrightarrow> x : HTT \<and> xs : HTT" | by (simp_all add: data_defs HTT_rews1)
lemmas HTT_rews = HTT_rews1 HTT_rews2 | lemma | HTT_rews2 | CCL | src/CCL/Hered.thy | [] | [
"HTT",
"add",
"and",
"inl",
"inr",
"one",
"succ",
"xs",
"zero"
] | https://github.com/isabelle-prover/mirror-isabelle | acb41fc3ab1760eee1b9ed0edd0df566941bf513 |
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