hcju/isabelleps huggingface dataset:
Rep_formula_sup: fixes x :: "'a formula"   and y :: "'a formula" shows "Rep_formula (x \<squnion> y) = Rep_formula x \<union> Rep_formula y"
minimal: fixes X' :: "'v set" assumes "X' \<subset> X" shows "integral\<^sup>N (count_space X') (weight \<Gamma>) \<le> integral\<^sup>N (count_space (\<^bold>E `` X')) (weight \<Gamma>)"
rt_fresher_imp_nsqn_le: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2"   and "ip \<in> kD rt1"   and "ip \<in> kD rt2" shows "nsqn rt1 ip \<le> nsqn rt2 ip"
assms_6: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"
simplex_extremal_le_exists: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "finite S"   and "x \<in> convex hull S"   and "y \<in> convex hull S" shows "\<exists>u\<in>S. \<exists>v\<in>S. norm (x - y) \<le> norm (u - v)"
insert_def: fixes a :: "'c"   and B :: "'c set" shows "{a} \<union> B = _Collect x (x = a) \<union> B"
measurableI: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and N :: "'e measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> f x \<in> space N"   and "\<And>A. A \<in> sets N \<Longrightarrow> f -` A \<inter> space M \<in> sets M" shows "f \<in> M \<rightarrow>\<^sub>M N"
between_norm: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "between (a, b) x = (norm (x - a) *\<^sub>R (b - x) = norm (b - x) *\<^sub>R (x - a))"
y2_prop: shows "q < y2 \<and> (case z of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f < 0)"
laststate_def: fixes s :: "nat \<Rightarrow> 'b" shows "laststate s \<equiv> s (Sequence.last s)"
grid_partition: fixes p :: "(nat \<times> int) list"   and d :: "nat" shows "grid p {d} = {p} \<union> grid (child p left d) {d} \<union> grid (child p right d) {d}"
simps_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'c btree" assumes "split split" shows "height_up\<^sub>i (T\<^sub>i t) = height t"
the_i_def: shows "the_i \<equiv> \<lambda>y. THE x. x < length U \<and> U ! x = Var y"
map_simp: fixes f :: "'a" assumes "arr f" shows "\<Omega>.map f = C.unity"
distinct_upt: fixes i :: "nat"   and j :: "nat" shows "distinct [i..<j]"
set_option2list: fixes k :: "'c option" shows "set (option2list k) = option2set k"
f_def: shows "f = (\<lambda>p a. signof p * (\<Prod>i\<in>{0..<n} - {a}. Sum $$ (i, p i)) * pderiv (Sum $$ (a, p a)))"
DBM_val_bounded_len'2: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and vs :: "nat list"   and c :: "'c" assumes "u \<turnstile>\<^bsub>v,n\<^esub> m"   and "0 \<notin> set vs"   and "v c \<le> n"   and "\<forall>k\<in>set vs. 0 < k \<longrightarrow> k \<le> n \<and> (\<exists>c. v c = k)" shows "dbm_entry_val u None (Some c) (len m 0 (v c) vs)"
eq_supteq: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s = t" shows "s \<unrhd> t"
SUP_eq_const: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and I :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Sup (f ` I) = x"
thread_ok_ts_change: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and ts' :: "'c \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and ta :: "('c, 'h, 'i) new_thread_action" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_ok ts ta = thread_ok ts' ta"
Cons_1: assumes "poly_inv p" shows "poly_inv (poly_subst \<sigma> p)"
ln_ge_zero: fixes x :: "real" assumes "1 \<le> x" shows "0 \<le> ln x"
f1: fixes b :: "'b"   and a :: "'a"   and ba :: "'b" shows "inner b (scale a ba) = cnj_class.cnj (a * inner ba b)"
uint16_test_bit_def: fixes x :: "uint16"   and n :: "integer" shows "uint16_test_bit x n = (if n < 0 \<or> 15 < n then undefined bit x n else bit x (nat_of_integer n))"
