hcju/isabelleps huggingface dataset:
is_pole_basic: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and z :: "complex"   and n :: "nat" assumes "f holomorphic_on A"   and "open A"   and "z \<in> A"   and "f z \<noteq> 0"   and "0 < n" shows "is_pole (\<lambda>w. f w / (w - z) ^ n) z"
sinvar_all_edges_normal_form_not_refl_SR_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool" shows "SecurityInvariant_withOffendingFlows.sinvar_all_edges_normal_form_not_refl_SR sinvar P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(s, r)\<in>edges G. s \<noteq> r \<longrightarrow> P (nP s) s (nP r) r)"
model_def: fixes e :: "nat \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd list \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'd list \<Rightarrow> bool"   and ps :: "('e, 'f) form list"   and p :: "('e, 'f) form" shows "(e,f,g,ps \<Turnstile> p) = (list_all (eval e f g) ps \<longrightarrow> eval e f g p)"
rtrancl_refl: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" shows "r\<^sup>*\<^sup>* a a"
span_insert_subset: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set"   and r :: "'c" assumes "module scale"   and "module.span scale A \<subseteq> module.span scale B"   and "r \<in> module.span scale B" shows "module.span scale (insert r A) \<subseteq> module.span scale B"
smult_0_left: fixes p :: "'a poly" shows "smult (0::'a) p = 0"
simps_1: fixes d :: "nat" shows "ladder_shift_j d [] = []"
return_pmf_in_lotteries_on: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "return_pmf x \<in> lotteries_on A"
solve_factored_ratfps_def: fixes p :: "'b poly"   and cs :: "('b \<times> nat) list" shows "solve_factored_ratfps p cs = (let n = length cs in case poly_pfd_simple p cs of (a, zs) \<Rightarrow> (a, zip_with (\<lambda>zs (c, n). (sum_list (map2 (\<lambda>z j. inverse_irred_power_poly z (n + 1 - j)) zs [0..<Suc n]), c)) zs cs))"
sum: fixes g :: "'c \<Rightarrow> real"   and A :: "'c set"   and x :: "'b" shows "sum g A *\<^sub>R x = (\<Sum>x\<in>A. g x *\<^sub>R x)"
set_shadow_root_impl: shows "set_shadow_root = set_shadow_root"
ifex_sat_untouched_False: fixes v :: "'a"   and i :: "'a ifex"   and a :: "'a \<Rightarrow> bool" assumes "v \<notin> ifex_var_set i"   and "ifex_sat i = Some a" shows "a v = False"
satPB_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "satPB (int \<xi> \<circ> \<rho>) \<Phi> = satPB \<xi> (substPB \<rho> \<Phi>)"
da_elim_cases_13: fixes Env :: "env"   and B :: "lname set"   and jump :: "jump"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Jmp jump\<rangle>\<guillemotright> A" obtains "jump = Ret \<longrightarrow> Result \<in> B" and "nrm A = UNIV" and "brk A = (case jump of Break l \<Rightarrow> \<lambda>k. if k = l then B else UNIV | _ \<Rightarrow> \<lambda>k. UNIV)"
nth_root_nat_ge: fixes k :: "nat"   and x :: "nat"   and n :: "nat" assumes "0 < k"   and "x ^ k \<le> n" shows "x \<le> nth_root_nat k n"
dur_zero_tail: fixes xs :: "('c \<Rightarrow> real) stream"   and \<omega> :: "('c \<Rightarrow> real) set stream"   and i :: "nat"   and j :: "nat"   and x :: "'c" assumes "\<forall>i. xs !! i \<in> \<omega> !! i"   and "\<forall>k\<ge>i. k \<le> j \<longrightarrow> zero x (\<omega> !! k)"   and "x \<in> \<X>"   and "i \<le> j"   and "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)" shows "dur xs j = dur xs i"
source_stability: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f" assumes "wb_lens x" shows "\<exists>v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
