hcju/isabelleps huggingface dataset:
semInt_prSubst: fixes Delta :: "('c, 'd, 'e, 'f, 'g) signature"   and SEM :: "('c, 'd, 'e, 'f, 'g, 'h) semDom" assumes "FixSyn TYPE('b) Delta"   and "FixSyn.wlsSEM Delta SEM" shows "FixSyn.prSubst Delta (FixSyn.semInt Delta SEM) SEM"
evalRecvArgs_pState_inv': fixes rargs :: "recvArg list"   and xs :: "integer list"   and g :: "gState\<^sub>I"   and p :: "pState"   and g' :: "gState\<^sub>I"   and p' :: "pState"   and prog :: "program" assumes "evalRecvArgs rargs xs g p = (g', p')"   and "pState_inv prog p" shows "pState_inv prog p'"
list_of_vec_rw_map: fixes m :: "'b vec" shows "list_of_vec m = map (($) m) [0..<dim_vec m]"
lsc_imp_liminf: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal"   and x :: "nat \<Rightarrow> 'b" assumes "lsc_at x0 f"   and "x \<longlonglongrightarrow> x0" shows "f x0 \<le> liminf (f \<circ> x)"
leadsTo_Trans: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set"   and C :: "'c set" assumes "F \<in> A \<longmapsto> B"   and "F \<in> B \<longmapsto> C" shows "F \<in> A \<longmapsto> C"
Fs: fixes F :: "int poly" assumes "F \<in># factors_of_factor_tree (Factor_Node f l r)" shows "Mp F = F \<and> monic F"
simps_2: fixes v :: "'a tree23"   and va :: "'a"   and vb :: "'a tree23s" shows "join_all (TTs v va vb) = join_all (join_adj (TTs v va vb))"
prog_step_preserves_valid: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid ts" shows "valid (ts[i := (p', is @ is', \<theta>, sb @ [Prog\<^sub>s\<^sub>b p p' is'], \<D>, \<O>, \<R>)])"
sub_branch_mem: fixes ps :: "('b, 'c) fm list"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and f :: "'c \<Rightarrow> 'd" assumes "(ps, i) \<in>. branch" shows "(sub_list f ps, f i) \<in>. sub_branch f branch"
inf_sup_distrib2_lgroup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "inf (sup b c) a = sup (inf b a) (inf c a)"
p: fixes x :: "nat" shows "c * b ^ x * of_nat x ^ deg \<le> p"
h2: shows "\<forall>n<dim_vec input_vec. List.member (find_nonzeros_from_input_vec input_vec) n \<longrightarrow> input_vec $ n \<noteq> 0"
separating_network_cut_of_sep: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "('a \<times> 'a) set" assumes "separating (web_of_network \<Delta>) S"   and "source \<Delta> \<noteq> sink \<Delta>" shows "separating_network \<Delta> (fst ` \<E>\<^bsub>web_of_network \<Delta>\<^esub> S)"
Prop1B: fixes c :: "IMP" assumes "secure c" shows " \<Turnstile> c : Sec (\<lambda>(r, t). \<exists>s. s , c \<Down> r  \<and>  s \<approx> t ) "
that: shows "mcont lub ord lub_spmf (ord_spmf (=)) f"
HMA_admits_SNF: shows "(Mod_Type_Connect.HMA_M ===> (=)) admits_SNF_JNF admits_SNF_HA"
law_restrict_is_ring: shows "ring (law_restrict (law_restrict R))"
boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> inf b c" obtains "a \<le> b" and "a \<le> c"
chine_composite_in_hom_2: shows "\<guillemotleft>\<mu>\<nu>\<pi>.chine : Chn (local.dom \<mu> \<star> local.dom \<nu> \<star> local.dom \<pi>) \<rightarrow>\<^sub>C Chn (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>)\<guillemotright>"
minus_minus: fixes x :: "int \<times> int" assumes "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> x) = x"
stake_scons: fixes i :: "Integer"   and x :: "'a"   and xs :: "[:'a:]" assumes "le\<cdot>1\<cdot>i = TT" shows "stake\<cdot>i\<cdot>(x :# xs) = x :# stake\<cdot>(i - 1)\<cdot>xs"
monotoneI: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)" shows "monotone orda ordb f"
SYN_def: shows "SYN \<equiv> range mid \<union> {ack}"
