hcju/isabelleps huggingface dataset:
that: fixes t :: "('b, 'a) Term.term" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t \<in> set S"   and "v \<in> fv t" shows "thesis"
mult1I: fixes M :: "'c multiset"   and a :: "'c"   and M0 :: "'c multiset"   and N :: "'c multiset"   and K :: "'c multiset"   and r :: "('c \<times> 'c) set" assumes "M = add_mset a M0"   and "N = M0 + K"   and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" shows "(N, M) \<in> mult1 r"
sublinearD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real"   and s :: "'b" assumes "sublinear t"   and "sound P"   and "sound Q"   and "0 \<le> a"   and "0 \<le> b"   and "0 \<le> c" shows "a * t P s + b * t Q s \<ominus> c \<le> t (\<lambda>s'. a * P s' + b * Q s' \<ominus> c) s"
assms: shows "lambda C -quasi_isometry_on {A..B} c"
integral_emptyI_1: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" assumes "b \<le> a" shows "integral {a..b} f = (0::'a)"
cet_val': fixes t :: "pdf_type"   and v :: "val"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "t = val_type v" shows "\<Gamma> \<turnstile>\<^sub>c CVal v : t"
codomain_unique: fixes b :: "'a"   and f :: "'a"   and b' :: "'a" assumes "b \<in> codomains f"   and "b' \<in> codomains f" shows "b = b'"
2: shows "fst (Spair s) \<Turnstile> (fold fst (map Spair ss) \<Turnstile> Cs) = fold fst (map Spair (ss @ [s])) \<Turnstile> Cs"
inv_one: shows "inv\<^bsub>add_monoid R\<^esub> \<zero> = \<zero>"
3: fixes ys :: "nat list"   and ba :: "nat list"   and x :: "nat"   and xs :: "nat list"   and s :: "nat" assumes "ys \<in> fst ` set (c2.gen_check ba)"   and "Max (set ba) < x" shows "cond1 ba ys (x # xs) s = False"
ipresIGAbs_def: fixes h :: "'u \<Rightarrow> 'v"   and hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'varSort, 'sort, 'aa, 'ab, 'u, 'w, 'ac) model_scheme"   and MOD' :: "('ad, 'ae, 'varSort, 'af, 'ag, 'ab, 'v, 'x, 'ah) model_scheme" shows "ipresIGAbs h hA MOD MOD' \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> hA (igAbs MOD xs x X) = igAbs MOD' xs x (h X)"
2: shows "(\<lambda>i. interval_lebesgue_integral lborel (ereal (g (l i))) (ereal (g (u i))) f) \<longlonglongrightarrow> LBINT x=a..b. g' x *\<^sub>R f (g x)"
modulo_tree_def: fixes x :: "'a tree"   and y :: "'a tree" shows "x mod y = pure (mod) \<diamondop> x \<diamondop> y"
big_step_mono_prog: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<pi>' :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi> \<subseteq>\<^sub>m \<pi>'"   and "\<pi>: (c, s) \<Rightarrow> t" shows "\<pi>': (c, s) \<Rightarrow> t"
bij_lens_weak: fixes x :: "'b \<Longrightarrow> 'c" assumes "bij_lens x" shows "weak_lens x"
le_lt_convs_5: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Eq) = (x = y)"
H_eq_1: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "MC_syntax.recurrent K s = (MC_syntax.H K s s = 1)"
that: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "homeomorphism T T f g"   and "\<And>i. i \<in> K \<Longrightarrow> f (x i) = y i"   and "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> S"   and "bounded {x. \<not> (f x = x \<and> g x = x)}" shows "thesis"
commute: fixes a :: "'e"   and b :: "'e" shows "a \<sqinter> b = b \<sqinter> a"
M: shows "\<forall>m\<in>M. \<forall>f T. Fun f T \<sqsubseteq> m \<longrightarrow> P f T"
nextss_closed_dfs: fixes g :: "(node \<times> node) list"   and xs :: "node list" shows "nextss g (dfs g xs []) \<subseteq> set (dfs g xs [])"
funcset_id: fixes A :: "'a set" shows "(\<lambda>x. x) \<in> A \<rightarrow> A"
char_matrix_def: fixes A :: "'c mat"   and e :: "'c" shows "char_matrix A e = A + - e \<cdot>\<^sub>m 1\<^sub>m (dim_row A)"
rsquarefree_card_degree: fixes p :: "complex poly" assumes "p \<noteq> 0" shows "rsquarefree p = (card {x. poly p x = 0} = degree p)"
effectI: fixes c :: "'a Heap"   and h :: "heap"   and r :: "'a"   and h' :: "heap" assumes "execute c h = Some (r, h')" shows "effect c h h' r"
antisym: fixes b :: "'c word"   and a :: "'c word" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