evp: shows "\<rho> \<turnstile> e \<Down> v'"
iso_class_eqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.isomorphic f g" shows "VoV.iso_class f = VoV.iso_class g"
compat: shows "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"
gen_cond_def: fixes s :: "('a, 'd) simple_state_scheme" shows "a.c.gen_cond s \<equiv> (a.a.V0 \<subseteq> a.c.gen_discovered s \<longrightarrow> \<not> gds_is_empty_stack a.c.gds s) \<and> \<not> gds_is_break a.c.gds s"
chambersubcomplex_image: shows "folding.ChamberSubcomplex (f \<turnstile> X)"
inf_commute: fixes x :: "'c"   and y :: "'c" shows "inf x y = inf y x"
terms_def: shows "terms = map2 (\<lambda>cs hfvs. (cs \<bullet>\<currency>\<currency> hfvs) ((1::'f) \<delta>\<delta> hs ! i)) css hfvss"
igiiu: shows "InvGlobalIncomingInfoUpright (shd s)"
mset_lc_eq: shows "mset (L # C) = mset (L # C')"
a_r_coset_defs_1: fixes G :: "('c, 'd) ring_scheme" shows "(+>\<^bsub>G\<^esub>) = (#>\<^bsub>add_monoid G\<^esub>)"
Basis: fixes s :: "'c set"   and S :: "'c set set" assumes "s \<in> S" shows "generate_topology S s"
i_shrink_eq_NoMsg_iAll_conv: fixes k :: "nat"   and s :: "nat \<Rightarrow> 'a message_af"   and t :: "nat" assumes "0 < k" shows "((s \<div> k) t = NoMsg) = (\<box> t1 [t * k\<dots>,k - Suc 0]. s t1 = NoMsg)"
parAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<sim>\<^sup>s\<^sub>l P \<parallel> Q \<parallel> R"
vector_mult_component: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec"   and y :: "('b, 'c) Finite_Cartesian_Product.vec"   and i :: "'c" shows "(x * y) $h i = x $h i * y $h i"
comp_bijective_complement: fixes y :: "'a"   and x :: "'a" assumes "bijective y" shows "- x * y = - (x * y)"
Conj_rep_eq: fixes xset :: "('a, 'b, 'c, 'd) formula set['a]" assumes "finite (supp xset)" shows "Rep_formula (Conj xset) = Conj\<^sub>\<alpha> (map_bset Rep_formula xset)"
prems_8: shows "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p"
aforms_of_ivl_leI2: fixes en :: "real"   and xsn :: "real"   and ysn :: "real" assumes "en \<le> 1"   and "xsn \<le> ysn" shows "(xsn + ysn) / 2 + (ysn - xsn) * en / 2 \<le> ysn"
r_result1_diverg_phi: fixes i :: "nat"   and x :: "nat"   and t :: "nat" assumes "\<phi> i x \<up>" shows "eval r_result1 [t, i, x] \<down>= 0"
inject_1: fixes x1 :: "('e, 'f, 'g) Tree set['e]"   and y1 :: "('e, 'f, 'g) Tree set['e]" shows "(tConj x1 = tConj y1) = (x1 = y1)"
Retr_incl: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" shows "Sretr theta \<subseteq> ZOretr theta"
assms_8: shows "i < row_length A1 * row_length B1"
P_exist: fixes h :: "complex \<Rightarrow> complex" assumes "\<exists>z'. h \<midarrow>z\<rightarrow> z'"   and "isolated_singularity_at h z"   and "\<exists>\<^sub>F w in at z. h w \<noteq> 0" shows "\<exists>n g r. P h n g r"
semiattractive_extend: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "semiattractive A r"   and "semiattractive B r"   and "\<forall>a\<in>A. \<forall>b\<in>B. r a b \<and> \<not> r b a" shows "semiattractive (A \<union> B) r"
g: fixes F :: "'c filter"   and G :: "('c \<Rightarrow> real) set"   and H :: "('c \<Rightarrow> real) set"   and g :: "'c \<Rightarrow> real" assumes "landau_function_family_pair F G H g" shows "g \<in> G"
simps_2: fixes x :: "nat"   and \<rho> :: "(nat \<times> val) list" shows "E (EVar x) \<rho> = {v. \<exists>v'. lookup \<rho> x = Some v' \<and> v \<sqsubseteq> v'}"
Ici_subset_Ioi_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less" shows "(ord.atLeast less_eq a \<subseteq> ord.greaterThan less b) = less b a"