ipurge_fail_aux_t_form_def: fixes Y :: "('b, 'c) ipurge_rec" shows "ipurge_fail_aux_t_form Y \<equiv> case ListOp Y of None \<Rightarrow> False | Some ys \<Rightarrow> List Y = []"
pr_weak_trans: fixes fn :: "'f \<times> nat"   and gm :: "'f \<times> nat"   and hk :: "'f \<times> nat" assumes "pr_weak' fn gm"   and "pr_weak' gm hk" shows "pr_weak' fn hk"
stake_add: fixes m :: "nat"   and s :: "'d stream"   and n :: "nat" shows "stake m s @ stake n (sdrop m s) = stake (m + n) s"
a_r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier R"   and "x \<in> carrier R" shows "H +> x \<subseteq> carrier R"
cdi_is_cd_icdi: fixes k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat"   and i :: "nat" assumes "k icd\<^bsup>\<pi>\<^esup>\<rightarrow> j" shows "(k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i) = (j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<or> i = j)"
label_stab_map_inj_on_vertices: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0" shows "inj_on (\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) (\<Union> X)"
Procs_defined: shows "Procs \<subseteq> dom \<Gamma>"
fp_cop_F_induct: fixes P :: "'a set \<Rightarrow> bool"   and ds :: "'b set" assumes "P {}"   and "\<And>fp. P fp \<Longrightarrow> P (cop_F ds fp)" shows "P (fp_cop_F ds)"
stepsI: fixes x :: "'a"   and xs :: "'a list" assumes "x \<rightarrow> hd xs"   and "steps xs" shows "steps (x # xs)"
R_def: fixes i :: "nat" shows "R i = (if i = 0 then A else if i = 1 then - A else {})"
hyps_10: shows "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"
fold_set_union: fixes A :: "'a\<^sub>m set"   and B :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option" assumes "finite A"   and "finite B" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (A \<union> B) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z A) B"
hyps: fixes b :: "(nat \<times> int) list" assumes "d < length b"   and "odd i"   and "lv b d + l' = l"   and "i < (ix b d + 1) * 2 ^ l'"   and "(ix b d - 1) * 2 ^ l' < i" shows "b[d := (l, i)] \<in> grid b {d}"
tautology_imply: fixes q :: "'b fm"   and ps :: "'b fm list" assumes "tautology q" shows "tautology (imply ps q)"
conv: shows "convergent_prod (\<lambda>p. if prime p then inverse ((1::'a) - fds_nth f p / nat_power p s) else (1::'a))"
apd_d_def: fixes x :: "('b \<times> 'b) set" shows "rel_antidomain_kleene_algebra.apd_d x = rel_ad (rel_ad x)"
L_split: fixes x :: "'a" shows "x * L \<le> x * bot \<squnion> L"
desargues_config_3D_coplanar_6_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, C, \<alpha>, \<beta>, \<gamma>} = 3"
right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> UNIV"   and "y \<in> UNIV"   and "z \<in> UNIV" shows "(y + x = z + x) = (y = z)"
coinduct_strong: fixes R :: "('a, 'b) resource \<Rightarrow> ('a, 'b) resource \<Rightarrow> bool"   and resource :: "('a, 'b) resource"   and resource' :: "('a, 'b) resource" assumes "R resource resource'"   and "\<And>resource resource'. R resource resource' \<Longrightarrow> rel_fun (=) (rel_spmf (rel_prod (=) (\<lambda>resource resource'. R resource resource' \<or> resource = resource'))) (run_resource resource) (run_resource resource')" shows "resource = resource'"
stop_\<I>_uniform: fixes A :: "'h set"   and B :: "'i set" shows "stop_\<I> (\<I>_uniform A B) = (if B = {} then \<bottom> else \<I>_uniform A (insert None (Some ` B)))"
1: shows "\<forall>i. is_zero_row i A \<longrightarrow> \<not> (\<exists>j>i. \<not> is_zero_row j A)"
ldistinct_2: shows "\<forall>x. x \<in> Y \<longrightarrow> ldistinct x"
