hcju/isabelleps huggingface dataset:
step_2_main_inv: fixes A :: "'a mat"   and j :: "nat" assumes "A \<in> carrier_mat n n"   and "local.inv_all local.uppert A"   and "local.inv_all local.diff_ev A"   and "local.ev_blocks_part j A" shows "local.inv_all local.uppert (step_2_main n j A) \<and> local.inv_all local.diff_ev (step_2_main n j A) \<and> local.ev_blocks (step_2_main n j A)"
strict_implies_order: fixes a :: "'a word"   and b :: "'a word" assumes "a <s b" shows "a \<le>s b"
hom_compose: fixes h :: "'a \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme"   and i :: "'d \<Rightarrow> 'f"   and I :: "('f, 'g) monoid_scheme" assumes "h \<in> hom G H"   and "i \<in> hom H I" shows "compose (carrier G) i h \<in> hom G I"
3: shows "\<forall>x\<in>(\<lambda>f. Some (match_no_prereq f p)) ` m. x \<noteq> None"
Quotient_ltln\<^sub>C: shows "Quotient (\<sim>\<^sub>C) abs_ltln\<^sub>C rep_ltln\<^sub>C cr_ltln\<^sub>C"
continuous_imp_square_integrable: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "continuous_on {a..b} f" shows "f square_integrable {a..b}"
less_iff_diff_less_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less a b = less (minus a b) zero"
size_2: fixes x2 :: "nat" shows "size_tv (Indet x2) = 0"
\<Phi>_simp: fixes yx :: "'d \<times> 'c" assumes "\<Phi>\<Psi>.DopxC.ide yx" shows "\<Phi> yx = \<Phi>\<Psi>.S.mkArr (\<Phi>\<Psi>.HomC.set (F (fst yx), snd yx)) (\<Phi>\<Psi>.HomD.set (fst yx, G (snd yx))) (inD \<circ> \<phi> (fst yx) \<circ> \<Phi>\<Psi>.\<psi>C (F (fst yx), snd yx))"
update_def: fixes f :: "('c \<Rightarrow> 'd list) \<times> (('d list \<Rightarrow> 'd list) \<Rightarrow> 'c \<Rightarrow> 'c)"   and upd :: "'d \<Rightarrow> 'd" shows "update f upd \<equiv> Circus_Syntax.list_update (snd f) upd"
alg_mult_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and x :: "'c" assumes "ring R" shows "ring.alg_mult R p x = (if p = [] then 0 else if x \<in> carrier R then GREATEST n. [\<one>\<^bsub>R\<^esub>, \<ominus>\<^bsub>R\<^esub> x] [^]\<^bsub>poly_ring R\<^esub> n pdivides\<^bsub>R\<^esub> p else 0)"
construct_lhs_vector_cleaner: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list" assumes "p \<noteq> 0" shows "w_vec p qs signs = vec_of_list (map (\<lambda>s. rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))) signs)"
wf1: shows "mat (row_length A1) (length A1) A1"
l11_4_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and A' :: "'a"   and C' :: "'a"   and D' :: "'a"   and F' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.Out Bet B A' A"   and "Tarski_neutral_dimensionless.Out Bet B C' C"   and "Tarski_neutral_dimensionless.Out Bet E D' D"   and "Tarski_neutral_dimensionless.Out Bet E F' F"   and "Cong B A' E D'"   and "Cong B C' E F'" shows "Cong A' C' D' F'"
a: shows "\<I> k \<subseteq> \<J> (Suc k) 0"
unapp_tm_def: fixes t :: "('c, 'd) tm" shows "unapp t = (case t of Hd x \<Rightarrow> Map.empty x | tm.App t u \<Rightarrow> Some (t, u))"
dist_vs2: shows "distinct (ram2 # vs2 @ [ram1])"
strong_induct: fixes P :: "'a \<Rightarrow> fm \<Rightarrow> bool"   and c :: "'a"   and fm :: "fm" assumes "\<And>x1 x2 c. P c (x1 IN x2)"   and "\<And>x1 x2 c. P c (x1 EQ x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (x1 OR x2)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Neg x)"   and "\<And>x1 x2 c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (SyntaxN.Ex x1 x2)" shows "P c fm"
