hcju/isabelleps huggingface dataset:
spheremap_lemma1: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "subspace S"   and "subspace T"   and "dim S < dim T"   and "S \<subseteq> T"   and "f differentiable_on sphere (0::'a) 1 \<inter> S" shows "f ` (sphere (0::'a) 1 \<inter> S) \<noteq> sphere (0::'a) 1 \<inter> T"
instrs_append: fixes xs :: "'b memref list"   and ys :: "'b memref list" shows "instrs (xs @ ys) = instrs xs @ instrs ys"
index_less_size_conv: fixes xs :: "'e list"   and x :: "'e" shows "(index xs x < length xs) = (x \<in> set xs)"
rsA: shows "r @@ s \<in> A\<^sup>\<star>"
sq_mtx_transpose_def: shows "sq_mtx_transpose \<equiv> map_fun to_vec to_mtx Finite_Cartesian_Product.transpose"
P: fixes as :: "'a list"   and bs :: "'a list" assumes "xs @ ys = as @ bs" shows "set as \<inter> set bs = {}"
card_calM: shows "card calM = p ^ a * m choose p ^ a"
coinduct: fixes X :: "('n, 'o, 'p, 'q, 'r, 's) U \<Rightarrow> ('t, 'u, 'v, 'w, 'x, 's) U \<Rightarrow> bool"   and x1 :: "('n, 'o, 'p, 'q, 'r, 's) U"   and x2 :: "('t, 'u, 'v, 'w, 'x, 's) U"   and L1 :: "'n \<Rightarrow> 't \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'v \<Rightarrow> bool"   and Contra1 :: "'q \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2 :: "'r \<Rightarrow> 'x \<Rightarrow> bool" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>x y. x1 = x \<and> x2 = y \<and> rel_G (\<lambda>uu_ uua_. X uu_ uua_ \<or> rel_U L1 Co1 Co2 Contra1 Contra2 uu_ uua_) L1 Co1 Co2 Contra1 Contra2 (D_U x) (D_U y)" shows "rel_U L1 Co1 Co2 Contra1 Contra2 x1 x2"
scast_of_nat: fixes x :: "nat" shows "scast (word_of_nat x) = word_of_nat x"
n: shows "\<forall>m\<ge>n. r \<le> real m"
a: shows "\<forall>u v \<sigma>. put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
supp_fun_eqvt: fixes f :: "'b" assumes "eqvt f" shows "supp f = {}"
cod_inv: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.cod (C.VV.inv f) = C.VV.dom f"
polynomial_product: fixes m :: "nat"   and a :: "nat \<Rightarrow> 'a"   and n :: "nat"   and b :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "\<And>i. m < i \<Longrightarrow> a i = (0::'a)"   and "\<And>j. n < j \<Longrightarrow> b j = (0::'a)" shows "(\<Sum>i\<le>m. a i * x ^ i) * (\<Sum>j\<le>n. b j * x ^ j) = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. a k * b (r - k)) * x ^ r)"
assoc_is_natural_1: 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 \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a> \<mu> \<nu> \<tau> = V (H \<mu> (H \<nu> \<tau>)) (\<a> (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
filter_mset_empty_conv: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset" shows "(filter_mset P M = {#}) = (\<forall>L\<in>#M. \<not> P L)"
vec_norm_zero: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(vec_norm v = 0) = (v = 0\<^sub>v n)"
gx0_def: shows "gx0 \<equiv> max x\<^sub>1 (SOME gx0. \<forall>x\<ge>gx0. gc2 * g' (real x) \<le> g x \<and> 0 < f x \<and> 0 \<le> g' (real x))"
smult_mat_def: fixes a :: "'c"   and A :: "'c mat" shows "a \<cdot>\<^sub>m A \<equiv> map_mat ((*) a) A"
funpow_dist_least: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "n < funpow_dist f x y" shows "(f ^^ n) x \<noteq> y"
flow_trans: fixes s :: "real"   and x0 :: "'a"   and t :: "real" assumes "s \<in> rev.existence_ivl0 x0"   and "t \<in> rev.existence_ivl0 (rev.flow0 x0 s)" shows "rev.flow0 x0 (s + t) = rev.flow0 (rev.flow0 x0 s) t"