word_and_1_bl: fixes x :: "'b word" shows "x AND 1 = of_bl [x !! 0]"
Quotient3_refl2: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and r :: "'b"   and s :: "'b" assumes "Quotient3 R Abs Rep"   and "R r s" shows "R s s"
vwalk_consE: fixes a :: "'a"   and p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "vwalk (a # p) G"   and "p \<noteq> []" obtains "a \<rightarrow>\<^bsub>G\<^esub> hd p" and "vwalk p G"
reachableI_app_iso: fixes G :: "('e, 'f) pre_digraph"   and u :: "'e"   and v :: "'e"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"   and "pre_digraph.digraph_isomorphism G hom" shows "iso_verts hom u \<rightarrow>\<^sup>*\<^bsub>app_iso hom G\<^esub> iso_verts hom v"
empty_rep_def: shows "empty_rep = (\<infinity>, -\<infinity>)"
get_edge_simps_3: fixes f :: "'a"   and d :: "'b"   and t :: "'a" shows "getTo (f, d, t) = t"
deterministic_init_def: fixes I :: "'b \<Rightarrow> 'c pmf" shows "deterministic_init I = (\<forall>init. card (set_pmf (I init)) = 1)"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = a + b" shows "b \<le> a"
a_Pair_inv_P: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP\<^sub>1 :: "Syntax.term"   and eP\<^sub>2 :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, Syntax.Pair eP\<^sub>1 eP\<^sub>2, eV : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" obtains e\<^sub>1 :: "Syntax.term" and eV\<^sub>1 :: "Syntax.term" and e\<^sub>2 :: "Syntax.term" and eV\<^sub>2 :: "Syntax.term" where "e = Syntax.Pair e\<^sub>1 e\<^sub>2" and "eV = Syntax.Pair eV\<^sub>1 eV\<^sub>2" and "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : \<tau>\<^sub>1" and "\<Gamma> \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : \<tau>\<^sub>2"
OrdInd2_lemma: fixes i :: "name"   and A :: "fm" shows "{OrdP (Var i), A(i::=Zero), SyntaxN.All i (OrdP (Var i) IMP A IMP A(i::=SUCC (Var i)))} \<turnstile> A"
compare_strict_1: fixes y :: "'a" shows "compare\<cdot>\<bottom>\<cdot>y = \<bottom>"
inf: shows "infinite {n. birkhoff_sum f n x \<le> 0}"
dual_all: fixes P :: "'c dual \<Rightarrow> bool" shows "(\<forall>x. P (dual x)) = (\<forall>x'. P x')"
filter_compliant_stateful_ACS_accu_no_side_effects: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and accu :: "('b \<times> 'b) list"   and edgesList :: "('b \<times> 'b) list" assumes "valid_reqs (get_ACS M)"   and "wf_graph G"   and "\<forall>F\<in>get_offending_flows (get_ACS M) \<lparr>nodes = nodes G, edges = edges G \<union> backflows (edges G)\<rparr>. F \<subseteq> backflows (edges G) - edges G"   and "set accu \<union> set edgesList \<subseteq> edges G"   and "\<forall>a\<in>set accu. a \<notin> backflows (edges G)" shows "filter_compliant_stateful_ACS_accu G M accu edgesList = rev (filter (\<lambda>e. e \<notin> backflows (edges G)) edgesList) @ accu"
mu_bound_Gramian_determinant: fixes l :: "nat"   and k :: "nat" assumes "l < k"   and "k < length (map of_int_hom.vec_hom fs)" shows "(fs.gs.\<mu> k l)\<^sup>2 \<le> fs.gs.d l * \<parallel>map of_int_hom.vec_hom fs ! k\<parallel>\<^sup>2"
insert_def: fixes a :: "'c"   and v :: "'d"   and xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.insert a v xs = add (Some (Node a v [])) xs"
OO_Grp_alt: fixes A :: "'g set"   and f :: "'g \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f" shows "(BNF_Def.Grp A f)\<inverse>\<inverse> OO BNF_Def.Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
space_PiM: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "space (Pi\<^sub>M I M) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
