hcju/isabelleps huggingface dataset:
invfun_set: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and C :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "C \<subseteq> B" shows "f ` (f\<inverse>\<^bsub>B,A\<^esub>) ` C = C"
less_append_swap: fixes x :: "nat list"   and y :: "nat list"   and u :: "nat list"   and v :: "nat list" assumes "x @ y <\<^sub>v u @ v"   and "length x = length u" shows "y @ x <\<^sub>v v @ u"
ev: shows "\<forall>\<^sub>F x in net. f x $ i = b"
cd1_sub: shows "CCDD' \<subseteq> CC \<union> DD"
bi_unique_alt_def: fixes A :: "'b \<Rightarrow> 'e \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
setcomp_dot1: fixes P :: "real \<Rightarrow> bool"   and i :: "'a" shows "{z. P (z \<bullet> (i, 0::'b))} = {(x, y). P (x \<bullet> i)}"
reach_Suc_empty: fixes w :: "'c stream"   and n :: "nat"   and A :: "('c, 'd) nba" assumes "w !! n \<notin> alphabet A" shows "reach A w (Suc n) = {}"
enabled_ex: fixes F :: "'a \<Rightarrow> state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled (\<exists>x. F x) = (\<exists>x. Enabled F x)"
streams_sset: fixes s :: "'d stream"   and A :: "'d set" assumes "s \<in> streams A" shows "sset s \<subseteq> A"
scale_left_commute: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "x \<in> UNIV" shows "a *\<^sub>R b *\<^sub>R x = b *\<^sub>R a *\<^sub>R x"
exteq_def: fixes f :: "recf"   and g :: "recf" shows "f \<simeq> g \<equiv> arity f = arity g \<and> (\<forall>xs. length xs = arity f \<longrightarrow> eval f xs = eval g xs)"
Inf_Sec_def: fixes S :: "Sec set" shows "Inf S = (if Low \<in> S then Low else High)"
simps_4: fixes e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (EApp e1 e2) \<rho> = {v3. \<exists>f v2 v2' v3'. VFun f \<in> E e1 \<rho> \<and> v2 \<in> E e2 \<rho> \<and> (v2', v3') \<in> set f \<and> v2' \<sqsubseteq> v2 \<and> v3 \<sqsubseteq> v3'}"
star_def: shows "RegExp2NAe.star = (\<lambda>(q, d, f). ([], \<lambda>a s. case s of [] \<Rightarrow> if a = None then {True # q} else {} | left # s \<Rightarrow> if left then True ## d a s \<union> (if f s \<and> a = None then {True # q} else {}) else {}, \<lambda>s. case s of [] \<Rightarrow> True | left # s \<Rightarrow> left \<and> f s))"
renumber': fixes a :: "'d"   and f :: "nat \<Rightarrow> 'd"   and r :: "('d \<times> 'd) set"   and b :: "'d" assumes "\<forall>i. (a, f i) \<in> r\<^sup>* \<and> (f i, f (Suc i)) \<in> r"   and "(a, b) \<in> r\<^sup>*"   and "b = f 0" shows "\<exists>f. f 0 = a \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
simps_88: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x71 :: "\<I>"   and x72 :: "'c formula" shows "(case Next x71 x72 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f7 x71 x72"
uniform_distributed_expectation: fixes X :: "'a \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes "distributed M lborel X (\<lambda>x. ennreal (indicat_real {a..b} x / content {a..b}))" shows "expectation X = (a + b) / 2"
p_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "ego.t_stop \<le> x" shows "ego.p y \<le> ego.p x"
starlike_imp_contractible: fixes S :: "'b set" assumes "starlike S" shows "contractible S"
image_set: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "f ` set xs = set (map f xs)"
map_of_tabulate_simp: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and x :: "'d" shows "map_of (map (\<lambda>x. (x, f x)) xs) x = (if x \<in> set xs then Some (f x) else None)"
