hcju/isabelleps huggingface dataset:
quot_Exists_imp_Pf: fixes e :: "(name, hf) finfun"   and \<alpha> :: "fm"   and \<beta> :: "fm"   and i :: "name" assumes "Pf \<lbrakk>\<guillemotleft>\<alpha> IMP \<beta>\<guillemotright>\<rbrakk>e"   and "atom i \<sharp> \<beta>" shows "Pf \<lbrakk>\<guillemotleft>SyntaxN.Ex i \<alpha> IMP \<beta>\<guillemotright>\<rbrakk>e"
headR_def: fixes t :: "('c, 'd) FingerTree" shows "headR t = FingerTreeStruc.headR (Rep_FingerTree t)"
app_mono: fixes p :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and \<tau> :: "(ty list \<times> ty err list) option"   and \<tau>' :: "(ty list \<times> ty err list) option"   and i :: "instr"   and m :: "nat"   and rT :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>' \<tau>'"   and "app i P m rT pc mpc xt \<tau>'" shows "app i P m rT pc mpc xt \<tau>"
advantage2_def: fixes \<G> :: "'b cyclic_group"   and x :: "nat"   and \<A> :: "'b \<Rightarrow> (nat \<times> nat) spmf" assumes "dis_log_alt \<G>" shows "dis_log_alt.advantage2 \<G> x \<A> \<equiv> spmf (dis_log_alt.dis_log2 \<G> x \<A>) True"
rtf_init_n_def: fixes u :: "nat" shows "rtf_init_n u \<equiv> if u \<in> V - {s, t} then adjacent_nodes u else {}"
set_mset_add_mset_insert: fixes a :: "'d"   and A :: "'d multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
cfs_add_nat_pow: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "p \<in> carrier P" shows "([n] \<cdot>\<^bsub>P\<^esub> p) k = [n] \<cdot> p k"
labeled_family_saturation_lifting: fixes NL :: "('f \<times> 'l) set" assumes "saturated NL" shows "no_labels.empty_ord.saturated (fst ` NL)"
star_left_induct_mult_iff: fixes x :: "'a"   and y :: "'a" shows "(x * y \<le> y) = (x\<^sup>\<star> * y \<le> y)"
get_parent_ok: fixes h :: "(_) heap"   and ptr :: "(_) node_ptr" assumes "DocumentClass.type_wf h"   and "DocumentClass.known_ptrs h"   and "ptr |\<in>| node_ptr_kinds h" shows "h \<turnstile> ok get_parent ptr"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
comparable: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
rel_restrictI: fixes x :: "'a"   and R :: "'a set"   and y :: "'a"   and E :: "('a \<times> 'a) set" assumes "x \<notin> R"   and "y \<notin> R"   and "(x, y) \<in> E" shows "(x, y) \<in> rel_restrict E R"
h2: shows "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
OstrowskiTr8: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x)" shows "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x))\<^bsup>\<hyphen> K\<^esup>)"
rep_eq: fixes x :: "'a reg_lan" shows "ewp x = l_ewp (Rep_reg_lan x)"
isLastAssertedCharacterization_2: fixes literal :: "Literal"   and literalList :: "Literal list"   and valuation :: "Literal list" assumes "isLastAssertedLiteral literal (oppositeLiteralList literalList) valuation" shows "literalTrue literal valuation"
word_and_mask_or_conv_and_mask: fixes n :: "'a word"   and index :: "nat" assumes "n !! index" shows "n AND mask index OR (1 << index) = n AND mask (index + 1)"
inj_map_strong: fixes x :: "'a \<times> 'b"   and xa :: "'a \<times> 'b"   and f1 :: "'a \<Rightarrow> 'c"   and f1a :: "'a \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'd"   and f2a :: "'b \<Rightarrow> 'd" assumes "\<And>z1 z1a. \<lbrakk>z1 \<in> Basic_BNFs.fsts x; z1a \<in> Basic_BNFs.fsts xa; f1 z1 = f1a z1a\<rbrakk> \<Longrightarrow> z1 = z1a"   and "\<And>z2 z2a. \<lbrakk>z2 \<in> Basic_BNFs.snds x; z2a \<in> Basic_BNFs.snds xa; f2 z2 = f2a z2a\<rbrakk> \<Longrightarrow> z2 = z2a"   and "map_prod f1 f2 x = map_prod f1a f2a xa" shows "x = xa"
