phanerozoic/Isabelle-Stdlib · Datasets at Hugging Face

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
Int_greatest : "\<lbrakk>C<=A; C<=B\<rbrakk> \<Longrightarrow> C <= A Int B" by (blast intro: subsetI IntI dest: subsetD)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_greatest
monoI : "(\<And>A B. A <= B \<Longrightarrow> f(A) <= f(B)) \<Longrightarrow> mono(f)" unfolding mono_def by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	monoI
monoD : "\<lbrakk>mono(f); A <= B\<rbrakk> \<Longrightarrow> f(A) <= f(B)" unfolding mono_def by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	monoD
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	src	[ "FOL" ]	src/CCL/Set.thy	mono_Un
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	src	[ "FOL" ]	src/CCL/Set.thy	mono_Int
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	src	[ "FOL" ]	src/CCL/Set.thy	mem_rews
Int_absorb : "A Int A = A" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_absorb
Int_commute : "A Int B = B Int A" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_commute
Int_assoc : "(A Int B) Int C = A Int (B Int C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_assoc
Int_Un_distrib : "(A Un B) Int C = (A Int C) Un (B Int C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_Un_distrib
subset_Int_eq : "(A<=B) \<longleftrightarrow> (A Int B = A)" by (blast intro: equalityI elim: equalityE)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	subset_Int_eq
Un_absorb : "A Un A = A" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_absorb
Un_commute : "A Un B = B Un A" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_commute
Un_assoc : "(A Un B) Un C = A Un (B Un C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_assoc
Un_Int_distrib : "(A Int B) Un C = (A Un C) Int (B Un C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_Int_distrib
Un_Int_crazy : "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_Int_crazy
subset_Un_eq : "(A<=B) \<longleftrightarrow> (A Un B = B)" by (blast intro: equalityI elim: equalityE)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	subset_Un_eq
Compl_disjoint : "A Int Compl(A) = {x. False}" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_disjoint
Compl_partition : "A Un Compl(A) = {x. True}" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_partition
double_complement : "Compl(Compl(A)) = A" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	double_complement
Compl_Un : "Compl(A Un B) = Compl(A) Int Compl(B)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_Un
Compl_Int : "Compl(A Int B) = Compl(A) Un Compl(B)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_Int
Compl_UN : "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_UN
Compl_INT : "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))" by (blast intro: equalityI) (Halmos, Naive Set Theory, page 16.)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_INT
Un_Int_assoc_eq : "((A Int B) Un C = A Int (B Un C)) \<longleftrightarrow> (C<=A)" by (blast intro: equalityI elim: equalityE)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_Int_assoc_eq
Union_Un_distrib : "Union(A Un B) = Union(A) Un Union(B)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Union_Un_distrib
Union_disjoint : "(Union(C) Int A = {x. False}) \<longleftrightarrow> (ALL B:C. B Int A = {x. False})" by (blast intro: equalityI elim: equalityE)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Union_disjoint
Inter_Un_distrib : "Inter(A Un B) = Inter(A) Int Inter(B)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Inter_Un_distrib
UN_eq : "(UN x:A. B(x)) = Union({Y. EX x:A. Y=B(x)})" by (blast intro: equalityI) (Look: it has an EXISTENTIAL quantifier)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	UN_eq
INT_eq : "(INT x:A. B(x)) = Inter({Y. EX x:A. Y=B(x)})" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	INT_eq
Int_Union_image : "A Int Union(B) = (UN C:B. A Int C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_Union_image
Un_Inter_image : "A Un Inter(B) = (INT C:B. A Un C)" by (blast intro: equalityI)	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_Inter_image
Union_mono : "A<=B \<Longrightarrow> Union(A) <= Union(B)" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Union_mono
Inter_anti_mono : "B <= A \<Longrightarrow> Inter(A) <= Inter(B)" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Inter_anti_mono
UN_mono : "\<lbrakk>A <= B; \<And>x. x:A \<Longrightarrow> f(x)<=g(x)\<rbrakk> \<Longrightarrow> (UN x:A. f(x)) <= (UN x:B. g(x))" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	UN_mono
