hcju/isabelleps huggingface dataset:
independent_sets_def: fixes Gr :: "nat set \<times> nat set set" shows "independent_sets Gr \<equiv> {vs. vs \<subseteq> uverts Gr \<and> all_edges vs \<inter> uedges Gr = {}}"
OneAncestor_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" shows "OneAncestor F G = (\<forall>A\<in>F - {Root F G}. \<exists>!s. s \<in> \<Union> (States ` (F - {A})) \<and> A \<in> the (G s))"
Disjunctive_def: shows "Apply.Disjunctive = {x. \<forall>X. x (Sup X) = Sup (x ` X)}"
Rep_atoms_inject: fixes x :: "'a atoms"   and y :: "'a atoms" shows "(Rep_atoms x = Rep_atoms y) = (x = y)"
meet_closed: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lower_semilattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqinter>\<^bsub>L\<^esub> y \<in> carrier L"
alpha2_def: shows "alpha2 \<equiv> $pc2 = #a \<and> #0 < $sem \<and> pc2$ = #b \<and> sem$ = $sem - #1 \<and> Unchanged (x, y, pc1)"
hseq_char: fixes \<nu> :: "'b"   and \<mu> :: "'b" shows "hseq \<nu> \<mu> = (arr \<mu> \<and> arr \<nu> \<and> src \<nu> = trg \<mu>)"
WT_res: fixes \<eta> :: "nat" shows "\<I>_res \<eta> \<oplus>\<^sub>\<I> \<I>_common' \<eta> \<turnstile>res res \<eta> \<surd>"
ex1: fixes C :: "'b formula set set" assumes "pcp (C::'b formula set set)" shows "\<exists>C'::'b formula set set. (C::'b formula set set) \<subseteq> C' \<and> pcp C' \<and> subset_closed C'"
eval_notfree_unchanged_1: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and V :: "char list" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"   and "V \<notin> fv e" shows "l' V = l V"
hyps_1: fixes s :: "'a" assumes "\<Gamma>\<turnstile>mark_guards f c1 \<down> Normal s" shows "\<Gamma>\<turnstile>c1 \<down> Normal s"
span_explicit: fixes b :: "'b set" shows "m1.span b = {\<Sum>a\<in>t. r a *a a |t r. finite t \<and> t \<subseteq> b}"
IDE_implies_ARR: fixes f :: "'c local.term set" assumes "IDE f" shows "ARR f"
fin_long_chain_def: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and Q :: "'a set" shows "[f[x .. y ..  z]Q] \<equiv> x \<noteq> y \<and> x \<noteq> z \<and> y \<noteq> z \<and> finite Q \<and> long_ch_by_ord f Q \<and> f 0 = x \<and> y \<in> Q \<and> f (card Q - 1) = z"
coplanar_perm_23: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet D C B A"
minSetOfComponentsTestL2p3: shows "minSetOfComponents level2 {data1, data10, data11} = {sS1, sS2, sS3}"
butlast_conv_take: fixes xs :: "'a list" shows "butlast xs = take (length xs - 1) xs"
convol_expand_snd: fixes f :: "'a \<Rightarrow> 'b \<times> 'c"   and g :: "'a \<Rightarrow> 'b" assumes "fst \<circ> f = g" shows "BNF_Def.convol g (snd \<circ> f) = f"
inj_finite_image_eq_finite: fixes f :: "'a \<Rightarrow> 'b"   and Z :: "'a set" assumes "inj_on f Z" shows "finite (f ` Z) = finite Z"
CD_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Y.CD_on ds X = (\<Union>d\<in>ds. Y.Cd d X)"
butlast_power: fixes n :: "nat"   and xs :: "'a list" shows "(butlast ^^ n) xs = take (length xs - n) xs"
w_in_hom_2: shows "\<guillemotleft>TTfgh_TfTgh.chine : TTfgh_TfTgh.chine \<rightarrow> TTfgh_TfTgh.chine\<guillemotright>"
hyps_1: shows "\<Gamma> \<turnstile> e1 : t1"
nn_cond_expl_is_cond_exp: fixes X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "\<forall>w\<in>space M. 0 \<le> X w"   and "integrable M X"   and "disc_fct Y"   and "point_measurable M (space N) Y" shows "\<forall>A\<in>sets N. LINT w|M. X w * indicat_real (Y -` A \<inter> space M) w = LINT w|M. expl_cond_expect M Y X w * indicat_real (Y -` A \<inter> space M) w"
lens_inv_def: fixes x :: "'e \<Longrightarrow> 'd" shows "inv\<^sub>L x = \<lparr>lens_get = create\<^bsub>x\<^esub>, lens_put = \<lambda>\<sigma>. get\<^bsub>x\<^esub>\<rparr>"
l3_trans_def: shows "l3_trans \<equiv> \<Union>m M KE Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B KE \<union> l3_step3 Ra A B m \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
