hcju/isabelleps huggingface dataset:
deg_pm_hom_components: fixes q1 :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q2 :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t1 :: "'b \<Rightarrow>\<^sub>0 nat"   and t2 :: "'b \<Rightarrow>\<^sub>0 nat" assumes "q1 \<in> hom_components p"   and "q2 \<in> hom_components p"   and "t1 \<in> keys q1"   and "t2 \<in> keys q2" shows "(deg_pm t1 = deg_pm t2) = (q1 = q2)"
permutes_empty: fixes p :: "'a \<Rightarrow> 'a" shows "(p permutes {}) = (p = id)"
invertible_inverse_invertible: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "invertible (inverse u)"
comp_arr_inv': fixes f :: "'d" assumes "D.iso f" shows "f \<cdot>\<^sub>D D.inv f = D.cod f"
image_Collect_subsetI: fixes P :: "'q \<Rightarrow> bool"   and f :: "'q \<Rightarrow> 'r"   and B :: "'r set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
that: fixes \<pi>'' :: "Syntax.term list" assumes "\<pi> = \<pi>'' @ X" shows "thesis"
last_snoc: fixes xs :: "'e list"   and x :: "'e" shows "last (xs @ [x]) = x"
denote_dim_pdo: fixes dims :: "nat list"   and S :: "com"   and \<rho> :: "complex mat" assumes "state_sig.well_com dims S"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "partial_density_operator \<rho>" shows "state_sig.denote dims S \<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims) \<and> partial_density_operator (state_sig.denote dims S \<rho>)"
invertible_implies_full_rank: fixes A :: "(('b, 'c) vec, 'c) vec" assumes "invertible A" shows "rank A = nrows A"
1: shows "wf\<^sub>s\<^sub>t X (map (\<lambda>d. \<forall>[]\<langle>\<or>\<noteq>: [(pair (t, s), pair d)]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D) @ A)"
cases: fixes P :: "'a prog"   and a1 :: "char list"   and a2 :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> a1 sees_methods a2" obtains D :: "char list" and fs :: "fdecl list" and ms :: "'a mdecl list" and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" where "a1 = Object" and "a2 = Mm__" and "class P Object = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "Mm__ = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms__"   | C :: "char list" and D :: "char list" and fs :: "fdecl list" and ms :: "'a mdecl list" and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" and Mm' :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" where "a1 = C" and "a2 = Mm'" and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D sees_methods Mm" and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)"
assms_1: fixes i :: "nat" shows "p i * i \<le> n"
lossless_sample_primes_excl: fixes P :: "nat" shows "lossless_spmf (sample_primes_excl {P})"
eq_inf_lproject_visible: fixes u :: "'action stream"   and v :: "'action stream" assumes "u =\<^sub>I v" shows "lproject visible (llist_of_stream u) = lproject visible (llist_of_stream v)"
convex_on_realI: fixes A :: "real set"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real" assumes "connected A"   and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<le> y\<rbrakk> \<Longrightarrow> f' x \<le> f' y" shows "convex_on A f"
simps_2: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "exec_plan s (a # as) = exec_plan (state_succ s a) as"
theFTOne_1: shows "theFTOne cl dl = theFT dl - theFT cl \<union> theFT cl"
start_remove_removes: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "cnt x (start_remove xs x ys) \<le> Suc (cnt x ys)"