INT_anti_mono : "\<lbrakk>B <= A; \<And>x. x:A \<Longrightarrow> f(x) <= g(x)\<rbrakk> \<Longrightarrow> (INT x:A. f(x)) <= (INT x:A. g(x))" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	INT_anti_mono
Un_mono : "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Un B <= C Un D" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Un_mono
Int_mono : "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Int B <= C Int D" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Int_mono
Compl_anti_mono : "A <= B \<Longrightarrow> Compl(B) <= Compl(A)" by blast	lemma	src	[ "FOL" ]	src/CCL/Set.thy	Compl_anti_mono
one :: "i" where "one == true"	definition	src	[ "CCL" ]	src/CCL/Term.thy	one
inl :: "i\<Rightarrow>i" where "inl(a) == <true,a>"	definition	src	[ "CCL" ]	src/CCL/Term.thy	inl
inr :: "i\<Rightarrow>i" where "inr(b) == <false,b>"	definition	src	[ "CCL" ]	src/CCL/Term.thy	inr
split :: "[i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "split(t,f) == case(t, bot, bot, f, \<lambda>u. bot)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	split
fst :: "i\<Rightarrow>i" where "fst(t) == split(t, \<lambda>x y. x)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	fst
snd :: "i\<Rightarrow>i" where "snd(t) == split(t, \<lambda>x y. y)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	snd
thd :: "i\<Rightarrow>i" where "thd(t) == split(t, \<lambda>x p. split(p, \<lambda>y z. z))"	definition	src	[ "CCL" ]	src/CCL/Term.thy	thd
zero :: "i" where "zero == inl(one)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	zero
succ :: "i\<Rightarrow>i" where "succ(n) == inr(n)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	succ
ncase :: "[i,i,i\<Rightarrow>i]\<Rightarrow>i" where "ncase(n,b,c) == when(n, \<lambda>x. b, \<lambda>y. c(y))"	definition	src	[ "CCL" ]	src/CCL/Term.thy	ncase
letrec :: "[[i,i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec(h, b) == b(\<lambda>x. fix(\<lambda>f. lam x. h(x,\<lambda>y. f`y))`x)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	letrec
letrec2 :: "[[i,i,i\<Rightarrow>i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec2 (h, f) == letrec (\<lambda>p g'. split(p,\<lambda>x y. h(x,y,\<lambda>u v. g'(<u,v>))), \<lambda>g'. f(\<lambda>x y. g'(<x,y>)))"	definition	src	[ "CCL" ]	src/CCL/Term.thy	letrec2
letrec3 :: "[[i,i,i,i\<Rightarrow>i\<Rightarrow>i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i\<Rightarrow>i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec3 (h, f) == letrec (\<lambda>p g'. split(p,\<lambda>x xs. split(xs,\<lambda>y z. h(x,y,z,\<lambda>u v w. g'(<u,<v,w>>)))), \<lambda>g'. f(\<lambda>x y z. g'(<x,<y,z>>)))" syntax "_letrec" :: "[idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ be _/ in _)\<close> [0,0,0,60] 60) "_letrec2" :: "[idt,idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ _ be _/ in _)\<close> [0,0,0,0,60] 60) "_letrec3" :: "[idt,idt,idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ _ _ be _/ in _)\<close> [0,0,0,0,0,60] 60) syntax_consts "_letrec" == letrec and "_letrec2" == letrec2 and "_letrec3" == letrec3 parse_translation \<open> let	definition	src	[ "CCL" ]	src/CCL/Term.thy	letrec3
nrec :: "[i,i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "nrec(n,b,c) == letrec g x be ncase(x, b, \<lambda>y. c(y,g(y))) in g(n)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	nrec
nil :: "i" (\<open>[]\<close>) where "[] == inl(one)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	nil
cons :: "[i,i]\<Rightarrow>i" (infixr \<open>$\<close> 80) where "h$t == inr(<h,t>)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	cons
lcase :: "[i,i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "lcase(l,b,c) == when(l, \<lambda>x. b, \<lambda>y. split(y,c))"	definition	src	[ "CCL" ]	src/CCL/Term.thy	lcase
lrec :: "[i,i,[i,i,i]\<Rightarrow>i]\<Rightarrow>i" where "lrec(l,b,c) == letrec g x be lcase(x, b, \<lambda>h t. c(h,t,g(t))) in g(l)"	definition	src	[ "CCL" ]	src/CCL/Term.thy	lrec
napply :: "[i\<Rightarrow>i,i,i]\<Rightarrow>i" (\<open>(\<open>notation=\<open>mixfix napply\<close>\<close>_ ^ _ ` _)\<close> [56,56,56] 56) where "f ^n` a == nrec(n,a,\<lambda>x g. f(g))" lemmas simp_can_defs = one_def inl_def inr_def and simp_ncan_defs = if_def when_def split_def fst_def snd_def thd_def lemmas simp_defs = simp_can_defs simp_ncan_defs lemmas ind_can_defs = zero_def succ_def nil_def cons_def and ind_ncan_defs = ncase_def nrec_def lcase_def lrec_def lemmas ind_defs = ind_can_defs ind_ncan_defs lemmas data_defs = simp_defs ind_defs napply_def and genrec_defs = letrec_def letrec2_def letrec3_def	definition	src	[ "CCL" ]	src/CCL/Term.thy	napply
