hcju/isabelleps huggingface dataset:
cauchy_in\<^sub>N_def: fixes N :: "'b quasinorm"   and u :: "nat \<Rightarrow> 'b" shows "cauchy_in\<^sub>N N u = (\<forall>e>0. \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. Norm N (u n - u m) < e)"
us_gs: shows "u # us = map gso (rev [0..<j])"
NN1: fixes e :: "real" assumes "0 < e" shows "(real k * \<delta> / 2) ^ NN e < e"
subst_typ_simulates_tsubstT_gen: fixes T :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubstT T \<rho> = subst_typ (map (\<lambda>(x, y). ((x, y), \<rho> x y)) (SOME l. distinct l \<and> tvsT T \<subseteq> set l)) T"
f_shrink_last_def: fixes xs :: "'d list"   and k :: "nat" shows "xs \<div>\<^sub>l k \<equiv> f_aggregate xs k last"
reflp_ge_eq: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "reflp R" shows "(=) \<le> R"
LT11: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> G) \<longrightarrow> F \<leadsto> G \<or> H"
The_onI: fixes a :: "'c"   and U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> U"   and "P a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> x = a" shows "P (the (THE x on U. P x))"
sum_group_def: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "sum_group I G \<equiv> subgroup_generated (product_group I G) {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"
vD_iD_gives_kD_2: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> iD rt" shows "ip \<in> kD rt"
hm_isEmpty_def: fixes ht :: "('c, 'd) hashtable" shows "hm_isEmpty ht \<equiv> return (the_size ht = 0)"
A: shows "path_connected {x. \<exists>i\<in>Basis. x \<bullet> i < a \<bullet> i}"
shiftr_int_def: fixes x :: "int"   and n :: "nat" shows "x >> n = x div 2 ^ n"
c1_1: shows "A1 \<in> carrier_mat nr1 n1"
delivered_to_node: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and ii :: "nat"   and R\<^sub>i :: "nat set"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state" assumes "s \<in> reachable (pnet np \<langle>ii; R\<^sub>i\<rangle>) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np \<langle>ii; R\<^sub>i\<rangle>)" shows "i = ii"
clst_eq: shows "N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts) = Q"
e: shows "\<^bold>|w\<^bold>| - Suc (i + p) + p = \<^bold>|w\<^bold>| - Suc i"
assms_2: shows "b_e = Binop_i t iop \<or> b_e = Binop_f t fop \<or> b_e = Relop_i t irop \<or> b_e = Relop_f t frop"
bernoulli: fixes p :: "real"   and M :: "bool stream measure" assumes "infinite_coin_toss_space p M" shows "M = bernoulli_stream p"
nn_integral_indicator_singleton: fixes y :: "'a"   and M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "{y} \<in> sets M" shows "set_nn_integral M {y} f = f y * emeasure M {y}"
GSubspace_is_Subspace: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "RModule UNIV (\<sharp>\<cdot>) U \<and> U \<subseteq> V"
lset_conv_lnth: fixes xs :: "'a llist" shows "lset xs = {lnth xs n |n. enat n < llength xs}"
prj_in_hom_2: shows "\<guillemotleft>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 : HH\<mu>H\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<nu>\<pi>.apex\<guillemotright>"
bounded_coeffs_imp_fds_abs_converges': fixes f :: "'b fds"   and s0 :: "'b"   and s :: "'b" assumes "Bseq (\<lambda>n. fds_nth f n * nat_power n s0)"   and "1 - s0 \<bullet> (1::'b) < s \<bullet> (1::'b)" shows "fds_abs_converges f s"
LFOi_refine: fixes li :: "'a list"   and l :: "'b set"   and A :: "('a \<times> 'b) set"   and R :: "('d \<times> 'c) set"   and fi :: "'a \<Rightarrow> 'd \<Rightarrow> 'd nres"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s0i :: "'d"   and s0 :: "'c"   and I :: "'b set \<Rightarrow> 'c \<Rightarrow> bool" assumes "(li, l) \<in> \<langle>A\<rangle>list_set_rel"   and "\<And>x xi s si. \<lbrakk>(xi, x) \<in> A; (si, s) \<in> R\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)"   and "(s0i, s0) \<in> R" shows "nfoldli li (\<lambda>_. True) fi s0i \<le> \<Down> R (FOREACH\<^bsup>I\<^esup> l f s0)"
carrier_singleton_group: fixes a :: "'b" shows "carrier (singleton_group a) = {a}"
cyclic_group_def: fixes G :: "('d, 'e) cyclic_group_scheme" shows "Cyclic_Group.cyclic_group G \<equiv> Group.group G \<and> cyclic_group_axioms G"