lessThan_strict_subset_iff: fixes m :: "'c"   and n :: "'c" shows "({..<m} \<subset> {..<n}) = (m < n)"
tl_Nil: fixes xs :: "'c list" shows "(tl xs = []) = (xs = [] \<or> (\<exists>x. xs = [x]))"
ideD_1: fixes a :: "'a1" assumes "A1.ide a" shows "A1.arr a"
simps_3: fixes p :: "fm" shows "qfree (NOT p) = qfree p"
A3b: fixes y :: "i" assumes "AxiomsSet7" shows "cod y \<cdot> y \<cong> y"
DA_delta_is_lift_NA_delta: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and w :: "'d list"   and Q :: "'c set" shows "DA.delta (na2da A) w Q = \<Union> (NA.delta A w ` Q)"
classes_above_sees_methods: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C sees_methods Mm" shows "P' \<turnstile> C sees_methods Mm"
distrPUTLG2: fixes PL :: "('c \<Rightarrow> 'd decision option) list"   and x :: "'c"   and P :: "'c \<Rightarrow> 'd decision option"   and PUT :: "'c \<Rightarrow> 'd decision option" assumes "PL \<noteq> []"   and "x \<in> dom P"   and "list2policy PL x = P x"   and "disjDomGD PL"   and "PUT x = P x" shows "PUTListGD PUT x PL"
apps_betasE: fixes r :: "dB"   and rs :: "dB list"   and s :: "dB" assumes "r \<degree>\<degree> rs \<rightarrow>\<^sub>\<beta> s" obtains r' :: "dB" where "r \<rightarrow>\<^sub>\<beta> r'" and "s = r' \<degree>\<degree> rs"   | rs' :: "dB list" where "rs => rs'" and "s = r \<degree>\<degree> rs'"   | t :: "dB" and u :: "dB" and us :: "dB list" where "r = Abs t" and "rs = u # us" and "s = t[u/0] \<degree>\<degree> us"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto><Rel> Q"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (<\<nu>y>R, <\<nu>y>S) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto><Rel'> <\<nu>x>Q"
nth_root_nat_aux1: fixes k :: "nat"   and n :: "nat" assumes "0 < k" shows "{m. m ^ k \<le> n} \<subseteq> {..n}"
mkt_stocks_assets: fixes Mkt :: "('d, 'c) discrete_market" shows "stk_strict_subs (stocks Mkt)"
zero_order_3: fixes n :: "'d multiset" shows "\<not> n \<subset># {#}"
farkas_check: fixes s' :: "('i, 'a) state"   and s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set" assumes "check s' = s"   and "\<U> s"   and "\<not> \<U> s'"   and "\<nabla> s'"   and "\<triangle> (\<T> s')"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s'"   and "\<diamond> s'"   and "index_valid as s'" shows "\<exists>C. farkas_coefficients_atoms_tableau (snd ` as) (\<T> s') C"
embedS_iff: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "embed r r' f" shows "embedS r r' f = (f ` Field r \<subset> Field r')"
ring_n_exp_K_exp: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and x :: "'a"   and m :: "nat" assumes "distinct_pds K n P"   and "x \<in> carrier (O\<^bsub>K P n\<^esub>)" shows "x^\<^bsup>K m\<^esup> = x^\<^bsup>O\<^bsub>K P n\<^esub> m\<^esup>"
inj: shows "inj (\<lambda>x. x /\<^sub>R norm (a - b))"
closure_of_mono: fixes S :: "'b set"   and T :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> T" shows "X closure_of S \<subseteq> X closure_of T"
distinct_45: fixes x51 :: "trm"   and x52 :: "POPLmarkRecord.type"   and x6 :: "(char list \<times> trm) list" shows "x51 \<bullet>\<^sub>\<tau> x52 \<noteq> Rcd x6"
vquot_fm_empty: fixes A :: "fm" shows "\<lfloor>A\<rfloor>{} = \<guillemotleft>A\<guillemotright>"
gauss_sum_1_mod_square_eq_k: assumes "\<forall>k>0. separable k" shows "(cmod (gauss_sum 1))\<^sup>2 = real n"