that_2: fixes i :: "nat" assumes "a = aa"   and "prefix (b @ [i]) bb"   and "hyps (nodeOf (a, b)) ba = Some (in_port_at (a, b) i)" shows "thesis"
assms: fixes i :: "nat" assumes "i < m" shows "c i \<le> c' i"
formulasD: fixes x :: "'a formula"   and S :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "x \<in> formulas S"   and "\<forall>i\<in>S. (i \<in> A) = (i \<in> B)" shows "(A \<in> Rep_formula x) = (B \<in> Rep_formula x)"
Plus_def: fixes A :: "'a set"   and B :: "'b set" shows "A <+> B = Sum_Type.Inl ` A \<union> Sum_Type.Inr ` B"
vars_mult_Var: fixes x :: "nat"   and p :: "int mpoly" shows "vars (Var x * p) = (if p = 0 then {} else insert x (vars p))"
sumlist_append: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> carrier G"   and "set ys \<subseteq> carrier G" shows "sumlist (xs @ ys) = sumlist xs \<oplus> sumlist ys"
of_int_of_nat_eq: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
nb\<^sub>2: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set ops" shows "\<forall>(v, a)\<in>set (sas_plus_operator.precondition_of op). \<forall>(v', a')\<in>set (sas_plus_operator.precondition_of op). v \<noteq> v' \<or> a = a'"
Un_ac_3: fixes A :: "'b set"   and B :: "'b set" shows "A \<union> B = B \<union> A"
m_def: shows "m \<equiv> The (is_icdi n \<pi>)"
card_Diff_subset: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "B \<subseteq> A" shows "card (A - B) = card A - card B"
While: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> val" shows "(WHILE b DO c, s) \<rightarrow> (IF b THEN c;; WHILE b DO c ELSE SKIP, s)"
eq_fin_word: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and p :: "'state" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "path w\<^sub>1 p" shows "path w\<^sub>2 p"
that: assumes "a \<in> sset u"   and "b \<in> sset v" shows "thesis"
nominal_ts_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool" shows "nominal_ts satisfies transition \<equiv> (\<forall>P \<phi> p. satisfies P \<phi> \<longrightarrow> satisfies (p \<bullet> P) (p \<bullet> \<phi>)) \<and> (\<forall>P \<alpha>Q p. transition P \<alpha>Q \<longrightarrow> transition (p \<bullet> P) (p \<bullet> \<alpha>Q))"
freshOutputAction_2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<noteq> b"
filterlim_sequentially_iff_filterlim_real: fixes f :: "'a \<Rightarrow> nat"   and F :: "'a filter" shows "filterlim f sequentially F = (LIM x F. real (f x) :> at_top)"
circle_y_bounds_2: fixes x :: "real" assumes "- 1 / 2 \<le> x \<and> x \<le> 1 / 2" shows "circle_y x \<le> 1 / 2"
add_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus a c) (plus b c)"
before_in_setD1: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "x < y in xs" shows "x \<in> set xs"
vars_mpoly_of_poly: fixes i :: "nat"   and p :: "'a poly" shows "vars (mpoly_of_poly i p) \<subseteq> {i}"
negligible_finite: fixes s :: "'c set" assumes "finite s" shows "negligible s"
leg_simps_2: shows "Maps.dom SPN_fgh.\<nu>.leg0 = SPN_fgh.\<nu>.apex"
Sup_\<omega>: shows "\<Squnion> (elts \<omega>) = \<omega>"
coeff_mult: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i\<in>{..n}. coeff P p i \<otimes> coeff P q (n - i))"
