hcju/isabelleps huggingface dataset:
left_adjoint_is_ide: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.is_left_adjoint f" shows "Span.ide f"
intros_9: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and t :: "state"   and rs\<^sub>2 :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1 @ rs\<^sub>2, Undecided\<rangle> \<Rightarrow> t'"
injective: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "injective_map \<alpha> S T" shows "inj_on \<alpha> S"
chart_eucl_simps_3: shows "apply_chart chart_eucl = (\<lambda>x. x)"
card_farey_fractions_asymptotics'': shows "(\<lambda>N. real (card (farey_fractions N))) \<sim>[sequentially] (\<lambda>N. 3 / pi\<^sup>2 * (real N)\<^sup>2)"
exhaust: fixes y :: "'a \<times> 'c" obtains a :: "'a" and b :: "'c" where "y = (a, b)"
equivalence_pseudofunctor_left: shows "equivalence_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F.right_map F.right_cmp"
mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "delta_mono x"   and "delta_gt d y z"   and "(0::'a) \<le> x" shows "delta_gt d (x * y) (x * z)"
lexl_not_refl: fixes r :: "('g \<times> 'g) set"   and x :: "'g list" assumes "irrefl r" shows "(x, x) \<notin> lex r"
intro: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "\<And>\<phi> \<psi> \<nu>. eq \<phi> \<psi> \<Longrightarrow> eq (af_letter \<phi> \<nu>) (af_letter \<psi> \<nu>)" shows "af_congruent_axioms eq"
extend_indets_zero_iff: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "(extend_indets p = 0) = (p = 0)"
dbl_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit0 k)"
sum_mult_def: fixes R :: "('d, 'e) Ring_scheme"   and A :: "'d set"   and B :: "'d set" shows "sum_mult R A B = {x. \<exists>n. \<exists>f\<in>{j. j \<le> n} \<rightarrow> set_mult R A B. \<Sigma>\<^sub>e R f n = x}"
ran_restrictD: fixes y :: "'d"   and m :: "'e \<Rightarrow> 'd option"   and A :: "'e set" assumes "y \<in> ran (m |` A)" shows "\<exists>x\<in>A. m x = Some y"
transfer_Ifun: fixes f :: "('c \<Rightarrow> 'd) star"   and F :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'c star"   and X :: "nat \<Rightarrow> 'c" assumes "f \<equiv> star_n F"   and "x \<equiv> star_n X" shows "f \<star> x \<equiv> star_n (\<lambda>n. F n (X n))"
div_div_div_same: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and d :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times d b"   and "dvd.dvd times b a" shows "divide (divide a d) (divide b d) = divide a b"
push_down_configuration_token_run: fixes x :: "nat"   and q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "x \<le> n \<and> y \<le> n \<and> token_run x n = token_run y n"
proves'_def: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term" shows "\<Theta>,\<Gamma> \<tturnstile> t \<equiv> wf_theory \<Theta> \<and> (\<forall>h\<in>\<Gamma>. wf_term (sig \<Theta>) h \<and> \<turnstile>\<^sub>\<tau> h : propT) \<and> \<Theta>,\<Gamma> \<turnstile> t"
two_comparisons_into_case_order_6: fixes x :: "'b"   and y :: "'b"   and Q :: "'d"   and R :: "'d"   and P :: "'d" shows "(if c1.lt x y then Q else if c1.lt y x then R else P) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
dims_tensors_from_net: fixes T :: "real tensor"   and m :: "real mat convnet" assumes "T \<in> set\<^sub>v (tensors_from_net m)" shows "dims T = input_sizes m"
finite_sorted_distinct_unique: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" shows "\<exists>!xs. set xs = A \<and> linorder.sorted less_eq xs \<and> distinct xs"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr ((F \<circ> G) f)"
lesseq_lesseq_r: fixes a :: "'a"   and b :: "'a" shows "lesseq_r a b = (a \<le> b)"
Union_iff: fixes A :: "'c"   and C :: "'c set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
non_simp_dec: fixes w :: "'b list"   and G :: "'b list set" assumes "w \<notin> \<BB> G"   and "w \<noteq> \<epsilon>"   and "w \<in> G" obtains us :: "'b list list" where "us \<in> lists G\<^sub>+" and "1 < \<^bold>|us\<^bold>|" and "concat us = w"