tauChainPar2: fixes Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q" shows "\<Psi> \<rhd> P \<parallel> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> P \<parallel> Q'"
mwc_sarp_def: fixes f :: "'b set \<Rightarrow> 'b set" shows "mwc_sarp f = (\<nexists>Xs. 1 < length Xs \<and> distinct (map f Xs) \<and> (\<forall>i. f (Xs !% i) \<subset> Xs !% i \<inter> Xs !% (i + 1)))"
convex_linear_vimage: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'b set" assumes "linear f"   and "convex S" shows "convex (f -` S)"
abstract_primitive_preserves_not_has_disc_negated: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b common_primitive match_expr"   and sel_f :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "\<forall>a. \<not> disc (Extra a)"   and "\<not> has_disc_negated disc neg m" shows "\<not> has_disc_negated disc neg (abstract_primitive sel_f m)"
MST_valid_validn: fixes K :: "nat" assumes "MST_valid" shows "MST_validn K"
cod_char: fixes f :: "'c local.term set" shows "cod f = (if arr f then mkarr (COD f) else null)"
trcl_rev_induct: fixes c :: "'a"   and w :: "'b list"   and c' :: "'a"   and S :: "('a \<times> 'b \<times> 'a) set"   and P :: "'a \<Rightarrow> 'b list \<Rightarrow> 'a \<Rightarrow> bool" assumes "(c, w, c') \<in> trcl S"   and "\<And>c. P c [] c"   and "\<And>c w c' e c''. \<lbrakk>(c, w, c') \<in> trcl S; (c', e, c'') \<in> S; P c w c'\<rbrakk> \<Longrightarrow> P c (w @ [e]) c''" shows "P c w c'"
trace_nil: fixes step :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and c' :: "'b" shows "traces.trace step c [] c' = (c = c')"
fresh_vsubstEnv_updEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and y1 :: "'var" assumes "goodEnv rho"   and "freshEnv ys y rho" shows "rho &[y1 // y]_ys = rho [y \<leftarrow> Var ys y1]_ys"
altinserta: shows "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) a = A"
set_map_3: fixes f1 :: "'f \<Rightarrow> 'd"   and f2 :: "'g \<Rightarrow> 'e"   and f3 :: "'h \<Rightarrow> 'c"   and v :: "('f, 'g, 'h) F2" shows "F2set3 (F2map f1 f2 f3 v) = f3 ` F2set3 v"
\<tau>move2_BlockSomeD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and T :: "ty"   and v :: "'addr val"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk {V:T=\<lfloor>v\<rfloor>; e} (Suc (Suc pc)) xcp" shows "\<tau>move2 P h stk e pc xcp"
minimal_polynomial_is_unique: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R"   and "(algebraic over K) x" shows "\<exists>!p. p \<in> carrier (K [X]) \<and> pirreducible K p \<and> local.eval p x = \<zero> \<and> lead_coeff p = \<one>"
nonzero_deg_pos: fixes p :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>" shows "0 \<le> deg R S X p"
adapt_stoch_proc_def: fixes F :: "'c \<Rightarrow> 'd measure"   and X :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and N :: "'e measure" shows "adapt_stoch_proc F X N = (\<forall>t. X t \<in> F t \<rightarrow>\<^sub>M N)"
l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs ShadowRootClass.type_wf ShadowRootClass.known_ptr"
nonempty_X: shows "(X::'b::type set) \<noteq> {}"
assms: shows "ac \<in>\<^sub>\<circ> r \<circ>\<^sub>\<bullet> s"
append_eq_append_conv: fixes xs :: "'d list"   and ys :: "'d list"   and us :: "'d list"   and vs :: "'d list" assumes "length xs = length ys \<or> length us = length vs" shows "(xs @ us = ys @ vs) = (xs = ys \<and> us = vs)"
