hcju/isabelleps huggingface dataset:
maps_to_nE: fixes n :: "int"   and f :: "nat \<Rightarrow> int"   and k :: "nat" assumes "maps_to_n n f" shows "f k \<in> {0..n}"
block_complement_elem_iff: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and ps :: "'b set"   and bl :: "'b set" assumes "incidence_system point_set block_collection"   and "ps \<subseteq> point_set" shows "(ps \<subseteq> incidence_system.block_complement point_set bl) = (\<forall>x\<in>ps. x \<notin> bl)"
ts_okD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and ts :: "'a \<Rightarrow> ('b \<times> 'd \<Rightarrow>f nat) option"   and m :: "'c"   and t :: "'a"   and x :: "'b"   and ln :: "'d \<Rightarrow>f nat" assumes "ts_ok P ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "P t x m"
assms_2: shows "j < card {j. j < dim_col A \<and> j \<in> J}"
equiv_trans_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "equiv_trans t u = (le_trans t u \<and> le_trans u t)"
prj_in_hom_5: shows "\<guillemotleft>Prj\<^sub>1\<^sub>0 : \<mu>\<nu>\<pi>.chine \<rightarrow>\<^sub>C \<nu>.apex\<guillemotright>"
assms_4: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and U :: "'x set" assumes "(h, U) \<in> set ps" shows "h \<in> ideal F"
similar_mat_wit_pow: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit (A ^\<^sub>m k) (B ^\<^sub>m k) P Q"
karatsuba_main: fixes f :: "'b list"   and n :: "nat"   and g :: "'b list"   and m :: "nat" shows "karatsuba_main f n g m = poly_of_list f * poly_of_list g"
digraph_mapI: fixes M :: "('a \<times> 'a) pre_map" assumes "\<And>a. a \<notin> parcs G \<Longrightarrow> edge_rev M a = a"   and "\<And>a. a \<in> parcs G \<Longrightarrow> edge_rev M a \<noteq> a"   and "\<And>a. a \<in> parcs G \<Longrightarrow> edge_rev M (edge_rev M a) = a"   and "\<And>a. a \<in> parcs G \<Longrightarrow> fst (edge_rev M a) = snd a"   and "edge_succ M permutes parcs G"   and "\<And>v. \<lbrakk>v \<in> pverts G; out_arcs (with_proj G) v \<noteq> {}\<rbrakk> \<Longrightarrow> cyclic_on (edge_succ M) (out_arcs (with_proj G) v)" shows "digraph_map (with_proj G) M"
DAnMT: shows "dom (C DenyAll) \<noteq> {}"
has_prod_def: fixes f :: "nat \<Rightarrow> 'e"   and p :: "'e" shows "f has_prod p \<equiv> raw_has_prod f 0 p \<or> (\<exists>i q. p = (0::'e) \<and> f i = (0::'e) \<and> raw_has_prod f (Suc i) q)"
reachable_apath: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G" shows "(u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v) = (\<exists>p. pre_digraph.apath G u p v)"
edges_conv_Edges: fixes f :: "face" assumes "distinct (vertices f)" shows "\<E> f = Edges (vertices f) \<union> (if vertices f = [] then {} else {(last (vertices f), hd (vertices f))})"
interval_of_plus: fixes a :: "'c"   and b :: "'c" shows "interval_of (a + b) = interval_of a + interval_of b"
inject_2: fixes x21 :: "'h"   and x22 :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and x23 :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option"   and y21 :: "'h"   and y22 :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and y23 :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "(qOp x21 x22 x23 = qOp y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
sep_conj_false_right: fixes P :: "'a \<Rightarrow> bool" shows "P ** sep_false = sep_false"
function_uminus_add_l: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F" shows "function_uminus S R f \<oplus>\<^bsub>F\<^esub> f = \<zero>\<^bsub>F\<^esub>"
