hcju/isabelleps huggingface dataset:
simps_38: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "While x61 x62 \<noteq> x31@[x32]"
lem1: fixes e :: "real" assumes "0 < e" shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
F_def: shows "F \<equiv> \<lambda>lmap xs. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> LCons (f x) (lmap xs)"
if_split: fixes P :: "'h \<Rightarrow> bool"   and Q :: "bool"   and x :: "'h"   and y :: "'h" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
simps_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and e :: "String.literal"   and ts :: "Formula.trm list"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Pred e ts) j = (case P e of None \<Rightarrow> j | Some k \<Rightarrow> k)"
GNI_def: fixes \<T> :: "trace set" shows "GNI \<T> \<equiv> \<forall>\<tau>\<^sub>1\<in>\<T>. \<forall>\<tau>\<^sub>2\<in>\<T>. \<exists>\<tau>\<^sub>3\<in>\<T>. high_in \<tau>\<^sub>3 = high_in \<tau>\<^sub>1 \<and> low_in \<tau>\<^sub>3 = low_in \<tau>\<^sub>2 \<and> low_out \<tau>\<^sub>3 = low_out \<tau>\<^sub>2"
pinv: shows "T \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg I), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). anycast (Q \<sigma> \<sigma>') a)"
case_1: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "sig \<Rightarrow> 'a"   and x1 :: "vname" shows "(case fid x1 of fid x \<Rightarrow> f1 x | mid x \<Rightarrow> f2 x) = f1 x1"
CR3_def: fixes \<tau> :: "ty" shows "CR3 \<tau> \<equiv> \<forall>t. NEUT t \<and> CR3_RED t \<tau> \<longrightarrow> t \<in> RED \<tau>"
idempotent_def: fixes r :: "('b \<times> 'b) set" shows "idempotent r \<equiv> r O r = r"
Riemann_lebesgue: fixes f :: "real \<Rightarrow> real" assumes "f absolutely_integrable_on {- pi..pi}" shows "Fourier_coefficient f \<longlonglongrightarrow> 0"
qc_wf_below: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set"   and A :: "'a set" assumes "s O ns \<subseteq> (s \<union> ns)\<^sup>* O s"   and "wf_below s A" shows "wf_below (ns\<^sup>* O s O ns\<^sup>*) A"
insert_Diff_if: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "insert x A\<setminus>B = (if x \<in> B then A\<setminus>B else insert x (A\<setminus>B))"
is_red_monomial_monomial_setD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_monomial p"   and "is_monomial_set B"   and "is_red B p" shows "red B p 0"
induct: fixes x1 :: "nat"   and x2 :: "com"   and P :: "nat \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile> x2"   and "\<And>l. P l SKIP"   and "\<And>x a l. \<lbrakk>sec a \<le> sec x; l \<le> sec x\<rbrakk> \<Longrightarrow> P l (x ::= a)"   and "\<And>l c\<^sub>1 c\<^sub>2. \<lbrakk>l \<turnstile> c\<^sub>1; P l c\<^sub>1; l \<turnstile> c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (c\<^sub>1;; c\<^sub>2)"   and "\<And>b l c\<^sub>1 c\<^sub>2. \<lbrakk>max (sec b) l \<turnstile> c\<^sub>1; P (max (sec b) l) c\<^sub>1; max (sec b) l \<turnstile> c\<^sub>2; P (max (sec b) l) c\<^sub>2\<rbrakk> \<Longrightarrow> P l (IF b THEN c\<^sub>1 ELSE c\<^sub>2)"   and "\<And>b c. \<lbrakk>sec b = 0; 0 \<turnstile> c; P 0 c\<rbrakk> \<Longrightarrow> P 0 (WHILE b DO c)" shows "P x1 x2"
children_eq2_h: fixes ptr' :: "(_) object_ptr" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr" shows "|h \<turnstile> get_child_nodes ptr'|\<^sub>r = |h2 \<turnstile> get_child_nodes ptr'|\<^sub>r"
len_lexext_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "len_lexext gt (xs @ [x]) xs"