is_arg_min_arg_min_nat: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "P x" shows "is_arg_min m P (arg_min m P)"
Gauss_Jordan_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k A k = snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])"
defAss_dominating: fixes n :: "'node"   and g :: "'g"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)" shows "defAss g n v = (\<exists>m\<in>set (\<alpha>n g). dominates g m n \<and> v \<in> allDefs g m)"
flow0_transversal_segment_monotone_step_reverse: fixes a :: "'a"   and b :: "'a"   and t1 :: "real"   and t2 :: "real"   and x :: "'a"   and t :: "real" assumes "transversal_segment a b"   and "t1 \<le> t2"   and "{t1..t2} \<subseteq> existence_ivl0 x"   and "flow0 x t1 \<in> {a<--<b}"   and "flow0 x t2 \<in> {a<--<flow0 x t1}"   and "\<And>t. t \<in> {t1<..<t2} \<Longrightarrow> flow0 x t \<notin> {a<--<b}"   and "t < t1"   and "t \<in> existence_ivl0 x" shows "flow0 x t \<notin> {a<--<flow0 x t1}"
assms_3: fixes x :: "'a" assumes "x \<in> X" shows "{x} \<in> sets M"
nvalid_restrict_to_nvalid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and M :: "'d set"   and n :: "nat"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>|\<^bsub>M\<^esub>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
cobounded2: fixes b :: "'b::semilattice_sup"   and a :: "'b::semilattice_sup" shows "(b::'b::semilattice_sup) \<le> (a::'b::semilattice_sup) \<squnion> b"
HFail_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "HFail s s' p = (Fail s s' p \<and> HNextPart s s')"
conseq: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>l s. P' l s \<longrightarrow> P l s"   and "\<turnstile>\<^sub>t {P} c {Q}"   and "\<forall>l s. Q l s \<longrightarrow> Q' l s" shows "\<turnstile>\<^sub>t {P'} c {Q'}"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a = inf a b)"
simps_12: fixes i :: "int"   and a :: "Cooper.num" shows "zlfm (NDvd i a) = (if i = 0 then zlfm (NEq a) else let (c, r) = zsplit0 a in if c = 0 then NDvd \<bar>i\<bar> r else if 0 < c then NDvd \<bar>i\<bar> (CN 0 c r) else NDvd \<bar>i\<bar> (CN 0 (- c) (Neg r)))"
that: fixes x :: "'a" assumes "rep_nat_term x = (t, 0)" shows "thesis"
CeqEnd: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "a \<noteq> []"   and "x \<in> dom (Cp (list2FWpolicy a))" shows "Cp (list2FWpolicy (b @ a)) x = Cp (list2FWpolicy a) x"
high_in_def: fixes \<tau> :: "trace" shows "high_in \<tau> \<equiv> the (initial \<tau> ''highIn'')"
range_Some: fixes A :: "'a set"   and a :: "'a" assumes "local.MaxR.MaxR_opt A = Some a" shows "a \<in> A \<inter> Field r"
is_qfN_def: fixes A :: "int"   and N :: "int" shows "is_qfN A N = (\<exists>x y. A = x\<^sup>2 + N * y\<^sup>2)"
nn_integral_cmult_indicator: fixes A :: "'c set"   and M :: "'c measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>+ x. c * indicator A x \<partial>M = c * emeasure M A"
eventually_all_ge_at_top: fixes P :: "'a \<Rightarrow> bool" assumes "eventually P at_top" shows "\<forall>\<^sub>F x in at_top. \<forall>y\<ge>x. P y"
vp_apow_ideal: fixes v :: "'a \<Rightarrow> ant"   and n :: "ant" assumes "valuation K v"   and "0 \<le> n" shows "ideal (Vr K v) (vp K v\<^bsup> Vr K v n\<^esup>)"
leadsETo_cancel1: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and B :: "'d set"   and A' :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] (B \<union> A')"   and "F \<in> B leadsTo[CC] B'" shows "F \<in> A leadsTo[CC] (B' \<union> A')"
dom_simp: fixes f :: "'b \<times> 'b" assumes "B.VV.arr f" shows "B.VV.dom f = B.VxV.dom f"
strict_dom_def: fixes i :: "nat"   and j :: "nat" shows "strict_dom i j \<equiv> let res = dom_kildall start ! j in i \<in> set res"
prod_D_F_eq_prod_A: shows "prod id D * prod id F = prod id A"
DomainE: fixes a :: "'b"   and r :: "('b \<times> 'c) set" assumes "a \<in> Domain r" obtains b :: "'c" where "(a, b) \<in> r"
ground_heads_nonempty: fixes \<zeta> :: "('s, 'v) hd" shows "local.ground_heads \<zeta> \<noteq> {}"
csa2: shows "set (characterize_consistent_signs_at_roots_copr p qs2) \<subseteq> set signs2"