legal_rules_finite: fixes A :: "'a \<Rightarrow> nat option"   and Q :: "'b set" assumes "finite_alphabet A"   and "finite Q" shows "finite (finite_alphabet.legal_rules A Q)"
option_rel_def: fixes R :: "('b \<times> 'c) set" shows "\<langle>R\<rangle>option_rel \<equiv> {(Some a, Some a') |a a'. (a, a') \<in> R} \<union> {(None, None)}"
abs_arctan: fixes x :: "real" shows "\<bar>arctan x\<bar> = arctan \<bar>x\<bar>"
eval_Prim: fixes f :: "'c" assumes "C.arr f" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<rbrace> = V f"
S\<^sub>r: shows "S\<^sub>r \<subseteq> S"
fermat_poly: fixes A :: "'b poly"   and n :: "nat"   and B :: "'b poly"   and C :: "'b poly" assumes "A ^ n + B ^ n + C ^ n = 0"   and "Gcd {A, B, C} = 1"   and "A \<noteq> 0"   and "B \<noteq> 0"   and "C \<noteq> 0"   and "\<exists>p\<in>{A, B, C}. pderiv (p ^ n) \<noteq> 0" shows "n \<le> 2"
bind_has_density: fixes M :: "'c measure"   and N :: "'d measure"   and f :: "'c \<Rightarrow> 'd measure"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" assumes "sigma_finite_measure M"   and "sigma_finite_measure N"   and "space M \<noteq> {}"   and "\<And>x. x \<in> space M \<Longrightarrow> has_density (f x) N (g x)"   and "(\<lambda>(x, y). g x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "has_density (M \<bind> f) N (\<lambda>y. \<integral>\<^sup>+ x. g x y \<partial>M)"
bij_def: fixes f :: "'d \<Rightarrow> 'e" shows "bij f = (inj f \<and> surj f)"
inA: fixes x :: "nat \<times> nat"   and y :: "nat" assumes "P x y" shows "x \<in> A"
successive_dest_tail: fixes xs :: "'b list"   and zs :: "'b list"   and y :: "'b"   and z :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "xs = zs @ [y, z]"   and "successive P xs" shows "successive P (zs @ [y]) \<and> \<not> P z \<or> successive P zs \<and> \<not> P y"
gt_chop: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "is_App t"   and "chop t \<ge>\<^sub>t s" shows "t >\<^sub>t s"
WhileFReds: fixes b :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and c :: "char list exp" assumes "P \<turnstile> \<langle>b,s,b\<^sub>0\<rangle> \<rightarrow>* \<langle>false,s',b'\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s,b\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,s',b'\<rangle>"
lookup_empty: fixes k :: "'c" shows "DAList.lookup DAList.empty k = None"
compactin_sing: fixes X :: "'b topology"   and a :: "'b" shows "compactin X {a} = (a \<in> topspace X)"
zero_less_norm_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(0 < norm x) = (x \<noteq> zero)"
hd_word: fixes a :: "'b"   and ws :: "'b list" shows "a # ws = [a] \<cdot> ws"
is_iso_classI: fixes f :: "'a" assumes "B.ide f" shows "B.is_iso_class \<lbrakk>f\<rbrakk>\<^sub>B"
mult_vm_inv: fixes x :: "complex \<times> complex"   and y :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" assumes "x = y *\<^sub>v\<^sub>m A"   and "mat_det A \<noteq> 0" shows "y = x *\<^sub>v\<^sub>m mat_inv A"
mod_dist_3: fixes P :: "'b \<Rightarrow> assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> (\<exists>\<^sub>Ax. P x)) = (\<exists>x. h \<Turnstile> P x)"
card_lists_length_eq: fixes A :: "'a set"   and n :: "nat" assumes "finite A" shows "card {xs. set xs \<subseteq> A \<and> length xs = n} = card A ^ n"