summable: fixes x :: "'a" assumes "x \<in> A" shows "f x abs_summable_on B' x"
CollectI: fixes P :: "'a::type \<Rightarrow> bool"   and a :: "'a::type" assumes "(P::'a::type \<Rightarrow> bool) (a::'a::type)" shows "(a::'a::type) \<in> {x::'a::type. (P::'a::type \<Rightarrow> bool) x}"
results_gpv_Fail: fixes \<Gamma> :: "('e, 'f) \<I>" shows "results_gpv \<Gamma> Fail = {}"
delayed_other_s_eq: fixes t :: "real" assumes "0 \<le> t" shows "delayed_safe_distance.other.s t = other.s (t + \<delta>)"
IH: fixes y :: "('a \<times> 'a option) list" assumes "length y - length A < length xa - length A"   and "insert_ops A"   and "insert_ops y"   and "set A \<subseteq> set y"   and "list_order A x y" shows "list_order y x y"
hyps: fixes S :: "nat set" assumes "S \<subseteq> {..<length as}" shows "map_pmf (\<lambda>l. nths l S) (bv (length as)) = bv (length (nths as S))"
verticesFrom_Def_2: fixes vs :: "'a list"   and v :: "'a" shows "rotate_to vs v \<equiv> v # snd (splitAt v vs) @ fst (splitAt v vs)"
digraph_isomorphism_def: fixes hom :: "('a, 'b, 'c, 'd) digraph_isomorphism" shows "digraph_isomorphism hom \<equiv> wf_digraph G \<and> inj_on (iso_verts hom) (pre_digraph.verts G) \<and> inj_on (iso_arcs hom) (pre_digraph.arcs G) \<and> (\<forall>a\<in>pre_digraph.arcs G. iso_verts hom (tail G a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head G a) = iso_head hom (iso_arcs hom a))"
terminates_Normal_elim_cases_11: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>Catch c1 c2 \<down> Normal s" obtains "\<Gamma>\<turnstile>c1 \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s'"
nn_integral_def_finite: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "integral\<^sup>N M f = Sup (integral\<^sup>S M ` {g. simple_function M g \<and> g \<le> f \<and> (\<forall>x. g x < top)})"
lm043: fixes XX :: "'c set set"   and X :: "'c set"   and x :: "'c set" assumes "\<Union> XX \<subseteq> X"   and "x \<in> XX"   and "x \<noteq> {}" shows "x \<inter> X \<noteq> {}"
monoid: shows "Group.monoid (S\<lparr>carrier := h ` carrier R, zero := h \<zero>\<rparr>)"
not_switch: fixes a :: "'b"   and x :: "'b" assumes "NOT a = x" shows "a = NOT x"
Diagonalize_Tensor_Arr_Unity: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t \<^bold>\<otimes> \<^bold>\<I>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
simps_2: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "OPT2 [a] [x, y] = [(0, [])]"
allSet: fixes P :: "'a \<Rightarrow> 'b set"   and Q :: "'b \<Rightarrow> bool"   and rb :: "'b"   and d :: "'a" assumes "\<forall>a. \<forall>r\<in>P a. Q r"   and "rb \<in> P d" shows "Q rb"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and r :: "('b \<times> 'b) set" shows "((a1, a2) \<in> r\<^sup>+) = ((\<exists>a b. a1 = a \<and> a2 = b \<and> (a, b) \<in> r) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r\<^sup>+ \<and> (b, c) \<in> r))"
cancel_separate_inj_converse: fixes x :: "'a" assumes "is_inj x" shows "x\<^sup>\<star> ; x\<^sup>T\<^sup>\<star> = x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>"
cnf2bhv_mono: fixes n :: "nat"   and n' :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'" shows "dynamic_component.cnf2bhv tCMP active c t n \<le> dynamic_component.cnf2bhv tCMP active c t n'"
assms: shows "\<langle>a, b\<rangle> \<in> elts (VSigma A B)"
