hcju/isabelleps huggingface dataset:
introduce_parallel_with_rely: fixes i :: "'a::refinement_lattice"   and j\<^sub>0 :: "'a::refinement_lattice"   and j\<^sub>1 :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (j\<^sub>0::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (j\<^sub>1::'a::refinement_lattice)" shows "((c::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice)) // (i::'a::refinement_lattice) \<sqsubseteq> (j\<^sub>1::'a::refinement_lattice) \<iinter> c // ((j\<^sub>0::'a::refinement_lattice) \<parallel> i) \<parallel> j\<^sub>0 \<iinter> d // (j\<^sub>1 \<parallel> i)"
ideal_pow_ideal: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and n :: "nat" assumes "Ring R"   and "ideal R I" shows "ideal R (I \<^bsup>\<diamondsuit>R n\<^esup>)"
keys_diff: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and b :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (a - b) \<subseteq> keys a \<union> keys b"
ring_record_simps_12: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
choiceSetI: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A"   and "\<And>y. y \<in> A \<Longrightarrow> x \<^bsub>r\<^esub>\<preceq> y" shows "x \<in> choiceSet A r"
assms_2: fixes x :: "Core.term" assumes "x \<in> set Hs" shows "term_ok thy x"
not_weak: fixes x :: "'b"   and y :: "'b" shows "(\<not> x \<le> y) = (y < x)"
simps_8: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b"   and f3 :: "'b"   and x1 :: "'c" shows "(case flat_complete_lattice.Value x1 of flat_complete_lattice.Value x \<Rightarrow> f1 x | flat_complete_lattice.Bot \<Rightarrow> f2 | flat_complete_lattice.Top \<Rightarrow> f3) = f1 x1"
max_add_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus (ord.max less_eq x y) z = ord.max less_eq (plus x z) (plus y z)"
hr_comp_prod_conv: fixes Ra :: "'e \<Rightarrow> 'c \<Rightarrow> assn"   and Rb :: "'f \<Rightarrow> 'd \<Rightarrow> assn"   and Ra' :: "('e \<times> 'a) set"   and Rb' :: "('f \<times> 'b) set" shows "hr_comp (Ra \<times>\<^sub>a Rb) (Ra' \<times>\<^sub>r Rb') = hr_comp Ra Ra' \<times>\<^sub>a hr_comp Rb Rb'"
simp_num_pair_l: fixes t :: "MIR.num"   and n :: "int"   and t' :: "MIR.num"   and n' :: "int" assumes "numbound0 t"   and "0 < n"   and "simp_num_pair (t, n) = (t', n')" shows "numbound0 t' \<and> 0 < n'"
remove_max_max: fixes l :: "'b list"   and m :: "'b"   and l' :: "'b list" assumes "l \<noteq> []"   and "(m, l') = remove_max l" shows "m = Max (SelectionSort_Functional.set l)"
emeasure_UN_countable: fixes I :: "'b set"   and X :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M"   and "countable I"   and "disjoint_family_on X I" shows "emeasure M (\<Union> (X ` I)) = \<integral>\<^sup>+ i. emeasure M (X i) \<partial>count_space I"
convergent_ereal: fixes X :: "nat \<Rightarrow> 'a" shows "convergent X = (limsup X = liminf X)"
removeAll_append: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "removeAll x (xs @ ys) = removeAll x xs @ removeAll x ys"
word_ao_dist: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(x OR y) AND z = x AND z OR y AND z"
Acts: fixes act :: "('b \<times> 'b) set"   and F :: "'b program"   and s :: "'b"   and s' :: "'b" assumes "act \<in> Acts F"   and "s \<in> reachable F"   and "(s, s') \<in> act" shows "s' \<in> reachable F"
decomps\<^sub>e\<^sub>s\<^sub>t_preserves_model_c: fixes D :: "('fun, 'var) extstrand_step list"   and A :: "('fun, 'var) extstrand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and M\<^sub>0 :: "('fun, 'var) Term.term set" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) \<I>"   and "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> A" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (A @ D)"