bis_Frel: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and B1' :: "'h set"   and B2' :: "'i set"   and s1' :: "'h \<Rightarrow> ('f, 'h, 'i) F1"   and s2' :: "'i \<Rightarrow> ('g, 'h, 'i) F2"   and R1 :: "('d \<times> 'h) set"   and R2 :: "('e \<times> 'i) set" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2 = ((R1 \<subseteq> B1 \<times> B1' \<and> R2 \<subseteq> B2 \<times> B2') \<and> (\<forall>b1 b1'. (b1, b1') \<in> R1 \<longrightarrow> F1rel (=) (in_rel R1) (in_rel R2) (s1 b1) (s1' b1')) \<and> (\<forall>b2 b2'. (b2, b2') \<in> R2 \<longrightarrow> F2rel (=) (in_rel R1) (in_rel R2) (s2 b2) (s2' b2')))"
abs_eq: fixes x :: "'b list" shows "fset_of_list x = Abs_fset (set x)"
simps_3: fixes v :: "'a" shows "local.le_rexp (Atom v) Zero = False"
inres_leof_mono: fixes m :: "'c nres"   and m' :: "'c nres"   and x :: "'c" assumes "m \<le>\<^sub>n m'"   and "nofail m"   and "inres m x" shows "inres m' x"
comp_def: fixes y :: "'a"   and x :: "'a" shows "V y x \<equiv> if x \<in> Collect arr \<and> x = y then x else null"
elims: fixes x :: "'b extN"   and xa :: "'b extN"   and y :: "'b extN" assumes "x + xa = y" obtains "xa__ = extN.Bot" and "xa = extN.Bot" and "ya__ = extN.N 0"   | uu_ :: "'b" where "xa__ = extN.Bot" and "xa = extN.Val uu_" and "ya__ = extN.N 1"   | y :: "nat" where "xa__ = extN.Bot" and "xa = extN.N yb__" and "ya__ = extN.N yb__"   | "xa__ = extN.Bot" and "xa = extN.Top" and "ya__ = extN.N 1"   | uv_ :: "'b" where "xa__ = extN.Val uv_" and "xa = extN.Bot" and "ya__ = extN.N 1"   | uw_ :: "'b" and ux_ :: "'b" where "xa__ = extN.Val uw_" and "xa = extN.Val ux_" and "ya__ = extN.N 2"   | uy_ :: "'b" and y :: "nat" where "xa__ = extN.Val uy_" and "xa = extN.N yc__" and "ya__ = extN.N (yc__ + 1)"   | uz_ :: "'b" where "xa__ = extN.Val uz_" and "xa = extN.Top" and "ya__ = extN.N 2"   | x :: "nat" where "xa__ = extN.N xb__" and "xa = extN.Bot" and "ya__ = extN.N xb__"   | x :: "nat" and va_ :: "'b" where "xa__ = extN.N xc__" and "xa = extN.Val va_" and "ya__ = extN.N (xc__ + 1)"   | x :: "nat" and y :: "nat" where "xa__ = extN.N xd__" and "xa = extN.N yd__" and "ya__ = extN.N (xd__ + yd__)"   | x :: "nat" where "xa__ = extN.N x" and "xa = extN.Top" and "ya__ = extN.N (x + 1)"   | "xa__ = extN.Top" and "xa = extN.Bot" and "ya__ = extN.N 1"   | vb_ :: "'b" where "xa__ = extN.Top" and "xa = extN.Val vb_" and "ya__ = extN.N 2"   | y :: "nat" where "xa__ = extN.Top" and "xa = extN.N y" and "ya__ = extN.N (y + 1)"   | "xa__ = extN.Top" and "xa = extN.Top" and "ya__ = extN.N 2"
sub_a_plus_e: shows "closed_segment (a + complex_of_real kde) (a + complex_of_real e) \<inter> (path_image p \<union> closed_segment (a - complex_of_real d) (a - complex_of_real kde)) \<subseteq> {a + complex_of_real e}"
bij_betw_the_inv_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "bij_betw (the_inv_into A f) B A"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c2 :: "com"   and x :: "nat"   and t :: "char list \<Rightarrow> int"   and y :: "nat"   and c1 :: "com" assumes "\<not> bval b s"   and "(c2, s) \<Rightarrow> x \<Down> t"   and "y = x + 1" shows "(IF b THEN c1 ELSE c2, s) \<Rightarrow> y \<Down> t"
weight_spmf_le_1: fixes p :: "'a spmf" shows "weight_spmf p \<le> 1"
A: shows "(lambda * 1) (C * 1 + 0) -quasi_isometry (to_Bonk_Schramm_extension \<circ> f)"
c2: fixes as :: "'a list"   and a :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "ps = as @ [a]"   and "ls = xs @ [x]"   and "x = a"   and "\<not> suffix as xs" shows "thesis"