uint_word_arith_bintrs_6: fixes a :: "'b word" shows "uint (word_pred a) = take_bit LENGTH('b) (uint a - 1)"
spectral_radius_jnf_norm_bound_le_1: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "spectral_radius A \<le> 1"   and "\<exists>n_as. jordan_nf A n_as" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (n - 1))"
next_plane0_incr_ExcessNotAt: fixes g' :: "graph"   and p :: "nat"   and g :: "graph" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g" shows "ExcessNotAt g None \<le> ExcessNotAt g' None"
set_Bit_eq: fixes w :: "'b word"   and n :: "nat" shows "setBit w n = ring_bit_operations_class.set_bit n w"
bind_sample1: fixes p :: "'p pmf"   and f :: "'p \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
numeral_fds: fixes n :: "num" shows "numeral n = fds_const (numeral n)"
h6: shows "List.member l2 m \<longrightarrow> (\<exists>n. (l1 @ l2) ! (n + length l1) = m)"
not_imp: fixes P :: "bool"   and Q :: "bool" shows "(\<not> (P \<longrightarrow> Q)) = (P \<and> \<not> Q)"
assms: shows "SPLIT (x1, y1) (x2, y2)"
eq_norm: fixes x :: "'b" shows "Rep (Abs x) = norm x"
that_2: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D)))"
optimal_paths_good: shows "ProofStepInduct G v0 v1 paths' P_new' sep_size"
f_Max\<^sub>0_mono_1: fixes y :: "nat"   and T :: "nat \<Rightarrow> nat"   and x :: "nat"   and m :: "nat" assumes "y \<le> T x" shows "f_Max\<^sub>0 (T(x := y)) m \<le> f_Max\<^sub>0 T m"
put\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes document_ptr :: "(_) document_ptr"   and document :: "('o, 'm, 'n, 'v option) RDocument_scheme"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put document_ptr document h = h'" shows "document_ptr |\<in>| document_ptr_kinds h'"
cases: fixes a :: "'b t_type"   and X :: "'b t_type" assumes "a \<in> t_ins_set X" obtains "a = X"   | x :: "'b" and y :: "'b" and yl :: "'b bintree" and yr :: "'b bintree" and ts :: "'b bintree list" where "a = \<lparr>t_type.folding = False, item = xa__, subtrees = yl__ # Branch ya__ yl__ yr__ # ts__\<rparr>" and "\<lparr>t_type.folding = False, item = xa__, subtrees = Branch ya__ yl__ yr__ # ts__\<rparr> \<in> t_ins_set X" and "xa__ \<le> ya__"   | x :: "'b" and y :: "'b" and yl :: "'b bintree" and yr :: "'b bintree" and ts :: "'b bintree list" where "a = \<lparr>t_type.folding = False, item = xb__, subtrees = yra__ # Branch yb__ yla__ yra__ # tsa__\<rparr>" and "\<lparr>t_type.folding = False, item = xb__, subtrees = Branch yb__ yla__ yra__ # tsa__\<rparr> \<in> t_ins_set X" and "\<not> xb__ \<le> yb__"   | x :: "'b" and ts :: "'b bintree list" where "a = \<lparr>t_type.folding = True, item = xc__, subtrees = Branch xc__ Leaf Leaf # tsb__\<rparr>" and "\<lparr>t_type.folding = False, item = xc__, subtrees = Leaf # tsb__\<rparr> \<in> t_ins_set X"   | x :: "'b" and xt :: "'b bintree" and y :: "'b" and yl :: "'b bintree" and yr :: "'b bintree" and ts :: "'b bintree list" where "a = \<lparr>t_type.folding = True, item = xd__, subtrees = Branch yc__ xta__ yrb__ # tsc__\<rparr>" and "\<lparr>t_type.folding = True, item = xd__, subtrees = xta__ # Branch yc__ ylb__ yrb__ # tsc__\<rparr> \<in> t_ins_set X" and "xd__ \<le> yc__"   | x :: "'b" and xt :: "'b bintree" and y :: "'b" and yl :: "'b bintree" and yr :: "'b bintree" and ts :: "'b bintree list" where "a = \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr>" and "\<lparr>t_type.folding = True, item = x, subtrees = xt # Branch y yl yr # ts\<rparr> \<in> t_ins_set X" and "\<not> x \<le> y"
mult_scalar_not_zero: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "p \<odot> q \<noteq> 0"
fmfilter_cong: fixes m :: "('d, 'e) fmap"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q m"
intros_2: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> s[a/x]"
trim_def: shows "trim = dropWhile (\<lambda>c. c \<in> set [CHR '' '', CHR ''\<newline>'', CHR 0x09, CHR 0x0D])"