min_lvar_not_in_bounds_Some': fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat" shows "min_lvar_not_in_bounds s = Some x\<^sub>i \<longrightarrow> x\<^sub>i \<in> lvars (\<T> s) \<and> \<not> in_bounds x\<^sub>i \<langle>\<V> s\<rangle> (\<B> s) \<and> (\<forall>x\<in>lvars (\<T> s). x < x\<^sub>i \<longrightarrow> in_bounds x \<langle>\<V> s\<rangle> (\<B> s))"
dist_bcontfun_def: shows "dist \<equiv> map_fun apply_bcontfun (map_fun apply_bcontfun id) (\<lambda>f g. SUP x. dist (f x) (g x))"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>x. norm (h x) \<le> norm x * K"
exists_neighboring_violation_pair: fixes t :: "('a, 'b, 'c) event list" assumes "trace init t final"   and "0 < count_violations t" shows "\<exists>i j. i < j \<and> postrecording_event t i \<and> prerecording_event t j \<and> (\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)) \<and> j < length t"
simps_2: fixes t1 :: "lam"   and t2 :: "lam"   and y :: "name"   and t' :: "lam" shows "App t1 t2[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
SDG_to_CFG_set_def: fixes S :: "'node SDG_node set" shows "\<lfloor>S\<rfloor>\<^bsub>CFG\<^esub> \<equiv> {m. CFG_node m \<in> S}"
rtrancl_mono: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s" shows "r\<^sup>* \<subseteq> s\<^sup>*"
cases: fixes x :: "'a extended" obtains xa :: "'a" where "x = Fin xa"   | "x = \<infinity>"   | "x = -\<infinity>"
simps: fixes a :: "('sf, 'sc, 'sz) formula" shows "FRadmit a = ((\<exists>\<theta>1 \<theta>2. a = Geq \<theta>1 \<theta>2) \<or> (\<exists>p args. a = $\<phi> p args) \<or> (\<exists>\<phi>. a = ! \<phi> \<and> FRadmit \<phi>) \<or> (\<exists>\<phi> \<psi>. a = (\<phi> && \<psi>) \<and> FRadmit \<phi> \<and> FRadmit \<psi>) \<or> (\<exists>\<phi> x. a = Exists x \<phi> \<and> FRadmit \<phi>) \<or> (\<exists>\<alpha> \<phi>. a = (\<langle> \<alpha> \<rangle> \<phi>) \<and> PRadmit \<alpha> \<and> FRadmit \<phi>))"
ac_Diff_iff: fixes c :: "'a"   and A :: "'a antichain"   and B :: "'a antichain" shows "(c \<in>\<^sub>A A - B) = (c \<in>\<^sub>A A \<and> c \<notin>\<^sub>A B)"
converse_relcompp: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and s :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "(r OO s)\<inverse>\<inverse> = s\<inverse>\<inverse> OO r\<inverse>\<inverse>"
compat_S_NS: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "compat_pair S NS" shows "S O NS \<subseteq> S"
Un_Int_eq_2: fixes S :: "'k set"   and T :: "'k set" shows "(S \<union> T) \<inter> T = T"
simps_7: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (Gt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
simps_7: fixes doctype' :: "char list \<Rightarrow> char list"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "doctype_update doctype' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype' doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr>"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<sim>\<^sup>s\<^sub>l P \<oplus> Q \<oplus> R"
zero_le_dist: fixes x :: "'b"   and y :: "'b" shows "0 \<le> dist x y"
simps: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and uu :: "nat"   and r :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2" shows "WBT.split_min \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 \<langle>l, (a, uu), r\<rangle> = (if l = \<langle>\<rangle> then (a, r) else let (x, l') = WBT.split_min \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 l in (x, WBT.balanceL \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 l' a r))"