length3D: fixes xs :: "'a list" assumes "|xs| = 3" shows "\<exists>x y z. xs = [x, y, z]"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<nu> \<star> \<pi>)"
acc_point_cres: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "acc_point s (Suc k) = const_res (Suc k) (res_seq s k)"
mem_diff: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "loc_reachp lc c' mds' mem'"   and "\<forall>x\<in>mds' AsmNoWrite. mem' x = mem'' x" shows "loc_reachp lc c' mds' mem''"
stoch_proc_subalg_nat_filt: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "borel_adapt_stoch_proc nat_filtration X" shows "subalgebra (nat_filtration n) (stoch_proc_filt M X borel n)"
ang_vec_plus_pi2: fixes z1 :: "complex"   and z2 :: "complex" assumes "\<angle> z1 z2 \<le> 0" shows "\<downharpoonright>\<angle> z1 z2 + pi\<downharpoonleft> = \<angle> z1 z2 + pi"
simps_2: fixes literal :: "Literal"   and valuation :: "Literal list" shows "val2form (literal # valuation) = [literal] # val2form valuation"
assms_2: shows "i < dim_vec (b @\<^sub>v c)"
igSubstAbsIPresIGWlsAbs_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstAbsIPresIGWlsAbs MOD \<equiv> \<forall>ys Y y us s A. isInBar (us, s) \<and> igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A)"
collinear_imp_coplanar: fixes s :: "'a set" assumes "collinear s" shows "coplanar s"
inj_imp_image_ring_inv_iso: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme" assumes "inj f" shows "inv f \<in> ring_iso (image_ring f R) R"
select_convs_3: fixes lasso_reach :: "'e list"   and lasso_va :: "'e"   and lasso_cysfx :: "'e list"   and more :: "'f" shows "lasso_cysfx \<lparr>lasso_reach = lasso_reach, lasso_va = lasso_va, lasso_cysfx = lasso_cysfx, \<dots> = more\<rparr> = lasso_cysfx"
ivl_disj_un_21: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..<m} \<union> {m..u} = {l..u}"
fnt_carrier_fnt_nbt: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "rational_preference carrier relation"   and "finite carrier" shows "\<forall>x\<in>carrier. finite (no_better_than x carrier relation)"
eqU: shows "\<Union> \<V> = \<Union> \<F>"
simps: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c \<times> 'e"   and a2 :: "'d \<times> 'f" shows "rel_prod R1 R2 a1 a2 = (\<exists>a b c d. a1 = (a, c) \<and> a2 = (b, d) \<and> R1 a b \<and> R2 c d)"
assms_2: shows "f $$ fls_subdegree f * y = (1::'a)"
u: shows "y \<notin> \<Union> (Fvars ` F)"
ide_in_hom: fixes a :: "'j" shows "J.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>J a\<guillemotright>"
locally_path_connected_space_alt: fixes X :: "'c topology" shows "locally_path_connected_space X = neighbourhood_base_of (\<lambda>U. openin X U \<and> path_connectedin X U) X"
eint_ex_split: fixes P :: "eint \<Rightarrow> bool" shows "(\<exists>c. P c) = (P \<infinity> \<or> (\<exists>x. P (eint x)))"
induct: fixes P1 :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and P3 :: "'opSym \<Rightarrow> ('index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) \<Rightarrow> bool"   and P4 :: "'opSym \<Rightarrow> ('bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option) \<Rightarrow> bool"   and P2 :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool"   and x1 :: "'sort"   and x2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x3 :: "'varSort \<times> 'sort"   and x4 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and x5 :: "'opSym"   and x6 :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and