Gromov_extension_continuous_inside: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous (at x within S) f" shows "continuous (at (to_Gromov_completion x) within to_Gromov_completion ` S) (Gromov_extension f)"
lift_into_qmsg: fixes A :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and R :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "A \<Turnstile> (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) global P"   and "\<And>\<xi>. U \<xi> \<xi>"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'"   and "\<And>\<sigma> \<sigma>' m. \<lbrakk>\<forall>j. U (\<sigma> j) (\<sigma>' j); R \<sigma> m\<rbrakk> \<Longrightarrow> R \<sigma>' m"   and "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). U (\<sigma> i) (\<sigma>' i))" shows "A \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile> (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) global P"
PiE_ext: fixes x :: "'a \<Rightarrow> 'b"   and k :: "'a set"   and s :: "'a \<Rightarrow> 'b set"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> Pi\<^sub>E k s"   and "y \<in> Pi\<^sub>E k s"   and "\<And>i. i \<in> k \<Longrightarrow> x i = y i" shows "x = y"
subcls_irrefl: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<prec>\<^sub>C D"   and "ws_prog G" shows "C \<noteq> D"
Ideal2_def: fixes m2 :: "bool \<times> bool"   and m1 :: "bool \<times> bool"   and out2 :: "bool" shows "and_secret_sharing.Ideal2 m2 m1 out2 = S2_and m2 out2 \<bind> (\<lambda>view2. out2 m2 m1 out2 \<bind> (\<lambda>out2. return_spmf (view2, out2)))"
assoc_props_2: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "trg \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> = trg f"
disjnt_def: fixes A :: "'e set"   and B :: "'e set" shows "disjnt A B = (A \<inter> B = {})"
1: shows "{z \<in> A. z \<le> y} \<subset> {z \<in> A. z \<le> x}"
partial_magma_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "partial_magma C \<equiv> \<exists>!n. \<forall>f. C n f = n \<and> C f n = n"
fps_compose_def: fixes a :: "'a fps"   and b :: "'a fps" shows "a oo b = Abs_fps (\<lambda>n. \<Sum>i = 0..n. a $ i * b ^ i $ n)"
trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "R i x y"   and "R i y z" shows "R i x z"
ring_simprules_3: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
segments_of_aform_strict: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1<..<1}"   and "length (half_segments_of_aform X) \<noteq> 1" shows "list_all (\<lambda>seg. ccw' (fst seg) (snd seg) (aform_val e X)) (segments_of_aform X)"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 |h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child)|\<^sub>r)|\<^sub>r"
compress_pterm_eq: fixes t :: "pterm" shows "compress_pterm t = t"
is_div_op_def: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "local.is_div_op div_op = (\<forall>a b. b dvd a \<longrightarrow> div_op a b * b = a)"
ereal_inverse_antimono: fixes x :: "ereal"   and y :: "ereal" assumes "0 \<le> x"   and "x \<le> y" shows "inverse y \<le> inverse x"
2: shows "open (UNIV - \<real>\<^sub>\<ge>\<^sub>0)"
not_empty_group_by_drop: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "nat"   and xs :: "'b list" shows "\<forall>x\<in>set (group_by f (drop l xs)). x \<noteq> []"
a_rcosetsI: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H +> x \<in> a_rcosets H"
insertI1: fixes a :: "'d"   and B :: "'d set" shows "a \<in> insert a B"
bisim1_pc_length_compE2: fixes P :: "'a prog"   and h :: "'heap"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option" assumes "P,E,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "pc \<le> length (compE2 E)"
if_distribR: fixes b :: "bool"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(if b then f else g) x = (if b then f x else g x)"
smult_l_null: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "(0::'a) \<cdot>\<^sub>v x = 0\<^sub>v n"