prv_scnj_mono_imp: fixes F1 :: "'fmla set"   and F2 :: "'fmla set" assumes "F1 \<subseteq> fmla"   and "F2 \<subseteq> fmla"   and "finite F1"   and "finite F2"   and "\<forall>\<phi>2\<in>F2. \<exists>\<phi>1\<in>F1. prv (imp \<phi>1 \<phi>2)" shows "prv (imp (scnj F1) (scnj F2))"
Abs_var_inverse: fixes y :: "atom" assumes "y \<in> {a. sort_of a \<in> range sort_of_ty}" shows "Rep_var (Abs_var y) = y"
map_index'_map_zip: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd \<Rightarrow> 'c"   and xs :: "'d list" shows "map_index' n f xs = map2 f [n..<n + length xs] xs"
assms_3: fixes x :: "'a" assumes "x \<in> xs" shows "degree (f x) = 0"
Quotient3_rep_abs: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and r :: "'b" assumes "Quotient3 R Abs Rep"   and "R r r" shows "R (Rep (Abs r)) r"
pcr_cr_eq: shows "pcr_ACI_rexp (=) = cr_ACI_rexp"
f10: shows "fst (fst (get_curr_win () s1)) = fst (fst (get_curr_win () s2))"
union_of_inc: fixes P :: "'c set set \<Rightarrow> bool"   and S :: "'c set"   and Q :: "'c set \<Rightarrow> bool" assumes "P {S}"   and "Q S" shows "(P union_of Q) S"
set_less_eq_linorder: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "class.linorder (ord.set_less_eq less_eq) (ord.set_less less_eq)"
discI: fixes list :: "'d list"   and x21 :: "'d"   and x22 :: "'d list" assumes "list = x21 # x22" shows "list \<noteq> []"
fac_body_strict: fixes r :: "nat discr Box \<rightarrow> nat discr Box" shows "fac_body\<cdot>r\<cdot>\<bottom> = \<bottom>"
1: shows "e_length (e_take (gap i t) (block i t)) = Suc (gap i t - 1)"
aligned_mask_disjoint: fixes a :: "'b word"   and n :: "nat"   and b :: "'b word" assumes "is_aligned a n"   and "b \<le> mask n" shows "a AND b = 0"
rsd_winners_map: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and alts :: "'a set"   and f :: "'c \<Rightarrow> 'b"   and agents :: "'c list" shows "rsd_winners R alts (map f agents) = rsd_winners (R \<circ> f) alts agents"
null_uni_facts_1: fixes v :: "i"   and x :: "\<nu>" shows "[Null (x\<^sup>P) \<^bold>\<rightarrow> \<^bold>\<box>Null (x\<^sup>P) in v]"
supplement2_Legendre': fixes p :: "int" assumes "prime p"   and "p \<noteq> 2" shows "Legendre 2 p = (if p mod 8 = 1 \<or> p mod 8 = 7 then 1 else - 1)"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<approx> <\<nu>x>P \<parallel> Q"
Rel_OR_simps_1: fixes x0 :: "'g"   and x1 :: "'a"   and w0 :: "'h" shows "(((x0, x1), Inl w0) \<in> Rel_OR) = ((x0, w0) \<in> Rel0 \<and> x1 \<in> valid_pub1)"
satisfy_equation_def: fixes p :: "real poly"   and qs :: "real poly list"   and subset_list :: "nat list list"   and sign_list :: "rat list list" shows "satisfy_equation p qs subset_list sign_list = (M_mat sign_list subset_list *\<^sub>v w_vec p qs sign_list = v_vec p qs subset_list)"
homeomorphic_map_imp_homeomorphic_space: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphic_map X Y f" shows "X homeomorphic_space Y"
submodule_subset1: fixes H :: "'a set"   and h :: "'a" assumes "submodule R M H"   and "h \<in> H" shows "h \<in> carrier M"
edom_mono: fixes x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "x \<sqsubseteq> y" shows "edom x \<subseteq> edom y"
spanE: fixes p :: "'b"   and B :: "'b set" assumes "p \<in> ideal B" obtains A :: "'b set" and q :: "'b \<Rightarrow> 'b" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b * b)"
