hcju/isabelleps huggingface dataset:
sorted_wrt_generate': fixes A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2) (generate' A B a b)"
simps_2: fixes n :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "stake (Suc n) \<sigma> = \<sigma> 0 # \<sigma>(1 \<rightarrow> n)"
True: shows "(0::'a) \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` A)"
cases: fixes a1 :: "nat list"   and a2 :: "nat list" assumes "a1 \<lhd> a2" obtains "a1 = []" and "a2 = []"   | "is" :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "a1 = i # is" and "a2 = d # ds" and "is \<lhd> ds" and "i < d"
ex_countable_subset_image_inj: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set"   and P :: "'b set \<Rightarrow> bool" shows "(\<exists>T. countable T \<and> T \<subseteq> f ` S \<and> P T) = (\<exists>T. countable T \<and> T \<subseteq> S \<and> inj_on f T \<and> P (f ` T))"
inf_le2: fixes x :: "'e multiset"   and y :: "'e multiset" shows "x \<inter># y \<subseteq># y"
closedin_diff: fixes U :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin U S"   and "openin U T" shows "closedin U (S - T)"
Units_one_closed: shows "\<one> \<in> Units (mult_of R)"
assms_3: shows "finite X \<longrightarrow> i < card X \<and> j < card X"
B: fixes pa :: "'b"   and xsa :: "'a list"   and ls :: "'a list"   and eq :: "'a"   and gs :: "'a list" shows "snd (snd (group_part_aux pa xsa (ls, eq, gs))) = rev (filter (\<lambda>x. pa < f x) xsa) @ gs"
consistency: fixes p :: "'a" shows "Cn {p} \<inter> Cn {.\<not> p} = Cn {}"
initial_ps_is_initial_ps2_with_empty_S: shows "initial_ps = initial_ps2 {||}"
iso_test_correct: fixes Fs\<^sub>1 :: "nat list list"   and Fs\<^sub>2 :: "nat list list" assumes "pre_iso_test Fs\<^sub>1"   and "pre_iso_test Fs\<^sub>2" shows "iso_test Fs\<^sub>1 Fs\<^sub>2 = Fs\<^sub>1 \<simeq> Fs\<^sub>2"
da_Super: fixes B :: "lname set"   and A :: "assigned"   and Env :: "env" assumes "This \<in> B"   and "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>" shows "Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> A"
if_elim: fixes b :: "'BExp"   and p :: "('Var, 'AExp, 'BExp) Stmt"   and q :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stmt.If b p q, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains "c' = p" and "mem' = mem" and "mds' = mds" and "eval\<^sub>B mem b"   | "c' = q" and "mem' = mem" and "mds' = mds" and "\<not> eval\<^sub>B mem b"
spec2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and j :: "'a"   and i :: "'b" assumes "\<forall>j i. P j i" shows "P j i"
floor_log2_div2: fixes n :: "nat" assumes "2 \<le> n" shows "\<lfloor>log 2 (real n)\<rfloor> = \<lfloor>log 2 (real (n div 2))\<rfloor> + 1"
IH_1: assumes "Oadmit \<sigma> ODE1 U" shows "osafe (Osubst ODE1 \<sigma>)"
power_mod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (power.power one times (modulo a b) n) b = modulo (power.power one times a n) b"
not_image_at_top_poly_part_circlepath: fixes p :: "complex poly"   and b :: "complex"   and z0 :: "complex"   and st :: "real"   and tt :: "real" assumes "0 < degree p" shows "\<forall>\<^sub>F r in at_top. b \<notin> path_image (poly p \<circ> part_circlepath z0 r st tt)"
create_opt_sol_in_LP_1: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and m :: "nat"   and n :: "nat"   and X :: "(nat, rat) mapping"   and x :: "rat vec"   and y :: "rat vec" assumes "rat_LP A b c m n"   and "create_optimal_solutions A b c = Inr X"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "[A*\<^sub>vx]\<le>b"