numeral_poly: fixes n :: "num" shows "numeral n = [:numeral n:]"
pmf_of_set_def: shows "local.pmf_of_set \<equiv> embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
PairE_lemma: fixes p :: "'h \<times> 'i" shows "\<exists>x y. p = (x, y)"
elims_3: fixes x :: "'v"   and xa :: "'v list" assumes "\<not> adj_path x xa" obtains v :: "'v" where "x = vb__" and "xa = []" and "vb__ \<notin> V"   | v :: "'v" and u :: "'v" and us :: "'v list" where "x = v" and "xa = u # us" and "\<not> (adjacent v u \<and> adj_path u us)"
WhileLowCompositionality: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c' :: "com"   and b :: "expr" assumes "nonInterference \<Gamma> c'"   and "\<Gamma> \<turnstile> b : Low"   and "\<Gamma>,Low \<turnstile> c'" shows "nonInterference \<Gamma> (while (b) c')"
span_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f" shows "span (f ` S) = f ` span S"
hyps_6: assumes "list_all is_InSet (unlabel A)" shows "list_all is_InSet (unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A))"
not_lnull_lappend_startE: fixes w :: "'b llist" assumes "\<not> lnull w" obtains a :: "'b" and v :: "'b llist" where "w = <a> $ v"
Sup_to_Inf_var: fixes X :: "'a set" shows "\<Squnion> X = - \<Sqinter> (uminus ` X)"
IH: fixes y :: "nat"   and t :: "tm"   and u :: "tm" assumes "y < n"   and "size t + size u < y" shows "Sigma_fm (t SUBS u) \<and> Sigma_fm (t IN u)"
rec_newleft3_def: shows "rec_newleft3 = Cn 2 rec_add [Cn 2 rec_mult [Cn 2 (constn 2) [recf.id 2 0], recf.id 2 0], Cn 2 rec_mod [recf.id 2 1, Cn 2 (constn 2) [recf.id 2 0]]]"
neg_0_equal_iff_equal: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.group_add minus plus zero uminus" shows "(zero = uminus a) = (zero = a)"
rp: shows "cnf (nnf F) \<turnstile> \<box>"
in_MD: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> M" shows "le x y"
nprt_move_matrix: fixes A :: "'a matrix"   and j :: "int"   and i :: "int" shows "nprt (move_matrix A j i) = move_matrix (nprt A) j i"
sI: shows "(\<lambda>x. generate G {x}) ` B = (\<lambda>x. generate G {x}) ` A"
dec_interp_not_Inl: fixes n :: "nat"   and FO :: "nat set"   and x :: "('a \<times> bool list) list"   and i :: "nat"   and p :: "nat" assumes "dec_interp n FO x ! i = Inl p"   and "i \<notin> FO"   and "i < n" shows "False"
poly_y_x_poly_lift: fixes p :: "'a poly" shows "poly_y_x (poly_lift p) = [:p:]"
tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_vars_subset': fixes F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "\<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ` set (tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F D)) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s D"
rplusinf_ex: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "isrlfm p"   and "Ifm (a # bs) (plusinf p)" shows "\<exists>x. Ifm (x # bs) p"
rel_optionI: fixes x :: "'a option"   and y :: "'b option"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Option.is_none x = Option.is_none y"   and "\<lbrakk>\<not> Option.is_none x; \<not> Option.is_none y\<rbrakk> \<Longrightarrow> P (the x) (the y)" shows "rel_option P x y"
alphaRes: fixes c :: "name"   and P :: "pi"   and a :: "name" assumes "c \<sharp> P" shows "<\<nu>a>P = <\<nu>c>([(a, c)] \<bullet> P)"
sum_mult_iOp_closed: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "x \<in> sum_mult R A B" shows "-\<^sub>a x \<in> sum_mult R A B"