rep_eq: fixes x :: "'a coreflexive"   and xa :: "'a coreflexive" shows "(x < xa) = (Rep_coreflexive x < Rep_coreflexive xa)"
dist: shows "local.dist_degree_factorize_main v w d res = facts"
induct: fixes P :: "(node \<times> node) list \<Rightarrow> node list \<Rightarrow> node list \<Rightarrow> bool"   and a0 :: "(node \<times> node) list"   and a1 :: "node list"   and a2 :: "node list" assumes "\<And>g ys. P g [] ys"   and "\<And>g x xs ys. \<lbrakk>List.member ys x \<Longrightarrow> P g xs ys; \<not> List.member ys x \<Longrightarrow> P g (nexts g x @ xs) (x # ys)\<rbrakk> \<Longrightarrow> P g (x # xs) ys" shows "P a0 a1 a2"
prems_7: shows "set uus \<inter> Fvars \<phi> = {}"
card_of_Func_squared: fixes A :: "'c set" shows "|Func UNIV A| =o |A \<times> A|"
frag_extend_minus: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_extend f (- x) = - frag_extend f x"
a_star_AndR: fixes M :: "trm"   and M' :: "trm"   and N :: "trm"   and N' :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname" assumes "M \<longrightarrow>\<^sub>a* M'"   and "N \<longrightarrow>\<^sub>a* N'" shows "AndR <a>.M <b>.N c \<longrightarrow>\<^sub>a* AndR <a>.M' <b>.N' c"
ordered_powerprod_axioms: shows "ordered_powerprod (\<preceq>) (\<prec>)"
maxim_greatest: fixes r :: "('d \<times> 'd) set"   and A :: "'d set"   and x :: "'d" assumes "wo_rel r"   and "finite A"   and "x \<in> A"   and "A \<subseteq> Field r" shows "(x, wo_rel.maxim r A) \<in> r"
SUP2_E: fixes B :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set"   and b :: "'b"   and c :: "'c" assumes "(\<Squnion> (B ` A)) b c" obtains x :: "'d" where "x \<in> A" and "B x b c"
h3: assumes "\<exists>(a', b', c')\<in>set les. a' \<noteq> 0 \<and> 4 * a' * c' \<le> b'\<^sup>2 \<and> (\<forall>(d, e, f)\<in>set les. \<exists>y'>(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a'). \<forall>x\<in>{(sqrt (b'\<^sup>2 - 4 * a' * c') - b') / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0)" shows "\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0"
iMODb_geD: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m, c ]" shows "r \<le> x"
DBM_reset'_sound': fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and cs :: "'c list"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and u'' :: "'c \<Rightarrow> 'd" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"   and "u'' \<turnstile>\<^bsub>v,n\<^esub> M" shows "\<exists>ts. set_clocks cs ts u \<turnstile>\<^bsub>v,n\<^esub> M"
simps: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<mu>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g (\<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>) = \<mu> \<Psi>"
symmetry_principle: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and H :: "circline"   and M :: "moebius" assumes "circline_symmetric z1 z2 H" shows "circline_symmetric (moebius_pt M z1) (moebius_pt M z2) (moebius_circline M H)"
east_west_tangents_incident_far_north_2: shows "proj2_incident far_north (polar west)"
cases: fixes x :: "'a \<times> 'b Filters.filter" obtains xa :: "'a" and y :: "'b Filters.filter" where "x = (xa, y)"
g_to_sorted_list_def: fixes m :: "'s" shows "g_to_sorted_list m \<equiv> reverse_iterateo m (#) []"
coprime_cong_mult: fixes a :: "'c"   and b :: "'c"   and m :: "'c"   and n :: "'c" assumes "[a = b] (mod m)"   and "[a = b] (mod n)"   and "coprime m n" shows "[a = b] (mod m * n)"
simps_1: fixes \<sigma> :: "'e \<Rightarrow> ('c monom \<times> 'd) list" shows "monom_list_subst \<sigma> [] = one_poly"