positive_if_decomp: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "\<exists>M. M * adjoint M = A" shows "positive A"
return_time_implicit_functionE: fixes S :: "'a set"   and s :: "'a \<Rightarrow> real"   and x :: "'a"   and Ds :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" assumes "returns_to {x \<in> S. s x = 0} x"   and "closed S"   and "\<And>x. (s has_derivative blinfun_apply (Ds x)) (at x)"   and "\<And>x. isCont Ds x"   and "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"   and "\<forall>\<^sub>F x in at (poincare_map {x \<in> S. s x = 0} x). s x = 0 \<longrightarrow> x \<in> S"   and "x \<notin> S \<or> s x \<noteq> 0" obtains e' :: "real" where "0 < e'" and "\<And>y. y \<in> ball x e' \<Longrightarrow> returns_to {x \<in> S. s x = 0} y" and "\<And>y. y \<in> ball x e' \<Longrightarrow> s (flow0 y (return_time {x \<in> S. s x = 0} y)) = 0" and "continuous_on (ball x e') (return_time {x \<in> S. s x = 0})" and "\<And>y. y \<in> ball x e' \<Longrightarrow> Ds (poincare_map {x \<in> S. s x = 0} y) o\<^sub>L flowderiv y (return_time {x \<in> S. s x = 0} y) o\<^sub>L embed2_blinfun \<in> invertibles_blinfun" and "\<And>U v saa. \<lbrakk>\<And>saa. saa \<in> U \<Longrightarrow> s (flow0 saa (v saa)) = 0; return_time {x \<in> S. s x = 0} x = v x; continuous_on U v; saa \<in> U; x \<in> U; U \<subseteq> ball x e'; connected U; open U\<rbrakk> \<Longrightarrow> return_time {x \<in> S. s x = 0} saa = v saa" and "(return_time {x \<in> S. s x = 0} has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)))) o\<^sub>L (Ds (poincare_map {x \<in> S. s x = 0} x) o\<^sub>L Dflow x (return_time {x \<in> S. s x = 0} x)))) (at x)"
assum_after_body: fixes P :: "'b com"   and pre :: "'b set"   and b :: "'b set"   and rely :: "('b \<times> 'b) set"   and guar :: "('b \<times> 'b) set"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and ys :: "('b com option \<times> 'b) list" assumes "\<Turnstile> P sat [pre \<inter> b, rely, guar, pre]"   and "(Some P, s) # xs \<in> cptn_mod"   and "fst (last ((Some P, s) # xs)) = None"   and "s \<in> b"   and "(Some (While b P), s) # (Some (Seq P (While b P)), s) # map (lift (While b P)) xs @ ys \<in> assum (pre, rely)" shows "(Some (While b P), snd (last ((Some P, s) # xs))) # ys \<in> assum (pre, rely)"
false_eqvt: fixes pi :: "('a \<times> 'a) list" shows "pi \<bullet> False = False"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case x2\<inverse> of x\<^sup>+ \<Rightarrow> f1 x | x\<inverse> \<Rightarrow> f2 x) = f2 x2"
floor_less_one: fixes x :: "'d" shows "(\<lfloor>x\<rfloor> < 1) = (x < (1::'d))"
step: fixes gpv :: "('a, 'b, 'c) gpv"   and out :: "'b"   and c :: "('a, 'b, 'c) rpv" assumes "X gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma> \<and> (\<forall>input\<in>responses_\<I> \<Gamma> out. X (c input) \<or> \<Gamma> \<turnstile>g c input \<surd>)"
var_def: fixes R :: "('d, 'e) ring_scheme" shows "X\<^bsub>R\<^esub> = [\<one>\<^bsub>R\<^esub>, \<zero>\<^bsub>R\<^esub>]"
isolation: fixes x :: "'a" shows "x\<^sup>\<star> + x\<^sup>\<infinity> \<cdot> (0::'a) = x\<^sup>\<infinity>"
simps_2: fixes T :: "ty"   and n :: "nat" shows "ty_of_htype (Array_type T n) = T\<lfloor>\<rceil>"
val_of_T: fixes as :: "'s list"   and os :: "'t list"   and s :: "'s"   and o\<^sub>1 :: "'t" assumes "length as = length os" shows "val_of s as os = emeasure (T (s, o\<^sub>1)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}"