add_eq_conv_diff: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and N :: "'a multiset" shows "(a, M = b, N) = (M = N \<and> a = b \<or> M = b, N - (a, {#}) \<and> N = a, M - (b, {#}))"
span_neg: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set" assumes "module scale"   and "x \<in> module.span scale S" shows "- x \<in> module.span scale S"
lless_imp_le: fixes R :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "x \<sqsubset>\<^bsub>R\<^esub> y" shows "x \<sqsubseteq>\<^bsub>R\<^esub> y"
disc_2: fixes x2 :: "'b" shows "\<not> isl (Inr x2)"
ring_hom_memE_1: fixes h :: "'d \<Rightarrow> 'e"   and R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme"   and x :: "'d" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R" shows "h x \<in> carrier S"
aform_err_ivl_err: fixes e :: "nat \<Rightarrow> real"   and ivl' :: "real interval" shows "aform_err e (ivl_err ivl') = set_of ivl'"
po_on_imp_antisymp_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "po_on P A" shows "antisymp_on P A"
A4': fixes q :: "'p"   and a :: "'p"   and b :: "'p"   and c :: "'p" shows "\<exists>x. B q a x \<and> a x \<equiv> b c"
vsubset_vinsert_leftI: fixes A :: "V"   and B :: "V"   and a :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "a \<in>\<^sub>\<circ> B" shows "vinsert a A \<subseteq>\<^sub>\<circ> B"
p_natintpow_prod: fixes m :: "int"   and n :: "nat" assumes "0 \<le> m" shows "[p] \<cdot> \<one> [^] n \<otimes> [p] \<cdot> \<one> [^] m = [p] \<cdot> \<one> [^] (int n + m)"
encompasses_cases: fixes seg :: "(real \<times> real) \<times> real \<times> real"   and x :: "real \<times> real" shows "0 \<le> det3 (fst seg) (snd seg) x \<or> 0 \<le> det3 (fst (snd seg, fst seg)) (snd (snd seg, fst seg)) x"
Guard_noFaultStuckD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and F :: "'f set" assumes "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)"   and "f \<notin> F" shows "s \<in> g"
entails_def: fixes N1 :: "('f \<times> 'l) set"   and N2 :: "('f \<times> 'l) set" shows "(N1 \<Turnstile>\<inter>\<G>L N2) = (\<forall>q\<in>Q. entails_q q (\<Union> (\<G>_F_L_q q ` N1)) (\<Union> (\<G>_F_L_q q ` N2)))"
resourced_adequacy: fixes e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and r :: "C"   and S :: "Vars.var list" assumes "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>r \<noteq> \<bottom>" shows "\<exists>\<Delta> v. \<Gamma> : e \<Down>\<^bsub>S\<^esub> \<Delta> : v"
sorted_descending_Cons: fixes x :: "'c"   and xs :: "'c list" shows "sorted_descending (x # xs) = (sorted_descending xs \<and> (\<forall>y\<in>set xs. y \<le> x))"
1: fixes b :: "int" assumes "b gchoose k = 0" shows "b < int k"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) atm" shows "satL \<xi> (Pos a) = satA \<xi> a"
allE: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k" assumes "\<forall>x. P x" obtains "P x"
function_uminus_def: fixes S :: "'a set"   and R :: "('b, 'c) ring_scheme"   and a :: "'a \<Rightarrow> 'b" shows "function_uminus S R a = (\<lambda>x\<in>S. \<ominus>\<^bsub>R\<^esub> a x)"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('w \<Rightarrow> 'w) \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme"   and v :: "'w" assumes "DocumentClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
dist_c': shows "(\<Sum>z\<in>outcomes. pmf p z * (c * u z)) = c * (\<Sum>z\<in>outcomes. pmf p z * u z)"
r_zero: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub> = x"