letB : "\<not> t=bot \<Longrightarrow> let x be t in f(x) = f(t)" apply (unfold let_def) apply (erule rev_mp) apply (rule_tac t = "t" in term_case) apply simp_all done	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letB
letBabot : "let x be bot in f(x) = bot" unfolding let_def by (rule caseBbot)	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letBabot
letBbbot : "let x be t in bot = bot" apply (unfold let_def) apply (rule_tac t = t in term_case) apply (rule caseBbot) apply simp_all done	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letBbbot
applyB : "(lam x. b(x)) ` a = b(a)" by (simp add: apply_def)	lemma	src	[ "CCL" ]	src/CCL/Term.thy	applyB
applyBbot : "bot ` a = bot" unfolding apply_def by (rule caseBbot)	lemma	src	[ "CCL" ]	src/CCL/Term.thy	applyBbot
fixB : "fix(f) = f(fix(f))" apply (unfold fix_def) apply (rule applyB [THEN ssubst], rule refl) done	lemma	src	[ "CCL" ]	src/CCL/Term.thy	fixB
letrecB : "letrec g x be h(x,g) in g(a) = h(a,\<lambda>y. letrec g x be h(x,g) in g(y))" apply (unfold letrec_def) apply (rule fixB [THEN ssubst], rule applyB [THEN ssubst], rule refl) done lemmas rawBs = caseBs applyB applyBbot method_setup beta_rl = \<open> Scan.succeed (fn ctxt => let val ctxt' = ctxt \|> Context_Position.set_visible false \|> Simplifier.add_simps @{thms rawBs} \|> Simplifier.set_loop (fn _ => stac ctxt @{thm letrecB}); in SIMPLE_METHOD' (CHANGED o simp_tac ctxt') end) \<close>	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letrecB
ifBtrue : "if true then t else u = t" and ifBfalse: "if false then t else u = u" and ifBbot: "if bot then t else u = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	ifBtrue
whenBinl : "when(inl(a),t,u) = t(a)" and whenBinr: "when(inr(a),t,u) = u(a)" and whenBbot: "when(bot,t,u) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	whenBinl
splitB : "split(<a,b>,h) = h(a,b)" and splitBbot: "split(bot,h) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	splitB
fstB : "fst(<a,b>) = a" and fstBbot: "fst(bot) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	fstB
sndB : "snd(<a,b>) = b" and sndBbot: "snd(bot) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	sndB
thdB : "thd(<a,<b,c>>) = c" and thdBbot: "thd(bot) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	thdB
ncaseBzero : "ncase(zero,t,u) = t" and ncaseBsucc: "ncase(succ(n),t,u) = u(n)" and ncaseBbot: "ncase(bot,t,u) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	ncaseBzero
nrecBzero : "nrec(zero,t,u) = t" and nrecBsucc: "nrec(succ(n),t,u) = u(n,nrec(n,t,u))" and nrecBbot: "nrec(bot,t,u) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	nrecBzero
lcaseBnil : "lcase([],t,u) = t" and lcaseBcons: "lcase(x$xs,t,u) = u(x,xs)" and lcaseBbot: "lcase(bot,t,u) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	lcaseBnil
lrecBnil : "lrec([],t,u) = t" and lrecBcons: "lrec(x$xs,t,u) = u(x,xs,lrec(xs,t,u))" and lrecBbot: "lrec(bot,t,u) = bot" unfolding data_defs by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	lrecBnil
letrec2B : "letrec g x y be h(x,y,g) in g(p,q) = h(p,q,\<lambda>u v. letrec g x y be h(x,y,g) in g(u,v))" unfolding data_defs letrec2_def by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letrec2B
letrec3B : "letrec g x y z be h(x,y,z,g) in g(p,q,r) = h(p,q,r,\<lambda>u v w. letrec g x y z be h(x,y,z,g) in g(u,v,w))" unfolding data_defs letrec3_def by beta_rl+	lemma	src	[ "CCL" ]	src/CCL/Term.thy	letrec3B
napplyBzero : "f^zero`a = a" and napplyBsucc: "f^succ(n)`a = f(f^n`a)" unfolding data_defs by beta_rl+ lemmas termBs = letB applyB applyBbot splitB splitBbot fstB fstBbot sndB sndBbot thdB thdBbot ifBtrue ifBfalse ifBbot whenBinl whenBinr whenBbot ncaseBzero ncaseBsucc ncaseBbot nrecBzero nrecBsucc nrecBbot lcaseBnil lcaseBcons lcaseBbot lrecBnil lrecBcons lrecBbot napplyBzero napplyBsucc	lemma	src	[ "CCL" ]	src/CCL/Term.thy	napplyBzero
term_injs : "(inl(a) = inl(a')) \<longleftrightarrow> (a=a')" "(inr(a) = inr(a')) \<longleftrightarrow> (a=a')" "(succ(a) = succ(a')) \<longleftrightarrow> (a=a')" "(a$b = a'$b') \<longleftrightarrow> (a=a' \<and> b=b')" by (inj_rl applyB splitB whenBinl whenBinr ncaseBsucc lcaseBcons)	lemma	src	[ "CCL" ]	src/CCL/Term.thy	term_injs
term_porews : "inl(a) [= inl(a') \<longleftrightarrow> a [= a'" "inr(b) [= inr(b') \<longleftrightarrow> b [= b'" "succ(n) [= succ(n') \<longleftrightarrow> n [= n'" "x$xs [= x'$xs' \<longleftrightarrow> x [= x' \<and> xs [= xs'" by (simp_all add: data_defs ccl_porews)	lemma	src	[ "CCL" ]	src/CCL/Term.thy	term_porews