index_unit_vec_3: fixes n :: "nat"   and i :: "nat" shows "dim_vec (unit_vec n i) = n"
d_def: shows "d \<equiv> case bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j) of (p, q, u, v, d) \<Rightarrow> d"
simple_packet_set_toSet_alt: fixes ms :: "'a simple_match list" shows "simple_packet_set_toSet ms = (\<Union>m\<in>set ms. {p. simple_matches m p})"
restrict_apply: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'b" shows "restrict f A x = (if x \<in> A then f x else undefined)"
exhaust_sel: fixes kripke :: "('b, 'c) kripke" obtains "kripke = Kripke (\<pi> kripke) (\<K> kripke)"
HPT_Delta_target: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "t \<in> TS"   and "target t \<in> States A"   and "A \<in> SAs (HA ST)" shows "t \<in> Delta A"
ms1: fixes e :: "lam"   and Es :: "ctx list" shows "<e,Es> \<mapsto>* <e,Es>"
scale_right_distrib: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c"   and y :: "'c" assumes "module scale" shows "scale a (x + y) = scale a x + scale a y"
assms_1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> I"   and "b \<in> I"   and "x < a"   and "a \<le> b" shows "f a \<le> f b"
phl_while: fixes p :: "'a"   and s :: "'a"   and w :: "'a"   and x :: "'a" assumes "\<lparr>p\<rparr>s\<lparr>s \<cdot> p \<cdot> s\<rparr>"   and "\<lparr>p\<rparr>w\<lparr>w \<cdot> p \<cdot> w\<rparr>"   and "\<lparr>p \<cdot> s\<rparr>x\<lparr>x \<cdot> p\<rparr>" shows "\<lparr>p\<rparr>(s \<cdot> x)\<^sup>\<infinity> \<cdot> w\<lparr>(s \<cdot> x)\<^sup>\<infinity> \<cdot> w \<cdot> (p \<cdot> w)\<rparr>"
composable: shows "src (cod \<nu>) = trg (cod \<pi>)"
LLs_LLq: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> atrm"   and "t2 \<in> atrm" shows "LLs t1 t2 = cnj (LLq t1 t2) (neg (eql t1 t2))"
E3: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> F \<longrightarrow> \<diamond>F"
games_option_stable: fixes games :: "ZF set"   and g :: "ZF"   and opt :: "ZF" assumes "games = fixgames games"   and "g \<in> games"   and "left_option g opt \<or> right_option g opt" shows "opt \<in> games"
2: shows "\<forall>v\<in>set_of_vector X. g v = (0::'a)"
dsg1: fixes x :: "'a" shows "domain_bdlo_monoid.d (1::'a) (\<cdot>) (\<sqinter>) \<top> x \<cdot> x = x"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'c llist" assumes "enat m < n" shows "ltake n xs $ m = xs $ m"
set_category_axioms: shows "set_category (\<cdot>\<acute>) \<AA>"
simps_2: fixes f :: "('fsym, 'tp) efsym"   and Tl :: "('fsym, 'tp) efsym trm list" shows "GE.tpOf (Fn f Tl) = GE_resOf f"
in_hom_def: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "CC.in_hom f a b \<equiv> a \<in> CC.domains f \<and> b \<in> CC.codomains f"
xfer_push_bounds: fixes f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat"   and B1 :: "nat"   and B2 :: "nat"   and p :: "pr_operation list" assumes "\<forall>p'. ((f, l), p', f', l') \<in> trcl pr_algo_lts' \<longrightarrow> length (filter is_SAT_PUSH' p') \<le> B1"   and "\<forall>p'. ((f, l), p', f', l') \<in> trcl pr_algo_lts' \<longrightarrow> length (filter is_NONSAT_PUSH' p') \<le> B2"   and "((f, l), p, f', l') \<in> trcl pr_algo_lts" shows "length (filter is_PUSH p) \<le> B1 + B2"
that: fixes m :: "nat" assumes "F \<subseteq> dgrad_set d m" shows "thesis"