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "exp0 list"   and a5 :: "'b state \<times> (v list, v) result" assumes "evaluate_list a1 a2 a3 a4 a5" obtains ck :: "bool" and env :: "v sem_env" and s :: "'b state" where "a1 = cka__" and "a2 = enva__" and "a3 = sa__" and "a4 = []" and "a5 = (sa__, Rval [])"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and vs :: "v list" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = ckb__" and "a2 = envb__" and "a3 = s1__" and "a4 = ea__ # es__" and "a5 = (s3__, Rval (v1__ # vs))" and "evaluate ckb__ envb__ s1__ ea__ (s2__, Rval v1__) \<and> evaluate_list ckb__ envb__ s2__ es__ (s3__, Rval vs)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and err :: "v error_result" and s :: "'b state" and s' :: "'b state" where "a1 = ckc__" and "a2 = envc__" and "a3 = s" and "a4 = eb__ # esa__" and "a5 = (s', Rerr err__)" and "evaluate ckc__ envc__ s eb__ (s', Rerr err__)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and err :: "v error_result" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = ck" and "a2 = env" and "a3 = s1" and "a4 = e # es" and "a5 = (s3, Rerr err)" and "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rerr err)"
ts_i: shows "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
fixp_strong_induct: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less"   and "ccpo.admissible Sup less_eq P"   and "monotone less_eq less_eq f"   and "P (Sup {})"   and "\<And>x. \<lbrakk>less_eq x (ccpo.fixp Sup less_eq f); P x\<rbrakk> \<Longrightarrow> P (f x)" shows "P (ccpo.fixp Sup less_eq f)"
S3EnvUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [ENext p]_(e p) \<and> $S3 rmhist p \<longrightarrow> unchanged e p"
of_int_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "0 \<le> z" shows "less_eq zero (ring_1.of_int one plus zero minus z)"
Next: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula"   and first :: "bool"   and nts :: "nat list"   and I :: "\<I>" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'"   and "first = (Monitor.progress \<sigma> \<phi>' j = 0)"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [Monitor.progress \<sigma> \<phi>' j - 1..<j] nts" shows "wf_mformula \<sigma> j n R (MNext I \<phi> first nts) (formula.Next I \<phi>')"
y: shows "\<langle>x, y\<rangle> \<in> elts f"
irreducible_cong: fixes a :: "'a"   and a' :: "'a" assumes "irreducible G a"   and "a \<sim> a'"   and "a \<in> carrier G"   and "a' \<in> carrier G" shows "irreducible G a'"
sel_1: fixes z :: "quat" shows "quat.Re (Quaternions.cnj z) = quat.Re z"
fresh_def: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "atom"   and x :: "'b" assumes "class.pt permute" shows "pt.fresh permute a x \<equiv> a \<notin> pt.supp permute x"
select_convs_1: fixes empty :: "'b"   and lookup :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'e" shows "MapOps.empty \<lparr>MapOps.empty = empty, lookup = lookup, update = update, \<dots> = more\<rparr> = empty"
ex_br_conj_iff: fixes y :: "'b"   and a :: "'b \<Rightarrow> 'a"   and I :: "'b \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x. (y, x) \<in> br a I \<and> P x) = (I y \<and> P (a y))"
unitaryD1: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "unitary U" shows "U* o\<^sub>C\<^sub>L U = id_cblinfun"
Standard_starfun_iff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a star" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "((*f* f) x \<in> Standard) = (x \<in> Standard)"
abs_eq: fixes xa :: "nat"   and x :: "nat" shows "(natural_of_nat xa < natural_of_nat x) = (xa < x)"
\<eta>_def: shows "\<eta> \<equiv> \<eta>.map"
Game_def: fixes L :: "game zet"   and R :: "game zet" shows "Game L R \<equiv> Abs_game (Opair (zimplode (zimage Rep_game L)) (zimplode (zimage Rep_game R)))"
prj0_in_hom': fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "\<guillemotleft>prj0 f g : local.dom (prj0 f g) \<rightarrow> local.dom g\<guillemotright>"
coprime_iff_gcd_eq_1: fixes a :: "'a"   and b :: "'a" shows "coprime a b = (gcd a b = (1::'a))"