to_fun_cases: fixes y :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" assumes "y \<in> UNIV" obtains x :: "('d, 'c) sq_matrix" where "y = to_fun x"
simps_1: fixes p :: "real"   and M :: "bool stream measure"   and u :: "real"   and d :: "real"   and v :: "real" assumes "infinite_coin_toss_space p M" shows "geom_rand_walk u d v 0 = (\<lambda>w. v)"
x: shows "atom x \<sharp> (r, d, d', y, z)"
aux_lemma: fixes k :: "nat"   and Usa :: "'a list" assumes "k \<le> n"   and "local.independent K Usa"   and "set Usa \<subseteq> E"   and "length Usa = k" shows "\<exists>Vs. length (Vs @ Usa) = n \<and> local.independent K (Vs @ Usa) \<and> Span K (Vs @ Usa) = E"
edges_are_in_V_1: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w" shows "v \<in> V"
monotone_ltl: shows "monotone (\<sqsubseteq>) (\<sqsubseteq>) ltl"
simps: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c list" shows "Graph_Defs.steps E a = ((\<exists>x. a = [x]) \<or> (\<exists>x y xs. a = x # y # xs \<and> E x y \<and> Graph_Defs.steps E (y # xs)))"
shows_attr_def: fixes av :: "char list \<times> char list" shows "shows_attr av = shows (fst av) \<circ> shows_string ([CHR ''='', CHR 0x22] @ snd av @ [CHR 0x22])"
interp_p: fixes x :: "complex"   and y :: "complex" assumes "(x, y) \<in> set (zip (map (unity_root k) (map int [0..<k])) ws)" shows "poly p x = y"
check_instr_pc: fixes ins :: "'addr instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc :: "nat"   and frs :: "'addr frame list"   and pc' :: "nat" assumes "check_instr ins P h stk loc C M pc frs"   and "ins_jump_ok ins pc'" shows "check_instr ins P h stk loc C M pc' frs"
alpha_refl: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "X #= X"
rreq_rrep_fresh_any_step_invariant: fixes i :: "nat" shows "paodv i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>((\<xi>, uu_), a, uu_). anycast (rreq_rrep_fresh (rt \<xi>)) a)"
rsquarefree_def: fixes p :: "'a poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. order a p = 0 \<or> order a p = 1))"
Abs_fset_inverse: fixes y :: "'c set" assumes "y \<in> {A. finite A}" shows "fset (Abs_fset y) = y"
WTrtCons: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> e : T"   and "P,E,h \<turnstile> es [:] Ts" shows "P,E,h \<turnstile> e # es [:] T # Ts"
currentLevelPrefixToLevelEq: fixes level :: "nat"   and M :: "('a \<times> bool) list" assumes "level \<le> currentLevel M" shows "currentLevel (prefixToLevel level M) = level"
arg_cnj_pi: fixes z :: "complex" assumes "arg z = pi" shows "arg (cnj z) = pi"
proj_fm: fixes x :: "'i"   and n :: "nat"   and m :: "'i \<Rightarrow> 'e" assumes "x \<in> J n" shows "(fm n m)\<^sub>F (Utn x) = m x"
subst_cls_eq_grounding_of_cls_subset_eq: fixes D :: "'a literal multiset"   and \<sigma> :: "'s"   and C :: "'a literal multiset" assumes "D \<cdot> \<sigma> = C" shows "\<G>_F C \<subseteq> \<G>_F D"
sort_set_unique_rel: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list"   and y :: "'b list" assumes "irreflp r \<and> transp r"   and "set x = set y"   and "sorted_wrt r x"   and "sorted_wrt r y" shows "x = y"
TBtheorem3a: fixes P :: "specID"   and E :: "Expression"   and Q :: "specID"   and PQ :: "specID" assumes "\<not> ine P E"   and "\<not> ine Q E"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ" shows "\<not> ine PQ E"
add_point_def: fixes pa :: "'a"   and p :: "'a" shows "add_point_sys.add_point pa p \<equiv> insert p (add_point pa)"
pair_vec_vecset_def: fixes A :: "(('d, 'f) vec \<times> ('e, 'g) vec set) option" shows "pair_vec_vecset A = (if Option.is_none A then None else Some (vec_to_iarray (fst (the A)), vec_to_iarray ` snd (the A)))"
AE_E3: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a measure" assumes "almost_everywhere M P" obtains N :: "'a set" where "\<And>x. x \<in> space M - N \<Longrightarrow> P x" and "N \<in> null_sets M"
ih: fixes m :: "nat" assumes "2 ^ n * q < 2 ^ m" shows "q < 2 ^ (m - n)"
prime_elem_multiplicity_prod_distrib: fixes p :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "prime_elem p"   and "(0::'a) \<notin> f ` A"   and "finite A" shows "multiplicity p (prod f A) = (\<Sum>x\<in>A. multiplicity p (f x))"