g_diff: shows "g C1_differentiable_on {0..1} - S"
homotopic_paths_subset: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c"   and t :: "'c set" assumes "homotopic_paths s p q"   and "s \<subseteq> t" shows "homotopic_paths t p q"
intros: fixes N :: "'a literal multiset set"   and M :: "'a literal multiset set" assumes "N - M \<subseteq> concls_of (sr_ext.inferences_from M)"   and "M - N \<subseteq> sr.Rf N" shows "sr_ext.derive M N"
append_subset_2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs = set (ys @ zs)" shows "set zs \<subseteq> set xs"
typeII_regions_integral: shows "integral s (partial_vector_derivative (\<lambda>x. F x \<bullet> j) i) = one_chain_line_integral F {j} one_chain_typeII"
hext_trans: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and h'' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "h \<unlhd> h'"   and "h' \<unlhd> h''" shows "h \<unlhd> h''"
tendsto_at_top_withinE: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and e :: "real" assumes "s \<noteq> {}"   and "(f \<longlongrightarrow> l) (at_top_within s)"   and "0 < e" obtains x0 :: "'b" where "x0 \<in> s" and "\<And>x. x \<in> {x0..} \<inter> s \<Longrightarrow> dist (f x) l < e"
extends_to_bicategory: shows "bicategory (\<bullet>) (\<star>) assoc unit src trg"
is_unit_right_imp_coprime: fixes b :: "'a"   and a :: "'a" assumes "is_unit b" shows "coprime a b"
mat_hom_one: fixes n :: "nat" shows "mat\<^sub>h (1\<^sub>m n) = 1\<^sub>m n"
infinite_imp_nonempty: fixes S :: "'a set" assumes "infinite S" shows "S \<noteq> {}"
ker_def: shows "ker = kernel G (G\<lparr>carrier := h ` carrier G\<rparr>) h"
Union_path_component: fixes S :: "'a set" shows "\<Union> {path_component_set S x |x. x \<in> S} = S"
is_process_REP_Bot: shows "is_process ({(s, X). front_tickFree s}, {d. front_tickFree d})"
assms: fixes x :: "'a" assumes "x \<in> set xs"   and "P x" shows "False"
algebra_iff_Un: fixes \<Omega> :: "'a set"   and M :: "'a set set" shows "algebra \<Omega> M = (M \<subseteq> Pow \<Omega> \<and> {} \<in> M \<and> (\<forall>a\<in>M. \<Omega> - a \<in> M) \<and> (\<forall>a\<in>M. \<forall>b\<in>M. a \<union> b \<in> M))"
inf_le2: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (inf_prod x y) y"
satisfiable_prefix_implies_\<nu>LTL: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<phi> \<in> \<nu>LTL"   and "\<nexists>i. af \<phi> (prefix i w) \<sim>\<^sub>C false\<^sub>n" shows "w \<Turnstile>\<^sub>n \<phi>"
is_lasso_prpl_pre_conv: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and prpl :: "('Q \<times> nat) list \<times> ('Q \<times> nat) list" shows "graph.is_lasso_prpl_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) prpl = (snd prpl \<noteq> [] \<and> graph.is_lasso_pre (b_graph.to_gbg_ext (degeneralize_ext ma) m) (lasso_of_prpl prpl))"
hyps: shows "\<Psi> \<rhd> P \<doteq> Q"
assms: shows "\<forall>i<n. f i \<le> 1"
is_ccompare_def: fixes uu :: "'c itself" shows "is_ccompare uu = (ID ccompare \<noteq> None)"
bezout_gcd_nat': fixes b :: "nat"   and a :: "nat" shows "\<exists>x y. b * y \<le> a * x \<and> a * x - b * y = gcd a b \<or> a * y \<le> b * x \<and> b * x - a * y = gcd a b"
height_1_iff: fixes t :: "('b \<times> bal) tree" assumes "avl t" shows "(height t = Suc 0) = (\<exists>x. t = \<langle>\<langle>\<rangle>, (x, Bal), \<langle>\<rangle>\<rangle>)"