mstate_ext_def: fixes mstate_i :: "nat"   and mstate_m :: "'a mformula"   and mstate_n :: "nat"   and more :: "'b" shows "\<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons mstate_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple mstate_i (Record.iso_tuple_cons Record.tuple_iso_tuple mstate_m mstate_n)) more"
select_correct_factor_int_poly_def: fixes init :: "'a"   and p :: "int poly" shows "local.select_correct_factor_int_poly init p \<equiv> let qs = factors_of_int_poly p; polys = map (\<lambda>q. (q, root_info q)) qs; ((q, ri), l, r) = local.select_correct_factor init polys in real_alg_2' ri q l r"
simps_2: fixes A :: "complex Matrix.mat"   and v :: "nat" shows "A \<otimes>\<^bsup>Suc (Suc v)\<^esup> = A \<Otimes> A \<otimes>\<^bsup>Suc v\<^esup>"
simps_1: shows "(\<not>\<^sub>b \<zero>) = \<one>"
projection_concatenation_commute: fixes l1 :: "'a list"   and l2 :: "'a list"   and X :: "'a set" shows "(l1 @ l2) \<upharpoonleft> X = l1 \<upharpoonleft> X @ l2 \<upharpoonleft> X"
PRE_D2: fixes Q :: "'a \<Rightarrow> bool"   and x :: "'a"   and S1 :: "('b \<times> 'a) set"   and S :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Q x \<and> (\<forall>y. (y, x) \<in> S1 \<longrightarrow> S x \<longrightarrow> P x y) \<longrightarrow> comp_PRE S1 Q P S x"
mset_eq_perm: fixes xs :: "'d list"   and ys :: "'d list" shows "(mset xs = mset ys) = (xs <~~> ys)"
pseudo_divmod_main_list_1_is_divmod_poly_one_main_list: fixes q :: "'a list"   and f :: "'a list"   and g :: "'a list"   and n :: "nat" shows "pseudo_divmod_main_list (1::'a) q f g n = divmod_poly_one_main_list q f g n"
sunflower_card_core_lift: fixes E :: "'b set"   and k :: "nat"   and c :: "nat"   and r :: "nat"   and F :: "'b set set"   and s :: "nat" assumes "finite E"   and "\<And>G. \<lbrakk>\<forall>A\<in>G. finite A \<and> card A \<le> k; c < card G\<rbrakk> \<Longrightarrow> \<exists>S\<subseteq>G. sunflower S \<and> card S = r"   and "\<forall>A\<in>F. A \<subseteq> E \<and> s \<le> card A \<and> card A \<le> k"   and "(card E choose s) * c < card F"   and "s \<noteq> 0"   and "r \<noteq> 0" shows "\<exists>S\<subseteq>F. sunflower S \<and> card S = r \<and> s \<le> card (\<Inter> S)"
swap\<^sub>C_def: shows "swap\<^sub>C = map_converter swap_sum swap_sum id id 1\<^sub>C"
prems_1: shows "(s, x # xs) \<in> conv ars"
assoc_simps_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "src\<^sub>D \<a>\<^sub>D[f, g, h] = src\<^sub>D h"
update_simps_13: fixes V :: "(nat, 'a) mapping"   and s :: "('b, 'a) state" shows "\<V> (\<V>_update V s) = V"
that: fixes f :: "(nat \<Rightarrow> real) \<Rightarrow> 'n"   and g :: "'n \<Rightarrow> nat \<Rightarrow> real" assumes "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g" shows "thesis"
RBT_impl_of: fixes t :: "('a, 'b) RBT.rbt" shows "RBT (impl_of t) = t"
gf_G_eq_gf_U_2: fixes x :: "'s"   and z :: "'a" assumes "convergence_G x x z" shows "gf_U x x z \<noteq> (1::'a)"
match_final: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'d"   and s1 :: "'b"   and s2 :: "'c" assumes "backward_simulation step1 step2 final1 final2 order match"   and "match i s1 s2"   and "final2 s2" shows "final1 s1"
funcsetI: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f \<in> A \<rightarrow> B"
assms_3: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F {b} \<gamma>"
app_subst_Fun: fixes S :: "nat \<Rightarrow> typ"   and t1 :: "typ"   and t2 :: "typ" shows "$ S (t1 -> t2) = $ S t1 -> $ S t2"