zero_less_double_add_iff_zero_less_single_add: 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 a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (plus a a) = less zero a"
tObsC_initial_2: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h"   and s :: "('b, 'e, 'f) BEState" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC" shows "FiniteDetBroadcastEnvironment.tObsC envObsC (tInit s) = tInit (envObsC (es s))"
lens_comp_left_id: fixes X :: "'d \<Longrightarrow> 'e" shows "1\<^sub>L ;\<^sub>L X = X"
digit_from_digits: fixes base :: "nat"   and n :: "nat"   and d :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "digits base"   and "\<And>j. j < n \<Longrightarrow> d j < base"   and "0 < n"   and "i < n" shows "digits.digit base (digits.from_digits base n d) i = d i"
Sup_image_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 Y :: "'a set"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "Y \<noteq> {}" shows "Sup (sup x ` Y) = sup x (Sup Y)"
invrules_2: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = U - (S \<union> L)"
smith_Suc_k: shows "Smith_normal_form_upt_k (diagonal_to_Smith A bezout) (Suc (min (nrows A) (ncols A) - 2))"
eigenpair_def: fixes k :: "complex"   and v :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "eigenpair k v H = (v \<noteq> vec_zero \<and> H *\<^sub>m\<^sub>v v = k *\<^sub>s\<^sub>v v)"
var: fixes \<Gamma> :: "(name, value) fmap"   and name :: "name"   and val :: "value"   and C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> value) list" assumes "fmlookup \<Gamma> name = Some val" shows "special_constants.veval C_info rs \<Gamma> (Svar name) val"
natLeq_ofilter_less: fixes n :: "nat" shows "ofilter natLeq {0..<n}"
fls_prpart0: shows "fls_prpart 0 = 0"
quotientfun_equality: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and b :: "'c" assumes "refl_on A r"   and "f respects r"   and "X \<in> A // r"   and "b \<in> f ` X" shows "quotientfun f X = b"
restrict_complement_singleton_eq: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c" shows "f |` (- {x}) = f(x := None)"
\<gamma>: shows "\<guillemotleft>\<gamma> : f\<^sup>* \<Rightarrow> w\<guillemotright> \<and> \<nu> \<cdot> \<r>[r] \<cdot> (r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sup>*] \<cdot> (\<rho> \<star> f\<^sup>*) = g \<star> \<gamma> \<and> \<epsilon> = \<theta>' \<cdot> (f \<star> \<gamma>)"
lc_not_0: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "lc p \<noteq> (0::'c)"
min_absorb1: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "min x y = x"
cos_diff: fixes x :: "'b"   and y :: "'b" shows "cos (x - y) = cos x * cos y + sin x * sin y"
AT_fps_0_nth: fixes f :: "nat \<Rightarrow> real"   and n :: "nat" shows "AT_fps f 0 $ n = (if n = 0 then - f 0 else f (n - 1) - f n)"
fmsubset_restrict_fset: fixes S :: "'c fset"   and m :: "('c, 'd) fmap" shows "fmrestrict_fset S m \<subseteq>\<^sub>f m"
bdia_def: fixes x :: "'a"   and y :: "'a" shows "\<langle>x| y = ar ar (y \<cdot> x)"
span_eq_iff: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c set" assumes "module scale" shows "(module.span scale s = s) = module.subspace scale s"
simps_61: fixes x101 :: "Term.var"   and x102 :: "expr"   and x21 :: "ty"   and x22 :: "expr" shows "x101:=x102 \<noteq> New x21[x22]"
takefill_replicate: fixes fill :: "'b"   and n :: "nat"   and m :: "nat" shows "takefill fill n (replicate m fill) = replicate n fill"
imageE: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "b \<in> f ` A" obtains x :: "'e" where "b = f x" and "x \<in> A"
distinct_12: fixes x41 :: "bexp"   and x42 :: "com"   and x43 :: "com"   and x21 :: "char list"   and x22 :: "aexp" shows "IF x41 THEN x42 ELSE x43 \<noteq> x21 ::= x22"
min_def: fixes a :: "'b"   and b :: "'b" shows "min a b = (if a \<le> b then a else b)"
integ_rev_exist_b: shows "\<forall>(k1, \<gamma>1)\<in>one_chain1. \<forall>b\<in>basis. line_integral_exists F {b} (reversepath (snd (f (k1, \<gamma>1))))"