sup_aci_3: fixes x :: "'c multiset"   and y :: "'c multiset"   and z :: "'c multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
t: fixes s :: "'a" assumes "P s"   and "\<not> b s" shows "Q s"
update_by_fun_gr_raw_eq_update_by_fun_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a"   and k :: "'b"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_raw.oalist_inv rep_key_order xs" shows "oalist_raw.update_by_fun_gr_raw rep_key_order k f xs = oalist_raw.update_by_fun_raw rep_key_order k f xs"
bot_unique: fixes bot :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.order_bot bot less_eq less" shows "less_eq a bot = (a = bot)"
remove_from_disconnected_nodes_removes: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and disc_nodes' :: "(_) node_ptr list" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "h \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes'" shows "node_ptr \<notin> set disc_nodes'"
M_def: shows "M = {monom_mult (punit.lc (rep_list p) / punit.lc (rep_list g)) (punit.lt (rep_list p) - punit.lt (rep_list g)) g |g. g \<in> dgrad_sig_set' (length fs) d \<and> is_sig_red (\<prec>\<^sub>t) (=) {g} p}"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "min b (min a c) = min a (min b c)"
CMP: shows "GEN_OP cmpk cmpk' (Rv \<rightarrow> Rv \<rightarrow> comp_res_rel)"
eMGF_implies_complete: fixes e :: "expr"   and P :: "assn"   and Q :: "vassn" assumes "\<forall>Z. {} |\<turnstile>\<^sub>e MGT\<^sub>e e Z"   and "\<Turnstile>\<^sub>e {P} e {Q}" shows "{} \<turnstile>\<^sub>e {P} e {Q}"
remove_vertex_E': fixes v :: "'a"   and w :: "'a"   and x :: "'a" assumes "v \<rightarrow>\<^bsub>remove_vertex x\<^esub> w" shows "v \<rightarrow> w"
l_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_def: fixes type_wf :: "(_) heap \<Rightarrow> bool" shows "l_get\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf \<equiv> l_type_wf\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t type_wf"
is_sentence_b1Xsuffix: shows "is_sentence (b1 @ [X] @ suffix)"
cases: fixes a :: "trm" assumes "valuep a" obtains T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda>:T__. t__)"   | T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda><:T. t)"   | fs :: "(char list \<times> trm) list" where "a = Rcd fs" and "\<forall>(l, t)\<in>set fs. valuep t"
less_gexp_antisym: fixes x :: "'a gexp"   and y :: "'a gexp" shows "(x < y) = (\<not> y < x \<and> x \<noteq> y)"
c_aux_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "c_aux a b c = c \<sqinter> e_aux a b c \<squnion> d_aux a b c"
distrib: fixes F :: "'b literal list list"   and \<Gamma> :: "'b formula multiset"   and G :: "'b literal list list" assumes "image_mset disj_of_clause (mset F) + \<Gamma> \<Rightarrow>\<^sub>n"   and "image_mset disj_of_clause (mset G) + \<Gamma> \<Rightarrow>\<^sub>n" shows "mset (concat (map (\<lambda>c. map (\<lambda>d. disj_of_clause c \<^bold>\<or> disj_of_clause d) G) F)) + \<Gamma> \<Rightarrow>\<^sub>n"
cong_y1: shows "[int a * int x1 + int b = int a' * int x1 + int b'] (mod int q)"
cr_def: shows "cr \<equiv> smult (content_ff q) r"
has_vector_derivative: fixes f :: "'a \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a"   and F :: "real filter" assumes "bounded_linear f"   and "(g has_vector_derivative g') F" shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
left_tree_disjoint: fixes s :: "'a"   and t :: "'a" assumes "s \<rightarrow> t" shows "left_tree s t \<inter> left_tree t s = {}"
assms_1: shows "HPhase1or2ReadThen s s' q d p"
transp_successively_remove1: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c list"   and a :: "'c" assumes "transp f"   and "successively f l" shows "successively f (remove1 a l)"
idem: fixes a :: "'a" shows "a \<sqinter> a = a"
range_disjointed_sets': fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sigma_sets \<Omega> E" shows "range (disjointed A) \<subseteq> sigma_sets \<Omega> E"