sigma_finite_pair_measure: fixes A :: "'b measure"   and B :: "'c measure" assumes "sigma_finite_measure A"   and "sigma_finite_measure B" shows "sigma_finite_measure (A \<Otimes>\<^sub>M B)"
assms_2: shows "\<forall>(j, p)\<in>{(i, t, s)} \<union> set D. \<forall>(k, q)\<in>{(i, t, s)} \<union> set D. (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"
BIT_inv_initial: fixes x :: "nat"   and y :: "nat" assumes "x \<noteq> y" shows "BIT_inv (map_pmf (Pair [x, y]) (fst BIT [x, y])) x [x, y]"
Dup_def: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" shows "Dup p i branch xs \<equiv> \<forall>(v, v')\<in>xs. is_at p i branch v v' \<and> is_elsewhere p i branch xs"
poly_mult_degree_eq: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p1"   and "polynomial\<^bsub>K [X]\<^esub> K p2" shows "degree (UP.poly_mult p1 p2) = (if p1 = [] \<or> p2 = [] then 0 else degree p1 + degree p2)"
fv_transaction_unfold: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" shows "fv_transaction T = fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T)"
wfp_on_imp_accessible_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wfp_on P A" shows "\<forall>x\<in>A. accessible_on P A x"
index_oddpart_1: fixes m :: "nat"   and i :: "nat" assumes "odd m" shows "index (2 ^ i * m) = i"
IH1: assumes "\<Gamma> \<turnstile> e1 : Low"   and "x \<in> fv e1" shows "\<Gamma> x = Some Low"
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"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" assumes "arrow_of_spans_of_maps_to_tabulation_data V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 w"   and "tabulation V H \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1" shows "arrow_of_spans_of_maps_to_tabulation V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 w"
iuf: shows "ea \<in> fst ` set (routing_port_ranges tbl wordinterval_UNIV) \<and> eb = wordinterval_Union (map snd (filter (\<lambda>x. ea = fst x) (routing_port_ranges tbl wordinterval_UNIV)))"
adm_subst: fixes t :: "'b \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool" assumes "cont t"   and "adm P" shows "adm (\<lambda>x. P (t x))"
ZObis_converse: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "((c, d) \<in> theta\<inverse> \<union> ZObis) = ((d, c) \<in> theta \<union> ZObis)"
ground_heads_arity: fixes f :: "'s"   and \<zeta> :: "('s, 'v) hd" assumes "f \<in> local.ground_heads \<zeta>" shows "arity_hd \<zeta> \<le> arity_sym f"
complement_part_not_part: fixes x :: "i"   and y :: "i" assumes "x \<noteq> u"   and "P y (\<midarrow> x)" shows "\<not> P y x"
max_norm_def: fixes A :: "((real, 'c) vec, 'b) vec" shows "\<parallel>A\<parallel>\<^sub>m\<^sub>a\<^sub>x = Max {\<bar>A $ i $ j\<bar> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
lub_in_Field: fixes lub :: "'b set \<Rightarrow> 'b"   and leq :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and M :: "'b set" assumes "bourbaki_witt_fixpoint lub leq f"   and "M \<in> Chains leq"   and "M \<noteq> {}" shows "lub M \<in> Field leq"
oalist_inv_map2_val_raw: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ys :: "('a \<times> 'f) list \<times> 'o"   and g :: "('a \<times> 'e) list \<times> 'o \<Rightarrow> ('a \<times> 'g) list \<times> 'o"   and h :: "('a \<times> 'f) list \<times> 'o \<Rightarrow> ('a \<times> 'g) list \<times> 'o"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "oalist_inv xs"   and "oalist_inv ys"   and "map2_val_compat' g"   and "map2_val_compat' h" shows "oalist_inv (map2_val_raw f g h xs ys)"