Compl: fixes x :: "'a"   and A :: "('b \<times> 'c) set" assumes "x \<in> space M"   and "A \<in> sets (N \<Otimes>\<^sub>M L)" shows "emeasure (f x \<Otimes>\<^sub>M g x) (space N \<times> space L - A) = emeasure (f x) (space (f x)) * emeasure (g x) (space (g x)) - emeasure (f x \<Otimes>\<^sub>M g x) A"
abs_eq_iff: fixes x :: "int extended \<times> int extended"   and y :: "int extended \<times> int extended" shows "(abs_ivl x = abs_ivl y) = eq_ivl x y"
assms_2: shows "set G \<union> {p} \<notin> C"
e_def: shows "e = (if i = 0 \<and> j = k - 1 \<and> cmod la = 1 \<and> k = m then la ^ off else 0)"
Image_subset: fixes r :: "('b \<times> 'c) set"   and A :: "'b set"   and B :: "'c set"   and C :: "'b set" assumes "r \<subseteq> A \<times> B" shows "r `` C \<subseteq> B"
CommuteWithIntervalCut2_def: fixes "fun" :: "'a set \<Rightarrow> 'a set" shows "CommuteWithIntervalCut2 fun \<equiv> \<forall>t fun2 I. fun2 = (\<lambda>I. I \<down>< t) \<or> fun2 = (\<lambda>I. I \<down>> t) \<longrightarrow> fun (fun2 I) = fun2 (fun I)"
absorb2: fixes a :: "'a upred"   and b :: "'a upred" assumes "b \<sqsubseteq> a" shows "(a \<or> b) = b"
hom_decompD: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "hom_decomp ps"   and "(h, U) \<in> set ps" shows "homogeneous h"
minimal_inD2: fixes P :: "'a set"   and x :: "'a"   and z :: "'a" assumes "minimal_in P x"   and "z \<in> P" shows "\<not> x \<succ> z"
assms_3: shows "isMinimalState state F0 decisionVars learnL"
segment_to_point_exists: fixes S :: "'b set"   and a :: "'b" assumes "closed S"   and "S \<noteq> {}" obtains b :: "'b" where "b \<in> S" and "open_segment a b \<inter> S = {}"
norm_eq_zero: fixes x :: "'a" shows "(norm x = 0) = (x = (0::'a))"
decoupling: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and G :: "'b program" assumes "F \<in> A \<longmapsto> B"   and "G \<in> stable B"   and "decoupled F G" shows "F \<squnion> G \<in> A \<longmapsto> B"
IH: fixes i :: "nat" assumes "S \<noteq> []"   and "i < length S" shows "set_of (S ! i) \<subseteq> set_of (interval_list_union S)"
mono_body_fixp: fixes F :: "('h \<Rightarrow> 'i Heap) \<Rightarrow> 'h \<Rightarrow> 'i Heap" assumes "\<And>x. mono_Heap (\<lambda>f. F f x)" shows "heap.fixp_fun F = F (heap.fixp_fun F)"
rhs: shows "(\<forall>a b. (\<lambda>x. if x \<in> s then f x else (0::'a)) integrable_on cbox a b) \<and> (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball (0::'n) B \<subseteq> cbox a b \<and> ball (0::'n) B \<subseteq> cbox c d \<longrightarrow> norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else (0::'a)) - integral (cbox c d) (\<lambda>x. if x \<in> s then f x else (0::'a))) < e)"
len_lexext_trans_strong: fixes zs :: "'b list"   and ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "len_lexext gt zs ys"   and "len_lexext gt ys xs" shows "len_lexext gt zs xs"
attractor_unfolding: fixes p :: "Player"   and W :: "'a set" shows "G'.attractor p W = W \<union> G'.attractor p W \<union> G'.directly_attracted p (G'.attractor p W)"
simps_1: fixes x :: "vrs"   and y :: "ty" shows "vrs_of (VarB x y) = {x}"
triangle_left: shows "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]"
lex_lengthD: fixes x :: "'a list"   and y :: "'a list"   and P :: "('a \<times> 'a) set" assumes "(x, y) \<in> lex P" shows "length x = length y"
f: shows "steps0 (Suc 0, l, r) tp stpa = (s', l', r')"
mapl_F_def: fixes l1 :: "'n \<Rightarrow> 'x"   and l2 :: "'o \<Rightarrow> 'y"   and l3 :: "'p \<Rightarrow> 'z" shows "mapl_F l1 l2 l3 = map_F l1 l2 l3 id id id id id id"
simps_1: fixes c :: "int \<times> int" shows "polybound0 (C c) = True"