trans :: "i set \<Rightarrow> o" (transitivity predicate) where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"	definition	src	[ "CCL" ]	src/CCL/Trancl.thy	trans
id :: "i set" (the identity relation) where "id == {p. EX x. p = <x,x>}"	definition	src	[ "CCL" ]	src/CCL/Trancl.thy	id
relcomp :: "[i set,i set] \<Rightarrow> i set" (infixr \<open>O\<close> 60) (composition of relations) where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"	definition	src	[ "CCL" ]	src/CCL/Trancl.thy	relcomp
rtrancl :: "i set \<Rightarrow> i set" (\<open>(\<open>notation=\<open>postfix ^\<close>\<close>_^)\<close> [100] 100) where "r^* == lfp(\<lambda>s. id Un (r O s))"	definition	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl
trancl :: "i set \<Rightarrow> i set" (\<open>(\<open>notation=\<open>postfix ^+\<close>\<close>_^+)\<close> [100] 100) where "r^+ == r O rtrancl(r)"	definition	src	[ "CCL" ]	src/CCL/Trancl.thy	trancl
transI : "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)" unfolding trans_def by blast	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	transI
transD : "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r" unfolding trans_def by blast	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	transD
idI : "<a,a> : id" apply (unfold id_def) apply (rule CollectI) apply (rule exI) apply (rule refl) done	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	idI
idE : "\<lbrakk>p: id; \<And>x. p = <x,x> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" apply (unfold id_def) apply (erule CollectE) apply blast done	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	idE
compI : "\<lbrakk><a,b>:s; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c> : r O s" unfolding relcomp_def by blast (proof requires higher-level assumptions or a delaying of hyp_subst_tac)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	compI
compE : "\<lbrakk>xz : r O s; \<And>x y z. \<lbrakk>xz = <x,z>; <x,y>:s; <y,z>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" unfolding relcomp_def by blast	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	compE
compEpair : "\<lbrakk><a,c> : r O s; \<And>y. \<lbrakk><a,y>:s; <y,c>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" apply (erule compE) apply (simp add: pair_inject) done lemmas [intro] = compI idI and [elim] = compE idE	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	compEpair
comp_mono : "\<lbrakk>r'<=r; s'<=s\<rbrakk> \<Longrightarrow> (r' O s') <= (r O s)" by blast	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	comp_mono
rtrancl_fun_mono : "mono(\<lambda>s. id Un (r O s))" apply (rule monoI) apply (rule monoI subset_refl comp_mono Un_mono)+ apply assumption done	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_fun_mono
rtrancl_unfold : "r^* = id Un (r O r^)" by (rule rtrancl_fun_mono [THEN rtrancl_def [THEN def_lfp_Tarski]]) (Reflexivity of rtrancl*)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_unfold
rtrancl_refl : "<a,a> : r^" apply (subst rtrancl_unfold) apply blast done (Closure under composition with r*)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_refl
rtrancl_into_rtrancl : "\<lbrakk><a,b> : r^; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^" apply (subst rtrancl_unfold) apply blast done (rtrancl of r contains r)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_into_rtrancl
r_into_rtrancl : "<a,b> : r \<Longrightarrow> <a,b> : r^*" apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl]) apply assumption done	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	r_into_rtrancl
rtrancl_full_induct : "\<lbrakk><a,b> : r^; \<And>x. P(<x,x>); \<And>x y z. \<lbrakk>P(<x,y>); <x,y>: r^; <y,z>: r\<rbrakk> \<Longrightarrow> P(<x,z>)\<rbrakk> \<Longrightarrow> P(<a,b>)" apply (erule def_induct [OF rtrancl_def]) apply (rule rtrancl_fun_mono) apply blast done (nice induction rule)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_full_induct
rtrancl_induct : "\<lbrakk><a,b> : r^; P(a); \<And>y z. \<lbrakk><a,y> : r^; <y,z> : r; P(y)\<rbrakk> \<Longrightarrow> P(z) \<rbrakk> \<Longrightarrow> P(b)" (by induction on this formula) apply (subgoal_tac "ALL y. <a,b> = <a,y> \<longrightarrow> P(y)") (now solve first subgoal: this formula is sufficient) apply blast (now do the induction) apply (erule rtrancl_full_induct) apply blast apply blast done (transitivity of transitive closure!! -- by induction.)	lemma	src	[ "CCL" ]	src/CCL/Trancl.thy	rtrancl_induct