x7 :: "'opSym"   and x8 :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "\<And>xs x. P1 (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; P3 delta inp; wlsBinp delta binp; P4 delta binp\<rbrakk> \<Longrightarrow> P1 (stOf delta) (Op delta inp binp)"   and "\<And>delta inp. \<lbrakk>wlsOpS delta; \<And>i. arOf delta i = None \<and> inp i = None \<or> (\<exists>s X. arOf delta i = Some s \<and> inp i = Some X \<and> wls s X \<and> P1 s X)\<rbrakk> \<Longrightarrow> P3 delta inp"   and "\<And>delta binp. \<lbrakk>wlsOpS delta; \<And>i. barOf delta i = None \<and> binp i = None \<or> (\<exists>us s A. barOf delta i = Some (us, s) \<and> binp i = Some A \<and> wlsAbs (us, s) A \<and> P2 (us, s) A)\<rbrakk> \<Longrightarrow> P4 delta binp"   and "\<And>xs s X x. \<lbrakk>isInBar (xs, s); wls s X; P1 s X\<rbrakk> \<Longrightarrow> P2 (xs, s) (Abs xs x X)" shows "(wls x1 x2 \<longrightarrow> P1 x1 x2) \<and> (wlsAbs x3 x4 \<longrightarrow> P2 x3 x4) \<and> (wlsInp x5 x6 \<longrightarrow> P3 x5 x6) \<and> (wlsBinp x7 x8 \<longrightarrow> P4 x7 x8)"
not_self_fresh: fixes x :: "'a" shows "atom x \<sharp> x = False"
rtrancl_into_trancl1: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
lengthFT_def: fixes tr :: "'a list \<times> 'b" shows "lengthFT tr \<equiv> Suc (length (fst tr))"
uncountable_minus_countable: fixes A :: "'a set"   and B :: "'a set" assumes "uncountable A"   and "countable B" shows "uncountable (A - B)"
assms: shows "B\<^sub>\<real> hyp2_rep p v hyp2_rep q"
residual_block_size: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>R" shows "int (card b) = \<k> - \<Lambda>"
primes_at_top_def: shows "primes_at_top = inf sequentially (principal {p. prime p})"
Units_inv_inv: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "inv (inv x) = x"
satisfies_bounded_nFAnd: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile>\<^sub>b nFAnd \<phi> \<psi>) = (\<AA> \<Turnstile>\<^sub>b FAnd \<phi> \<psi>)"
as_3: shows "as \<bullet>\<cdot> Tpows_v = (0::'b)"
thms_notE: fixes H :: "'b pl set"   and p :: "'b pl"   and q :: "'b pl" assumes "H |- p -> false"   and "H |- p" shows "H |- q"
wary_cases_apps\<^sub>h: fixes t :: "('s, 'v) tm" assumes "wary t" obtains \<zeta> :: "('s, 'v) hd" and ss :: "('s, 'v) tm list" where "t = apps (Hd \<zeta>) ss" and "\<And>sa. sa \<in> set ss \<Longrightarrow> wary sa" and "of_nat (length ss) \<le> arity_hd\<^sub>h \<zeta>"
distinct_9: shows "X5.Xd1' \<noteq> X5.Xd2"
total_leq_lit: fixes A :: "('a, 'b) Term.term literal set" shows "Restricted_Predicates.total_on leq_lit A"
emeasure_count_space_density_singleton: fixes x :: "'c"   and A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "x \<in> A"   and "has_density M (count_space A) f" shows "emeasure M {x} = f x"
Amin_memTr: fixes f :: "nat \<Rightarrow> ant"   and n :: "nat" shows "f \<in> {i. i \<le> n} \<rightarrow> Z\<^sub>\<infinity> \<longrightarrow> Amin n f \<in> Z\<^sub>\<infinity>"
prems_11: shows "InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
mi: shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> P x = MP"
solve_factored_ratfps_roots: fixes p :: "'a poly"   and cs :: "('a \<times> nat) list" shows "map snd (snd (solve_factored_ratfps p cs)) = map fst cs"
jkbpC_init: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and s :: "'e"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "PreEnvironmentJView jkbp jview" shows "(tInit s \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview) = (s \<in> set envInit)"