continuous_on_generated_topo_iff: fixes T1 :: "'c topology"   and S :: "'d set set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map T1 (topology_generated_by S) f = ((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> \<Union> S)"
odd_square_cong_4_nat: fixes x :: "nat" assumes "odd x" shows "[x\<^sup>2 = 1] (mod 4)"
fresherI2: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "nsqn\<^sub>r r = nsqn\<^sub>r r'"   and "\<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r" shows "r \<sqsubseteq> r'"
l4_17: 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 P :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Cong A P A Q"   and "Cong B P B Q" shows "Cong C P C Q"
trace_refine_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<sqsubseteq>\<^sub>T Q \<equiv> T Q \<subseteq> T P"
obtain_comp_block_orig: fixes bl1 :: "'a set" assumes "bl1 \<in># \<B>\<^sup>C" obtains bl2 :: "'a set" where "bl2 \<in># \<B>" and "bl1 = bl2\<^sup>c"
cases: fixes x :: "'b list" obtains "xb__ = []"   | x :: "'b" where "xb__ = [xc__]"   | x :: "'b" and y :: "'b" and xs :: "'b list" where "xb__ = x # y # xs"
intros: fixes p :: "perm"   and x :: "'e" shows "hull_relp (p \<bullet> x) x"
ih: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and s' :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>2"   and "s' \<leadsto> s" shows "\<Gamma> \<turnstile> s' is t : T\<^sub>2"
61162: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> x \<sqinter> y) = - x"
star_right_induct_mult_iff: fixes y :: "'a"   and x :: "'a" shows "(y \<sqinter> x \<le> y) = (y \<sqinter> top \<le> y)"
simps_1: shows "FingerTreeStruc.gmft Empty = (0::'a)"
subj: shows "\<forall>a gc. gc \<in> set (jkbp a) \<longrightarrow> subjective a (guard gc)"
simps_4: fixes x2 :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" shows "Assume x2 \<noteq> Skip"
crsp_steps2: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and as :: "nat"   and stp :: "nat"   and am :: "nat list" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (0, lm) (Suc 0, l, r) ires"   and "as < length ap"   and "abc_steps_l (0, lm) ap stp = (as, am)" shows "\<exists>stpa\<ge>stp. crsp ly (as, am) (steps0 (Suc 0, l, r) tp stpa) ires"
mat_assoc_test_1: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "A * B * (C * D) = A * B * C * D"
list_update_code_3: fixes x :: "'a"   and xs :: "'a list"   and i :: "nat"   and y :: "'a" shows "(x # xs)[Suc i := y] = x # xs[i := y]"
mat_assoc_test_10: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "trace (A * B * C) = trace (B * C * A)"
greaterThanLessThan_iff: fixes i :: "'a Word.word"   and l :: "'a Word.word"   and u :: "'a Word.word" shows "(i \<in> signed.greaterThanLessThan l u) = (l <s i \<and> i <s u)"
assms: shows "(a, b) \<in> set (arcs i j xs)"
simps_15: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = {x21} \<union> set x22"
lm099: fixes x :: "'c"   and X :: "'c set"   and Y :: "'c set" assumes "x \<in> X" shows "1 \<in> (X <|| Y) `` {x}"
assms_2: shows "x \<notin>\<^sub>\<circ> B"
at_within_def: fixes "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and s :: "'e set" assumes "class.topological_space open" shows "topological_space.at_within open a s = inf (topological_space.nhds open a) (principal (s - {a}))"
mpoly_coeff_Const: fixes c :: "'a"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Const c) m = (if m = 0 then c else (0::'a))"
Alist_inverse: fixes y :: "('a \<times> 'b) list" assumes "y \<in> {xs. (distinct \<circ> map fst) xs}" shows "alist.impl_of (Alist y) = y"
Associativity: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" shows "(\<Psi> \<otimes> \<Psi>') \<otimes> \<Psi>'' \<simeq> \<Psi> \<otimes> \<Psi>' \<otimes> \<Psi>''"
sets_embed_measure: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj f" shows "sets (embed_measure M f) = {f ` A |A. A \<in> sets M}"
ARR_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f"   and "free_monoidal_category.ARR C g"   and "monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C g)" shows "free_monoidal_category.ARR C (free_monoidal_category.comp C f g)"