ind_in_set_id: fixes A :: "nat set" assumes "finite A" shows "ind_in_set A ` A = {0..<card A}"
nonempty_subseteq_mset_eq_singleton: fixes M :: "'b multiset"   and x :: "'b" assumes "M \<noteq> {#}"   and "M \<subseteq># {#x#}" shows "M = {#x#}"
l_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas_axioms: shows "l_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas DocumentClass.type_wf"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and sl :: "name"   and sl' :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name"   and t :: "tm"   and u :: "tm" assumes "atom l \<sharp> (s, k, sl, sl', m, n, sm, sm', sn, sn')"   and "atom sl \<sharp> (s, sl', m, n, sm, sm', sn, sn')"   and "atom sl' \<sharp> (s, m, n, sm, sm', sn, sn')"   and "atom m \<sharp> (s, n, sm, sm', sn, sn')"   and "atom n \<sharp> (s, sm, sm', sn, sn')"   and "atom sm \<sharp> (s, sm', sn, sn')"   and "atom sm' \<sharp> (s, sn, sn')"   and "atom sn \<sharp> (s, sn')"   and "atom sn' \<sharp> s" shows "SeqQuoteP t u s k = LstSeqP s k (HPair t u) AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (Var sl EQ Zero AND Var sl' EQ Zero OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND Var sl EQ Eats (Var sm) (Var sn) AND Var sl' EQ Q_Eats (Var sm') (Var sn')))))))))))"
satisfies_T0: shows "is_left_adjoint r\<^sub>0s\<^sub>1.p\<^sub>0"
single_valued_confluent: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "single_valued r"   and "(x, y) \<in> r\<^sup>*"   and "(x, z) \<in> r\<^sup>*" shows "(y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
dependent_zero: fixes A :: "'d set" assumes "(0::'d) \<in> A" shows "cdependent A"
Diff_Int_distrib: fixes C :: "'a set"   and A :: "'a set"   and B :: "'a set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
\<D>_2: fixes K :: "'a set" assumes "K \<in> \<D>" shows "K \<subseteq> cbox a b"
power_add: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "a ^ (m + n) = a ^ m * a ^ n"
mono_restriction_is_mono: fixes f :: "'a set \<Rightarrow> 'c" assumes "mono f" shows "mono (\<lambda>S. f (S \<inter> V))"
allVars_def: fixes u :: "'g \<Rightarrow> ('node, 'val set) mapping"   and p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" shows "uninst_code.ssa.allVars u p g \<equiv> \<Union>n\<in>set (\<alpha>n g). uninst_code.ssa.allDefs p g n \<union> uninst_code.ssa.allUses u p g n"
dotprod_less_left: fixes b :: "nat list"   and w :: "nat list"   and v :: "nat list" assumes "length b = length w"   and "0 \<notin> set b"   and "v <\<^sub>v w" shows "v \<bullet> b < w \<bullet> b"
a_conditions_def: fixes M :: "('c, 'd) insert_message set"   and a :: "'c extended \<Rightarrow> 'c sort_key extended" shows "a_conditions M a = (a \<turnstile> < a \<stileturn> \<and> (\<forall>m. m \<in> M \<longrightarrow> a (P m) < a (S m) \<and> a \<lbrakk>I m\<rbrakk> = \<lbrakk>\<Psi> (a (P m), a (S m)) (I m)\<rbrakk>))"
nth_list_of: fixes xs :: "'a llist" assumes "lfinite xs" shows "(!) (list_of xs) = ($) xs"
elims_2: fixes x :: "ereal"   and xa :: "ereal" assumes "x < xa" obtains x :: "real" and y :: "real" where "xa__ = ereal xb__" and "xa = ereal y" and "xb__ < y"   | x :: "real" where "xa__ = ereal x" and "xa = \<infinity>"   | r :: "real" where "xa__ = - \<infinity>" and "xa = ereal r"   | "xa__ = - \<infinity>" and "xa = \<infinity>"