mult_minus_right: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.ring plus zero minus uminus times" shows "times a (uminus b) = uminus (times a b)"
trivial_group_subgroup_generated: fixes S :: "'a set" assumes "S \<subseteq> {\<one>}" shows "trivial_group (subgroup_generated G S)"
simps_2: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
mm_equiv_low_eq: fixes c\<^sub>1 :: "'Com"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "'Com"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
split_nat: fixes P :: "nat \<Rightarrow> bool"   and i :: "int" shows "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
pcr_tllist_def: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_tllist T Ta \<equiv> rel_prod (llist_all2 T) Ta OO cr_tllist"
nn_integral_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M = (SUP i. integral\<^sup>N M (f i))"
summable_ignore_initial_segment: fixes f :: "nat \<Rightarrow> 'e"   and k :: "nat" assumes "summable f" shows "summable (\<lambda>n. f (n + k))"
f_def: shows "f \<equiv> restrict1 (canonical_retraction A D \<circ> canonical_retraction B C) (\<Union> A)"
space_state_measure: fixes V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "space (state_measure V \<Gamma>) = (\<Pi>\<^sub>E y\<in>V. type_universe (\<Gamma> y))"
trace_outer_prod: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "trace (outer_prod v w) = inner_prod w v"
hcomp_comm: fixes P :: "interface"   and Q :: "interface" shows "(P \<otimes> Q) = (Q \<otimes> P)"
powr_powr_swap: fixes x :: "real"   and a :: "real"   and b :: "real" shows "(x powr a) powr b = (x powr b) powr a"
vsv: fixes r :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "vsv r"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r" shows "b = c"
flow_continuous_on: fixes x0 :: "'a" shows "continuous_on (existence_ivl0 x0) (flow0 x0)"
fls_divide_X': fixes f :: "'a fls" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X = fls_shift 1 f"
intros_6: fixes iop :: "binop_i"   and c1 :: "i32"   and c2 :: "i32" assumes "app_binop_i iop c1 c2 = None" shows "\<lparr>[$C ConstInt32 c1, $C ConstInt32 c2, $Binop_i T_i32 iop]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
component_of_term_of_pair: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and k :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.component_of_term pair_of_term (term_of_pair (t, k)) = k"
poincare_between_y_axis_0uv: fixes x :: "real"   and y :: "real" assumes "- 1 < x"   and "x < 1"   and "x \<noteq> 0"   and "- 1 < y"   and "y < 1"   and "y \<noteq> 0" shows "poincare_between 0\<^sub>h (of_complex (\<i> * cor x)) (of_complex (\<i> * cor y)) = (x < 0 \<and> y < 0 \<and> y \<le> x \<or> 0 < x \<and> 0 < y \<and> x \<le> y)"
left_unique_fun: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "left_total A"   and "left_unique B" shows "left_unique (rel_fun A B)"
distinct_9: fixes x2 :: "'d" shows "Abrupt x2 \<noteq> Stuck"
abss_def: shows "abss \<equiv> \<lambda>(l, u). if u \<in> V then (l, [u]\<^sub>\<R>) else (l, - V)"
extends_to_prebicategory_with_homs_and_units: shows "prebicategory_with_homs_and_units (\<cdot>) (\<star>) \<a> \<i>' src trg"
rec_mult_def: shows "rec_mult = PR Z (CN rec_add [recf.Id 3 1, recf.Id 3 2])"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod ((G \<circ> FG.\<epsilon>) f) = G (A.cod f)"