de_Morgan_conj: fixes x :: "'c"   and y :: "'c" shows "NOT (x AND y) = NOT x OR NOT y"
wf_mbuf2'_def: fixes \<sigma> :: "(char list \<times> 'b list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'b list set"   and \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula"   and buf :: "'b option list set list \<times> 'b option list set list" shows "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf = wf_mbuf2 (min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)) (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j) (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) buf"
bottom_right_bottom: fixes a :: "'b" shows "gcd a (1::'b) = (1::'b)"
bounded_generate_check_def: fixes C\<^sub>2 :: "nat list \<Rightarrow> nat \<Rightarrow> bool"   and B\<^sub>2 :: "nat"   and C\<^sub>1 :: "nat list \<Rightarrow> nat list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool"   and B\<^sub>1 :: "nat list \<Rightarrow> nat" shows "bounded_generate_check C\<^sub>2 B\<^sub>2 C\<^sub>1 B\<^sub>1 \<equiv> bounded_gen_check C\<^sub>2 B\<^sub>2 \<and> bounded_generate_check_axioms C\<^sub>2 C\<^sub>1 B\<^sub>1"
true_cls_mono: fixes C :: "'b literal multiset"   and D :: "'b literal multiset"   and I :: "'b set" assumes "set_mset C \<subseteq> set_mset D"   and "I \<Turnstile> C" shows "I \<Turnstile> D"
distinct_step: fixes s :: "'b list"   and r :: "'b"   and as :: "nat \<times> nat list" shows "distinct (step s r as) = distinct s"
simps: fixes a1 :: "cld"   and a2 :: "fd list" shows "class_fields a1 a2 = (\<exists>dcl cl fds meth_defs. a1 = cld_def dcl cl fds meth_defs \<and> a2 = fds)"
sup_state_opt_any_None: fixes P :: "'c prog"   and s :: "(ty list \<times> ty err list) option" shows "P \<turnstile> s \<le>' None = (s = None)"
is_expansion_aux_imp_smallomega: fixes basis :: "(real \<Rightarrow> real) list"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and b' :: "real \<Rightarrow> real"   and r :: "real" assumes "basis_wf basis"   and "is_expansion_aux xs f basis"   and "filterlim b' at_top at_top"   and "trimmed_ms_aux xs"   and "\<forall>g\<in>set basis. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (b' x))"   and "r < 0" shows "f \<in> \<omega>(\<lambda>x. b' x powr r)"
simps_2: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and m :: "'a common_primitive match_expr"   and rs :: "'a common_primitive rule list"   and allowed :: "'a word set"   and denied1 :: "'a word set" shows "no_spoofing_algorithm iface ipassmt (Rule m Accept # rs) allowed denied1 = no_spoofing_algorithm iface ipassmt rs (allowed \<union> get_exists_matching_src_ips iface m) denied1"
FP_rule_def: shows "FP_rule \<equiv> ([(pusall || \<langle> Game hgid1 \<rangle> \<langle> Game hgidd \<rangle> TT) \<rightarrow> \<langle> Game hgidd \<rangle> TT], \<langle> Game hgid1** \<rangle> pusall \<rightarrow> \<langle> Game hgidd \<rangle> TT)"
cs\<^sub>0_def: shows "cs\<^sub>0 \<equiv> ((replicate N False)[0 := True], 0::'a, \<lambda>_. INIT)"
all_inQ: shows "\<forall>a\<in>\<E>. a \<in> Q"
objI_trg: fixes a :: "'a" assumes "arr a"   and "I.map a = a" shows "obj a"
norm_unsigned_equal: fixes w :: "bit list" assumes "length (norm_unsigned w) = length w" shows "norm_unsigned w = w"
non_winning_moves_remains_non_winning_Odd: fixes p :: "'a list"   and \<sigma> :: "'a list \<Rightarrow> 'a"   and m :: "'a" assumes "odd (length p)"   and "\<forall>\<sigma>. \<not> strategy_winning_by_Odd \<sigma> p" shows "\<not> strategy_winning_by_Odd \<sigma> (p @ [m])"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (x + xa) = (\<lambda>xb. x *\<^sub>V xb + xa *\<^sub>V xb)"
T_bal_tree: fixes n :: "nat"   and xs :: "'b tree" assumes "n = size xs" shows "T_bal_tree n xs = 4 * n + 3"
card_berlekamp_basis_number_factors: shows "card (set (berlekamp_basis u)) = card P"
fds_shift_zero: fixes c :: "'b" shows "fds_shift c 0 = 0"
simps_2: fixes \<A> :: "'c ltl set" shows "\<A> \<Turnstile>\<^sub>P false = False"
supp_except_fun: fixes s :: "'a \<Rightarrow> 'b"   and V :: "'a set" shows "supp_fun (except_fun s V) = supp_fun s - V"