Int_greatest : "\<lbrakk>C<=A; C<=B\<rbrakk> \<Longrightarrow> C <= A Int B" by (blast intro: subsetI IntI dest: subsetD)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_greatest

monoI : "(\<And>A B. A <= B \<Longrightarrow> f(A) <= f(B)) \<Longrightarrow> mono(f)" unfolding mono_def by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

monoI

monoD : "\<lbrakk>mono(f); A <= B\<rbrakk> \<Longrightarrow> f(A) <= f(B)" unfolding mono_def by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

monoD

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

src

[ "FOL" ]

src/CCL/Set.thy

mono_Un

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

src

[ "FOL" ]

src/CCL/Set.thy

mono_Int

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

src

[ "FOL" ]

src/CCL/Set.thy

mem_rews

Int_absorb : "A Int A = A" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_absorb

Int_commute : "A Int B = B Int A" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_commute

Int_assoc : "(A Int B) Int C = A Int (B Int C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_assoc

Int_Un_distrib : "(A Un B) Int C = (A Int C) Un (B Int C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_Un_distrib

subset_Int_eq : "(A<=B) \<longleftrightarrow> (A Int B = A)" by (blast intro: equalityI elim: equalityE)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

subset_Int_eq

Un_absorb : "A Un A = A" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_absorb

Un_commute : "A Un B = B Un A" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_commute

Un_assoc : "(A Un B) Un C = A Un (B Un C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_assoc

Un_Int_distrib : "(A Int B) Un C = (A Un C) Int (B Un C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_Int_distrib

Un_Int_crazy : "(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_Int_crazy

subset_Un_eq : "(A<=B) \<longleftrightarrow> (A Un B = B)" by (blast intro: equalityI elim: equalityE)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

subset_Un_eq

Compl_disjoint : "A Int Compl(A) = {x. False}" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_disjoint

Compl_partition : "A Un Compl(A) = {x. True}" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_partition

double_complement : "Compl(Compl(A)) = A" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

double_complement

Compl_Un : "Compl(A Un B) = Compl(A) Int Compl(B)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_Un

Compl_Int : "Compl(A Int B) = Compl(A) Un Compl(B)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_Int

Compl_UN : "Compl(UN x:A. B(x)) = (INT x:A. Compl(B(x)))" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_UN

Compl_INT : "Compl(INT x:A. B(x)) = (UN x:A. Compl(B(x)))" by (blast intro: equalityI) (*Halmos, Naive Set Theory, page 16.*)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_INT

Un_Int_assoc_eq : "((A Int B) Un C = A Int (B Un C)) \<longleftrightarrow> (C<=A)" by (blast intro: equalityI elim: equalityE)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_Int_assoc_eq

Union_Un_distrib : "Union(A Un B) = Union(A) Un Union(B)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Union_Un_distrib

Union_disjoint : "(Union(C) Int A = {x. False}) \<longleftrightarrow> (ALL B:C. B Int A = {x. False})" by (blast intro: equalityI elim: equalityE)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Union_disjoint

Inter_Un_distrib : "Inter(A Un B) = Inter(A) Int Inter(B)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Inter_Un_distrib

UN_eq : "(UN x:A. B(x)) = Union({Y. EX x:A. Y=B(x)})" by (blast intro: equalityI) (*Look: it has an EXISTENTIAL quantifier*)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

UN_eq

INT_eq : "(INT x:A. B(x)) = Inter({Y. EX x:A. Y=B(x)})" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

INT_eq

Int_Union_image : "A Int Union(B) = (UN C:B. A Int C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_Union_image

Un_Inter_image : "A Un Inter(B) = (INT C:B. A Un C)" by (blast intro: equalityI)

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_Inter_image

Union_mono : "A<=B \<Longrightarrow> Union(A) <= Union(B)" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Union_mono

Inter_anti_mono : "B <= A \<Longrightarrow> Inter(A) <= Inter(B)" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Inter_anti_mono

UN_mono : "\<lbrakk>A <= B; \<And>x. x:A \<Longrightarrow> f(x)<=g(x)\<rbrakk> \<Longrightarrow> (UN x:A. f(x)) <= (UN x:B. g(x))" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

UN_mono

INT_anti_mono : "\<lbrakk>B <= A; \<And>x. x:A \<Longrightarrow> f(x) <= g(x)\<rbrakk> \<Longrightarrow> (INT x:A. f(x)) <= (INT x:A. g(x))" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

INT_anti_mono

Un_mono : "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Un B <= C Un D" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Un_mono

Int_mono : "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Int B <= C Int D" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Int_mono

Compl_anti_mono : "A <= B \<Longrightarrow> Compl(B) <= Compl(A)" by blast

lemma

src

[ "FOL" ]

src/CCL/Set.thy

Compl_anti_mono

one :: "i" where "one == true"

definition

src

[ "CCL" ]

src/CCL/Term.thy

one

inl :: "i\<Rightarrow>i" where "inl(a) == <true,a>"

definition

src

[ "CCL" ]

src/CCL/Term.thy

inl

inr :: "i\<Rightarrow>i" where "inr(b) == <false,b>"

definition

src

[ "CCL" ]

src/CCL/Term.thy

inr

split :: "[i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "split(t,f) == case(t, bot, bot, f, \<lambda>u. bot)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

split

fst :: "i\<Rightarrow>i" where "fst(t) == split(t, \<lambda>x y. x)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

fst

snd :: "i\<Rightarrow>i" where "snd(t) == split(t, \<lambda>x y. y)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

snd

thd :: "i\<Rightarrow>i" where "thd(t) == split(t, \<lambda>x p. split(p, \<lambda>y z. z))"