cases: fixes a :: "'b tree"   and A :: "'b set" assumes "a \<in> trees A" obtains f :: "'b" and ts :: "'b tree list" where "a = Node f ts" and "f \<in> A" and "\<forall>t\<in>set ts. t \<in> trees A"
bij_betw_iff_bijections: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
IH: fixes \<rho> :: "complex mat" assumes "partial_density_operator \<rho> \<and> \<rho> \<in> carrier_mat d d" shows "positive (denote_while_n M0 M1 DS n \<rho>)"
wf_matchP_aux_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and I :: "\<I>"   and r :: "Formula.formula Regex.regex"   and aux :: "(nat \<times> (mregex, event_data option list set) mapping) list"   and ne :: "nat" shows "wf_matchP_aux \<sigma> V n R I r aux ne = (sorted_wrt (\<lambda>x y. fst y < fst x) aux \<and> (\<forall>t X. (t, X) \<in> set aux \<longrightarrow> ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<and> (case to_mregex r of (mr, \<phi>s) \<Rightarrow> \<forall>ms\<in>RPDs mr. qtable n (fv_regex r) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne - 1) (formula.MatchP (point (\<tau> \<sigma> (ne - 1) - t)) (from_mregex ms \<phi>s))) (lookup X ms))) \<and> (\<forall>t. ne \<noteq> 0 \<and> t \<le> \<tau> \<sigma> (ne - 1) \<and> enat (\<tau> \<sigma> (ne - 1) - t) \<le> right I \<and> (\<exists>i. \<tau> \<sigma> i = t) \<longrightarrow> (\<exists>X. (t, X) \<in> set aux)))"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n insert x A = Finite_Set.fold inf x A"
ptr_kinds_ptr_kinds_M: fixes ptr :: "'a"   and h :: "'b"   and ptr_kinds :: "'b \<Rightarrow> 'a fset" shows "(ptr \<in> set |h \<turnstile> l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds|\<^sub>r) = (ptr |\<in>| ptr_kinds h)"
sgn_finite: fixes a :: "ereal" shows "\<bar>sgn a\<bar> \<noteq> \<infinity>"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom t)"
fixp_preserves_mcont: fixes F :: "('d \<Rightarrow> 'a) \<Rightarrow> 'd \<Rightarrow> 'a"   and lub :: "'d set \<Rightarrow> 'd"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. monotone (fun_ord (\<le>)) (\<le>) (\<lambda>f. F f x)"   and "\<And>f. mcont lub ord Sup (\<le>) f \<Longrightarrow> mcont lub ord Sup (\<le>) (F f)" shows "mcont lub ord Sup (\<le>) (ccpo.fixp (fun_lub Sup) (fun_ord (\<le>)) F)"
cfs_X_plus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and n :: "nat" assumes "UP_cring R"   and "a \<in> carrier R" shows "X_poly_plus R a n = (if n = 0 then a else if n = 1 then \<one>\<^bsub>R\<^esub> else \<zero>\<^bsub>R\<^esub>)"
c: shows "properdiv_set 6 = {1, 2, 3}"
Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Not\<^sub>\<alpha> t\<^sub>\<alpha>)"
simps_2: fixes Ci :: "'a literal list \<times> nat"   and Cis :: "('a literal list \<times> nat) list" shows "remdups_clss (Ci # Cis) = (let Ci' = select_min_weight_clause Ci Cis in Ci' # remdups_clss (filter (\<lambda>(D, uu_). mset D \<noteq> mset (fst Ci')) (Ci # Cis)))"
before_in_setD2: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" assumes "x < y in xs" shows "y \<in> set xs"
abs_summable_complex_powr_iff: fixes s :: "complex" shows "summable (\<lambda>n. cmod (exp (complex_of_real (ln (real n)) * s))) = (Re s < - 1)"
independent_explicit_finite_subsets: fixes A :: "('c \<Rightarrow>\<^sub>0 'b) set" shows "phull.independent A = (\<forall>S\<subseteq>A. finite S \<longrightarrow> (\<forall>u. (\<Sum>v\<in>S. u v \<cdot> v) = 0 \<longrightarrow> (\<forall>v\<in>S. u v = (0::'b))))"