error_def: fixes exception :: "'b" shows "error exception = Prog (\<lambda>h. Inl exception)"
inject_1: fixes x21 :: "bool"   and x22 :: "expr"   and x23 :: "expr"   and x24 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x25 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x26 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x27 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y21 :: "bool"   and y22 :: "expr"   and y23 :: "expr"   and y24 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y25 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y26 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y27 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set" shows "(Assign x21 x22 x23 x24 x25 x26 x27 = Assign y21 y22 y23 y24 y25 y26 y27) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25 \<and> x26 = y26 \<and> x27 = y27)"
ccw'_sortedP_snd_segments_of_aform: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "1 < length (inl (snd X))" shows "ccw'.sortedP (lowest_vertex (fst X, nlex_pdevs (snd X))) (butlast (map snd (segments_of_aform X)))"
effect_to_assignments_i: fixes as :: "('a \<times> bool) list"   and op :: "'a strips_operator" assumes "as = effect_to_assignments op" shows "as = map (\<lambda>v. (v, True)) (add_effects_of op) @ map (\<lambda>v. (v, False)) (delete_effects_of op)"
isomorphic_implies_ide_1: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f g" shows "Span.ide f"
left_neutral_mow: fixes a :: "'ag" assumes "a \<in> U" shows "0\<^sub>o\<^sub>w +\<^sub>o\<^sub>w a = a"
lim_le: fixes f :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "convergent f"   and "\<And>n. f n \<le> x" shows "lim f \<le> x"
domIff: fixes a :: "'e"   and m :: "'e \<Rightarrow> 'f option" shows "(a \<in> dom m) = (m a \<noteq> None)"
t_twice_len_def: shows "t_twice_len = length t_twice div 2"
abs_eq: fixes xb :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and xa :: "'d"   and x :: "'d" shows "minor (of_fun xb) xa x = of_fun (\<lambda>k l. if k = xa \<and> l = x then 1::'c else if k = xa \<or> l = x then 0::'c else xb k l)"
rtf_init_neighbor_invar: shows "neighbor_invar c s t pp_init_f pp_init_l rtf_init_n"
is_natural_2: fixes f :: "'a" assumes "S.arr f" shows "R (S.cod f) \<cdot>\<^sub>S R f = R f"
hyp2_axiom6: shows "\<forall>a b. B\<^sub>K a b a \<longrightarrow> a = b"
T_ins3_Bal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins3 n d x t = Bal2 t'"   and "bal_i n (height t + d)" shows "real (T_ins3 n d x t) + \<Phi> t' - \<Phi> t \<le> (6 * e + 2) * real (height t + 1)"
hseqI': fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "C.hseq \<nu> \<mu>"
increasingD: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> 'c"   and x :: "'b set"   and y :: "'b set" assumes "increasing M f"   and "x \<subseteq> y"   and "x \<in> M"   and "y \<in> M" shows "f x \<le> f y"
except_decomp: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and U :: "'b set" shows "p = except p U + except p (- U)"
circle_cube_nice: shows "circle_cube = (\<lambda>(x, y). (d * x_coord x, (2 * y - 1) * d * circle_y (x_coord x)))"
\<theta>_def: fixes xb :: "('e, 'f) F \<times> 'g" shows "\<theta> xb = F id \<langle>id, \<lambda>a. snd xb\<rangle> (fst xb)"
inf_sup_ord_4: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and y :: "'f"   and x :: "'f" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
indep_in_not_basis_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "indep_in \<E> X"   and "\<not> basis_in \<E> X" shows "\<exists>x\<in>\<E> - X. indep_in \<E> (insert x X)"
sat_MatchP_pointD: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and t :: "nat"   and r :: "Formula.formula Regex.regex"   and I :: "\<I>" assumes "Formula.sat \<sigma> V v i (formula.MatchP (point t) r)"   and "mem t I" shows "Formula.sat \<sigma> V v i (formula.MatchP I r)"
Times_Plus_TimesLR_PlusL: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times_Plus (TimesLR (Plus r s) t) = Plus (Times_Plus (TimesLR r t)) (Times_Plus (TimesLR s t))"
rtranclp_induct: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P a"   and "\<And>y z. \<lbrakk>r\<^sup>*\<^sup>* a y; r y z; P y\<rbrakk> \<Longrightarrow> P z" shows "P b"