fg: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
respects_own_guarantees_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cm :: "'c \<times> (Mode \<Rightarrow> 'a set)" assumes "sifum_security_init dma \<C>_vars eval" shows "sifum_security_init.respects_own_guarantees dma \<C>_vars eval cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> sifum_security_init.doesnt_read_or_modify \<C>_vars eval (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> sifum_security_init.doesnt_modify dma eval (fst cm) x)"
Invariants_func_is_invariant_n: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and n :: "nat" assumes "qmpt M T"   and "f \<in> borel_measurable (qmpt.Invariants M T)"   and "x \<in> space M" shows "f ((T ^^ n) x) = f x"
assms_3: shows "maximal U (\<lambda>T. T \<subseteq> S \<and> lin_indpt T)"
take_nths_eq_single: fixes xs :: "'b list"   and i :: "nat"   and n :: "nat" assumes "length xs - i < 2 ^ n" shows "take_nths i n xs = take 1 (drop i xs)"
cod_def: fixes f :: "'a" shows "cod f = (if codomains f \<noteq> {} then SOME b. b \<in> codomains f else null)"
Proc_CFG_Entry_Exit: fixes prog :: "cmd" shows "prog \<turnstile> Entry -IEdge (\<lambda>s. False)\<^sub>\<surd>\<rightarrow>\<^sub>p Exit"
le_imp_neg_le: fixes 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 a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b" shows "less_eq (uminus b) (uminus a)"
currentD_OUT_eq_0: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f" shows "(d_OUT f x = 0) = (\<forall>y. f (x, y) = 0)"
RETURN_SPEC_conv: fixes r :: "'a" shows "return r = (spec x. x = r)"
congruent_triangle_sss_18: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "angle b1 c1 a1 = angle a2 c2 b2"
to_nat_split: fixes x :: "'a"   and y :: "'a" shows "(to_nat x = to_nat y) = (x = y)"
mpoly_induct: fixes P :: "'b mpoly \<Rightarrow> bool"   and p :: "'b mpoly" assumes "\<And>m a. P (MPoly_Type.monom m a)"   and "\<And>p1 p2 m a. \<lbrakk>P p1; P p2; p2 = MPoly_Type.monom m a; m \<notin> keys (mapping_of p1)\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
sep_add_zero_sym: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.sep_algebra plus zero sep_disj" shows "plus zero x = x"
swap_self: fixes a :: "'b"   and b :: "'b" shows "swap a b * swap a b = 1"
rename_union: fixes Sel :: "'at set"   and C :: "'at Literal set"   and D :: "'at Literal set" shows "rename_clause Sel (C \<union> D) = rename_clause Sel C \<union> rename_clause Sel D"
axioms_1: fixes G :: "('a, 'b) Graph_scheme"   and T :: "('c, 'd) Graph_scheme"   and bag :: "'c \<Rightarrow> 'a set" assumes "TreeDecomposition G T bag" shows "Graph G"
upper_circle_01_def: shows "upper_circle_01 = {x. cmod (x - (1 / 2 + complex_of_real (sqrt 3 / 6) * \<i>)) < sqrt 3 / 3}"
commute: fixes a :: "'b"   and b :: "'b" shows "a XOR b = b XOR a"
simps_14: fixes x :: "nat"   and a :: "floatarith"   and vs :: "real list" shows "isDERIV x (Ln a) vs = (isDERIV x a vs \<and> 0 < interpret_floatarith a vs)"
id_maximal_Exist: assumes "\<not> zeroring R" shows "\<exists>I. maximal_ideal R I"
length_take: fixes n :: "nat"   and xs :: "'c list" shows "length (take n xs) = ord_class.min (length xs) n"
theFTOne_2: shows "theFTOne dl cl = theFT cl - theFT dl \<union> theFT dl"
scale_right_distrib: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and y :: "'b" assumes "module scale" shows "scale a (x + y) = scale a x + scale a y"