one_le_integral_t: fixes x :: "'s" assumes "recurrent x" shows "1 \<le> U' x x"
op'_is: shows "op' = \<phi>\<^sub>O \<Psi> op"
no_oif_match_fbs: fixes fw :: "'b simple_rule list"   and rt :: "('b, 'd) routing_rule_scheme list"   and ifs :: "'e" assumes "no_oif_match fw" shows "list_all (\<lambda>m. oiface (fst (snd m)) = ifaceAny) (map (apfst of_nat) (annotate_rlen (lr_of_tran_fbs rt fw ifs)))"
listset_ConsE: fixes ys :: "'a list"   and x :: "'a set"   and xs :: "'a set list" assumes "ys \<in> listset (x # xs)" obtains y :: "'a" and ys' :: "'a list" where "y \<in> x" and "ys' \<in> listset xs" and "ys = y # ys'"
n_def: shows "n = \<lceil>(a - x) / p\<rceil>"
arr_char: fixes f :: "'a \<times> 'a" shows "S.VV.arr f = (S.arr (fst f) \<and> S.arr (snd f) \<and> S.src (fst f) = S.trg (snd f))"
l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr DocumentClass.type_wf DocumentClass.known_ptr Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Core_DOM_Functions.set_child_nodes Core_DOM_Functions.set_child_nodes_locs"
mat1: fixes env :: "v sem_env"   and e :: "exp0"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and e' :: "exp0"   and uu :: "pat"   and env' :: "(char list \<times> v) list"   and bv :: "(v, v) result" assumes "cupcake_evaluate_single env e (Rval v0)"   and "cupcake_match_result (c env) v0 pes Bindv = Rval (e', uu, env')"   and "cupcake_evaluate_single (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) e' bv" shows "cupcake_evaluate_single env (Mat e pes) bv"
shiftl1_of_bl: fixes bl :: "bool list" shows "shiftl1 (of_bl bl) = of_bl (bl @ [False])"
ends_in_segment_1: fixes a :: "'a"   and b :: "'a" shows "a \<in> {a--b}"
invariant_def: fixes P :: "'b \<Rightarrow> bool"   and succs :: "'b \<Rightarrow> 'b list" shows "invariant P succs \<equiv> \<forall>g g'. g' \<in> set (succs g) \<longrightarrow> P g \<longrightarrow> P g'"
length_emsapss_encode: fixes m :: "bit list"   and x :: "nat" assumes "emsapss_encode m x \<noteq> []" shows "length (emsapss_encode m x) = roundup x 8 * 8"
Suc_1: fixes l :: "nat" assumes "n < l" shows "map_pmf (\<lambda>y. y ! n) (bv l) = bernoulli_pmf (5 / 10)"
assms_1: fixes i :: "'x"   and j :: "'x" assumes "R i j"   and "j < n" shows "S i j"
smod_int_range: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "a smod b \<in> {- \<bar>b\<bar> + 1..\<bar>b\<bar> - 1}"
normalized_gauss_int_iff: fixes z :: "gauss_int" shows "(normalize z = z) = (z = 0 \<or> 0 < ReZ z \<and> 0 \<le> ImZ z)"
pmf_bind_pmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b pmf"   and x :: "'b" assumes "A \<noteq> {}"   and "finite A" shows "pmf (pmf_of_set A \<bind> f) x = (\<Sum>xaa\<in>A. pmf (f xaa) x) / real (card A)"
proper_sub_Hd: fixes s :: "('a, 'b) tm"   and \<zeta> :: "('a, 'b) hd" shows "\<not> proper_sub s (Hd \<zeta>)"
s: shows "Smith_normal_form_upt_k (diagonal_to_Smith_aux A [0..<k] bezout) k"
uv_5: shows "\<not> (u1, u2) \<approx>\<^sub>v (v1, v2)"
SKIP_def: shows "SKIP \<equiv> Abs_process ({(s, X). s = [] \<and> tick \<notin> X} \<union> {(s, X). s = [tick]}, {})"
reachable_def: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and xs :: "'c set" assumes "DFS succs is_node invariant ins memb empt" shows "DFS.reachable succs xs = {(x, y). y \<in> set (succs x)}\<^sup>* `` xs"
anonymise_prefs_from_table: fixes agents :: "'c set"   and alts :: "'d set"   and xs :: "('c \<times> 'd set list) list" assumes "prefs_from_table_wf agents alts xs" shows "anonymous_profile (prefs_from_table xs) = mset (map snd xs)"