exec_step_ind_Return_Last_Init: fixes P :: "jvm_prog"   and h :: "heap"   and stk\<^sub>0 :: "val list"   and loc\<^sub>0 :: "val list"   and C\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and sh :: "sheap" shows "exec_step_ind (StepI Return) P h stk\<^sub>0 loc\<^sub>0 C\<^sub>0 clinit pc ics [] sh (None, h, [], sh(C\<^sub>0 \<mapsto> (fst (the (sh C\<^sub>0)), Done)))"
lead_coeff_i_def': fixes x :: "'i list" shows "lead_coeff_i ops x = coeff_i ops x (degree_i x)"
assms_3: shows "Esigma n = Esigma m * (x + 1)"
finfun_fst_update: fixes f :: "('a, 'b \<times> 'c) finfun"   and a :: "'a"   and bc :: "'b \<times> 'c" shows "finfun_fst (finfun_update f a bc) = finfun_update (finfun_fst f) a (fst bc)"
lcm_ratfps_def: shows "lcm = normalization_euclidean_semiring_class.lcm"
fvt: fixes t :: "'trm" assumes "set (map snd txs) \<subseteq> var"   and "set (map fst txs) \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set (map fst txs)" shows "set (getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)) \<inter> FvarsT t = {}"
digraph_def: fixes G :: "('b, 'c) pre_digraph" shows "digraph G \<equiv> fin_digraph G \<and> loopfree_digraph G \<and> nomulti_digraph G"
mult_of_nat_commute: fixes x :: "nat"   and y :: "'a" shows "of_nat x * y = y * of_nat x"
ide_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.ide (trg\<^sub>D \<mu>)"
scaleC_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and r :: "complex"   and b :: "'b" shows "a *\<^sub>V r *\<^sub>C b = r *\<^sub>C (a *\<^sub>V b)"
mat_of_rows_rows: fixes A :: "'a mat" shows "mat_of_rows (dim_col A) (rows A) = A"
acc_limit_alt: fixes r :: "nat \<Rightarrow> 'Q" assumes "finite (range r)" shows "is_acc r = (\<forall>n<num_acc. limit r \<inter> accn n \<noteq> {})"
assms: shows "lappend xs ys \<sqsubseteq> zs"
distinct_fst_annotate_rlen: fixes l :: "'b list" shows "distinct (map fst (annotate_rlen l))"
fps_shift_cutoff: fixes n :: "nat"   and f :: "'a fps" shows "fps_shift n f * fps_X ^ n + fps_cutoff n f = f"
arr_dom_iff_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr (local.dom f) = arr f"
g_loop: fixes x :: "'v" shows "g (x, x) = 0"
assms: shows "\<Gamma> \<turnstile> Fst P : \<tau>"
rtrancl_image_idem: fixes R :: "('a \<times> 'a) set"   and s :: "'a set" shows "R\<^sup>* `` R\<^sup>* `` s = R\<^sup>* `` s"
IH_1: assumes "v \<notin> ifex_var_set t" shows "val_ifex t (a(v := any)) = val_ifex t a"
mod_less: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "a < b" shows "a mod b = a"
rel_trans: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a"   and z :: "'a \<times> 'a" assumes "x \<sim> y"   and "y \<sim> z"   and "x \<in> R \<times> S"   and "y \<in> R \<times> S"   and "z \<in> R \<times> S" shows "x \<sim> z"
irred_0: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "\<not> is_red F 0"
C0_def: shows "C0 = convex hull \<Union> (S ` I)"
between_mem_segment: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "between (a, b) x = (x \<in> closed_segment a b)"
eWlsBinp_simp2: fixes ebinp :: "'bindex \<Rightarrow> 'c withERR option"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and delta :: "'opSym" assumes "\<not> liftAll (\<lambda>eA. eA \<noteq> ERR) ebinp" shows "\<not> eWlsBinp MOD delta ebinp"
Y_small_def: shows "Y\<^sub>\<prec> = {#ys \<in># mset (chop 5 xs). median ys \<prec> M#}"
runit_ide_eq: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "\<r>[f] = \<lparr>Chn = \<p>\<^sub>1[Leg0 (Dom f), C.cod (Leg0 (Dom f))], Dom = \<lparr>Leg0 = \<p>\<^sub>0[Leg0 (Dom f), C.cod (Leg0 (Dom f))], Leg1 = Leg1 (Dom f) \<cdot> \<p>\<^sub>1[Leg0 (Dom f), C.cod (Leg0 (Dom f))]\<rparr>, Cod = Cod f\<rparr>"
negative_eq_positive: fixes n :: "nat"   and m :: "nat" shows "(- int n = int m) = (n = 0 \<and> m = 0)"
assms_4: fixes s :: "'a" assumes "s \<in> D" shows "V' s (f s) \<le> g s * V s"