ipurge_ref_distrib_union: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list"   and X :: "'c set"   and Y :: "'c set" shows "ipurge_ref I D u xs (X \<union> Y) = ipurge_ref I D u xs X \<union> ipurge_ref I D u xs Y"
order1: fixes A :: "'a tensor" assumes "order A \<le> 1" shows "cprank_max1 A"
infsetsum_nonneg: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "0 \<le> infsetsum f A"
minimum_elem_mem: fixes X :: "'a set"   and a :: "'a" assumes "X \<subseteq> carrier D"   and "minimum_elem D X a" shows "a \<in> X"
subset_box_1: fixes c :: "'b"   and d :: "'b"   and a :: "'b"   and b :: "'b" shows "(cbox c d \<subseteq> cbox a b) = ((\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i) \<longrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i))"
pi_eq: fixes h1 :: "real \<Rightarrow> complex"   and h2 :: "real \<Rightarrow> complex" assumes "valid_path h1"   and "valid_path h2"   and "\<forall>t\<in>{0..1}. cmod (h1 t - \<gamma> t) < d \<and> cmod (h2 t - \<gamma> t) < d"   and "pathstart h2 = pathstart h1"   and "pathfinish h2 = pathfinish h1" shows "path_image h1 \<subseteq> - cball z (e / 2) \<and> path_image h2 \<subseteq> - cball z (e / 2) \<and> (\<forall>f. f holomorphic_on - cball z (e / 2) \<longrightarrow> contour_integral h2 f = contour_integral h1 f)"
assms_3: shows "VV p** \<subseteq> G'.VV p**"
p: shows "\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A"
assms: shows "((=), (=)) \<in> R \<rightarrow> R \<rightarrow> bool_rel"
fv_heapToAssn: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "fv (heapToAssn \<Gamma>) = fv \<Gamma>"
elem_limit_simp: fixes x :: "'b"   and f :: "'b set \<Rightarrow> 'b set"   and X :: "'b set" shows "(x \<in> limit f X) = (\<exists>n. x \<in> funpower f n X)"
asIMOD_igSwapIGVar: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapIGVar (asIMOD SEM)"
borel_eq_countable_basis: fixes B :: "'c set set" assumes "countable B"   and "topological_basis B" shows "borel = sigma UNIV B"
valid5: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<upsilon> \<upsilon> X = true"
simps: fixes uu :: "'a extC" shows "uu\<^sup>\<star> = extC.Top"
carrier_vecD: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "dim_vec v = n"
distinct_37: fixes x5 :: "msg"   and x61 :: "msg"   and x62 :: "msg" shows "Hash x5 \<noteq> \<lbrace>x61, x62\<rbrace>"
monomial_set_is_GB: fixes G :: "('t \<Rightarrow>\<^sub>0 'd) set" assumes "is_monomial_set G" shows "is_Groebner_basis G"
BSD1: shows "BSD \<V>1 Tr\<^bsub>ES1\<^esub>"
collapse: fixes option :: "'a option" assumes "option \<noteq> None" shows "\<lfloor>the option\<rfloor> = option"
leadsTo_imp_extend_leadsTo: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" assumes "F \<in> A \<longmapsto> B" shows "extend h F \<in> extend_set h A \<longmapsto> extend_set h B"
left_mult_zero_mat: fixes A :: "'c mat"   and n :: "nat"   and nc :: "nat"   and nr :: "nat" assumes "A \<in> carrier_mat n nc" shows "0\<^sub>m nr n * A = 0\<^sub>m nr nc"
next_tame_comp: fixes g :: "graph"   and p :: "nat" assumes "tame g"   and "final g"   and "Seed\<^bsub>p\<^esub> [next_tame0\<^bsub>p\<^esub>]\<rightarrow>* g" shows "Seed\<^bsub>p\<^esub> [next_tame\<^bsub>p\<^esub>]\<rightarrow>* g"
cfun_eq_iff: fixes f :: "'e \<rightarrow> 'f"   and g :: "'e \<rightarrow> 'f" shows "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
