hcju/isabelleps huggingface dataset:
err_step_def: fixes n :: "nat"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and p :: "nat"   and t :: "'a err" shows "err_step n app step p t = (case t of Err \<Rightarrow> error n | OK \<tau> \<Rightarrow> if app p \<tau> then map_snd OK (step p \<tau>) else error n)"
numneg_nz: fixes a :: "MIR.num" assumes "nozerocoeff a" shows "nozerocoeff (numneg a)"
divide_le_eq_1_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less_eq (divide b a) one = less_eq a b"
simps_1: fixes uu :: "'c"   and uv :: "'c"   and F :: "'b" shows "Maps.Map (Maps.MkArr uu uv F) = F"
is_rerr_def: fixes \<xi> :: "'b state_scheme" shows "is_rerr \<xi> \<equiv> case msg \<xi> of Rerr dests' sip' \<Rightarrow> {\<xi>\<lparr>dests := dests', sip := sip'\<rparr>} | _ \<Rightarrow> {}"
edc_val: fixes v :: "val"   and vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "countable_type (val_type v)" shows "(vs, vs', \<Gamma>, \<delta>) \<turnstile>\<^sub>c Val v \<Rightarrow> map_vars Suc (branch_prob_cexpr (vs, vs', \<Gamma>, \<delta>)) *\<^sub>c \<langle>CVar 0 =\<^sub>c CVal v\<rangle>\<^sub>c"
elims: fixes x :: "(int \<times> (real \<Rightarrow> real \<times> real)) list"   and y :: "real \<Rightarrow> real \<times> real" assumes "rec_join x = y" obtains "x = []" and "y = (\<lambda>x. 0)"   | oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" where "x = [oneC__]" and "y = coeff_cube_to_path oneC__"   | oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" and v :: "int \<times> (real \<Rightarrow> real \<times> real)" and va :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" where "x = oneC # v # va" and "y = coeff_cube_to_path oneC +++ rec_join (v # va)"
chambersubcomplex_image: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexEndomorphism X f" shows "ChamberComplex.ChamberSubcomplex X (f \<turnstile> X)"
extend_cbox: fixes K :: "'a set" assumes "K \<in> \<D>" shows "\<exists>a b. extend K = cbox a b"
comp_red_monic_basis_is_monic_set: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "is_monic_set (set (comp_red_monic_basis xs))"
compatible_tail: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "compatible G H" shows "tail G = tail H"
smem_ind_def: fixes s :: "s"   and i :: "nat" shows "smem_ind s i = inst.mem (inst s ! i)"
assms_2: shows "x \<in>\<^sub>\<circ> X"
Lp_measurable_subset: fixes p :: "ennreal"   and M :: "'a measure" shows "lspace M p \<subseteq> borel_measurable M"
fst_apsnd: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'b \<times> 'd" shows "fst (apsnd f x) = fst x"
power_le_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a n) one"
cancel_le: fixes x :: "ordinal"   and y :: "ordinal" shows "(F x \<le> F y) = (x \<le> y)"
hd: shows "hd \<HH> = {\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>}"
list_all2_map_map: fixes xs :: "'a list"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> R (f x) (g x)" shows "list_all2 R (map f xs) (map g xs)"
assms: shows "rt1 \<approx>\<^bsub>dip\<^esub> rt2"
map_of_alist_some: fixes xs :: "('c \<times> 'b) list"   and k :: "'c"   and v :: "'b" assumes "map_of_alist xs\<langle>k\<rangle> = Some v" shows "(k, v) \<in> set xs"
hom_0_iff: fixes x :: "'b poly" shows "(map_poly to_fract x = 0) = (x = 0)"