induct: fixes P :: "'a poly \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a poly"   and a1 :: "nat" assumes "\<And>p k. \<lbrakk>\<And>x. \<lbrakk>k \<noteq> 0; x = root_unity k; x dvd p; p \<noteq> 0\<rbrakk> \<Longrightarrow> P (p div x) k; \<And>x. \<lbrakk>k \<noteq> 0; x = root_unity k; \<not> x dvd p\<rbrakk> \<Longrightarrow> P p (k - 1)\<rbrakk> \<Longrightarrow> P p k" shows "P a0 a1"
convex_rel_interior_closure: fixes S :: "'a set" assumes "convex S" shows "rel_interior (closure S) = rel_interior S"
image_eq_to_f: fixes f1 :: "'e \<Rightarrow> 'd"   and S1 :: "'e set"   and f2 :: "'f \<Rightarrow> 'd"   and S2 :: "'f set" assumes "f1 ` S1 = f2 ` S2" obtains f :: "'f \<Rightarrow> 'e" where "\<And>x. x \<in> S2 \<Longrightarrow> f x \<in> S1 \<and> f1 (f x) = f2 x"
less_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<prec> y"   and "y \<prec> z" shows "x \<prec> z"
times_iff: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V" shows "(\<langle>a, b\<rangle> \<in> elts (vtimes A B)) = (a \<in> elts A \<and> b \<in> elts B)"
reach_impl_access_eq: shows "\<forall>l. s1\<turnstile> l reachable_from x \<longrightarrow> s1@@l = s2@@l"
init_prefixE: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat"   and g :: "nat \<Rightarrow> nat option" assumes "prefix f n = prefix g n" shows "f \<triangleright> n = g \<triangleright> n"
split_intersecting_def: fixes X :: "'b set"   and Y :: "'b set" shows "split_intersecting X Y = SPEC (\<lambda>(R, S). X = R \<union> S \<and> X \<inter> Y \<subseteq> R \<and> S \<inter> Y = {})"
left_right_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and l :: "'c"   and r :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.left unapp (app l r) = l"
bit_uint_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (uint w) n = (n < LENGTH('b) \<and> bit w n)"
strict_reflclp: fixes x :: "'b"   and A :: "'b set"   and y :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<in> A"   and "y \<in> A"   and "transp_on P A"   and "irreflp_on P A" shows "strict P\<^sup>=\<^sup>= x y = P x y"
P'eqP'': shows "P' = P''[y::=u]"
return_spmf_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "(A ===> rel_spmf A) return_spmf return_spmf"
abs_eq: fixes x :: "'a" shows "- dual x = dual (- x)"
Q_def: shows "Q = wQ_of_wstate (llast Sts)"
expected_utility: fixes u :: "'alt \<Rightarrow> real" shows "measure_pmf.expectation q u = measure_pmf.expectation p u + \<epsilon> * (measure_pmf.expectation q' u - measure_pmf.expectation p' u)"
regular_event: fixes ev :: "('d, 'e, 'f) event" shows "regular_event ev \<equiv> isTrans ev \<or> isSend ev \<or> isRecv ev"
sorted_iff_nth_mono: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! i) (xs ! j))"
snd_fst_flip: fixes xy :: "'b \<times> 'a" shows "snd xy = (fst \<circ> (\<lambda>(x, y). (y, x))) xy"
eventually_principal_subs: fixes P :: "'a set \<Rightarrow> bool"   and As :: "'a set set"   and A :: "'a set" shows "eventually P (principal_subs As A) = (\<forall>x. x \<in> As \<and> x \<subseteq> A \<longrightarrow> P x)"
rel_pmf_antisym: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'c pmf" assumes "rel_pmf R p q"   and "rel_pmf R q p"   and "reflp R"   and "transp R"   and "antisymp R" shows "p = q"
funion_finsert_right: fixes A :: "'c fset"   and a :: "'c"   and B :: "'c fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
trans_not_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b \<noteq> Less"   and "cmp b c \<noteq> Less" shows "cmp a c \<noteq> Less"
assms_3: shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