RTrans: shows "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> R \<longmapsto> K\<lparr>N\<rparr> \<prec> R'"
k_def: shows "k = the (eval r_find_k [local.prefixes j t])"
assms_5: shows "R = Ax \<union> R1 \<union> p_e R2 M N \<union> R3"
pgwt_is_empty_synth: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana" shows "intruder_model.public_ground_wf_term arity public t = intruder_model.intruder_synth arity public {} t"
ok_shift: fixes i :: "nat"   and m :: "nat"   and r :: "mregex" shows "ok (i + m) (Monitor.shift r i) = ok m r"
simps_15: fixes x1 :: "'c" shows "set1_blindable\<^sub>m (Unblinded x1) = {x1}"
prems_\<iota>\<^sub>0_in: shows "set (prems_of \<iota>\<^sub>0) \<subseteq> \<Union> (\<G>_F ` M)"
that: fixes z :: "'a" assumes "poly p z = (0::'a)" shows "thesis"
sum_list_map2_plus: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs = length ys" shows "sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys"
hyps_4: fixes \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" assumes "fv (as @ \<Gamma>, body) \<subseteq> set L \<union> domA (as @ \<Gamma>)" shows "\<N>\<lbrakk> body \<rbrakk>\<^bsub>\<N>\<lbrace>as @ \<Gamma>\<rbrace>\<rho>\<^esub> \<sqsubseteq> \<N>\<lbrakk> z \<rbrakk>\<^bsub>\<N>\<lbrace>\<Delta>\<rbrace>\<rho>\<^esub>"
hyps_3: fixes P :: "'c multiset \<Rightarrow> 'a" assumes "P \<in> carrier L" shows "index_free P i"
valid_path_in_V: fixes G :: "('c, 'd) Graph_scheme"   and P :: "'c llist" assumes "Digraph G"   and "Digraph.valid_path G P" shows "lset P \<subseteq> V\<^bsub>G\<^esub>"
has_contour_integral_bound_circlepath: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and z :: "complex"   and r :: "real"   and B :: "real" assumes "(f has_contour_integral i) (circlepath z r)"   and "0 \<le> B"   and "0 < r"   and "\<And>x. cmod (x - z) = r \<Longrightarrow> cmod (f x) \<le> B" shows "cmod i \<le> B * (2 * pi * r)"
MemI: fixes t :: "tm"   and u :: "tm" shows "se_fm (t IN u)"
has_white_path_to_step_1: fixes x :: "'a"   and y :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and z :: "'a" assumes "(x has_white_path_to y) s"   and "(y points_to z) s"   and "white z s" shows "(x has_white_path_to z) s"
e22: shows "0 < e / 2 / (2 * B * real DIM('M)) ^ DIM('N)"
op_dg_is_arr: fixes \<CC> :: "V"   and b :: "V"   and a :: "V"   and f :: "V" shows "f : b \<mapsto>\<^bsub>op_dg \<CC>\<^esub> a = f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
order_on_defs_3: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> Relation.total_on A r"
absolutely_integrable_spike_eq: fixes S :: "'a set"   and T :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "(f absolutely_integrable_on T) = (g absolutely_integrable_on T)"
distinct_indices_state_def: fixes s :: "('c, 'd) state" shows "distinct_indices_state s = (\<forall>i x b x' b'. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x' = Some (i, b') \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x' = Some (i, b') \<longrightarrow> x = x' \<and> b = b')"
commute: fixes a :: "'d"   and b :: "'d" shows "a \<squnion> b = b \<squnion> a"
Syntax_with_Numerals_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set" shows "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num \<equiv> Generic_Syntax var trm fmla Var FvarsT substT Fvars subst \<and> Syntax_with_Numerals_axioms trm FvarsT num"
times_def: fixes a :: "'a G"   and b :: "'a G" shows "a * b \<equiv> Abs_G (Rep_G a - u + Rep_G b \<squnion> (0::'a))"