assms_1: shows "y \<in> vars_mset (subst_mset (subst x t) M)"
prems: shows "free_vars (\<integral>\<^sub>c e \<partial>t) \<subseteq> V"
dist_at2: fixes vs :: "'c list"   and a :: "'c list"   and r :: "'c"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "b = d"
bet2_le2__le2356_R2: 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 A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Bet A B C"   and "Bet A' B' C'"   and "Tarski_neutral_dimensionless.Le Bet Cong A B A' B'"   and "Tarski_neutral_dimensionless.Le Bet Cong A' C' A C" shows "Tarski_neutral_dimensionless.Le Bet Cong B' C' B C"
p_eq: shows "p = (det3 b c p / det3 a b c) *\<^sub>R a + (det3 c a p / det3 a b c) *\<^sub>R b + (det3 a b p / det3 a b c) *\<^sub>R c"
w_simps_3: shows "src TfHgh_HfHgh.chine = src (tab\<^sub>0 (g \<star> h) \<star> TfHgh.p\<^sub>0)"
Restr_Field: fixes r :: "('a \<times> 'a) set" shows "Restr r (Field r) = r"
correctCompositionKS_exprChannel_k_Q: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and key :: "Keys"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "kKS key \<notin> LocalSecrets PQ"   and "ch \<in> ins Q"   and "exprChannel ch (kE key)"   and "kKS key \<notin> specKeysSecrets PQ"   and "correctCompositionIn PQ" shows "ch \<in> ins PQ \<and> exprChannel ch (kE key)"
\<phi>_in_hom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and y :: "'d"   and f :: "'c"   and x :: "'c" assumes "right_adjoint_functor C D G"   and "partial_magma.ide D y"   and "partial_magma.in_hom C f (right_adjoint_functor.F C D G y) x" shows "partial_magma.in_hom D (right_adjoint_functor.\<phi> C D G y f) y (G x)"
responses_\<I>_empty_iff: fixes \<I> :: "('i, 'h) \<I>"   and out :: "'i" shows "(responses_\<I> \<I> out = {}) = (out \<notin> outs_\<I> \<I>)"
var: fixes t :: "'trm" assumes "set (map snd txs) \<subseteq> var"   and "set (r # map fst txs) \<subseteq> trm"   and "set [] \<subseteq> fmla"   and "t \<in> set (r # map fst txs)" shows "set (getFrN (map snd txs) (r # map fst txs) [] (length txs)) \<inter> set (map snd txs) = {}"
intro: fixes t :: "nat \<Rightarrow> real"   and t1 :: "real"   and p :: "nat"   and r :: "real"   and L :: "real"   and B :: "real" assumes "\<And>j. t j \<le> t1 \<Longrightarrow> grid.max_stepsize t j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (t1 - t 0) + 1) - 1))" shows "max_step_axioms t t1 p L B r"
ceiling_le_one: fixes x :: "'b" shows "(\<lceil>x\<rceil> \<le> 1) = (x \<le> (1::'b))"
mset_set_empty_iff: fixes A :: "'b set" shows "(mset_set A = {#}) = (A = {} \<or> infinite A)"
RedsInitRInit: fixes C :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list" assumes "P \<turnstile> \<langle>RI (C,C\<bullet>\<^sub>sclinit([])) ; Cs \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
labeling_consistent: fixes a :: "('c, 'ls, 'gs) global_config"   and b :: "('d, 'ls, 'gs) global_config" assumes "(a, b) \<in> rel_of_pred rel_gc" shows "s1.sa.L a = s2.sa.L b"
generateI: fixes E :: "'a set"   and H :: "'a set" assumes "subgroup E G"   and "H \<subseteq> E"   and "\<And>K. \<lbrakk>subgroup K G; H \<subseteq> K\<rbrakk> \<Longrightarrow> E \<subseteq> K" shows "E = generate G H"
\<psi>_\<phi>: fixes i :: "'b" assumes "i \<in> S" shows "\<psi> (\<phi> i) = i"