bij_betw_the_inv_into: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "bij_betw (the_inv_into A f) B A"
existence_ivl_cballs: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> UNIV"   and "x0 \<in> X" obtains t :: "real" and u :: "real" and L :: "real" where "\<And>y. y \<in> cball x0 u \<Longrightarrow> cball t0 t \<subseteq> (+) t0 ` existence_ivl0 y" and "\<And>s y. \<lbrakk>y \<in> cball x0 u; s \<in> cball t0 t\<rbrakk> \<Longrightarrow> flow0 y (s - t0) \<in> cball y u" and "L-lipschitz_on (cball t0 t \<times> cball x0 u) (\<lambda>(t, x). flow0 x (t - t0))" and "\<And>y. y \<in> cball x0 u \<Longrightarrow> cball y u \<subseteq> X" and "0 < t" and "0 < u"
assms_3: shows "left_forkee \<notin> RID\<^sub>G s\<^sub>1"
eint_add_cancel_fact: fixes c :: "eint"   and b :: "eint" assumes "c \<noteq> \<infinity>" shows "c + (b - c) = b"
interval_integral_FTC_finite: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and F :: "real \<Rightarrow> 'a" assumes "continuous_on {min a b..max a b} f"   and "\<And>x. \<lbrakk>min a b \<le> x; x \<le> max a b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {min a b..max a b})" shows "interval_lebesgue_integral lborel (ereal a) (ereal b) f = F b - F a"
orthog_proj_unique: fixes M :: "'b set"   and h :: "'b"   and r :: "'b"   and s :: "'b" assumes "closed_csubspace M"   and "h - r \<in> orthogonal_complement M \<and> r \<in> M"   and "h - s \<in> orthogonal_complement M \<and> s \<in> M" shows "r = s"
is_strict_def: fixes p :: "'h \<Rightarrow> ('i \<Rightarrow> 'j option) \<Rightarrow> 'g \<Rightarrow> 'd decision option" shows "is_strict p = (\<lambda>r d. \<forall>P1. r o\<^sub>f (p P1 \<emptyset> \<circ> d) = \<emptyset>)"
isoI: fixes f :: "'a \<times> 'a" assumes "local.iso (fst f)"   and "local.iso (snd f)" shows "CC.iso f"
shadow_root_ptr_kinds_commutes: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap" shows "(cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr |\<in>| object_ptr_kinds h) = (shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
record_simps_12: fixes g :: "'h \<Rightarrow> 'f"   and f :: "'i \<Rightarrow> 'g"   and x :: "('h, 'i) test1" shows "update_field_t11 g (update_field_t12 f x) = update_field_t12 f (update_field_t11 g x)"
Im_divide: fixes x :: "complex"   and y :: "complex" shows "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
inducts: fixes x1 :: "'a list"   and x2 :: "('b, 'c) sequent"   and P :: "'a list \<Rightarrow> ('b, 'c) sequent \<Rightarrow> bool" assumes "(x1, x2) \<in> Ax"   and "\<And>i. P [] ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>)"   and "P [] ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)" shows "P x1 x2"
abs_def: fixes x :: "int" shows "Preliminaries.abs x \<equiv> if 0 \<le> x then x else 0 - x"
obtain_shortest_path: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a"   and u :: "'a" obtains p :: "enat list" where "WGraph.path w s p u" and "WGraph.\<delta> w s u = sum_list p"
fcomp_apply: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'd"   and x :: "'e" shows "fcomp f g x = g (f x)"
arc_join: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "arc g1"   and "arc g2"   and "pathfinish g1 = pathstart g2"   and "path_image g1 \<inter> path_image g2 \<subseteq> {pathstart g2}" shows "arc (g1 +++ g2)"