definition

src

[ "CCL" ]

src/CCL/Term.thy

thd

zero :: "i" where "zero == inl(one)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

zero

succ :: "i\<Rightarrow>i" where "succ(n) == inr(n)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

succ

ncase :: "[i,i,i\<Rightarrow>i]\<Rightarrow>i" where "ncase(n,b,c) == when(n, \<lambda>x. b, \<lambda>y. c(y))"

definition

src

[ "CCL" ]

src/CCL/Term.thy

ncase

letrec :: "[[i,i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec(h, b) == b(\<lambda>x. fix(\<lambda>f. lam x. h(x,\<lambda>y. f`y))`x)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

letrec

letrec2 :: "[[i,i,i\<Rightarrow>i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec2 (h, f) == letrec (\<lambda>p g'. split(p,\<lambda>x y. h(x,y,\<lambda>u v. g'(<u,v>))), \<lambda>g'. f(\<lambda>x y. g'(<x,y>)))"

definition

src

[ "CCL" ]

src/CCL/Term.thy

letrec2

letrec3 :: "[[i,i,i,i\<Rightarrow>i\<Rightarrow>i\<Rightarrow>i]\<Rightarrow>i,(i\<Rightarrow>i\<Rightarrow>i\<Rightarrow>i)\<Rightarrow>i]\<Rightarrow>i" where "letrec3 (h, f) == letrec (\<lambda>p g'. split(p,\<lambda>x xs. split(xs,\<lambda>y z. h(x,y,z,\<lambda>u v w. g'(<u,<v,w>>)))), \<lambda>g'. f(\<lambda>x y z. g'(<x,<y,z>>)))" syntax "_letrec" :: "[idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ be _/ in _)\<close> [0,0,0,60] 60) "_letrec2" :: "[idt,idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ _ be _/ in _)\<close> [0,0,0,0,60] 60) "_letrec3" :: "[idt,idt,idt,idt,i,i]\<Rightarrow>i" (\<open>(\<open>indent=3 notation=\<open>mixfix letrec be in\<close>\<close>letrec _ _ _ _ be _/ in _)\<close> [0,0,0,0,0,60] 60) syntax_consts "_letrec" == letrec and "_letrec2" == letrec2 and "_letrec3" == letrec3 parse_translation \<open> let

definition

src

[ "CCL" ]

src/CCL/Term.thy

letrec3

nrec :: "[i,i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "nrec(n,b,c) == letrec g x be ncase(x, b, \<lambda>y. c(y,g(y))) in g(n)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

nrec

nil :: "i" (\<open>[]\<close>) where "[] == inl(one)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

nil

cons :: "[i,i]\<Rightarrow>i" (infixr \<open>$\<close> 80) where "h$t == inr(<h,t>)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

cons

lcase :: "[i,i,[i,i]\<Rightarrow>i]\<Rightarrow>i" where "lcase(l,b,c) == when(l, \<lambda>x. b, \<lambda>y. split(y,c))"

definition

src

[ "CCL" ]

src/CCL/Term.thy

lcase

lrec :: "[i,i,[i,i,i]\<Rightarrow>i]\<Rightarrow>i" where "lrec(l,b,c) == letrec g x be lcase(x, b, \<lambda>h t. c(h,t,g(t))) in g(l)"

definition

src

[ "CCL" ]

src/CCL/Term.thy

lrec

napply :: "[i\<Rightarrow>i,i,i]\<Rightarrow>i" (\<open>(\<open>notation=\<open>mixfix napply\<close>\<close>_ ^ _ ` _)\<close> [56,56,56] 56) where "f ^n` a == nrec(n,a,\<lambda>x g. f(g))" lemmas simp_can_defs = one_def inl_def inr_def and simp_ncan_defs = if_def when_def split_def fst_def snd_def thd_def lemmas simp_defs = simp_can_defs simp_ncan_defs lemmas ind_can_defs = zero_def succ_def nil_def cons_def and ind_ncan_defs = ncase_def nrec_def lcase_def lrec_def lemmas ind_defs = ind_can_defs ind_ncan_defs lemmas data_defs = simp_defs ind_defs napply_def and genrec_defs = letrec_def letrec2_def letrec3_def

definition

src

[ "CCL" ]

src/CCL/Term.thy

napply

letB : "\<not> t=bot \<Longrightarrow> let x be t in f(x) = f(t)" apply (unfold let_def) apply (erule rev_mp) apply (rule_tac t = "t" in term_case) apply simp_all done

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letB

letBabot : "let x be bot in f(x) = bot" unfolding let_def by (rule caseBbot)

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letBabot

letBbbot : "let x be t in bot = bot" apply (unfold let_def) apply (rule_tac t = t in term_case) apply (rule caseBbot) apply simp_all done