gker_hom_to_img: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "gker\<^bsub>F,Img\<^bsub>F,G\<^esub> f\<^esub> f = gker\<^bsub>F,G\<^esub> f"
zero_perm_eqvt: fixes p :: "perm" shows "p \<bullet> 0 = 0"
assms_1: shows "h \<turnstile> a_insert_node ptr new_child reference_child_opt \<rightarrow>\<^sub>h h2"
less_linear: fixes x :: "'c"   and y :: "'c" shows "x < y \<or> x = y \<or> y < x"
scale_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "'b" assumes "module scale" shows "scale a (scale b x) = scale (a * b) x"
ceiling_le_iff: fixes x :: "'c"   and z :: "int" shows "(\<lceil>x\<rceil> \<le> z) = (x \<le> of_int z)"
A: shows "\<forall>\<^sub>F x in F. g x \<noteq> 0"
fin_w: shows "finite (CAS \<times> (CC \<union> {C}) \<times> AAS \<times> AS)"
quasi_isometry_betweenD_3: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "lambda C -quasi_isometry_between X Y f"   and "y \<in> Y" shows "\<exists>x\<in>X. dist (f x) y \<le> C"
the_RES_inv: fixes m :: "'a nres" assumes "nofail m" shows "RES (Autoref_Misc.the_RES m) = m"
not_eq_extremum: fixes a :: "'d" shows "(a \<noteq> top) = (a < top)"
l_new_element_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
elims: fixes x :: "nat"   and y :: "('a, 'es, 'as) BEState Trace set" assumes "SPR.jkbpCn x = y" obtains "x = 0" and "y = rel_ext (\<lambda>uu_. \<exists>s. uu_ = tInit s \<and> s \<in> set envInit)"   | n :: "nat" where "x = Suc n" and "y = rel_ext (\<lambda>uu_. \<exists>t eact aact. uu_ = t \<leadsto> envTrans eact aact (tLast t) \<and> t \<in> SPR.jkbpCn n \<and> eact \<in> set (envAction (tLast t)) \<and> (\<forall>a. aact a \<in> set (jAction (SPR.MCn n) t a)))"
liftAll_cong: fixes inp :: "'a \<Rightarrow> 'b option"   and phi :: "'b \<Rightarrow> bool"   and phi' :: "'b \<Rightarrow> bool" assumes "\<And>i v. inp i = Some v \<Longrightarrow> phi v = phi' v" shows "liftAll phi inp = liftAll phi' inp"
AccSigleLoop: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet" shows "DAcc i C = {S} \<and> DAcc i S = {C} \<longrightarrow> Acc i C = {C, S}"
ll2f_None_Suc: fixes t :: "'b llist"   and i :: "nat" assumes "t !! i = None" shows "t !! Suc i = None"
fmap_def: shows "fmap = COERCE((udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'c \<rightarrow> udom\<cdot>'c, ('a \<rightarrow> 'b) \<rightarrow> 'a\<cdot>'c \<rightarrow> 'b\<cdot>'c)\<cdot>fmapU"
axioms_2: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
SUP_funpow_contracting: fixes f :: "'g \<Rightarrow> 'g"   and m :: "'g" assumes "cont f" shows "f (SUP i. (f ^^ i) m) \<le> (SUP i. (f ^^ i) m)"
alphaAbs_qAbs_iff_ex_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
types_stable_update: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and t :: "'BExp set"   and x :: "'Var" assumes "types_stable \<Gamma> \<S>"   and "type_stable \<S> t" shows "types_stable (\<Gamma>(x \<mapsto> t)) \<S>"
inv_image_def: fixes r :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c" shows "inv_image r f = {(x, y). (f x, f y) \<in> r}"
fds_shift_def: fixes c :: "'b"   and f :: "'b fds" shows "fds_shift c f = fds (\<lambda>n. fds_nth f n * nat_power n c)"
to_F_in2: fixes q :: "'q" assumes "q \<in> Q" shows "to_F \<iota> \<in> no_labels.Red_I_\<G>_q q (fst ` N)"
dvd_sum: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and d :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_semiring_1 times one plus zero"   and "\<And>a. a \<in> A \<Longrightarrow> dvd.dvd times d (f a)" shows "dvd.dvd times d (comm_monoid_add.sum plus zero f A)"