shadow_root_none: shows "h2 \<turnstile> get_shadow_root new_element_ptr \<rightarrow>\<^sub>r None"
assms: shows "s' = snd (fst (sub_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
assms_2: shows "enc (w, I) = enc (w', I')"
lex_def: fixes r :: "('b \<times> 'b) set" shows "lex r = \<Union> (range (lexn r))"
rep_eq: fixes x :: "'a fpxs" shows "fpxs_nth (- x) = (\<lambda>xa. - fpxs_nth x xa)"
AA26: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and B :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "'b \<Rightarrow> 'd" assumes "|~ \<langle>A\<rangle>_v = \<langle>B\<rangle>_w" shows "\<turnstile> \<diamond>\<langle>A\<rangle>_v = \<diamond>\<langle>B\<rangle>_w"
reorder_enttI: fixes A :: "assn"   and C :: "assn"   and B :: "assn"   and D :: "assn" assumes "A * true = C * true"   and "B * true = D * true" shows "A \<Longrightarrow>\<^sub>t B \<equiv> C \<Longrightarrow>\<^sub>t D"
powerset2up_induct: fixes C :: "'a set set"   and P :: "'a set \<Rightarrow> bool" assumes "finite C"   and "C \<noteq> {}"   and "\<And>M. M \<in> C \<Longrightarrow> P M"   and "\<And>M N. \<lbrakk>P M; P N\<rbrakk> \<Longrightarrow> P (M \<union> N)" shows "P (\<Union> C)"
INV_rule_from_inv_rule: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set" assumes "init T \<subseteq> I"   and "{I \<inter> reach T} TS.trans T {> I}" shows "reach T \<subseteq> I"
A2: shows "z \<le> n \<longrightarrow> c_nth (c_f_list f n x) (n - z) = f z x"
integrable_abs: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f" shows "integrable M (\<lambda>x. \<bar>f x\<bar>)"
ipath_sim2: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd"   and r :: "nat \<Rightarrow> 'c" assumes "g_rename_precond G f"   and "ipath (g_E (fr_rename_ext ecnv f G)) r" shows "ipath (g_E G) (g_rename_precond.fi G f \<circ> r)"
wf_interp_for_formula_any_Inl: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and p :: "nat"   and I :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_interp_for_formula \<Sigma> (w, Inl p # I) \<phi>" shows "\<forall>p. formula.wf_interp_for_formula \<Sigma> (w, Inl p # I) \<phi>"
Compl_subset_Compl_iff: fixes A :: "'c set"   and B :: "'c set" shows "(- A \<subseteq> - B) = (B \<subseteq> A)"
Suc: shows "below m \<times> below (2 * n) \<in> tiling domino"
v: shows "vertices (map_graph (R O on_graph B f) A) \<subseteq> vertices (map_graph_fn B f)"
simps: fixes w :: "'a list" shows "Lyndon w = (w \<noteq> \<epsilon> \<and> (\<forall>n. 0 < n \<and> n < \<^bold>|w\<^bold>| \<longrightarrow> w <lex rotate n w))"
the_\<nu>_props_1: shows "\<guillemotleft>w'.the_\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright>"
ca_of_c_of_aj_of: fixes A :: "'a" shows "CA_of A = C_of A + poss (Aj_of A)"
assms: fixes i :: "'c" assumes "i \<in> S" shows "f i \<in> borel_measurable M"
alphaAbs_qAbs_iff_alphaAbs_ex_distinct_qAFresh: fixes xs :: "'h"   and x :: "'i"   and X :: "('f, 'g, 'h, 'i, 'j) qTerm"   and xs' :: "'h"   and x' :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_distinct_qAFresh xs x X xs' x' X'"
dim_eliminate_entries_gen_2: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and v :: "nat \<Rightarrow> 'c"   and B :: "'c mat"   and i :: "nat"   and as :: "nat" shows "dim_col (eliminate_entries_gen minus times v B i as) = dim_col B"