PseudoHA_CompFun: fixes SA :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "CompFun (PseudoHA SA D) = EmptyMap (States SA)"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "u \<in> set (pawalk_verts u p)"
bet2_le2__le: fixes a :: "'p"   and P :: "'p"   and b :: "'p"   and A :: "'p"   and Q :: "'p"   and B :: "'p" assumes "Bet a P b"   and "Bet A Q B"   and "P a Le Q A"   and "P b Le Q B" shows "a b Le A B"
mirror_\<alpha>_\<rho>: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "(\<lambda>(t, k). (Inum (a # bs) t, k)) ` set (\<alpha>_\<rho> p) = (\<lambda>(t, k). (Inum (a # bs) t, k)) ` set (\<rho> (mirror p))"
hyps: fixes qs :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length ps = length qs" shows "tab(ps @ xs [\<mapsto>] qs) = tab(ps [\<mapsto>] qs, xs [\<mapsto>] [])"
Cons_1: assumes "distinct (map fst xs)" shows "distinct (map fst (map_pair f xs))"
fg_zeroTr: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set"   and z :: "'d"   and a :: "'d" assumes "fgmodule_condition R f i s A z"   and "a \<in> A" shows "\<zero> \<^sub>s\<cdot> a = z"
F_def: shows "F = {f \<in> borel_measurable M. integrable M (\<lambda>x. \<bar>f x\<bar> powr p)}"
call_no_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and rrs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m (Call chain) # rrs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
ka: shows "(\<lambda>(s, n). I (s, n) \<and> vars b \<subseteq> dom s) = I"
cobounded1: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.min a b \<preceq> a"
W_consts_2: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n true\<^sub>n"
oth_class_taut_5_k: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<equiv> \<^bold>\<not>\<phi> \<^bold>\<or> \<psi> in v]"
simps: fixes a1 :: "('a \<times> 'b) list"   and a2 :: "'a"   and a3 :: "'b" shows "srewrite_step a1 a2 a3 = ((\<exists>name rhs rest. a1 = (name, rhs) # rest \<and> a2 = name \<and> a3 = rhs) \<or> (\<exists>name name' rs rhs rhs'. a1 = (name', rhs') # rs \<and> a2 = name \<and> a3 = rhs \<and> name \<noteq> name' \<and> srewrite_step rs name rhs))"
continuous_on_eval_fds: fixes A :: "'a set"   and f :: "'a fds" assumes "A \<subseteq> {s. conv_abscissa f < ereal (s \<bullet> (1::'a))}" shows "continuous_on A (eval_fds f)"
nat_pow_of_inv: fixes a :: "'a"   and n :: "nat" assumes "a \<in> Units G" shows "inv a [^] n = inv (a [^] n)"
awalk_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "G.awalk u p v \<equiv> u \<in> verts G \<and> set p \<subseteq> arcs G \<and> G.cas u p v"
b': shows "local.prefixes (Suc (Suc k)) j \<down>= b'"
assm: shows "v=vl\<parallel>v1 \<and> v1=v2\<parallel>vr \<and> v2=vu--v3 \<and> v3=v'--vd \<and> |restrict v' f c| = 1"
results'_gpv_Pure: fixes generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat"   and gpv :: "('d, 'e, 'f) gpv"   and x :: "'d" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "x \<in> generat_pures generat" shows "x \<in> results'_gpv gpv"
r_neg2: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> carrier (R Quot I)"   and "y \<in> carrier (R Quot I)" shows "x \<oplus>\<^bsub>R Quot I\<^esub> (\<ominus>\<^bsub>R Quot I\<^esub> x \<oplus>\<^bsub>R Quot I\<^esub> y) = y"
simps_3: fixes r :: "('a \<times> 'a) set" shows "redexes (Spec r) = {Spec r}"
strict_trans1: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "a \<preceq> b"   and "b \<prec> c" shows "a \<prec> c"