simps_1: fixes \<nu> :: "'b"   and u :: "'b local.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle>\<^sub>0 \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = u"
lookup_eq_map_of_ap: fixes k :: "'b"   and subs :: "('b \<times> 'a) list" shows "lookup (\<lambda>x. x = k) subs = map_of subs k"
lzip_trans_fusion: fixes g :: "'e \<Rightarrow> ('c, 'e) step"   and h :: "'f \<Rightarrow> ('d, 'f) step"   and sg :: "'e"   and sh :: "'f" shows "lunstream (zip_raw g h) (sg, sh, None) = lzip (lunstream g sg) (lunstream h sh)"
inf_mono: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a c"   and "less_eq b d" shows "less_eq (inf a b) (inf c d)"
lookup: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "lookup x \<in> {f. finite {x. f x \<noteq> (0::'c)}}"
ct_suffix_singleton_any: fixes t :: "ct" shows "ct_suffix [TAny] [t]"
intros_6: fixes F :: "'b formula"   and G :: "'b formula" shows "(F \<^bold>\<and> G) \<^bold>\<rightarrow> G \<in> AX10"
psi_def: shows "psi \<equiv> initPsi \<and> \<box>[n1 \<or> n2]_vars \<and> Live2"
simps_2: shows "sklvl1.domain Resp = {xny, xni, xnr, xgnx, xgny, xsk, xEnd}"
approx_stc_eq: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "x \<approx> y" shows "stc x = stc y"
index_eq_index_conv: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "x \<in> set xs \<or> y \<in> set xs" shows "(index xs x = index xs y) = (x = y)"
max_wp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "maximal (wp a)"   and "maximal (wp b)" shows "maximal (wp (a ;; b))"
canonical_suffix: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "canonical_sequence A \<sigma>" shows "canonical_sequence A \<sigma> [k..]"
l: fixes s :: "'a" assumes "s \<in> cball (g t) u \<inter> g ` T" shows "L-lipschitz_on (cball x u \<inter> X) (f s)"
ismaxcoeff_mono: fixes t :: "Ferrack.num"   and c :: "int"   and c' :: "int" assumes "ismaxcoeff t c"   and "c \<le> c'" shows "ismaxcoeff t c'"
o_objects_exist_2: fixes v :: "i" shows "[\<^bold>\<box>(\<^bold>\<exists>x. \<lparr>O!,x\<^sup>P\<rparr>) in v]"
VInter_vempty: shows "\<Inter>\<^sub>\<circ> []\<^sub>\<circ> = []\<^sub>\<circ>"
qf3_cube_impl_cube_form: fixes a :: "int"   and b :: "int"   and w :: "int" assumes "coprime a b \<and> a\<^sup>2 + 3 * b\<^sup>2 = w ^ 3 \<and> odd w" shows "is_cube_form a b"
enforce_spmf_bot: shows "enforce_spmf \<bottom> = (\<lambda>_. return_pmf None)"
assms_3: shows "h = (\<lambda>(b, i). case f b of (d, fs) \<Rightarrow> (d ^ Suc i, map (\<lambda>f. (f, i)) fs))"
ord_resolve_counterex_reducing: fixes N :: "'a literal multiset set"   and DA :: "'a literal multiset"   and M :: "'a literal multiset set" assumes "{#} \<notin> N"   and "DA \<in> N"   and "\<not> gr.INTERP M N \<TTurnstile> DA"   and "\<And>C. \<lbrakk>C \<in> N; \<not> gr.INTERP M N \<TTurnstile> C\<rbrakk> \<Longrightarrow> DA \<le> C" obtains CAs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and E :: "'a literal multiset" where "set CAs \<subseteq> N" and "gr.INTERP M N \<TTurnstile>m mset CAs" and "\<And>CA. CA \<in> set CAs \<Longrightarrow> gr.production M N CA \<noteq> {}" and "gr.ord_resolve M CAs DA AAs As E" and "\<not> gr.INTERP M N \<TTurnstile> E" and "E < DA"
induct: fixes P :: "(int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>c\<^sub>0 c\<^sub>1. P (c\<^sub>0, c\<^sub>1) []"   and "\<And>c\<^sub>0 c\<^sub>1 uu_. P (c\<^sub>0, c\<^sub>1) [uu_]"   and "\<And>c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>x. \<lbrakk>x = find_closest_bf p\<^sub>0 (take 7 (v # va)); dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (v # va); \<And>x. \<lbrakk>x = find_closest_bf p\<^sub>0 (take 7 (v # va)); \<not> dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (p\<^sub>0, x) (v # va)\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
intro: fixes M :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. prob_space (M i)" shows "product_prob_space_axioms M"