HT'set_r_def: fixes r :: "((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'b) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and \<pi> :: "char list \<Rightarrow> com option"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> 'b) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> com) \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "HT'set_r r \<pi> \<Theta> \<equiv> \<forall>(n, P, c, Q)\<in>\<Theta>. HT' \<pi> (\<lambda>s. r n s \<and> P s) c Q"
not_in_mult: fixes var :: "nat"   and p :: "real mpoly"   and q :: "real mpoly" assumes "var \<notin> vars p"   and "var \<notin> vars q" shows "var \<notin> vars (p * q)"
deg_to_X_d: fixes d :: "nat" assumes "Idomain S" shows "deg_n R S X (X^\<^bsup>R d\<^esup>) = d"
r_nth_prim: shows "prim_recfn 2 r_nth"
simps_2: fixes x :: "'b"   and xl :: "'b bintree"   and xr :: "'b bintree" shows "t_multiset (Branch x xl xr) = {#x#} + t_multiset xl + t_multiset xr"
hmset_of_enat_le: fixes m :: "enat"   and n :: "enat" shows "(hmset_of_enat m \<le> hmset_of_enat n) = (m \<le> n)"
run_sim: fixes ra :: "nat \<Rightarrow> 'b" assumes "b.is_run ra" obtains rb :: "nat \<Rightarrow> 'a" where "a.is_run rb" and "(ra, rb) \<in> run_rel (R\<inverse>)"
Span_eq_eval_img: fixes K :: "'a set"   and x :: "'a"   and n :: "nat" assumes "subfield K R"   and "x \<in> carrier R" shows "Span K (exp_base x n) = (\<lambda>p. local.eval p x) ` {p \<in> carrier (K [X]). length p \<le> n}"
polynomial_of_mset_append: fixes xs :: "(char list multiset \<times> int) multiset"   and ys :: "(char list multiset \<times> int) multiset" shows "polynomial_of_mset (xs + ys) = polynomial_of_mset xs + polynomial_of_mset ys"
A1: fixes x :: "i"   and y :: "i" shows "(dom x \<cong> cod y \<^bold>\<leftarrow> E (x \<cdot> y)) \<^bold>\<and> (E (x \<cdot> y) \<^bold>\<leftarrow> dom x \<cong> cod y)"
Dickson_term: fixes d :: "'a \<Rightarrow> nat"   and K :: "'k set"   and m :: "nat" assumes "dickson_grading d"   and "finite K" shows "almost_full_on (adds\<^sub>t) {t. pp_of_term t \<in> dgrad_set d m \<and> component_of_term t \<in> K}"
finite_fv: fixes t :: "Core.term" shows "finite (fv t)"
sum_in_spanI: fixes q :: "'b \<Rightarrow> 'a"   and B :: "'b set" shows "(\<Sum>b\<in>B. q b *s b) \<in> local.span B"
sum: fixes s1 :: "'c"   and R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set"   and s2 :: "'c" assumes "s1 \<in> I \<cdot>\<^bsub>R\<^esub> J"   and "s2 \<in> I \<cdot>\<^bsub>R\<^esub> J" shows "s1 \<oplus>\<^bsub>R\<^esub> s2 \<in> I \<cdot>\<^bsub>R\<^esub> J"
mark_as_done_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and l :: "nat"   and u :: "nat"   and I :: "'b \<Rightarrow> node_state option" shows "GS.mark_as_done SBIP l u I \<equiv> WHILE\<^sub>T (\<lambda>(l, I). l < u) (\<lambda>(l, I). ASSERT (l < length (GS.S SBIP)) \<bind> (\<lambda>_. RETURN (Suc l, I(GS.S SBIP ! l \<mapsto> DONE)))) (l, I) \<bind> (\<lambda>(uu_, I). RETURN I)"
dom_subid_aux1'': fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> y \<le> y"
sep_implD: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a \<Rightarrow> bool"   and y :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_impl plus sep_disj x y h" shows "\<forall>h'. sep_disj h h' \<and> x h' \<longrightarrow> y (plus h h')"
filterlim_tendsto_add_at_bot_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F" shows "(LIM x F. f x + g x :> at_bot) = filterlim g at_bot F"