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letBbbot

applyB : "(lam x. b(x)) ` a = b(a)" by (simp add: apply_def)

lemma

src

[ "CCL" ]

src/CCL/Term.thy

applyB

applyBbot : "bot ` a = bot" unfolding apply_def by (rule caseBbot)

lemma

src

[ "CCL" ]

src/CCL/Term.thy

applyBbot

fixB : "fix(f) = f(fix(f))" apply (unfold fix_def) apply (rule applyB [THEN ssubst], rule refl) done

lemma

src

[ "CCL" ]

src/CCL/Term.thy

fixB

letrecB : "letrec g x be h(x,g) in g(a) = h(a,\<lambda>y. letrec g x be h(x,g) in g(y))" apply (unfold letrec_def) apply (rule fixB [THEN ssubst], rule applyB [THEN ssubst], rule refl) done lemmas rawBs = caseBs applyB applyBbot method_setup beta_rl = \<open> Scan.succeed (fn ctxt => let val ctxt' = ctxt |> Context_Position.set_visible false |> Simplifier.add_simps @{thms rawBs} |> Simplifier.set_loop (fn _ => stac ctxt @{thm letrecB}); in SIMPLE_METHOD' (CHANGED o simp_tac ctxt') end) \<close>

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letrecB

ifBtrue : "if true then t else u = t" and ifBfalse: "if false then t else u = u" and ifBbot: "if bot then t else u = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

ifBtrue

whenBinl : "when(inl(a),t,u) = t(a)" and whenBinr: "when(inr(a),t,u) = u(a)" and whenBbot: "when(bot,t,u) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

whenBinl

splitB : "split(<a,b>,h) = h(a,b)" and splitBbot: "split(bot,h) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

splitB

fstB : "fst(<a,b>) = a" and fstBbot: "fst(bot) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

fstB

sndB : "snd(<a,b>) = b" and sndBbot: "snd(bot) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

sndB

thdB : "thd(<a,<b,c>>) = c" and thdBbot: "thd(bot) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

thdB

ncaseBzero : "ncase(zero,t,u) = t" and ncaseBsucc: "ncase(succ(n),t,u) = u(n)" and ncaseBbot: "ncase(bot,t,u) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

ncaseBzero

nrecBzero : "nrec(zero,t,u) = t" and nrecBsucc: "nrec(succ(n),t,u) = u(n,nrec(n,t,u))" and nrecBbot: "nrec(bot,t,u) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

nrecBzero

lcaseBnil : "lcase([],t,u) = t" and lcaseBcons: "lcase(x$xs,t,u) = u(x,xs)" and lcaseBbot: "lcase(bot,t,u) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

lcaseBnil

lrecBnil : "lrec([],t,u) = t" and lrecBcons: "lrec(x$xs,t,u) = u(x,xs,lrec(xs,t,u))" and lrecBbot: "lrec(bot,t,u) = bot" unfolding data_defs by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

lrecBnil

letrec2B : "letrec g x y be h(x,y,g) in g(p,q) = h(p,q,\<lambda>u v. letrec g x y be h(x,y,g) in g(u,v))" unfolding data_defs letrec2_def by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letrec2B

letrec3B : "letrec g x y z be h(x,y,z,g) in g(p,q,r) = h(p,q,r,\<lambda>u v w. letrec g x y z be h(x,y,z,g) in g(u,v,w))" unfolding data_defs letrec3_def by beta_rl+

lemma

src

[ "CCL" ]

src/CCL/Term.thy

letrec3B

napplyBzero : "f^zero`a = a" and napplyBsucc: "f^succ(n)`a = f(f^n`a)" unfolding data_defs by beta_rl+ lemmas termBs = letB applyB applyBbot splitB splitBbot fstB fstBbot sndB sndBbot thdB thdBbot ifBtrue ifBfalse ifBbot whenBinl whenBinr whenBbot ncaseBzero ncaseBsucc ncaseBbot nrecBzero nrecBsucc nrecBbot lcaseBnil lcaseBcons lcaseBbot lrecBnil lrecBcons lrecBbot napplyBzero napplyBsucc

lemma

src

[ "CCL" ]

src/CCL/Term.thy

napplyBzero

term_injs : "(inl(a) = inl(a')) \<longleftrightarrow> (a=a')" "(inr(a) = inr(a')) \<longleftrightarrow> (a=a')" "(succ(a) = succ(a')) \<longleftrightarrow> (a=a')" "(a$b = a'$b') \<longleftrightarrow> (a=a' \<and> b=b')" by (inj_rl applyB splitB whenBinl whenBinr ncaseBsucc lcaseBcons)

lemma

src

[ "CCL" ]

src/CCL/Term.thy

term_injs

term_porews : "inl(a) [= inl(a') \<longleftrightarrow> a [= a'" "inr(b) [= inr(b') \<longleftrightarrow> b [= b'" "succ(n) [= succ(n') \<longleftrightarrow> n [= n'" "x$xs [= x'$xs' \<longleftrightarrow> x [= x' \<and> xs [= xs'" by (simp_all add: data_defs ccl_porews)

lemma

src

[ "CCL" ]

src/CCL/Term.thy

term_porews

trans :: "i set \<Rightarrow> o" (*transitivity predicate*) where "trans(r) == (ALL x y z. <x,y>:r \<longrightarrow> <y,z>:r \<longrightarrow> <x,z>:r)"