borel_measurable_Pair: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'e" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
lift_def: fixes bs :: "bool list"   and i :: "nat"   and P :: "nat fset" shows "lift bs i P = (if bs ! i then finsert 0 (upshift P) else upshift P)"
power_even_eq: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (2 * n) = power.power one times (power.power one times a n) 2"
fps_ln_0: fixes c :: "'a" shows "fps_ln c $ 0 = (0::'a)"
adjunction_data_in_bicategory_axioms: shows "adjunction_data_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
and_or_or_or_s: fixes w1 :: "word32"   and w2 :: "word32"   and w3 :: "word32"   and w4 :: "word32" assumes "get_S w1 = 0 \<and> w2 AND 128 = 0 \<and> w3 AND 128 = 0 \<and> w4 AND 128 = 0" shows "get_S (((w1 AND 4279238655 OR w2) OR w3) OR w4) = 0"
assms_3: shows "Idem\<^bsub>L\<^esub> f"
map_of_eq_None_iff: fixes xys :: "('d \<times> 'c) list"   and x :: "'d" shows "(table_of xys x = None) = (x \<notin> fst ` set xys)"
plus_1_eSuc_1: fixes q :: "enat" shows "1 + q = eSuc q"
restrict_indets_pp_zero: shows "restrict_indets_pp 0 = 0"
count_space_def: fixes \<Omega> :: "'c set" shows "count_space \<Omega> = measure_of \<Omega> (Pow \<Omega>) (\<lambda>A. if finite A then of_nat (card A) else \<infinity>)"
right_stoch_mat_def: fixes a :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "right_stoch_mat a = (\<forall>j. stoch_vec (column j a))"
map_scale_sum_distrib_right: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "sum f A \<cdot> p = (\<Sum>a\<in>A. f a \<cdot> p)"
hcomp_eqI: fixes f :: "'a"   and g :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "hseq f g"   and "f = f'"   and "g = g'" shows "f \<star> g = f' \<star>\<^sub>B g'"
f_def: shows "f = (\<lambda>x. real (nth_prime (Suc x)) / real (nth_prime x))"
zn: assumes "0 = w1 \<and> w2 = NEG_INF" shows "P w1 w2"
snth_Stream: fixes x :: "'a"   and s :: "'a stream"   and i :: "nat" shows "(x ## s) !! Suc i = s !! i"
dim_row_append_rows: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_row (A @\<^sub>r B) = dim_row A + dim_row B"
derangements_swap: fixes d :: "nat \<Rightarrow> nat"   and S :: "nat set"   and x :: "nat"   and y :: "nat" assumes "d \<in> derangements S"   and "x \<notin> S"   and "y \<notin> S"   and "x \<noteq> y" shows "Fun.swap x y d \<in> derangements (insert x (insert y S))"
init_class_fieldmap_def: 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 C :: "char list" shows "init_class_fieldmap P C \<equiv> map_of (map (\<lambda>(F, T). (F, default_val T)) (fst (snd (the (class P C)))))"
coeff_poly_from_roots: fixes A :: "'c set"   and k :: "nat"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "k \<le> card A" shows "coeff (\<Prod>x\<in>A. [:- f x, 1::'d:]) k = (- (1::'d)) ^ (card A - k) * (\<Sum>X | X \<subseteq> A \<and> card X = card A - k. prod f X)"
not_primepow_1_nat: shows "\<not> primepow 1"
full_checker_def: fixes spec0 :: "int mpoly"   and A :: "(nat, int mpoly) fmap"   and pac :: "(int mpoly, nat, nat) pac_step list" shows "full_checker spec0 A pac = normalize_poly_spec spec0 \<bind> (\<lambda>spec. remap_polys_change_all spec {} A \<bind> (\<lambda>(st, \<V>, A). if is_failed st then RETURN (st, \<V>, A) else SPEC ((\<subseteq>) (\<V> \<union> vars spec0)) \<bind> (\<lambda>\<V>. PAC_checker spec (\<V>, A) st pac)))"
simps: fixes r :: "A" shows "body\<cdot>r = A"
hom_connected_4: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "(\<nu> \<star> \<mu> \<noteq> null) = (\<nu> \<star> cod \<mu> \<noteq> null)"