obj_terminalI: fixes t :: "V"   and \<CC> :: "V" assumes "t \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<And>a. a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<Longrightarrow> Ex1 (is_arr \<CC> a t)" shows "obj_terminal \<CC> t"
count_add_mset: fixes b :: "'b"   and A :: "'b multiset"   and a :: "'b" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
succs_Throw: fixes \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" shows "succs Throw \<tau> pc = []"
steps_reaches1: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "Graph_Defs.steps E (x # xs @ [y])" shows "E\<^sup>+\<^sup>+ x y"
cong_mult_rcancel_nat: fixes k :: "nat"   and m :: "nat"   and a :: "nat"   and b :: "nat" assumes "coprime k m" shows "[a * k = b * k] (mod m) = [a = b] (mod m)"
correctness: fixes m0 :: "'grp"   and \<eta> :: "nat"   and m1 :: "'grp"   and \<sigma> :: "bool" assumes "m0 \<in> carrier (\<G> \<eta>)"   and "m1 \<in> carrier (\<G> \<eta>)" shows "sim_def.correctness \<eta> (m0, m1) \<sigma>"
lengthClosed: fixes p :: "(name \<times> name) list"   and xs :: "'d list" shows "length (p \<bullet> xs) = length xs"
I: fixes n :: "nat" assumes "N \<le> n" shows "1 - alpha < PS.prob {x \<in> space M. \<bar>(g x + birkhoff_sum f n x) / B n\<bar> < (L - 1) * a}"
set_tl_tss': shows "set (linearize (tl_queue tss')) \<subseteq> set (linearize tss)"
rel_eq: shows "rel = foldr (\<union>) (concat (map (\<lambda>(t, rel). if left I \<le> \<tau> \<sigma> ne - t then [lookup rel mr] else []) aux')) {}"
bidirected_subgraph_obtain: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G"   and "arcs H \<noteq> arcs G"   and "finite (arcs G)"   and "\<exists>rev. bidirected_digraph G rev"   and "\<exists>rev. bidirected_digraph H rev" obtains a :: "'d" and a' :: "'d" where "{a, a'} \<subseteq> arcs G - arcs H" and "a' \<noteq> a" and "tail G a' = head G a" and "head G a' = tail G a"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "bicategory_of_spans V H \<a> \<i> src trg" shows "maps_category V H \<a> \<i> src trg"
that: fixes c :: "'b"   and t :: "'a" assumes "c \<noteq> (0::'b)"   and "p = monomial c t" shows "thesis"
summable_imp_sums_bounded: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f" shows "bounded (range (\<lambda>n. sum f {..<n}))"
1: shows "\<forall>ys. length ys < length ps \<longrightarrow> real (time (closest_pair_rec_tm ys)) \<le> closest_pair_recurrence (length ys)"
abs_eq: fixes x :: "int" shows "signed (Word.Word x) = of_int (signed_take_bit (LENGTH('c) - Suc 0) x)"
emeasure_step: fixes K :: "'a measure"   and Aa :: "'a set"   and As :: "'a set list"   and L :: "'a \<Rightarrow> 'a stream measure" assumes "K \<in> space (prob_algebra M)"   and "Aa \<in> sets M"   and "As \<in> lists (sets M)"   and "L \<in> M \<rightarrow>\<^sub>M prob_algebra (stream_space M)" shows "emeasure (K \<bind> (\<lambda>y. L y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>)))) (scylinder (space M) (Aa # As)) = \<integral>\<^sup>+y\<in>Aa. emeasure (L y) (scylinder (space M) As)\<partial>K"
transform2: fixes P :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'b state \<times> 'b state" assumes "P = Q" shows "(\<tau> \<Turnstile> P) = (\<tau> \<Turnstile> Q)"
P_true: assumes "\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. P (\<lambda>_. x) \<tau> = true \<tau>" shows "oid_of (x \<tau>) \<notin> oid_of ` \<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>"