in_homE: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" obtains "D.arr f" and "D.dom f = a" and "D.cod f = b"
lang_def: fixes m :: "'b" shows "Automata.gba.lang (to_gba_ext m) \<equiv> Collect (Automata.gba.accept (to_gba_ext m))"
pmf_bind_spmf_None: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "pmf (p \<bind> f) None = pmf p None + (LINT x|measure_spmf p. pmf (f x) None)"
select_convs_4: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "S \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = S"
power_diff_1_eq: fixes x :: "'a"   and n :: "nat" shows "x ^ n - (1::'a) = (x - (1::'a)) * sum ((^) x) {..<n}"
ennreal_numeral_less_top: fixes i :: "num" shows "numeral i < top"
le_norm_dfp: shows "r / (r - cmod (p - a)) \<le> cmod (deriv f p)"
simps_7: fixes r :: "'a pfm"   and p :: "'a pfm" shows "static ([r]\<^sub>! p) = False"
add1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
cet_add_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : REAL" shows "\<Gamma> \<turnstile>\<^sub>c e1 +\<^sub>c e2 : REAL"
min_mult_distrib_left: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "p * min x y = (if (0::'a) \<le> p then min (p * x) (p * y) else max (p * x) (p * y))"
simps_4: fixes v :: "tree"   and va :: "tree" shows "branch [] (Branching v va) = False"
glbound_in_of_less_eq2: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "glbound_in_of P x y b" shows "y \<succeq> b"
FvarsT_Var: fixes x :: "'var" assumes "x \<in> var" shows "FvarsT (Var x) = {x}"
single_valued_inter2: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (S \<inter> R)"
compact_imp_complete: fixes s :: "'a set" assumes "compact s" shows "complete s"
gcd_diff1: fixes m :: "'a"   and n :: "'a" shows "gcd (m - n) n = gcd m n"
cl_f_list_map: fixes cl_f_list :: "(cl \<times> char list) list" shows "map (case_fd (\<lambda>cl f. f)) (map (\<lambda>(x, y). fd_def x y) cl_f_list) = map (\<lambda>(cl_XXX, f_XXX). f_XXX) cl_f_list"
rep_eq: fixes x :: "nat \<Rightarrow> bool" shows "Rep_uint16 (set_bits x) = set_bits x"
antisymmetric_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "antisymmetric B (\<sqsubseteq>)"
check_sat: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<not> \<U> (check s)" shows "\<Turnstile> (check s)"
Bex_set: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
assms_2: shows "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'"
L2_set_insert: fixes F :: "'a set"   and a :: "'a"   and f :: "'a \<Rightarrow> real" assumes "finite F"   and "a \<notin> F" shows "L2_set f (insert a F) = sqrt ((f a)\<^sup>2 + (L2_set f F)\<^sup>2)"
shr_conv: fixes ls :: "'b"   and ts :: "'c"   and m :: "'a"   and ws :: "'d"   and "is" :: "'e" shows "shr (ls, (ts, m), ws, is) = m"
SimInit: shows "Schema FIG.Init \<preceq>Sim Schema DFIG.Init"
fresh: shows "fresh \<notin> params p \<union> \<Union> (params ` set z') \<union> {c}"
epath_stl_smap_fst: fixes stepSs :: "(('sequent \<times> 'rule) \<times> 'a) stream" assumes "epath (smap fst stepSs)" shows "epath (smap fst (stl stepSs))"
sup_a_def: fixes a :: "'a"   and b :: "'a" shows "sup_a a b = (a l\<rightarrow> b) r\<rightarrow> b"
assms_1: shows "\<forall>i. f i \<in> A \<and> P (f i) (f (Suc i))"
dependent_biggerset_general: fixes S :: "('a, 'b) vec set" assumes "finite S \<Longrightarrow> vec.dim S < card S" shows "vec.dependent S"
sum_simp: fixes x :: "'e"   and A :: "'e set"   and R :: "('e, 'f) partial_object_scheme" assumes "x \<in> A"   and "A \<subseteq> carrier R" shows "x \<in> carrier R"