residualGraph_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and f :: "nat \<times> nat \<Rightarrow> 'b" shows "residualGraph c f \<equiv> \<lambda>(u, v). if (u, v) \<in> Graph.E c then c (u, v) - f (u, v) else if (v, u) \<in> Graph.E c then f (v, u) else (0::'b)"
star_subdist_var_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.B2_algebra plus times one zero less_eq less star" shows "less_eq x (star (plus x y))"
reciprocal_def: shows "reciprocal \<equiv> map_fun rep_complex_homo abs_complex_homo reciprocal_hcoords"
supp_set_elem_finite: fixes S :: "'c set"   and m :: "'c"   and y :: "atom" assumes "finite S"   and "m \<in> S"   and "y \<in> supp m" shows "y \<in> supp S"
distinct_89: fixes x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com"   and x111 :: "('d, 'e, 'f) com"   and x112 :: "('d, 'e, 'f) com" shows "While x61 x62 \<noteq> Catch x111 x112"
LPDs_aux_def: shows "LPDs_aux = saturate (\<lambda>S. S \<union> \<Union> (LPD ` S))"
residual_blocks_ss: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>R" shows "b \<subseteq> \<V>"
Abs: fixes x :: "name"   and ss :: "lam list"   and ss' :: "lam list"   and r :: "lam"   and r' :: "lam" assumes "x \<sharp> (ss, ss')"   and "r \<rightarrow>\<^sub>s r'"   and "ss [\<rightarrow>\<^sub>s] ss'" shows "(Lam [x].r) \<degree>\<degree> ss \<rightarrow>\<^sub>s (Lam [x].r') \<degree>\<degree> ss'"
diff_right_commute: fixes M :: "'a multiset"   and N :: "'a multiset"   and Q :: "'a multiset" shows "M - N - Q = M - Q - N"
distinct_map_fst: fixes x :: "'a"   and xs :: "('a \<times> 'b) list"   and y :: "'b" assumes "x \<notin> fst ` set xs"   and "distinct (map fst xs)" shows "(x, y) \<notin> set xs"
iMinI2: fixes k :: "'a"   and I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "k \<in> I"   and "\<And>x. x \<in> I \<Longrightarrow> P x" shows "P (iMin I)"
path_component_imp_homotopic_points: fixes S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "path_component S a b" shows "homotopic_loops S (linepath a a) (linepath b b)"
pmod_degree: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "q \<noteq> []" shows "p pmod q = [] \<or> degree (p pmod q) < degree q"
simps_15: fixes e :: "'a exp"   and e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (if (e) e\<^sub>1 else e\<^sub>2) A = (\<D> e A \<and> \<D> e\<^sub>1 (A \<squnion> \<A> e) \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e))"
length_output_deep_model': fixes m :: "real mat convnet"   and rs :: "nat list" assumes "remove_weights m = deep_model'_l rs" shows "dim_vec (tensors_from_net m) = rs ! 0"
gram_schmidt_result_4: fixes ws :: "'a vec list"   and us :: "'a vec list" assumes "set ws \<subseteq> carrier_vec n"   and "distinct ws"   and "lin_indpt (set ws)"   and "us = gram_schmidt n ws" shows "length us = length ws"
assms_4: shows "v \<preceq>\<^sub>F\<^sub>I w"
cycle_free_diag_equiv: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c" shows "(\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs) = cycle_free M n"
hcomplex_of_hypreal_mult_HComplex: fixes r :: "real star"   and x :: "real star"   and y :: "real star" shows "hcomplex_of_hypreal r * HComplex x y = HComplex (r * x) (r * y)"
af\<^sub>G_subsequence_U: fixes \<phi> :: "'b ltl"   and \<psi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and n :: "nat" shows "af\<^sub>G (\<phi> U \<psi>) (prefix (Suc n) w) = af\<^sub>G (\<phi> U \<psi>) (w [1 \<rightarrow> Suc n]) and af\<^sub>G \<phi> (prefix (Suc n) w) or af\<^sub>G \<psi> (prefix (Suc n) w)"