definition

src

[ "CCL" ]

src/CCL/Trancl.thy

trans

id :: "i set" (*the identity relation*) where "id == {p. EX x. p = <x,x>}"

definition

src

[ "CCL" ]

src/CCL/Trancl.thy

id

relcomp :: "[i set,i set] \<Rightarrow> i set" (infixr \<open>O\<close> 60) (*composition of relations*) where "r O s == {xz. EX x y z. xz = <x,z> \<and> <x,y>:s \<and> <y,z>:r}"

definition

src

[ "CCL" ]

src/CCL/Trancl.thy

relcomp

rtrancl :: "i set \<Rightarrow> i set" (\<open>(\<open>notation=\<open>postfix ^*\<close>\<close>_^*)\<close> [100] 100) where "r^* == lfp(\<lambda>s. id Un (r O s))"

definition

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl

trancl :: "i set \<Rightarrow> i set" (\<open>(\<open>notation=\<open>postfix ^+\<close>\<close>_^+)\<close> [100] 100) where "r^+ == r O rtrancl(r)"

definition

src

[ "CCL" ]

src/CCL/Trancl.thy

trancl

transI : "(\<And>x y z. \<lbrakk><x,y>:r; <y,z>:r\<rbrakk> \<Longrightarrow> <x,z>:r) \<Longrightarrow> trans(r)" unfolding trans_def by blast

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

transI

transD : "\<lbrakk>trans(r); <a,b>:r; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c>:r" unfolding trans_def by blast

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

transD

idI : "<a,a> : id" apply (unfold id_def) apply (rule CollectI) apply (rule exI) apply (rule refl) done

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

idI

idE : "\<lbrakk>p: id; \<And>x. p = <x,x> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" apply (unfold id_def) apply (erule CollectE) apply blast done

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

idE

compI : "\<lbrakk><a,b>:s; <b,c>:r\<rbrakk> \<Longrightarrow> <a,c> : r O s" unfolding relcomp_def by blast (*proof requires higher-level assumptions or a delaying of hyp_subst_tac*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

compI

compE : "\<lbrakk>xz : r O s; \<And>x y z. \<lbrakk>xz = <x,z>; <x,y>:s; <y,z>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" unfolding relcomp_def by blast

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

compE

compEpair : "\<lbrakk><a,c> : r O s; \<And>y. \<lbrakk><a,y>:s; <y,c>:r\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P" apply (erule compE) apply (simp add: pair_inject) done lemmas [intro] = compI idI and [elim] = compE idE

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

compEpair

comp_mono : "\<lbrakk>r'<=r; s'<=s\<rbrakk> \<Longrightarrow> (r' O s') <= (r O s)" by blast

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

comp_mono

rtrancl_fun_mono : "mono(\<lambda>s. id Un (r O s))" apply (rule monoI) apply (rule monoI subset_refl comp_mono Un_mono)+ apply assumption done

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_fun_mono

rtrancl_unfold : "r^* = id Un (r O r^*)" by (rule rtrancl_fun_mono [THEN rtrancl_def [THEN def_lfp_Tarski]]) (*Reflexivity of rtrancl*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_unfold

rtrancl_refl : "<a,a> : r^*" apply (subst rtrancl_unfold) apply blast done (*Closure under composition with r*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_refl

rtrancl_into_rtrancl : "\<lbrakk><a,b> : r^*; <b,c> : r\<rbrakk> \<Longrightarrow> <a,c> : r^*" apply (subst rtrancl_unfold) apply blast done (*rtrancl of r contains r*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_into_rtrancl

r_into_rtrancl : "<a,b> : r \<Longrightarrow> <a,b> : r^*" apply (rule rtrancl_refl [THEN rtrancl_into_rtrancl]) apply assumption done

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

r_into_rtrancl

rtrancl_full_induct : "\<lbrakk><a,b> : r^*; \<And>x. P(<x,x>); \<And>x y z. \<lbrakk>P(<x,y>); <x,y>: r^*; <y,z>: r\<rbrakk> \<Longrightarrow> P(<x,z>)\<rbrakk> \<Longrightarrow> P(<a,b>)" apply (erule def_induct [OF rtrancl_def]) apply (rule rtrancl_fun_mono) apply blast done (*nice induction rule*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_full_induct

rtrancl_induct : "\<lbrakk><a,b> : r^*; P(a); \<And>y z. \<lbrakk><a,y> : r^*; <y,z> : r; P(y)\<rbrakk> \<Longrightarrow> P(z) \<rbrakk> \<Longrightarrow> P(b)" (*by induction on this formula*) apply (subgoal_tac "ALL y. <a,b> = <a,y> \<longrightarrow> P(y)") (*now solve first subgoal: this formula is sufficient*) apply blast (*now do the induction*) apply (erule rtrancl_full_induct) apply blast apply blast done (*transitivity of transitive closure!! -- by induction.*)

lemma

src

[ "CCL" ]

src/CCL/Trancl.thy

rtrancl_induct