elims_2: fixes x :: "('c, 'd) strand_step" assumes "simple\<^sub>s\<^sub>t\<^sub>p x" obtains t :: "('c, 'd) Term.term" where "x = receive\<langle>t\<rangle>\<^sub>s\<^sub>t"   | v :: "'d" where "x = send\<langle>Var v\<rangle>\<^sub>s\<^sub>t"   | X :: "'d list" and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" where "x = \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t" and "\<exists>\<I>. ineq_model \<I> X F"
topcontinuous_at_atin: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" shows "topcontinuous_at X Y f x = (x \<in> topspace X \<and> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> limitin Y f (f x) (atin X x))"
peek_res_def: fixes Pf :: "vals \<Rightarrow> 'b assn" shows "peek_res Pf = (\<lambda>Y. Pf Y Y)"
mbr_fieldm_simp: fixes n :: "vname"   and f :: "qtname \<times> Decl.field" shows "mbr (fieldm n f) = fdecl (n, fld f)"
generated_ring_def: shows "generated_ring = {\<Union> C |C. C \<subseteq> M \<and> finite C \<and> disjoint C}"
mul_const_def: fixes p :: "int poly"   and c :: "int" shows "mul_const p c = coeff p 0 * c mod m"
encode1_def: fixes dims :: "nat list"   and vars :: "nat set"   and i :: "nat" shows "partial_state.encode1 dims vars i = digit_decode (partial_state.dims1 dims vars) (nths (digit_encode dims i) vars)"
alpha_Tree_S_transform_Tree: fixes t1 :: "('a, 'b, 'c) Tree"   and t2 :: "('a, 'b, 'c) Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "S_transform_Tree t1 = S_transform_Tree t2"
1: shows "[y\<^sup>P \<^bold>= (\<^bold>\<iota>x. \<phi> (x\<^sup>P)) in v]"
trc_aux: fixes t :: "'s Trace"   and aec :: "'s odlist"   and a :: "'a"   and cec :: "'s odlist"   and as :: "'a list" assumes "t \<in> Clock.jkbpC"   and "toSet aec = agent_abs a t"   and "toSet cec = common_abs t" shows "toSet (big_union (clock_commonKnowledge envObs cec as) (toList aec)) \<subseteq> toSet cec"
nat_of_natural_of_nat: fixes n :: "nat" shows "nat_of_natural (of_nat n) = n"
evalDdb_subst_env_ext_db: fixes e :: "db"   and s :: "db"   and \<rho> :: "nat \<rightarrow> ValD" shows "evalDdb (e<s/0>)\<cdot>\<rho> = evalDdb e\<cdot>(env_ext_db\<cdot>(evalDdb s\<cdot>\<rho>)\<cdot>\<rho>)"
b1: shows "b1 = p / (2::'a) + m - q / ((2::'a) * a)"
Y0_def: shows "Y0 = {v \<in> Y. v $ n = (0::'a)}"
Un_Int_eq_1: fixes S :: "'b set"   and T :: "'b set" shows "(S \<union> T) \<inter> S = S"
Lindelof_space_def: fixes X :: "'a topology" shows "Lindelof_space X \<equiv> \<forall>\<U>. (\<forall>U\<in>\<U>. openin X U) \<and> \<Union> \<U> = topspace X \<longrightarrow> (\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<U> \<and> \<Union> \<V> = topspace X)"
block_designI: fixes k :: "nat" assumes "\<And>bl. bl \<in># \<B>\<^sup>+ \<Longrightarrow> card bl = k" shows "block_design \<V>\<^sup>+ \<B>\<^sup>+ (int k)"
F_mndet1: fixes P :: "'a \<Rightarrow> 'a process" shows "F (mndet {} P) = {(s, X). s = []}"
right_neutral: fixes a :: "'d multiset" shows "a \<union># {#} = a"
Nf_def: shows "Nf = (\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x))"
map_simp_ide: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>'.map a = \<tau> a"
scc_root_scc_tree_rtrancl: fixes v :: "'v"   and scc :: "'v set"   and x :: "'v" assumes "scc_root s v scc"   and "x \<in> scc"   and "x \<in> dom (discovered s)" shows "(v, x) \<in> (tree_edges s)\<^sup>*"
greatest_eq: shows "Greatest_plus_one = (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (Gauss_Jordan_in_ij A Greatest_plus_one (mod_type_class.from_nat k)))"
heap_upds_okE: fixes \<Gamma> :: "(Vars.var \<times> 'a) list"   and S :: "stack_elem list"   and x :: "Vars.var" assumes "heap_upds_ok (\<Gamma>, S)"   and "x \<in> domA \<Gamma>" shows "x \<notin> upds S"