cap_simps_2: fixes x :: "'v" shows "cap (\<langle>x\<rangle>, vertex.SINK) = (if x \<in> A \<Gamma> then weight \<Gamma> x - d_OUT (h 0) x else 0)"
isomorphic_reduced_homology_by_singleton: fixes a :: "'a"   and X :: "'a topology"   and p :: "int" assumes "a \<in> topspace X" shows "reduced_homology_group p X \<cong> relative_homology_group p X {a}"
Plus_comm: fixes r :: "'b language"   and s :: "'b language" shows "Plus r s = Plus s r"
atLeastLessThan_empty_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.preorder less_eq less" shows "(ord.atLeastLessThan less_eq less a b = {}) = (\<not> less a b)"
foldA: shows "foldr (\<lambda>x. max (length x - Suc 0)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0 = foldr (\<lambda>x. max (length x)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0 - 1"
assms_5: shows "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"
vec_of_list_append: fixes xs :: "'b list"   and ys :: "'b list" shows "vec_of_list (xs @ ys) = vec_of_list xs @\<^sub>v vec_of_list ys"
has_integral_AE: fixes \<Omega> :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'d" assumes "AE x\<in>\<Omega> in lborel. f x = g x" shows "(f has_integral x) \<Omega> = (g has_integral x) \<Omega>"
subst_fm_UnionP: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and i :: "name"   and u :: "tm" shows "(UnionP x y z)(i::=u) = UnionP (subst i u x) (subst i u y) (subst i u z)"
VSEquality_def: shows "VSEquality = opt \<circ> QE_dnf opt (\<lambda>x. qe_eq_repeat) \<circ> VSLuckiest \<circ> opt"
Inf_apply: fixes A :: "('c \<Rightarrow> 'b) set"   and x :: "'c" shows "(\<Sqinter>A) x = (\<Sqinter>f\<in>A. f x)"
is_natural_1: fixes f :: "'a" assumes "A.arr f" shows "G f \<cdot>\<^sub>B \<tau> (A.dom f) = \<tau> f"
rightreg_scalar_mult_def: fixes R :: "'a set"   and r :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Ring1 R" shows "Ring1.rightreg_scalar_mult R r f = (\<lambda>x. if x \<in> R then f (x * r) else (0::'b))"
equiv_starrel_iff: fixes x :: "nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'b" shows "(starrel `` {x} = starrel `` {y}) = ((x, y) \<in> starrel)"
berl: shows "finite_field_factorization f'' = (c'', fs'')"
sublens_iff_subscene: fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y" shows "X \<subseteq>\<^sub>L Y = \<lbrakk>X\<rbrakk>\<^sub>\<sim> \<subseteq>\<^sub>S \<lbrakk>Y\<rbrakk>\<^sub>\<sim>"
alpha_illustration_1: fixes X :: "tyvrs"   and T :: "ty"   and Y :: "tyvrs" shows "(\<forall>X<:T. Tvar X) = (\<forall>Y<:T. Tvar Y)"
create_name_igba_impl_def: shows "create_name_igba_impl \<equiv> \<lambda>x. create_name_gba_impl x \<bind> (\<lambda>xa. gba_to_idx_ext_impl (def_hashmap_size TYPE(nat)) (=) bounded_hashcode_nat (\<lambda>xb. ()) xa \<bind> (\<lambda>xb. stat_set_data_nres (gen_card (\<lambda>x. foldli (id x)) (gi_V xa)) (gen_card (\<lambda>x. foldli (id x)) (gi_V0 xb)) (igbgi_num_acc xb) \<bind> (\<lambda>xc. RETURN xb)))"
translation_right_to_left: fixes e :: "'a allegorical_term"   and G :: "('a, 'b) labeled_graph"   and f :: "(nat \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "graph_homomorphism (translation e) G f"   and "(0, x) \<in> f"   and "(1, y) \<in> f" shows "(x, y) \<in> :G:\<lbrakk>e\<rbrakk>"
ccSUP_upper2: fixes A :: "'d set"   and i :: "'d"   and u :: "'a"   and f :: "'d \<Rightarrow> 'a" assumes "countable A"   and "i \<in> A"   and "u \<le> f i" shows "u \<le> Sup (f ` A)"