rep_abs_rsp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and x1 :: "'e"   and x2 :: "'e" assumes "Quotient3 R Abs Rep"   and "R x1 x2" shows "R x1 (Rep (Abs x2))"
ipset_from_cidr_lowest: fixes a :: "'a word"   and n :: "nat" shows "a \<in> ipset_from_cidr a n"
distinct_203: fixes x121 :: "expr"   and x122 :: "expr"   and x15 :: "expr" shows "x121;; x122 \<noteq> throw x15"
Red1ALength: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and xs :: "'addr val list" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>length,(h, xs)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val (Intg (word_of_nat n)),(h, xs)\<rangle>"
unbounded_complement_components: fixes S :: "complex set"   and C :: "complex set" assumes "SC_Chain S"   and "C \<in> components (- S)"   and "connected S"   and "if bounded S then connected (frontier S) else \<forall>C\<in>components (frontier S). \<not> bounded C" shows "\<not> bounded C"
multiset_heapify_of_list_tree: fixes l :: "'c list" shows "Heap.multiset (heapify (of_list_tree l)) = mset l"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_all_front_tickfree_traces2: fixes s :: "'b event list" assumes "front_tickFree s" shows "s \<in> T (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)"
normalize_monic: fixes p :: "'c poly" assumes "monic p" shows "normalize p = p"
sum_g': shows "(\<Sum>\<^sup>+ x. ennreal (g' x)) = integral\<^sup>N (count_space B) g"
hcomp_def: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "\<nu> \<circ> \<mu> \<equiv> if Span.arr \<mu> \<and> Span.arr \<nu> \<and> Span.src \<nu> = Span.trg \<mu> then \<lparr>Chn = Span.chine_hcomp \<nu> \<mu>, Dom = \<lparr>Leg0 = Leg0 (Dom \<mu>) \<odot> Maps.PRJ\<^sub>0 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)), Leg1 = Leg1 (Dom \<nu>) \<odot> Maps.PRJ\<^sub>1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))\<rparr>, Cod = \<lparr>Leg0 = Leg0 (Cod \<mu>) \<odot> Maps.PRJ\<^sub>0 (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)), Leg1 = Leg1 (Cod \<nu>) \<odot> Maps.PRJ\<^sub>1 (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>))\<rparr>\<rparr> else Span.null"
continuous_on_pbernpoly: fixes n :: "nat"   and A :: "real set" assumes "n \<noteq> 1" shows "continuous_on A (pbernpoly n)"
1: shows "\<l>\<^sub>D[F f] \<cdot>\<^sub>D D.inv (unit (trg\<^sub>C f) \<star>\<^sub>D F f) = F \<l>\<^sub>C[f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, f)"
lcos_m_assoc: fixes M :: "'a set set"   and g :: "'a set"   and h :: "'a set" assumes "M \<subseteq> carrier (G Mod N)"   and "g \<in> carrier (G Mod N)"   and "h \<in> carrier (G Mod N)" shows "g <#\<^bsub>G Mod N\<^esub> (h <#\<^bsub>G Mod N\<^esub> M) = g \<otimes>\<^bsub>G Mod N\<^esub> h <#\<^bsub>G Mod N\<^esub> M"
degree_one_imp_splitted: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "degree p = 1" shows "splitted p"
extremum: fixes a :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "\<delta> \<le> a"
stack_reduction_def: fixes k :: "stack"   and k' :: "stack" shows " k \<mapsto> k'  \<equiv> \<forall>t. t \<star> k \<mapsto> t \<star> k'"
dropWhile_def: shows "dropWhile \<equiv> \<lambda>uu uua. rec_list (\<lambda>P. []) (\<lambda>x xs xsa P. if P x then xsa P else x # xs) uua uu"