lens_obs_eq_def: fixes X :: "'b \<Longrightarrow> 'c"   and s\<^sub>1 :: "'c"   and s\<^sub>2 :: "'c" shows "(s\<^sub>1 \<simeq>\<^bsub>X\<^esub> s\<^sub>2) = (s\<^sub>1 = put\<^bsub>X\<^esub> s\<^sub>2 (get\<^bsub>X\<^esub> s\<^sub>1))"
runStateT_plusU: fixes a :: "udom\<cdot>('a, 'b) stateT"   and b :: "udom\<cdot>('a, 'b) stateT"   and s :: "'b" shows "runStateT\<cdot>(plusU\<cdot>a\<cdot>b)\<cdot>s = mplus\<cdot>(runStateT\<cdot>a\<cdot>s)\<cdot>(runStateT\<cdot>b\<cdot>s)"
subgroups_of_size_def: fixes p :: "nat" shows "Hgroup.subgroups_of_size p = {H. subgroup H H' \<and> card H = p}"
le_antisym: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "n \<le> m" shows "m = n"
edge_preserving_def: fixes h :: "('b \<times> 'c) set"   and e1 :: "('d \<times> 'b \<times> 'b) set"   and e2 :: "('d \<times> 'c \<times> 'c) set" shows "edge_preserving h e1 e2 \<equiv> \<forall>(k, v1, v2)\<in>e1. \<forall>v1' v2'. (v1, v1') \<in> h \<and> (v2, v2') \<in> h \<longrightarrow> (k, v1', v2') \<in> e2"
filt_equiv_def: fixes F :: "'d \<Rightarrow> 'e measure"   and M :: "'e measure"   and N :: "'e measure" shows "filt_equiv F M N = (sets M = sets N \<and> Filtration.filtration M F \<and> (\<forall>t A. A \<in> sets (F t) \<longrightarrow> (emeasure M A = 0) = (emeasure N A = 0)))"
interval_integrable_endpoints_reverse: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integrable lborel a b f = interval_lebesgue_integrable lborel b a f"
simps: fixes a :: "'c llist" shows "lfinite a = (a = LNil \<or> (\<exists>xs x. a = LCons x xs \<and> lfinite xs))"
source: fixes S :: "'procS" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R SourceTerm S"
InternalSets_Int: fixes X :: "'a star set"   and Y :: "'a star set" assumes "X \<in> InternalSets"   and "Y \<in> InternalSets" shows "X \<inter> Y \<in> InternalSets"
D_def: shows "D = 92 * lambda\<^sup>2 * (C + deltaG TYPE('b))"
ord_of_add: fixes i :: "nat"   and j :: "nat" shows "ord_of (i + j) = ord_of i @+ ord_of j"
mbhd_insert_def: fixes x :: "'b"   and l :: "'b list" shows "mbhd_insert x l \<equiv> if l = [] then [x] else if x = hd l then l else x # l"
is_unit_unit_factor: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "dvd.dvd times a one" shows "unit_factor a = a"
neq_4k1_k43: fixes m :: "nat"   and n :: "nat"   and m' :: "nat"   and n' :: "nat" shows "(4 * real m + 1) / (2 * 2 ^ n) \<noteq> (4 * real m' + 3) / (2 * 2 ^ n')"
dual_comp: fixes x :: "'a"   and y :: "'a" shows "(x * y) ^ o = x ^ o * y ^ o"
c: shows "\<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>'"
mset_le_subtract_left: fixes A :: "'c multiset"   and B :: "'c multiset"   and X :: "'c multiset" assumes "A + B \<subseteq># X" shows "B \<subseteq># X - A \<and> A \<subseteq># X"
borel_measurable_iff_halfspace_ge: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"
assms_4: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>"   and act :: "'c" assumes "P t\<^sub>\<alpha>" shows "P (Act\<^sub>\<alpha> act t\<^sub>\<alpha>)"
torder_on_porder_on_consistent_tranclp_antisym: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "porder_on A r"   and "torder_on B s"   and "order_consistent r s"   and "B \<subseteq> A" shows "antisymp (\<lambda>x y. r x y \<or> s x y)\<^sup>+\<^sup>+"
inf_mono: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e"   and d :: "'e" 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)"
simps_2: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and ts :: "('a, 'b) tm list" shows "apps s (t # ts) = apps (App s t) ts"