contractible_sphere: fixes a :: "'a"   and r :: "real" shows "contractible (sphere a r) = (r \<le> 0)"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(top = a \<sqinter> b) = (a = top \<and> b = top)"
extended_Gromov_product_inside_boundary: fixes e :: "'b"   and x :: "'b"   and y :: "'b Gromov_completion" shows "extended_Gromov_product_at e (to_Gromov_completion x) y = Inf {liminf (\<lambda>n. ereal (Gromov_product_at e x (v n))) |v. (\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> y}"
r_neg1: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H" shows "\<ominus>\<^bsub>H\<^esub> x \<oplus>\<^bsub>H\<^esub> (x \<oplus>\<^bsub>H\<^esub> y) = y"
gh: fixes i :: "'a set"   and x :: "'a" assumes "i \<in> F"   and "x \<in> (S \<union> UF) \<inter> (S \<union> (i - {a i}))" shows "g x = h i x"
y_def: shows "y = \<Squnion>{f x |x. x \<in> X}"
strict_trans2: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a <s b"   and "b \<le>s c" shows "a <s c"
sources_some_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "sources (some_src \<mu>) = sources \<mu>"
two_faces_subdivFace': fixes g :: "graph"   and f :: "face"   and u :: "nat"   and v :: "nat"   and n :: "nat"   and ovs :: "nat option list" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g"   and "2 \<le> |faces g|" shows "2 \<le> |faces (subdivFace' g f v n ovs)|"
trans_equiv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Equiv"   and "cmp b c = Equiv" shows "cmp a c = Equiv"
exhaust_sel: fixes G\<^sub>m :: "('e, 'f, 'g, 'h) G\<^sub>m" obtains "G\<^sub>m = G\<^sub>m (the_G\<^sub>m G\<^sub>m)"
Y_\<G>: fixes Y :: "'b ltln set"   and \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" assumes "Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"   and "\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>"   and "\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n (\<psi>\<^sub>1[Y]\<^sub>\<mu> or\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>)" shows "Y \<subseteq> \<G> \<phi> (suffix i w)"
last_message_hold_one: fixes x :: "'b message_af" shows "last_message_hold [x] = [x]"
higher_eq_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "(higher p v = higher q v) = (\<forall>u. v \<prec>\<^sub>t u \<longrightarrow> lookup p u = lookup q u)"
pos_stock_borel_adapted: shows "borel_adapt_stoch_proc G (prices Mkt risk_free_asset)"
cases: fixes a :: "hf" assumes "coding_hf a" obtains x :: "hf" where "a = x__" and "\<exists>i. x__ = ord_of i"   | x :: "hf" and y :: "hf" where "a = \<langle>x, y\<rangle>" and "coding_hf x" and "coding_hf y"
find_Some_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" shows "(find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
length_inf_prefix: fixes xs :: "'b list"   and ys :: "'b list" shows "length (xs \<sqinter> ys) = inf_prefix_length xs ys"
conf: shows "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]"
degree_le_max_degree: fixes A :: "('b poly, 'c) sq_matrix"   and i :: "'c"   and j :: "'c" shows "degree (to_fun A i j) \<le> max_degree A"
mod_star_conv: fixes A :: "assn"   and B :: "assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> A * B) = (\<exists>hr as1 as2. h = (hr, as1 \<union> as2) \<and> as1 \<inter> as2 = {} \<and> (hr, as1) \<Turnstile> A \<and> (hr, as2) \<Turnstile> B)"
openin_subtopology_diff_closed: fixes S :: "'b set"   and X :: "'b topology"   and T :: "'b set" assumes "S \<subseteq> topspace X"   and "closedin X T" shows "openin (subtopology X S) (S - T)"