neg_imp_zdiv_nonneg_iff: fixes b :: "int"   and a :: "int" assumes "b < 0" shows "(0 \<le> a div b) = (a \<le> 0)"
1: shows "transp (\<lambda>a\<^sub>1 a\<^sub>2. g a\<^sub>1 \<le> g a\<^sub>2)"
seg_betw: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> segment a b) = [[a x b]]"
use_spec_rule: fixes m :: "'a nres"   and \<Psi> :: "'a \<Rightarrow> bool"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "m \<le> SPEC \<Psi>"   and "m \<le> SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)" shows "m \<le> SPEC \<Phi>"
orthogonal: fixes x :: "'a vec" assumes "x \<in> local.span (gso ` {0..<i})" shows "gso i \<bullet> x = (0::'a)"
Sbis_raw_coind: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "theta \<subseteq> Sretr theta" shows "theta \<subseteq> Sbis"
fr_Send8: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "nReturn k t = []" shows "\<not> (\<exists>k<n. nSend k t \<noteq> [])"
li: shows "a \<otimes> (b \<otimes> inv (a \<otimes> b)) = \<one>"
lex_pm_trans: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and u :: "'a \<Rightarrow>\<^sub>0 'b" assumes "lex_pm s t"   and "lex_pm t u" shows "lex_pm s u"
disj_N: assumes "{..<N} \<subseteq> UNIV" shows "disjoint_family_on A {..<N}"
pcr_cr_eq: shows "pcr_finfun (=) (=) = cr_finfun"
abel_semigroup_axioms_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "abel_semigroup_axioms f \<equiv> \<forall>a b. f a b = f b a"
max_min_same_4: fixes y :: "'b"   and x :: "'b" shows "max y (min x y) = y"
split_sorted_2: fixes xs :: "('a btree \<times> 'a) list"   and p :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list" assumes "split xs p = (ls, (sub, sep) # rs)"   and "sorted_less (separators xs)" shows "p \<le> sep"
g_rename_precond_axioms: shows "g_rename_precond G f"
r_coset_subset_G: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set"   and x :: "'d" assumes "Group.monoid G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<subseteq> carrier G"
assms: fixes i :: "'a"   and j :: "'a" shows "D (\<lambda>t. A t $$ i $ j) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R A' t $$ i $ j) at t within T"
trans: fixes b :: "'t \<Rightarrow>\<^sub>0 'e"   and a :: "'t \<Rightarrow>\<^sub>0 'e"   and c :: "'t \<Rightarrow>\<^sub>0 'e" assumes "b \<preceq>\<^sub>p a"   and "c \<preceq>\<^sub>p b" shows "c \<preceq>\<^sub>p a"
Red1AAssStore: fixes s :: "'heap \<times> 'addr val list"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and w :: "'addr val"   and U :: "ty"   and P :: "'addr J1_prog"   and uf :: "bool"   and t :: "'thread_id" assumes "typeof_addr (hp s) a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "typeof\<^bsub>hp s\<^esub> w = \<lfloor>U\<rfloor>"   and "\<not> P \<turnstile> U \<le> T" shows "uf,P,t \<turnstile>1 \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val w,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ArrayStore),s\<rangle>"
infinite_ball: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "infinite (ball a r)"
existence_ivl_maximal_segment: fixes x :: "real \<Rightarrow> 'a"   and t :: "real"   and x0 :: "'a" assumes "(x solves_ode f) {t0--t} X"   and "x t0 = x0"   and "{t0--t} \<subseteq> T" shows "t \<in> existence_ivl t0 x0"
\<II>_def: shows "\<II> = \<I> (length Doc)"
right_neutral: fixes a :: "'b \<times> 'c" shows "times_prod' a one_prod = a"