zero_left: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> \<D>"   and "y \<in> \<D>"   and "x \<noteq> y"   and "x \<inter> {x. x \<bullet> i \<le> c} = y \<inter> {x. x \<bullet> i \<le> c}" shows "content (y \<inter> {x. x \<bullet> i \<le> c}) = 0"
eWlsInp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and einp :: "'index \<Rightarrow> 'a withERR option" assumes "eWlsInp MOD delta einp" shows "\<exists>inp. igWlsInp MOD delta inp \<and> einp = OKI inp"
boundedE: fixes b :: "'e multiset"   and c :: "'e multiset"   and a :: "'e multiset" assumes "b \<union># c \<subseteq># a" obtains "b \<subseteq># a" and "c \<subseteq># a"
elementary_monoidal_category_axioms: shows "elementary_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit\<^sub>F\<^sub>M\<^sub>C runit\<^sub>F\<^sub>M\<^sub>C assoc\<^sub>F\<^sub>M\<^sub>C"
not_matchrelD: fixes e :: "('d, 'e) Term.term \<times> ('d, 'f) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option" assumes "\<nexists>y. (({#e#}, \<sigma>), y) \<in> matchrel" shows "(\<exists>f ss x. e = (Fun f ss, Var x)) \<or> (\<exists>x t. e = (Var x, t) \<and> \<sigma> x \<noteq> None \<and> \<sigma> x \<noteq> Some t) \<or> (\<exists>f g ss ts. e = (Fun f ss, Fun g ts) \<and> (f \<noteq> g \<or> length ss \<noteq> length ts))"
indRelRSTPO_impl_SRel_and_TRel_weakly_reflect_barbs_1: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs (indRelRSTPO SRel TRel) (STCalWB SWB TWB)" shows "rel_weakly_reflects_barbs SRel SWB"
determines_span: assumes "ide h" shows "span_in_category (\<odot>) \<lparr>Leg0 = \<lbrakk>\<lbrakk>tab\<^sub>0 h\<rbrakk>\<rbrakk>, Leg1 = \<lbrakk>\<lbrakk>tab\<^sub>1 h\<rbrakk>\<rbrakk>\<rparr>"
bind_is_OK_I2: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog" assumes "h \<turnstile> ok f"   and "\<And>x h'. \<lbrakk>h \<turnstile> f \<rightarrow>\<^sub>r x; h \<turnstile> f \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> h' \<turnstile> ok g x" shows "h \<turnstile> ok Heap_Error_Monad.bind f g"
dimQ_def: fixes x :: "'c['d, 'e]" shows "dimQ x = Quantity.dim (fromQ x)"
that: assumes "a = b"   and "u \<le> v" shows "thesis"
telescopic_base_aux: fixes K :: "'a set"   and F :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "subfield F R"   and "dimension n K F"   and "dimension 1 F E" shows "dimension n K E"
strategy_attracts_extends_VVp: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and W :: "'a set"   and v0 :: "'a" assumes "strategy p \<sigma>"   and "strategy_attracts p \<sigma> S W"   and "v0 \<in> VV p"   and "v0 \<in> directly_attracted p S"   and "v0 \<notin> S" shows "\<exists>\<sigma>. strategy p \<sigma> \<and> strategy_attracts_via p \<sigma> v0 (insert v0 S) W"
mat_of_cblinfun_inj: shows "inj mat_of_cblinfun"
b_e_type_empty1: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "\<C> \<turnstile> [] : ts _> ts'" shows "ts = ts'"
residue_simple': fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex"   and c :: "complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s - {z}"   and "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c" shows "residue f z = c"
tensor_mat_minus1: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d1 d1"   and "m3 \<in> carrier_mat d2 d2" shows "tensor_mat (m1 - m2) m3 = tensor_mat m1 m3 - tensor_mat m2 m3"
insert_Diff1: fixes x :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "x \<in> B" shows "insert x A\<setminus>B = A\<setminus>B"