simps: fixes x :: "'a"   and a :: "'a list"   and b :: "'a list" shows "Challenge1.insert x (BUF a b) = BUF (a @ [x]) b"
nths_append: fixes l :: "'a list"   and l' :: "'a list"   and A :: "nat set" shows "nths (l @ l') A = nths l A @ nths l' {j. j + |l| \<in> A}"
UN_extend_simps_4: fixes A :: "'e \<Rightarrow> 'd set"   and C :: "'e set"   and B :: "'d set" shows "\<Union> (A ` C) \<inter> B = (\<Union>x\<in>C. A x \<inter> B)"
assms_3: shows "x \<in># \<Sum>\<^sub># (mset_set (mset_set ` P))"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "wave \<Gamma> f = (separating \<Gamma> (TER\<^bsub>\<Gamma>\<^esub> f) \<and> (\<forall>x. x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<longrightarrow> d_OUT f x = 0))"
in_rtrancl_UnI: fixes x :: "'ap \<times> 'ap"   and R :: "('ap \<times> 'ap) set"   and S :: "('ap \<times> 'ap) set" assumes "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>*" shows "x \<in> (R \<union> S)\<^sup>*"
OK_OK_conv: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty"   and T' :: "ty" shows "P \<turnstile> OK T \<le>\<^sub>\<top> OK T' = subtype P T T'"
trans_not_greater: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "compare cmp a b \<noteq> Greater"   and "compare cmp b c \<noteq> Greater" shows "compare cmp a c \<noteq> Greater"
simps_6: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Is :: "char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and es :: "char list set"   and SS :: "char list \<Rightarrow> char list set"   and b :: "bexp"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and LVQ :: "char list set"   and LVE :: "char list set" shows "vc ({((I, Is), S, E, es, SS)} WHILE b DO C) Q LVQ LVE = ((\<forall>s1 s2 l. s1 = s2 on Is \<longrightarrow> I l s1 = I l s2) \<and> (\<forall>y\<in>LVE \<union> LVQ. let Ss = SS y in \<forall>s1 s2. s1 = s2 on Ss \<longrightarrow> S s1 y = S s2 y) \<and> (\<forall>s1 s2. s1 = s2 on es \<longrightarrow> E s1 = E s2) \<and> (\<forall>l s. (I l s \<and> bval b s \<longrightarrow> pre C I l s \<and> 1 + preT C E s + time C s \<le> E s \<and> S s = S (postQ C s) on \<Union> (funStar SS ` (LVE \<union> LVQ))) \<and> (I l s \<and> \<not> bval b s \<longrightarrow> Q l s \<and> 1 \<le> E s \<and> S s = s on \<Union> (funStar SS ` (LVE \<union> LVQ)))) \<and> vc C I Is (es \<union> \<Union> (funStar SS ` LVE)))"
rep_eq: fixes x :: "perm"   and xa :: "('a, 'b, 'c, 'd) formula" shows "Rep_formula (x \<bullet> xa) = x \<bullet> Rep_formula xa"
zcount_image_zmset: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d zmultiset"   and x :: "'c" shows "zcount (Auxiliary.image_zmset f M) x = sum (zcount M) (f -` {x} \<inter> set_zmset M)"
hyps: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updTs ts TAS) (upd_invs I P TAS)"
dfa_of_pf_correctness: fixes n :: "nat"   and bss :: "bool list list"   and p :: "pf" assumes "list_all (is_alph n) bss" shows "dfa_accepts (dfa_of_pf n p) bss = eval_pf p (nats_of_boolss n bss)"
finsum_closed: fixes f :: "'h \<Rightarrow> 'c"   and A :: "'h set" assumes "f \<in> A \<rightarrow> carrier M" shows "finsum M f A \<in> carrier M"
absorb_iff2: fixes b :: "'c"   and a :: "'c" shows "(b \<le> a) = (a \<sqinter> b = b)"
vproduct_vsingletonD_2: fixes f :: "V"   and c :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {c}) A" shows "f\<lparr>c\<rparr> \<in>\<^sub>\<circ> A c"
length_0_conv: fixes xs :: "'c list" shows "(|xs| = 0) = (xs = [])"