vector_smult_lneg: fixes c :: "'a"   and x :: "('a, 'b) vec" shows "- c *s x = - (c *s x)"
intro: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "wide_subdigraph_axioms \<BB> \<CC>" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>"
relates_def: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b stream" shows "relates \<phi> s = \<phi> (shd s) (shd (stl s))"
eval: fixes a :: "'v \<Rightarrow> 'a" shows "eval_monom a m = a v ^ n"
strict_iff_order: fixes b :: "'b word"   and a :: "'b word" shows "(b <s a) = (b \<le>s a \<and> a \<noteq> b)"
split_list_at: fixes k :: "nat"   and D :: "'a list" assumes "k < length D" shows "D = take k D @ [D ! k] @ drop (Suc k) D"
count_roots_interval_rat: fixes p :: "int poly" assumes "square_free p" shows "root_info_cond (count_roots_interval_rat p) p"
LIM: fixes i :: "'a" assumes "isLim i"   and "\<And>j. j \<in> local.underS i \<Longrightarrow> P j" shows "P i"
select_convs_5: fixes Obj :: "'g set"   and Mor :: "'h set"   and Dom :: "'h \<Rightarrow> 'g"   and Cod :: "'h \<Rightarrow> 'g"   and Id :: "'g \<Rightarrow> 'h"   and Comp :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "Category.Id \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Id"
incl: fixes A :: "'c set"   and B :: "'c set set"   and b :: "'c set" assumes "Congruence.partition A B"   and "b \<in> B" shows "b \<subseteq> A"
IH: fixes gpv :: "('a, 'b, 'c) gpv"   and n :: "enat" assumes "X gpv n" shows "interaction_bound' gpv \<le> n"
P1: shows "Bet A B A0 \<and> A' B' C' CongA C B A0"
bigtheta_iff: shows "((\<lambda>_. 1) \<in> \<Theta>(\<lambda>x. \<Prod>g\<leftarrow>group_primfuns fs. (eval_primfun' \<circ> fst) g x powr snd g)) = list_all ((=) 0) (map snd (group_primfuns fs))"
DIAndCutP1_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndCutP1 vid1 pid1 = ([[$\<alpha> vid1]]Pc pid1)"
adjacent_finite: fixes v :: "'v" assumes "finite E" shows "finite {n. adjacent v n}"
insert_def: fixes e :: "'c"   and a :: "'d"   and q :: "('c, 'd) SkewBinomialHeap" shows "local.insert e a q \<equiv> Abs_SkewBinomialHeap (BsSkewBinomialHeapStruc.bs_insert e a (Rep_SkewBinomialHeap q))"
kernel_dim: shows "kernel_dim (char_matrix A lam) = local.dim"
lookup_def: fixes xs :: "'x" shows "lookup xs = lookup_raw (list_of_oalist xs)"
dist_les_2: fixes a :: "'b"   and list :: "['b]" shows "a : list \<notsqsubseteq> []"
ball_imageD: fixes f :: "'l \<Rightarrow> 'k"   and A :: "'l set"   and P :: "'k \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
wlsAbs_fresh_nchotomy: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and P :: "('index, 'bindex, 'varSort, 'var, 'opSym, 'sort) paramS" assumes "wlsAbs (xs, s) A"   and "wlsPar P" shows "\<exists>x X. A = Abs xs x X \<and> wls s X \<and> (\<forall>ys. x \<notin> varsOfS P ys) \<and> (\<forall>s'. \<forall>Y\<in>termsOfS P s'. fresh xs x Y) \<and> (\<forall>us s'. \<forall>B\<in>absOfS P (us, s'). freshAbs xs x B) \<and> (\<forall>rho\<in>envsOfS P. freshEnv xs x rho)"
Qg_carrier: fixes N :: "'a set" shows "carrier (G / N) = set_rcs G N"
fst_strip_comb_imp_eq_head_of: fixes t :: "Core.term"   and ts :: "Core.term list" shows "fst (strip_comb_imp (t, ts)) = head_of t"
TRANS: fixes X :: "freeExp"   and Y :: "freeExp"   and Z :: "freeExp" assumes "X \<sim> Y"   and "Y \<sim> Z" shows "X \<sim> Z"
that: fixes q :: "'alt pmf"   and i :: "'agent" assumes "q \<in> lotteries_on alts"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "i \<in> agents"   and "p \<prec>[SD (R i)] q" shows "thesis"