composite_pseudofunctor_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e" shows "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<equiv> (bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<and> bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C) \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
neg_0_less_iff_less: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (uminus a) = less a zero"
s_def: fixes i :: "nat" shows "s i = fst (f i)"
SemWhileF: fixes b :: "BExpr"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val"   and c :: "IMP" assumes "\<not> evalB b s"   and "t = s" shows " s , While b c \<rightarrow>\<^sub>1  t "
WTrt_binop_simps_7: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Add\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
leg1_simps_3: shows "src p\<^sub>1 = src p\<^sub>0"
epm_5: fixes e :: "'a"   and w :: "'a"   and v :: "'a" assumes "e \<le> w"   and "injective w"   and "w * v \<le> v"   and "e \<le> v * - v\<^sup>T"   and "vector v" shows "prim_P w v e = bot"
rotate1_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "rotate1 (map f xs) = map f (rotate1 xs)"
basis_inD_1: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> X" shows "indep_in \<E> X"
Abs_inject: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(Abs x = Abs y) = (x = y)"
base: shows "U (min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)) z"
in_new_spairs_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "(Inl (p, b) \<in> set (new_spairs bs p)) = (b \<in> set bs \<and> is_regular_spair p b)"
simps_4: fixes n :: "nat"   and k :: "nat" shows "Stirling (Suc n) (Suc k) = Suc k * Stirling n (Suc k) + Stirling n k"
is_OK_returns_heap_I: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog"   and h' :: "'aq" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'" shows "h \<turnstile> ok f"
finsum_addf: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. f x + g x) = sum f A + sum g A"
fsafe_Not: fixes p :: "('a, 'b, 'c) formula" assumes "fsafe p" shows "fsafe (! p)"
the_wall_betw_adj_fundchamber: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "(f, g) \<in> fundfoldpairs" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H C0 (Abs_induced_automorph f g `\<rightarrow> C0)) = {f \<turnstile> \<C>, g \<turnstile> \<C>}"
indep: shows "\<forall>s\<^sub>1 s\<^sub>2. Tr\<^sub>P n s\<^sub>1 \<inter> Tr\<^sub>r\<^sub>p n s\<^sub>2 = {}"
collect_locks'E: fixes l :: "'c"   and las :: "'c lock_actions" assumes "l \<in> collect_locks' las" obtains "must_acquire_lock (las $ l)"
card_eq_1_iff_ex1: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "(card A = 1) = (A = {x})"
simps: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" shows "(x # xs) ! n = (case n of 0 \<Rightarrow> x | Suc k \<Rightarrow> xs ! k)"
intros_2: fixes \<Gamma> :: "('a, 'b) web_scheme"   and x :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "weight \<Gamma> x \<le> d_IN f x" shows "x \<in> SAT \<Gamma> f"
to_list_correct_1: fixes m :: "('c, 'd) RBT.rbt" assumes "rm.invar m" shows "map_of (rm.to_list m) = rm.\<alpha> m"
disj_zero: fixes x :: "'b" shows "total_partial.disjoint x (0::'b)"
Bseq_iff3: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)"
cases: fixes a1 :: "pat"   and a2 :: "trm"   and a3 :: "trm list" assumes "\<turnstile> a1 \<rhd> a2 \<Rightarrow> a3" obtains T :: "POPLmarkRecord.type" and t :: "trm" where "a1 = PVar T" and "a2 = t" and "a3 = [t]"   | fps :: "(char list \<times> pat) list" and fs :: "(char list \<times> trm) list" and ts :: "trm list" where "a1 = PRcd fps" and "a2 = Rcd fs" and "a3 = ts" and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts"