homeomorphism_compose: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a"   and U :: "'c set"   and h :: "'b \<Rightarrow> 'c"   and k :: "'c \<Rightarrow> 'b" assumes "homeomorphism S T f g"   and "homeomorphism T U h k" shows "homeomorphism S U (h \<circ> f) (g \<circ> k)"
integrableI_bounded_set: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and B :: "real" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "emeasure M A < \<infinity>"   and "AE x\<in>A in M. norm (f x) \<le> B"   and "AE x in M. x \<notin> A \<longrightarrow> f x = (0::'c)" shows "integrable M f"
simps_2: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "mkdisjoint A (Suc n) = A (Suc n) - A n"
init: fixes m0 :: "'b nres"   and X :: "'b set"   and x :: "'b"   and cond :: "'b \<Rightarrow> bool"   and step :: "'b \<Rightarrow> 'b nres" assumes "m0 = RES X"   and "x \<in> X" shows "rwof m0 cond step x"
leadsETo_Un_Un: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "F \<in> B leadsTo[CC] B'" shows "F \<in> (A \<union> B) leadsTo[CC] (A' \<union> B')"
x: shows "\<forall>x\<in>S. d' dvd x"
sum_of_prideals2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>l\<le>n. f l \<in> carrier R" shows "f ` {i. i \<le> n} \<subseteq> sum_pr_ideals R f n"
mem_restr_def: fixes A :: "'c list set"   and x :: "'c option list" shows "mem_restr A x = (\<exists>y\<in>A. list_all2 (\<lambda>a b. a \<noteq> None \<longrightarrow> a = Some b) x y)"
issued_app: fixes s :: "event list"   and s' :: "event list" shows "Trace.issued (s @ s') = Trace.issued s \<union> Trace.issued s'"
card_le_Suc_iff: fixes n :: "nat"   and A :: "'c set" shows "(Suc n \<le> card A) = (\<exists>a B. A = insert a B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
abs_power2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a 2) = power.power one times a 2"
divide_le_eq_1_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq (divide b a) one = less_eq b a"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and order :: "order" shows "P (case order of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = (\<not> (order = Eq \<and> \<not> P f1 \<or> order = Lt \<and> \<not> P f2 \<or> order = Gt \<and> \<not> P f3))"
closed_3: shows "is_TComp_var_instance_closed \<Gamma> M"
action_cfg_corec: fixes s :: "'a"   and d :: "'b \<Rightarrow> 'a pmf"   and c :: "'b \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'b" shows "action (cfg_corec s d c x) = d x"
SubgroupD1: fixes H :: "'r set" assumes "R_scalars.Subgroup H" shows "Group H"
l_remove_child_wf2_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_remove_child_wf2 type_wf known_ptr known_ptrs remove_child heap_is_wellformed get_child_nodes remove \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_remove_child_wf2_axioms type_wf known_ptrs remove_child heap_is_wellformed get_child_nodes remove"
1: shows "f ` ({a} \<union> B) = g ` C"
1: shows "kauff_mat (ws \<circ> basic []) = rat_poly.matrix_mult (kauff_mat ws) (kauff_mat (basic []))"
exec_ta_satisfiable: fixes t :: "'thread_id"   and s :: "'addr option \<times> 'heap \<times> 'addr frame list"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "'addr option \<times> 'heap \<times> 'addr frame list" assumes "P,t \<turnstile> s -ta-jvm\<rightarrow> s'" shows "\<exists>s. exec_mthr.actions_ok s t ta"
sub_sharp_def: fixes R :: "('h, 'i) Ring_scheme"   and L :: "('e, 'h, 'j) Module_scheme"   and M :: "('f, 'h, 'k) Module_scheme"   and N :: "('g, 'h, 'l) Module_scheme"   and u :: "'f \<Rightarrow> 'g" shows "sub_sharp R L M N u = restrict (compos L u) (mHom R L M)"