proper_induct: fixes c :: "('a, 'b, 'c) cmd"   and phi :: "('a, 'b, 'c) cmd \<Rightarrow> bool" assumes "proper c"   and "phi Done"   and "\<And>atm. phi (Atm atm)"   and "\<And>c1 c2. \<lbrakk>phi c1; phi c2\<rbrakk> \<Longrightarrow> phi (c1 ;; c2)"   and "\<And>tst c. phi c \<Longrightarrow> phi (While tst c)"   and "\<And>ch c1 c2. \<lbrakk>phi c1; phi c2\<rbrakk> \<Longrightarrow> phi (Ch ch c1 c2)"   and "\<And>cl. \<lbrakk>properL cl; \<And>c. c \<in> set cl \<Longrightarrow> phi c\<rbrakk> \<Longrightarrow> phi (Par cl)"   and "\<And>cl. \<lbrakk>properL cl; \<And>c. c \<in> set cl \<Longrightarrow> phi c\<rbrakk> \<Longrightarrow> phi (ParT cl)" shows "phi c"
Sup_eq_RESE: fixes A :: "'b nres set"   and B :: "'b set" assumes "Sup A = RES B" obtains C :: "'b set set" where "A = RES ` C" and "B = \<Union> C"
norm_dot: fixes v :: "(nat \<Rightarrow> real) \<times> nat" shows "\<parallel>v\<parallel> = sqrt (v \<cdot> v)"
one_mod_2_pow_eq: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "modulo one (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = zero_neq_one.of_bool one zero (0 < n)"
subst_fv_dom_img_single_2: fixes v :: "'a"   and t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<notin> fv t"   and "\<sigma> v = t"   and "\<And>w. v \<noteq> w \<Longrightarrow> \<sigma> w = Var w" shows "range_vars \<sigma> = fv t"
trunc_bound_pdevsE: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and x :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" obtains err :: "'a" where "\<bar>err\<bar> \<le> snd (trunc_bound_pdevs p x)" and "pdevs_val e (fst (trunc_bound_pdevs p x)) = pdevs_val e x + err"
Star_1: fixes s :: "'a regex" assumes "\<forall>x\<in>atms r. safe Strict x \<longrightarrow> safe Lax x"   and "local.safe_regex Past Strict r"   and "s \<in> rpd test i r" shows "local.safe_regex Past Strict s"
to_fun_X_minus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "UP_cring.to_fun R (X_poly_minus R a) b = b \<ominus>\<^bsub>R\<^esub> a"
rel_refl: fixes R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R3a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "('a, 'b, 'c) generat" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x"   and "\<And>x. R3a x x" shows "rel_generat R1a R2a R3a x x"
comp_of_ords_of_le_lt: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "comp_of_ords (le_of_comp c) (lt_of_comp c) = c"
prime_nat_iff'': fixes p :: "nat" shows "prime p = (p \<noteq> 0 \<and> p \<noteq> 1 \<and> (\<forall>m. 0 < m \<and> m < p \<longrightarrow> coprime p m))"
finite_dimensional_vector_space_ow_axioms: shows "finite_dimensional_vector_space_ow U\<^sub>M\<^sub>_\<^sub>1 (+) (0::'b) (-) uminus (*s\<^sub>1) basis1"
in_inv_image: fixes x :: "'a"   and y :: "'a"   and r :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"
map_id_clunky: fixes xs :: "'a list"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "set xs = UNIV" shows "x \<in> fst ` set (map (\<lambda>x. (x, f x)) xs)"
Infinitesimal_HFinite_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Infinitesimal"   and "y \<in> HFinite" shows "x * y \<in> Infinitesimal"
mat_extension_hermitian: fixes m :: "'a mat" assumes "m \<in> carrier_mat ps_P.d1 ps_P.d1"   and "hermitian m" shows "hermitian (ps_P.mat_extension m)"
IMPLEFT_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set"   and Y :: "ntrm set" shows "pi \<bullet> IMPLEFT (A IMP B) X Y = IMPLEFT (A IMP B) (pi \<bullet> X) (pi \<bullet> Y)"