next_change_multiplicity'_def: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" shows "next_change_multiplicity' c0 c1 loc t n \<equiv> n \<noteq> 0 \<and> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t) \<and> c1 = c0\<lparr>c_pts := (c_pts c0)(loc := update_zmultiset (c_pts c0 loc) t n), c_work := (c_work c0)(loc := c_work c0 loc + frontier_changes (update_zmultiset (c_pts c0 loc) t n) (c_pts c0 loc))\<rparr>"
VUnion_vinsert: fixes x :: "V"   and A :: "V" shows "\<Union>\<^sub>\<circ> (vinsert x A) = x \<union>\<^sub>\<circ> \<Union>\<^sub>\<circ> A"
union_lcomm: fixes M :: "'f multiset"   and N :: "'f multiset"   and K :: "'f multiset" shows "M + (N + K) = N + (M + K)"
conf_f_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "conf_f P h \<equiv> \<lambda>(ST, LT) is (stk, loc, C, M, pc). P,h \<turnstile> stk [:\<le>] ST \<and> P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT \<and> pc < length is"
gcd_div_unit1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "gcd (b div a) c = gcd b c"
fold_combine_plus_def: shows "fold_combine_plus = fold_combine_plus.combine.F"
limsup_root_limit: fixes f :: "nat \<Rightarrow> 'b"   and l :: "ereal" assumes "(\<lambda>n. ereal (root n (norm (f n)))) \<longlonglongrightarrow> l" shows "limsup (\<lambda>n. ereal (root n (norm (f n)))) = l"
simps_5: fixes t :: "Core.term"   and u :: "Core.term" shows "fv (t $ u) = fv t \<union> fv u"
nnegI: fixes P :: "'a \<Rightarrow> 'b" assumes "\<And>x. (0::'b) \<le> P x" shows "nneg P"
zero_vec_def: shows "0 \<equiv> \<chi>i. (0::'b)"
associative_weak_composition_axioms: shows "associative_weak_composition (\<cdot>) (\<star>) \<a>"
section_retraction_of_iso_1: fixes f :: "'s"   and g :: "'s" assumes "S.iso f"   and "S.ide (g \<cdot>\<^sub>S f)" shows "S.inverse_arrows f g"
intro: fixes Arr :: "'a set"   and Dom :: "'a \<Rightarrow> 'b"   and Obj :: "'b set"   and Cod :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> Arr \<Longrightarrow> Dom x \<in> Obj"   and "\<And>x. x \<in> Arr \<Longrightarrow> Cod x \<in> Obj" shows "graph Obj Arr Dom Cod"
assms_3: shows "indexed_eval_aux Ps i m \<noteq> \<zero>"
psi_v_2: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> V" shows "\<exists>i. psi_v i = f"
Keys_Un: fixes A :: "('e \<Rightarrow>\<^sub>0 'f) set"   and B :: "('e \<Rightarrow>\<^sub>0 'f) set" shows "Keys (A \<union> B) = Keys A \<union> Keys B"
slice_kind_Upd: fixes a :: "'edge"   and S :: "'node SDG_node set"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = \<Up>f" shows "slice_kind S a = \<Up>id"
assms: shows "xs = [] \<or> ys = [] \<or> last xs \<noteq> hd ys"
"100_2": fixes k :: "nat" shows "snd ((p ## r) !! Suc k) = st_succ A (w !! k) (fst ((p ## r) !! Suc k)) (snd ((p ## r) !! k))"
right_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * b * c = a * (b * c)"
h2: shows "\<forall>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<longrightarrow> list_constr (fst elem1 @ map ((+) (length qs1)) (fst elem2)) (length (qs1 @ qs2))"
sup_inf_distrib2: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c"   and z :: "'c"   and x :: "'c" assumes "class.distrib_lattice inf less_eq less sup" shows "sup (inf y z) x = inf (sup y x) (sup z x)"
satisfies_T0: shows "is_left_adjoint u\<^sub>0"
h3: shows "isolate_variable_sparse p x (i + 1) = 0 \<longrightarrow> MPoly_Type.degree (f i) x = 0"
fin2: shows "2 * \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M \<noteq> \<infinity>"