closed_open_segment_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "closed_segment a b \<noteq> open_segment c d"
midpoint_distinct_1: fixes A :: "'p"   and B :: "'p"   and I :: "'p" assumes "A \<noteq> B"   and "I Midpoint A B" shows "I \<noteq> A \<and> I \<noteq> B"
refl_on_reflcl_Image: fixes B :: "'a set"   and A :: "('a \<times> 'a) set"   and C :: "'a set" assumes "refl_on B A"   and "C \<subseteq> B" shows "A\<^sup>= `` C = A `` C"
simps_5: fixes x :: "('c, 'd) Node"   and xs :: "('c, 'd) Node list"   and v :: "'d"   and va :: "('c, 'd) Digit"   and vb :: "('c, 'd) FingerTreeStruc"   and vc :: "('c, 'd) Digit" shows "app3 (Single x) xs (Deep v va vb vc) = nlcons x (lconsNlist xs (Deep v va vb vc))"
Ax: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset" shows "\<^bold>\<not> (Atom k), Atom k, \<Gamma> \<Rightarrow>\<^sub>n"
truncate_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" shows "Cring_Poly.truncate R f = f \<ominus>\<^bsub>UP R\<^esub> leading_term R f"
simps_4: fixes F :: "'b formula" shows "to_mini_formula (\<^bold>\<not> F) = to_mini_formula F \<^bold>\<rightarrow> \<bottom>"
da_elim_cases_29: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and T :: "ref_ty"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e InstOf T\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
while_SE_unfold: fixes b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "_while_SE b c = _if_SE b (c ;- _while_SE b c) (unit\<^sub>S\<^sub>E ())"
ct_1: fixes s :: "'s" shows "ct s \<in> K s"
runProc_pState_inv: fixes prog :: "program"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list"   and g :: "'a gState_scheme" assumes "pState_inv prog p" shows "pState_inv prog (snd (runProc name args prog g p))"
le_cong_r: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y .= z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq> z"
measurable_enn2ereal: shows "enn2ereal \<in> borel_measurable borel"
AnalBinds_Cons: fixes x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "FBinds ((x, e) # \<Gamma>)\<cdot>ae = (FBinds \<Gamma>\<cdot>ae)(x := fup\<cdot>(Texp e)\<cdot>(ae x))"
IH_1: fixes n :: "nat"   and R :: "'a list set"   and \<phi>' :: "'a MFOTL.formula" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'" shows "case meval n (\<tau> \<sigma> j) (\<Gamma> \<sigma> j) \<phi> of (xs, \<phi>\<^sub>n) \<Rightarrow> wf_mformula \<sigma> (Suc j) n R \<phi>\<^sub>n \<phi>' \<and> list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>')) [Monitor.progress \<sigma> \<phi>' j..<Monitor.progress \<sigma> \<phi>' (Suc j)] xs"
simps_4: fixes lam :: "lambda"   and \<beta> :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "\<A> (L lam) \<beta> ve = DC (lam, \<beta>)"
val_1mx_pow: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x)" shows "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x^\<^bsup>K Suc n\<^esup>)"
scale_sum_left: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and x :: "('a, 'b) vec" shows "sum f A *s x = (\<Sum>a\<in>A. f a *s x)"
l11_29_a: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "\<exists>Q. Tarski_neutral_dimensionless.InAngle Bet C A B Q \<and> Tarski_neutral_dimensionless.CongA Bet Cong A B Q D E F"
factorization_m_lead_coeff: fixes m :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "poly_mod_2 m"   and "poly_mod.factorization_m m f (c, fs)" shows "lead_coeff (poly_mod.Mp m f) = poly_mod.M m c"