inner_simps_4: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "(x - y) \<bullet> z = x \<bullet> z - y \<bullet> z"
wordinterval_empty_Empty_WordInterval: shows "wordinterval_empty Empty_WordInterval"
LatticeTriangle: fixes A :: "'a \<Rightarrow> bool"   and D :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool" shows "\<turnstile> (A \<leadsto> D \<or> B) \<and> (B \<leadsto> D) \<longrightarrow> (A \<leadsto> D)"
ddh_ext_def: fixes \<G> :: "'d cyclic_group" shows "ddh_ext \<G> \<equiv> Cyclic_Group.cyclic_group \<G>"
le__nlt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D" shows "\<not> Tarski_neutral_dimensionless.Lt Bet Cong C D A B"
assms: shows "tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D = []"
true_clss_set_mset: fixes I :: "'a set"   and CC :: "'a literal multiset multiset" shows "(I \<Turnstile>s set_mset CC) = (I \<Turnstile>m CC)"
interpretation_grounds: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "fv (t \<cdot> \<theta>) = {}"
rep_eq: fixes x :: "\<kappa> \<Rightarrow> \<o>" shows "IsProperInX x = (\<forall>x v. (\<exists>a. \<nu>\<upsilon> a = \<nu>\<upsilon> x \<and> eval\<o> (x (a\<^sup>P)) dj v) = eval\<o> (x (x\<^sup>P)) dj v)"
L_def: shows "L = 1 / epsilon * (1 + emeasure P (space M))"
map_def: fixes f :: "'j" shows "Fun_f.map f \<equiv> if J.arr f then f__ else C.null"
rel_envTE: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and M :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and m :: "('g, 'i) envT"   and m' :: "('h, 'j) envT" assumes "local.rel_envT R M m m'" obtains f :: "'g \<Rightarrow> 'i" and g :: "'h \<Rightarrow> 'j" where "m = EnvT f" and "m' = EnvT g" and "(R ===> M) f g"
x_2: shows "{0<..<p} = {y. \<exists>i. y = x ^ i mod p}"
vfsequence_vfinite_vcardI: fixes xs :: "V" assumes "vsv xs"   and "vfinite xs"   and "\<D>\<^sub>\<circ> xs = vcard xs" shows "vfsequence xs"
pts_symbolic_to_pts_symbolic_c_from_initial: fixes \<S>\<^sub>0 :: "('fun, 'var) strand_step list set"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(\<S>\<^sub>0, []) \<Rightarrow>\<^sup>\<bullet>\<^sup>* (\<S>, \<A>)"   and "\<I> \<Turnstile> \<langle>\<A>\<rangle>"   and "wf\<^sub>s\<^sub>t\<^sub>s' \<S>\<^sub>0 []"   and "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>\<^sub>0) \<union> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>\<^sub>0))"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<exists>\<A>\<^sub>d. \<A> = to_st (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d) \<and> (\<S>\<^sub>0, []) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>, \<A>\<^sub>d) \<and> \<I> \<Turnstile>\<^sub>c \<langle>to_st \<A>\<^sub>d\<rangle>"
add_commute: fixes a :: "('a \<times> 'a set) set"   and b :: "('a \<times> 'a set) set" shows "a \<union> b = b \<union> a"
NEx: shows "\<exists>x. \<^bold>\<not> (E x)"
intros_3: fixes xs :: "'c option list"   and P :: "'c option process"   and ys :: "'c option list"   and Y :: "'c option set"   and Q :: "'c option process" assumes "xs \<in> sentences P"   and "(ys, Y) \<in> failures Q"   and "ys \<noteq> []" shows "(xs @ ys, Y) \<in> seq_comp_failures P Q"
simps_1: fixes k :: "nat" shows "btree_assn k Leaf None = emp"
diff_pCons: fixes a :: "'c"   and p :: "'c poly"   and b :: "'c"   and q :: "'c poly" shows "pCons a p - pCons b q = pCons (a - b) (p - q)"