preserves_arr: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.arr f" shows "S.arr (S.src f)"
False: shows "\<nexists>t p. (WHILE b DO c, s) \<Rightarrow> p \<Down> t \<and> Q t < \<infinity>"
induct: fixes P :: "'a negation_type list \<Rightarrow> bool"   and a0 :: "'a negation_type list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (Neg x # xs)"   and "\<And>v xs. P xs \<Longrightarrow> P (Pos v # xs)" shows "P a0"
iMODb_iprev': fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "t \<in> [ r, mod m, c ]"   and "t < r + m * c" shows "iprev (t + m) [ r, mod m, c ] = t"
obj_self_composable_1: fixes a :: "'a" assumes "obj a" shows "a \<star> a \<noteq> null"
DomSolV_axioms: shows "DomSolV F \<delta>"
Rep_N_cases: fixes y :: "'b" assumes "y \<in> N" obtains x :: "'b N" where "y = Rep_N x"
exec_stop_def: shows "exec_stop = (\<lambda>\<sigma>. break_status \<sigma> \<or> return_status \<sigma>)"
cod_trg: fixes \<mu> :: "'a arrow_of_spans_data" assumes "arr \<mu>" shows "cod (trg \<mu>) = trg \<mu>"
polys_2: fixes qa :: "int poly"   and ria :: "root_info" assumes "(qa, ria) \<in> set polys" shows "qa \<noteq> 0"
compos_def: fixes A :: "('e, 'g) aGroup_scheme"   and g :: "'h \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'h" shows "compos A g f = compose (carrier A) g f"
assms_1: shows "x \<in>\<^sub>\<circ> vlrestriction D"
set_elem_equalityI: fixes A :: "'d set"   and B :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
mono_Max_commute: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "finite A"   and "A \<noteq> {}" shows "f (linorder.Max less_eq A) = Max (f ` A)"
scale_sum_right: fixes a :: "real"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "a *\<^sub>R sum f A = (\<Sum>x\<in>A. a *\<^sub>R f x)"
Ord_linear_le: fixes k :: "V"   and l :: "V" assumes "Ord k"   and "Ord l" obtains "k \<le> l"   | "l \<le> k"
simps_2: fixes i :: "nat"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and h :: "'b" shows "nth_derivative (Suc i) f x h = nth_derivative i (\<lambda>x. frechet_derivative f (at x) h) x h"
fun_relE2: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'c"   and Rv :: "('b \<times> 'c) set"   and t :: "'b"   and x :: "'a" assumes "(f, f') \<in> Id \<rightarrow> Rv"   and "t = f x" shows "(t, f' x) \<in> Rv"
set_update_subsetI: fixes xs :: "'b list"   and A :: "'b set"   and x :: "'b"   and i :: "nat" assumes "set xs \<subseteq> A"   and "x \<in> A" shows "set (xs[i := x]) \<subseteq> A"
t_or_f_a: fixes P :: "'c \<times> 'd alpha_d_scheme \<Rightarrow> bool"   and a :: "'c"   and b :: "'d alpha_d_scheme" assumes "P (a, b)" shows "P (a, b\<lparr>ok := True\<rparr>) \<or> P (a, b\<lparr>ok := False\<rparr>)"
sdiv_smod_id: fixes a :: "'b word"   and b :: "'b word" shows "a sdiv b * b + a smod b = a"
induced_automorphism_half_chamber_system_pgallery_map_g: fixes Cs :: "'a set list" assumes "order.greater_eq (g \<turnstile> folding_g.\<C>) (set Cs)"   and "pgallery Cs" shows "pgallery (\
 \<Turnstile> Cs)"
Inf_closed_cl_op_var: fixes X :: "'a set" assumes "\<forall>x\<in>X. x \<in> range cl_op" shows "\<Sqinter> X \<in> range cl_op"
exhaust: fixes y :: "'a ast_effect" obtains x1 :: "'a atom formula list" and x2 :: "'a atom formula list" where "y = Effect x1 x2"