spr_simAbs_list: fixes X :: "'s list set" shows "spr_simAbs ` fromList ` X = (\<lambda>ss. rel_ext (\<lambda>uu_. \<exists>s. uu_ = (ss, s) \<and> s \<in> ss)) ` set ` X"
iso_is_section: fixes f :: "'a" assumes "local.iso f" shows "section f"
is_ide: 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\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and w :: "'b" assumes "arrow_of_spans_of_maps V H \<a> \<i> src trg r\<^sub>0 r\<^sub>1 s\<^sub>0 s\<^sub>1 w" shows "partial_magma.ide V w"
per_union_def: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a" shows "per_union R a b \<equiv> R \<union> {(x, y). (x, a) \<in> R \<and> (y, b) \<in> R} \<union> {(y, x). (x, a) \<in> R \<and> (y, b) \<in> R}"
inj_comm_monoid_add_hom_axioms: shows "inj_comm_monoid_add_hom hom"
square_free_factorizationD_4: fixes p :: "'c poly"   and c :: "'c"   and bs :: "('c poly \<times> nat) list" assumes "square_free_factorization p (c, bs)"   and "p = 0" shows "c = (0::'c) \<and> bs = []"
antisym_conv1: fixes y :: "'a list"   and x :: "'a list" assumes "\<not> y <lex x" shows "(y \<le>lex x) = (x = y)"
normalize_def'_1: fixes p :: "'a list" shows "p = replicate (length p - length (local.normalize p)) \<zero> @ drop (length p - length (local.normalize p)) p"
P2: shows "Bet B A A0 \<and> Cong A A0 E D"
induced_automorphism_automorphism: shows "ChamberComplexAutomorphism X \
"
set_sort1: fixes f :: "'a list \<Rightarrow> 'a"   and xss :: "'a list list" shows "set (sort1 f xss) = set xss"
H: shows "\<Gamma> \<turnstile> (LET p = t\<^sub>1 IN t\<^sub>2) : T"
psi_def: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and x :: "nat" assumes "r1_bc s" shows "r1_bc.psi s j x \<equiv> Some (e_nth (r1_bc.prefixes s j (Suc x)) x)"
TAbs_type: fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and s :: "trm"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda><:S. s) : T"   and "\<Gamma> \<turnstile> T <: (\<forall><:U. U')" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
add_order_gt0: fixes a :: "'a" shows "order.greater (add_order a) 0 = (\<exists>n. order.greater n 0 \<and> a +^ n = (0::'a))"
pmf_set_map: fixes f :: "'c \<Rightarrow> 'd" shows "set_pmf \<circ> map_pmf f = (`) f \<circ> set_pmf"
uint_arith_simps_3: fixes a :: "'a word"   and b :: "'a word" shows "(a = b) = (uint a = uint b)"
lens_comp_right_id: fixes X :: "'d \<Longrightarrow> 'e" shows "X ;\<^sub>L 1\<^sub>L = X"
cases: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" assumes "(a1, a2) \<in> encoding.indRelL Enc" obtains S :: "'a" where "a1 = TargetTerm (Enc S)" and "a2 = SourceTerm S"
routing_ipassmt_def: fixes rt :: "'c routing_rule list" shows "routing_ipassmt rt \<equiv> map ((apfst Iface \<circ>\<circ> apsnd) cidr_split) (routing_ipassmt_wi rt)"
X_closed: shows "X_poly Zp \<in> carrier (UP Zp)"
flowD_outside: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "flow (residual_network f) g"   and "e \<notin> \<^bold>E\<^bsub>residual_network f\<^esub>" shows "g e = 0"
ladder_j_L'_thm: fixes index :: "nat" assumes "index < length L" shows "ladder_j L' index + length \<delta> = ladder_j L index"
true_cls_mset_def: fixes I :: "'a set"   and CC :: "'a literal multiset multiset" shows "(I \<Turnstile>m CC) = (\<forall>C\<in>#CC. I \<Turnstile> C)"
ex_in_conv: fixes A :: "'b::type set" shows "(\<exists>x::'b::type. x \<in> (A::'b::type set)) = (A \<noteq> {})"