proj_Eps: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "equiv A r"   and "X \<in> A // r" shows "Equiv_Relations.proj r (SOME x. x \<in> X) = X"
us: shows "us \<equiv> getFrN (map snd txs) (r # map fst txs) [] (length txs)"
const_1_sum_mod_2: assumes "const 1" shows "(f 0 + f 1) mod 2 = 0"
reachable_awalk: 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.awalk G u p v)"
collapse: fixes BinomialTree :: "('c, 'd) BinomialTree" shows "Node (val BinomialTree) (prio BinomialTree) (rank BinomialTree) (children BinomialTree) = BinomialTree"
ipresCons_imp_ipresSubst: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igSubstCls MOD"   and "igConsIPresIGWls MOD"   and "igFreshCls MOD" shows "ipresSubst h MOD"
single_add: fixes k :: "'c"   and a :: "'d"   and b :: "'d" shows "monomial (a + b) k = monomial a k + monomial b k"
map_sel: fixes f :: "'t \<Rightarrow> 'n"   and a :: "('t, 'o, 'p, 'q, 'r, 's) U" shows "D_U (mapl_U f a) = mapl_G (mapl_U f) f (D_U a)"
seq_char: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "seq g f = (arr f \<and> arr g \<and> Dom g = Cod f)"
fp_cop_F_allocation: fixes ds :: "'b set" shows "inj_on Xd (ManiPref.cop ds)"
sorted_list_of_set_Un: fixes A :: "'a set"   and B :: "'a set" assumes "A \<lless> B"   and "finite A"   and "finite B" shows "list_of (A \<union> B) = list_of A @ list_of B"
lang_lang_trad: shows "subst \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, S, \<bb>, S]|}\<rparr> {|[Inr (init \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, S, \<bb>, S]|}\<rparr>)]|} = to_language (lang_trad \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, S, \<bb>, S]|}\<rparr>)"
fempty_iff: fixes c :: "'e" shows "(c |\<in>| {||}) = False"
map_comp: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f1 :: "'j \<Rightarrow> 'g"   and f2 :: "'k \<Rightarrow> 'h"   and f3 :: "'l \<Rightarrow> 'i"   and v :: "('j, 'k, 'l) F2" shows "F2map g1 g2 g3 (F2map f1 f2 f3 v) = F2map (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
dense_repr_normalize: fixes p :: "'a list" shows "dense_repr (local.normalize p) = dense_repr p"
representsD2: fixes B :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b" assumes "represents B r x" shows "x = rep r"
k: fixes i :: "nat" shows "\<exists>g. \<Phi> g i id k"
of_int_0_le_iff: 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (ring_1.of_int one plus zero minus z) = (0 \<le> z)"
subdistl_eq: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<cdot> x + z \<cdot> (x + y) = z \<cdot> (x + y)"
small_VUnion: fixes A :: "V" shows "small {a. \<exists>x\<in>elts A. a \<in>\<^sub>\<circ> x}"
convergent_def: fixes "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd" assumes "class.topological_space open" shows "topological_space.convergent open X = (\<exists>L. filterlim X (topological_space.nhds open L) sequentially)"
is_normal_series: shows "normal_series (G\<lparr>carrier := H\<rparr>) \<HH>"
generalised_directly_subsumes_original: fixes t' :: "transition"   and t :: "transition"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and i :: "nat"   and r :: "nat"   and s' :: "nat"   and e1 :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s :: "nat" assumes "input_stored_in_reg t' t e = Some (i, r)"   and "initially_undefined_context_check (tm e) r s'"   and "no_illegal_updates t r" shows "directly_subsumes (tm e1) (tm e) s s' t' t"
simps_1: fixes x :: "char list" shows "ground\<^sub>t (\<epsilon> x) = False"
simulation_\<tau>Runs1: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tls1 :: "('h, 'g option) tllist" assumes "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "\<tau>trsys.\<tau>Runs trsys1 \<tau>move1 s1 tls1" shows "\<exists>tls2. \<tau>trsys.\<tau>Runs trsys2 \<tau>move2 s2 tls2 \<and> tllist_all2 tlsim (rel_option bisim) tls1 tls2"
segment_bound1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> closed_segment a b" shows "norm (x - a) \<le> norm (b - a)"