xor_inverse: fixes x :: "'b" shows "x \<oplus> x = \<bottom>"
l_get_dom_component_get_elements_by_class_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_elements_by_class_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
inorder_del: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c" assumes "t \<in> T h"   and "sorted1 (inorder t)" shows "inorder (del x t) = del_list x (inorder t)"
ideal_prod_one: fixes I :: "'a set set" assumes "ideal I (R Quot I)" shows "I \<cdot>\<^bsub>R Quot I\<^esub> carrier (R Quot I) = I"
af\<^sub>G_U: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and x :: "'a set"   and xs :: "'a set list" shows "af\<^sub>G (\<phi> U \<psi>) (x # xs) = af\<^sub>G (\<phi> U \<psi>) xs and af\<^sub>G \<phi> (x # xs) or af\<^sub>G \<psi> (x # xs)"
simps_1: fixes N :: "nat" shows "freediscrim (NONCE N) = 0"
SC_soundness: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Turnstile> \<Gamma> \<Rightarrow> \<Delta>"
a_all_ptrs_in_heap_def: fixes h :: "(_) heap" shows "local.a_all_ptrs_in_heap h = (\<forall>host shadow_root_ptr. h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h)"
swap_simp: fixes x :: "'c"   and y :: "'b" shows "prod.swap (x, y) = (y, x)"
trans: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm b a"   and "lex_pm c b" shows "lex_pm c a"
map_eq_Cons_conv: fixes f :: "'l \<Rightarrow> 'k"   and xs :: "'l list"   and y :: "'k"   and ys :: "'k list" shows "(map f xs = y # ys) = (\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys)"
v0_VV: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player" assumes "vm_path G P v0" shows "v0 \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<or> v0 \<in> (if p** = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"
inverse_transformations_axioms_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and \<tau>' :: "'d \<Rightarrow> 'e" shows "inverse_transformations_axioms A B \<tau> \<tau>' \<equiv> \<forall>a. partial_magma.ide A a \<longrightarrow> category.inverse_arrows B (\<tau> a) (\<tau>' a)"
weaken_local: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "other P I \<sigma> \<sigma>'"   and "\<And>\<xi> \<xi>'. P \<xi> \<xi>' \<Longrightarrow> Q \<xi> \<xi>'" shows "other Q I \<sigma> \<sigma>'"
zero_hom_0_axioms: shows "zero_hom_0 of_int"
multiset_induct': fixes P :: "'a multiset \<Rightarrow> bool"   and M :: "'a multiset" assumes "P {#}"   and "\<And>M x. P M \<Longrightarrow> P ({#x#} + M)" shows "P M"
i2_1: shows "flow0 x t2 = {a -- b}\<^bsub>i2\<^esub>"
aval_reg_some_superset: fixes r :: "nat \<Rightarrow>f value option"   and r' :: "nat \<Rightarrow>f value option"   and a :: "vname aexp"   and i :: "value list"   and v :: "value" assumes "\<forall>a. r $ a \<noteq> None \<longrightarrow> r $ a = r' $ a"   and "aval a (join_ir i r) = Some v" shows "aval a (join_ir i r') = Some v"
isolated_singularity_at_times: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and g :: "complex \<Rightarrow> complex" assumes "isolated_singularity_at f z"   and "isolated_singularity_at g z" shows "isolated_singularity_at (\<lambda>w. f w * g w) z"
ceiling_neg_numeral: fixes v :: "num" shows "\<lceil>- numeral v\<rceil> = - numeral v"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "partial_magma.ide C a" shows "yoneda_functor_fixed_object_axioms C a"