TBtheorem1b: fixes PQ :: "specID"   and M :: "chanID set"   and E :: "Expression"   and P :: "specID"   and Q :: "specID" assumes "ineM PQ M E"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ" shows "ineM P M E \<or> ineM Q M E"
right_total_rel_set: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_total A" shows "right_total (rel_set A)"
prod_lens_indep_out_var: fixes b :: "'d \<Longrightarrow> 'e"   and x :: "'f \<Longrightarrow> 'e"   and a :: "'g \<Longrightarrow> 'h" assumes "b \<bowtie> x" shows "a \<times> b \<bowtie> $x\<acute>"
ring_iso_imp_img_field: fixes R :: "('e, 'f) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and S :: "('g, 'h) ring_scheme" assumes "field R"   and "h \<in> ring_iso R S" shows "field (S\<lparr>zero := h \<zero>\<^bsub>R\<^esub>\<rparr>)"
projection_split_first: fixes \<tau> :: "'a list"   and X :: "'a set"   and x :: "'a"   and xs :: "'a list" assumes "\<tau> \<upharpoonleft> X = x # xs" shows "\<exists>\<alpha> \<beta>. \<tau> = \<alpha> @ [x] @ \<beta> \<and> \<alpha> \<upharpoonleft> X = []"
cor_add: fixes a :: "real"   and b :: "real" shows "cor (a + b) = cor a + cor b"
generalise_output_eq: fixes t :: "'a transition_scheme"   and r :: "nat"   and v :: "value"   and c :: "nat \<Rightarrow>f value option"   and p :: "nat"   and i :: "value list" assumes "Outputs t ! r = L v"   and "c $ p = Some v" shows "evaluate_outputs t i c = apply_outputs ((Outputs t)[r := V (R p)]) (join_ir i c)"
monoid_comm_monoidI: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.monoid G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
sup_aci_1: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = y \<squnion> x"
csqrt_principal: fixes z :: "complex" shows "0 < Re (csqrt z) \<or> is_imag (csqrt z) \<and> 0 \<le> Im (csqrt z)"
weak_respection_of_barbs_and_closures_3: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (Rel\<^sup>+) CWB"
assms_2: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"   and "P g" shows "P f"
supr_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wo_rel r" shows "wo_rel.supr r A \<equiv> minim r (Above r A)"
card_UNIV_literal_def: shows "card_UNIV_class.card_UNIV = Phantom(String.literal) 0"
zero_le_one: shows "(0::'a) \<le> (1::'a)"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "lookup uu [] = None"
lower_id_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(lower p v = p) = (p = 0 \<or> lt p \<prec>\<^sub>t v)"
assms_2: shows "theta \<subseteq> Wretr (theta \<union> Wbis)"
Cauchy_theorem_quadrisection: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and f :: "complex \<Rightarrow> complex"   and K :: "real"   and e :: "real" assumes "continuous_on (convex hull {a, b, c}) f"   and "dist a b \<le> K"   and "dist b c \<le> K"   and "dist c a \<le> K"   and "e * K\<^sup>2 \<le> cmod (contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f)" shows "\<exists>a' b' c'. a' \<in> convex hull {a, b, c} \<and> b' \<in> convex hull {a, b, c} \<and> c' \<in> convex hull {a, b, c} \<and> dist a' b' \<le> K / 2 \<and> dist b' c' \<le> K / 2 \<and> dist c' a' \<le> K / 2 \<and> e * (K / 2)\<^sup>2 \<le> cmod (contour_integral (linepath a' b') f + contour_integral (linepath b' c') f + contour_integral (linepath c' a') f)"
unique_sig_Base_foo: fixes sig :: "sig" assumes "tprg\<turnstile> mdecl (sig, snd Base_foo) declared_in Base" shows "sig = foo_sig"
assms: shows "parallel i j x \<down>= prod_encode (0, v)"
bound0_I: fixes p :: "fm"   and b :: "real"   and bs :: "real list"   and b' :: "real" assumes "bound0 p" shows "Ifm (b # bs) p = Ifm (b' # bs) p"