set_sort_oalist_aux: fixes xs :: "('a \<times> 'd) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "set (sort_oalist_aux ko xs) = set (fst xs)"
Inf_le_Sup: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "less_eq (Inf A) (Sup A)"
def_homo: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<delta> x->excludes\<^sub>S\<^sub>e\<^sub>t(y) = (\<delta> x and \<upsilon> y)"
A_def: shows "A = {f \<in> space (lborel_f (Suc n)). insertion f p = 0}"
borel_measurable_nn_integral: fixes M :: "'a measure"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> ennreal"   and N :: "'b measure" assumes "sigma_finite_measure M"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. integral\<^sup>N M (f x)) \<in> borel_measurable N"
PowI: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A" shows "A \<in> Pow B"
rev_apply_guards: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards (rev G) s = apply_guards G s"
inject_2: fixes x2 :: "mdecl"   and y2 :: "mdecl" shows "(mdecl x2 = mdecl y2) = (x2 = y2)"
abs_summable_on_zero_diff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "f abs_summable_on A"   and "\<And>x. x \<in> B - A \<Longrightarrow> f x = (0::'d)" shows "f abs_summable_on B"
lem1: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. f x * (a * g x)) = a * (\<Sum>x\<in>A. f x * g x)"
sets_restrict_space_iff: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(A \<in> sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> sets M)"
len_decomp: fixes xs :: "nat list"   and ys :: "nat list"   and y :: "nat"   and zs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and x :: "nat"   and z :: "nat" assumes "xs = ys @ y # zs" shows "linordered_monoid.len m x z xs = dbm_add (linordered_monoid.len m x y ys) (linordered_monoid.len m y z zs)"
\<rho>: fixes p :: "fm"   and i :: "int"   and bs :: "real list"   and d :: "int" assumes "iszlfm p (real_of_int i # bs)"   and "Ifm (real_of_int i # bs) p"   and "d_\<delta> p d"   and "0 < d"   and "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * i) \<noteq> Inum (real_of_int i # bs) e + real_of_int j" shows "Ifm (real_of_int (i - d) # bs) p"
intros_14: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> Stuck"
simps_14: fixes f1 :: "'a"   and f2 :: "num \<Rightarrow> 'a \<Rightarrow> 'a"   and f3 :: "num \<Rightarrow> 'a \<Rightarrow> 'a"   and x3 :: "num" shows "rec_num f1 f2 f3 (num.Bit1 x3) = f3 x3 (rec_num f1 f2 f3 x3)"
tm_def: fixes e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "tm e = snd |`| e"
functions_of: fixes A :: "'b set"   and B :: "'c set"   and X :: "'c set" assumes "finite A"   and "finite B"   and "X \<subseteq> B"   and "card X = card A" shows "functions_of A X \<in> {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_permutation A B"
nonEmpty: fixes bot :: "'a" assumes "class.bot bot" shows "\<exists>x. x \<noteq> bot"
quad_part_2_leq: fixes var :: "nat"   and L :: "real list"   and SQ :: "real"   and p :: "real mpoly"   and deg :: "nat"   and sq :: "real mpoly" assumes "var < length L"   and "0 \<le> SQ"   and "MPoly_Type.degree p var = deg"   and "\<forall>x. insertion (nth_default 0 (L[var := x])) sq = SQ" shows "aEval (Leq p) (L[var := sqrt SQ]) = aEval (Leq (quadratic_part_2 var sq p)) (L[var := sqrt SQ])"
in_trancl_imp: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v :: "'d \<times> nat"   and v' :: "'d \<times> nat" assumes "wf_sub_rel subs"   and "(v, v') \<in> subs\<^sup>+" shows "(v, v') \<in> subs"
simplex_map_compose: fixes p :: "nat"   and h :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "simplex_map p (h \<circ> g) = simplex_map p h \<circ> simplex_map p g"