cut_ge_not_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<ge> t \<noteq> {}) = (\<exists>x\<in>I. t \<le> x)"
INF_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Inf ((\<lambda>i. f) ` A) = f"
distinct_ParamDefs: fixes a :: "'a" assumes "valid_edge a" shows "distinct (ParamDefs (targetnode a))"
assms: shows "j \<in>\<^sub>\<circ> Vset \<alpha>"
131: fixes x :: "'a" shows "- x \<squnion> - - - x = - x"
cmp_simps_4: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (cmp (f, g)) = local.map f \<star>\<^sub>D local.map g"
dfs_code3'_correct: fixes v0i :: "'b"   and src :: "'c"   and Rv :: "('b \<times> 'c) set"   and vdi :: "'b"   and tgt :: "'c"   and succi :: "'b \<Rightarrow> 'b list"   and E :: "('c \<times> 'c) set"   and cmpk' :: "'c \<Rightarrow> 'c \<Rightarrow> comp_res"   and cmpk :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res" assumes "(v0i, src) \<in> Rv"   and "(vdi, tgt) \<in> Rv"   and "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"   and "eq_linorder cmpk'"   and "(cmpk, cmpk') \<in> Rv \<rightarrow> Rv \<rightarrow> comp_res_rel"   and "finite (E\<^sup>* `` {src})" shows "dfs_impl3' cmpk succi v0i vdi = ((src, tgt) \<in> E\<^sup>*)"
rep_eq: fixes x :: "integer" shows "Rep_uint (Uint x) = word_of_int (int_of_integer x)"
\<I>\<^sub>1_char: shows "\<I>\<^sub>1 = F \<I>\<^sub>C"
equiv_trans_equiv_utrans: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u" shows "equiv_utrans t u"
terminal_ttl: fixes xs :: "('g, 'f) tllist" shows "terminal (ttl xs) = terminal xs"
less_poly_def: fixes x :: "'a poly"   and y :: "'a poly" shows "(x < y) = pos_poly (y - x)"
Bossy''_CD_on_range: fixes X :: "'a set"   and X' :: "'a set" assumes "X \<subseteq> Bossy''.CD_on ds X'" shows "X \<subseteq> set [d1h1, d1h2, d1h3, d3h1, d3h2, d3h3]"
totatives_mod: fixes n :: "nat"   and x :: "nat" assumes "residues_nat n"   and "x \<in> totatives n" shows "x mod n = x"
has_field_derivative_cong_eventually: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and S :: "'a set"   and u :: "'a" assumes "\<forall>\<^sub>F x in at x within S. f x = g x"   and "f x = g x" shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative u) (at x within S)"
learn_lim_closed_subseteq: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and V :: "(nat \<Rightarrow> nat option) set" assumes "learn_lim \<psi> U s"   and "V \<subseteq> U" shows "learn_lim \<psi> V s"
monoid_axioms_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a" shows "monoid_axioms f z \<equiv> (\<forall>a. f z a = a) \<and> (\<forall>a. f a z = a)"
thm_2_3_6: fixes ord :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'b set"   and fs :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "set fs \<subseteq> P[X]"   and "pm_powerprod.is_GB_cofactor_bound ord (set fs) b" shows "ordered_term.is_Groebner_basis to_pair_unit fst ord (set (ordered_term.Macaulay_list ord ord_strict (pm_powerprod.deg_shifts ord X b fs)))"
mem_monad_approx: fixes u :: "'a star"   and x :: "'a star" assumes "u \<in> monad x" shows "x \<approx> u"
induct: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>k c. P k c 0"   and "\<And>k c n. \<lbrakk>P k c n; P k c n\<rbrakk> \<Longrightarrow> P k c (Suc n)" shows "P a0 a1 a2"