value_class_inhabitants: fixes typeId :: "CTypeId"   and P :: "Value \<Rightarrow> bool" shows "(\<forall>x. \<tau> x = CClassT typeId \<longrightarrow> P x) = (\<forall>a. P (objV typeId a))"
wlp_complete: fixes S :: "com"   and P :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P" shows "\<turnstile>\<^sub>p {wlp S P} S {P}"
negate_f: assumes "has_disc_negated disc False m" shows "\<forall>pts. \<not> has_disc_negated disc2 False (negate_f C pts)"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> ManiPref'.Cd d X) = (ManiPref'.Cd d X = {x})"
d_def: shows "d = min (cmod z / 2) (if Im z = 0 then Re z / 2 else \<bar>Im z\<bar> / 2)"
get_tag_name_locs_impl: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs" shows "get_tag_name_locs = l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs"
open_uniformity: fixes "open" :: "'d set \<Rightarrow> bool"   and uniformity :: "('d \<times> 'd) filter"   and U :: "'d set" assumes "class.open_uniformity open uniformity" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
strict_trans1: fixes b :: "'t \<Rightarrow>\<^sub>0 'c"   and a :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'t \<Rightarrow>\<^sub>0 'c" assumes "b \<preceq>\<^sub>p a"   and "c \<prec>\<^sub>p b" shows "c \<prec>\<^sub>p a"
a_4: fixes x :: "'a"   and y :: "'a" shows "ar x \<le> ar (y \<cdot> x)"
thread_not_empty: shows "\<not> thread_empty (execs (current s))"
subcocycle_integrable_2: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "subcocycle u" shows "random_variable borel (u n)"
to_fun_smult: fixes f :: "nat \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a" assumes "f \<in> carrier P"   and "b \<in> carrier R"   and "c \<in> carrier R" shows "to_fun (c \<odot>\<^bsub>P\<^esub> f) b = c \<otimes> to_fun f b"
cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) character_data_ptr"   and y :: "(_) character_data_ptr" shows "(cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r y) = (x = y)"
valuation_map: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v \<in> carrier K \<rightarrow> Z\<^sub>\<infinity>"
\<tau>': shows "(fst \<tau>1, snd \<tau>1 @ snd \<tau>2 @ snd \<tau>3) \<in> seq ars"
norm_triangle_lt: fixes x :: "'a"   and y :: "'a"   and e :: "real" assumes "norm x + norm y < e" shows "norm (x + y) < e"
Class_widen2: fixes P :: "'a prog"   and C :: "char list"   and T :: "ty" shows "subtype P (Class C) T = (\<exists>D. T = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D)"
equivalence_subset: fixes L :: "('a, 'b) eq_object_scheme"   and A :: "'a set" assumes "equivalence L"   and "A \<subseteq> carrier L" shows "equivalence (L\<lparr>carrier := A\<rparr>)"
dynC: shows "if stat then dynC = declclass f else dynC = obj_class (lookup_obj (snd s) a)"
delta_price_adapted: fixes cash_flow :: "bool stream \<Rightarrow> real"   and T :: "nat"   and N :: "bool stream measure"   and q :: "real" assumes "cash_flow \<in> borel_measurable (G T)"   and "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "borel_adapt_stoch_proc G (delta_price N cash_flow T)"
head_apps: fixes s :: "('a, 'b) tm"   and ss :: "('a, 'b) tm list" shows "head (apps s ss) = head s"
H_def: shows "H = geodesic_subsegment G a (dist x a) (dist y a)"
simps_2: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and i :: "char list" shows "(M, s \<Turnstile> Pro i) = \<pi> M s i"
span_eq: fixes S :: "'c set"   and T :: "'c set" shows "(ideal S = ideal T) = (S \<subseteq> ideal T \<and> T \<subseteq> ideal S)"