cases: fixes a :: "('a, 'b) trm" assumes "dsafe a" obtains i :: "'b" where "a = Var i__"   | r :: "real" where "a = Const r"   | args :: "'b \<Rightarrow> ('a, 'b) trm" and i :: "'a" where "a = $f ia__ args" and "\<And>i. dsafe (args i)"   | \<theta>\<^sub>1 :: "('a, 'b) trm" and \<theta>\<^sub>2 :: "('a, 'b) trm" where "a = Plus \<theta>\<^sub>1__ \<theta>\<^sub>2__" and "dsafe \<theta>\<^sub>1__" and "dsafe \<theta>\<^sub>2__"   | \<theta>\<^sub>1 :: "('a, 'b) trm" and \<theta>\<^sub>2 :: "('a, 'b) trm" where "a = Times \<theta>\<^sub>1 \<theta>\<^sub>2" and "dsafe \<theta>\<^sub>1" and "dsafe \<theta>\<^sub>2"   | \<theta> :: "('a, 'b) trm" where "a = Differential \<theta>" and "dfree \<theta>"   | i :: "'b" where "a = $' i"
confluentpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r\<^sup>*\<^sup>* x y; r\<^sup>*\<^sup>* x z\<rbrakk> \<Longrightarrow> \<exists>u. r\<^sup>*\<^sup>* y u \<and> r\<^sup>*\<^sup>* z u" shows "confluentp r"
IH1: fixes S :: "'node SDG_node set" assumes "n'' \<in> set ns"   and "n\<^sub>1 \<in> S" shows "n'' \<in> HRB_slice S"
IF_DIAGONAL: fixes G :: "Var \<Rightarrow> L"   and H :: "Var \<Rightarrow> L"   and p :: "L"   and q :: "L"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "\<forall>x. LEQ (G x) (H x) \<or> LEQ p (H x)"   and "\<nexists>x. LEQ p (H x) \<and> LEQ (H x) q"   and "X \<rhd> c1 : Sec (Q p H) (EQ G q) (EQ H q) \<Phi>"   and "X \<rhd> c2 : Sec (Q p H) (EQ G q) (EQ H q) \<Psi>" shows "X \<rhd> Iff b c1 c2 : Sec (Q p H) (EQ G q) (EQ H q) (\<lambda>(s, t). EQ H q s t)"
snth_SCons_Suc: fixes x :: "'a"   and xs :: "'a stream"   and n :: "nat" shows "(x ## xs) !! Suc n = xs !! n"
is_map_image: fixes M :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "is_map M" shows "is_map ((\<lambda>(a, b). (a, f a b)) ` M)"
\<sigma>: shows "\<sigma> \<in> space (state_measure V \<Gamma>)"
"3_2": assumes "x \<noteq> y"   and "rel x y"   and "transp rel"   and "\<And>x y. x \<noteq> y \<Longrightarrow> rel x y \<or> rel y x"   and "sorted_wrt rel xs"   and "sorted_wrt rel (y # ys)" shows "sorted_wrt rel (merge_wrt rel xs (y # ys))"
rev_predicate2D: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "P x y"   and "P \<le> Q" shows "Q x y"
lin: fixes k1 :: "'a"   and K :: "'a set"   and x :: "'a"   and k2 :: "'a" assumes "k1 \<in> K.simple_extension K x"   and "k2 \<in> K" shows "k1 \<otimes>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> x \<oplus>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> k2 \<in> K.simple_extension K x"
seq_lexord_linear: fixes R :: "('b \<times> 'b) set"   and x :: "'b seq"   and y :: "'b seq" assumes "\<forall>a b. (a, b) \<in> R \<or> a = b \<or> (b, a) \<in> R" shows "(x, y) \<in> seq_lexord R \<or> x = y \<or> (y, x) \<in> seq_lexord R"
vertical_composite_iso_inverse: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "natural_isomorphism A B F G \<tau>" shows "vertical_composite.map A B \<tau> (inverse_transformation.map A B G \<tau>) = F"
geodesic_segment_def: fixes G :: "'b set" shows "geodesic_segment G = (\<exists>x y. geodesic_segment_between G x y)"
interaction_bounded_by'_exec_gpv_count: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'a) gpv"   and n :: "nat"   and x :: "'e"   and s' :: "'d \<Rightarrow> 'a option"   and s :: "'d \<Rightarrow> 'a option"   and count :: "('d \<Rightarrow> 'a option) \<Rightarrow> nat" assumes "interaction_bounded_by' consider gpv n"   and "(x, s') \<in> set_spmf (exec_gpv random_oracle gpv s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (random_oracle s x); finite (dom s); consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> Suc (count s)"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (random_oracle s x); finite (dom s); \<not> consider x; x \<in> outs_\<I> \<I>_full\<rbrakk> \<Longrightarrow> count s' \<le> count s"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "finite (dom s)" shows "count s' \<le> n + count s"
assms_3: shows "geodesic_segment_between G (f a) (f b)"
a4_def: shows "a4 = is_a_proper_intersec P A B' A' B"
gfp_little_fusion: fixes f :: "'a \<Rightarrow> 'a"   and h :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "mono f"   and "h \<circ> f \<le> g \<circ> h" shows "h (gfp f) \<le> gfp g"