Abs_\<I>_inverse: fixes y :: "nat \<times> enat" assumes "y \<in> {(x, j). enat x \<le> j}" shows "Rep_\<I> (Abs_\<I> y) = y"
A: shows "padic_mult p f g (Suc (nat (padic_val p f + padic_val p g))) = f (Suc (nat (padic_val p f + padic_val p g))) \<otimes>\<^bsub>residue_ring (p ^ Suc (nat (padic_val p f + padic_val p g)))\<^esub> g (Suc (nat (padic_val p f + padic_val p g)))"
Hs: shows "map gs2.gso [0..<m] = map gs1.gso [0..<m]"
weakSimTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<rhd> Q \<leadsto><Rel'> R"   and "eqvt Rel''"   and "{(\<Psi>, P, R) |\<Psi> P R. \<exists>Q. (\<Psi>, P, Q) \<in> Rel \<and> (\<Psi>, Q, R) \<in> Rel'} \<subseteq> Rel''"   and "\<And>\<Psi>' R S. (\<Psi>', R, S) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> R \<leadsto><Rel> S" shows "\<Psi> \<rhd> P \<leadsto><Rel''> R"
pred_equals_eq2: fixes R :: "('b \<times> 'c) set"   and S :: "('b \<times> 'c) set" shows "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
assms_2: fixes y :: "'b" assumes "y \<in> set_pmf q" shows "set_pmf p \<inter> {x. R x y} \<noteq> {}"
tendsto_imp_filterlim_at_right: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> L) F"   and "\<forall>\<^sub>F x in F. L < f x" shows "filterlim f (at_right L) F"
cloBT: shows "closedin (top_of_set B) ({0} \<times> T)"
correctness: fixes G :: "('b, 'c) form list"   and p :: "('b, 'c) form" assumes "G \<turnstile> p" shows "\<forall>e f g. e,f,g,G \<Turnstile> p"
le_\<alpha>_iff: fixes k :: "nat"   and Gr :: "nat set \<times> nat set set" assumes "0 < k" shows "(enat k \<le> \<alpha> Gr) = (k \<in> card ` independent_sets Gr)"
osp_symmetry: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B C OSP P Q" shows "A B C OSP Q P"
case_one_not_learn: fixes j :: "nat" assumes "\<nexists>t. local.prefixes t j \<up>" shows "\<not> learn_lim \<phi> {\<psi> j} s"
assms_2: shows "x \<notin>\<^sub>\<circ> A"
scc_root_transfer': fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and s' :: "('c, 'f) state_scheme"   and r :: "'c"   and scc :: "'c set" assumes "DFS_invar G param s"   and "discovered s = discovered s'"   and "tree_edges s = tree_edges s'" shows "scc_root' (g_E G) s r scc = scc_root' (g_E G) s' r scc"
add_comm: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y = y \<oplus> x"
that: fixes N :: "nat set" assumes "N \<subseteq> M"   and "infinite N"   and "\<And>T. T \<subseteq> S \<Longrightarrow> decides \<F> T N" shows "thesis"
gt_sym_imp_hd: fixes g :: "'s"   and f :: "'s" assumes "g >\<^sub>s f" shows "Sym g >\<^sub>h\<^sub>d Sym f"
orthogonal_scaleR: fixes c :: "real"   and x :: "'b" assumes "c \<noteq> 0" shows "orthogonal (c *\<^sub>R x) = orthogonal x"
floor_le_round: fixes x :: "'a" shows "\<lfloor>x\<rfloor> \<le> round x"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form" shows "SingleSuccedent.mset ( ant \<Rightarrow>* suc) = ant \<oplus> suc"
NN_2: fixes e :: "real" assumes "0 < e" shows "- ln e / ln (real k * \<delta>) < real (NN e)"
assms_2: shows "range seq \<subseteq> dgrad_sig_set' (length fs) d"
wf: shows "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
divisors_nat_copy_def: fixes n :: "nat" shows "divisors_nat_copy n \<equiv> if n = 0 then [] else remdups_adj (sort (map prod_list (subseqs (prime_factorization_nat n))))"