verit_comp_simplify_29: shows "(0::'a) \<le> (1::'a)"
assms_3: shows "natural_transformation J (\<cdot>) D D' \<tau>"
cobounded1: fixes a :: "'e"   and b :: "'e" shows "a \<sqinter> b \<sqsubseteq> a"
rel_distinct_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y2 :: "'d" shows "\<not> rel_option R None (Some y2)"
sturm_rat: fixes p :: "rat poly" shows "sturm (real_of_rat_poly p) = map real_of_rat_poly (sturm_rat p)"
j_least: fixes a :: "nat"   and i :: "nat" assumes "coprime a p"   and "j \<le> i" shows "[a ^ (2 ^ i * m) = 1] (mod p)"
UNIV_not_singleton: fixes x :: "'d" shows "UNIV \<noteq> {x}"
path_singleton: fixes s :: "'lbl"   and l1 :: "'vtx"   and l2 :: "'vtx" assumes "s \<in>\<^sub>A weights l1 l2" shows "path l1 l2 [(l1, s, l2)]"
it_b_eq: fixes a :: "'a" assumes "a \<in> set as" shows "it_b a = foldli (bs a)"
simps_8: fixes c :: "poly"   and n :: "nat"   and p :: "poly" shows "polybound0 (CN c n p) = (n \<noteq> 0 \<and> polybound0 c \<and> polybound0 p)"
antisym_conv2: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "(\<not> y < x) = (x = y)"
scale: fixes r :: "real"   and b :: "'n" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
cond_mono: fixes P\<^sub>2 :: "('b, 'c) uexpr"   and P\<^sub>1 :: "('b, 'c) uexpr"   and Q\<^sub>2 :: "('b, 'c) uexpr"   and Q\<^sub>1 :: "('b, 'c) uexpr"   and b :: "'c upred" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 \<triangleleft> b \<triangleright> Q\<^sub>1 \<sqsubseteq> P\<^sub>2 \<triangleleft> b \<triangleright> Q\<^sub>2"
a: shows "a k \<in> S \<rightarrow> carrier_vec n"
foldSetD_determ_aux: fixes e :: "'a"   and A :: "'b set"   and n :: "nat"   and x :: "'a"   and y :: "'a" assumes "e \<in> D"   and "card A < n"   and "A \<subseteq> B"   and "(A, x) \<in> foldSetD D (\<cdot>) e"   and "(A, y) \<in> foldSetD D (\<cdot>) e" shows "y = x"
nontriv_solution_of_nat_of_nat: fixes a :: "nat"   and b :: "nat" shows "nontriv_solution (of_nat a, of_nat b) = nontriv_solution (a, b)"
mono_downclosed: fixes f :: "'c \<Rightarrow> 'd" assumes "mono f" shows "\<forall>Y. downclosed_set Y \<longrightarrow> downclosed_set (f -` Y)"
sum_list_map2_plus: fixes xs :: "'c list"   and ys :: "'c list" assumes "length xs = length ys" shows "sum_list (map2 (+) xs ys) = sum_list xs + sum_list ys"
dX_singular: fixes X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "PdXXX.dX X d = {x}"
plus_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.ordered_ab_semigroup plus zero less_eq"   and "less_eq y x" shows "less_eq (plus y z) (plus x z)"
Generic_Syntax_axioms: shows "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"
Some: shows "val_of e = \<lfloor>v\<rfloor>"
IH: assumes "Suc n \<le> deg R f" shows "deg R (f \<ominus>\<^bsub>P\<^esub> shift (Suc n) f \<otimes>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> Suc n) < Suc n"
agr: shows "Vagree (stateinterpol \<omega> \<omega>' S) (stateinterpol \<omega> \<omega>' (insert z S)) (- {z})"
list_empty_prefix: fixes xs :: "'b list"   and y :: "'b"   and zs :: "'b list"   and us :: "'b list" assumes "xs @ y # zs = y # us"   and "y \<notin> set xs" shows "xs = []"
stern_brocot_rationals: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "0 < n" shows "root (traverse_tree (mk_path m n) (pure rat_of \<diamondop> stern_brocot_recurse)) = Fract (int m) (int n)"
case_d4: fixes b :: "real"   and a :: "real"   and r :: "real"   and c :: "real"   and y' :: "real" assumes "b \<noteq> 2 * a * r"   and "\<not> b < 2 * a * r"   and "a * r\<^sup>2 - b * r + c = 0"   and "- r < y'" shows "\<exists>x\<in>{- r<..y'}. \<not> a * x\<^sup>2 + b * x + c < 0"
neg_less_0_iff_less: fixes a :: "'a" shows "(- a < (0::'a)) = ((0::'a) < a)"
ok_Join_commute: fixes F :: "'c program"   and G :: "'c program"   and H :: "'c program" shows "(F ok G \<and> F \<squnion> G ok H) = (G ok H \<and> F ok (G \<squnion> H))"
valuate_Var: fixes x :: "nat"   and v :: "nat \<Rightarrow> 'a" shows "Abstract_Linear_Poly.Var x \<lbrace> v \<rbrace> = v x"
intro: fixes v :: "complex Matrix.mat"   and n :: "nat" assumes "dim_col v = 1"   and "dim_row v = 2 ^ n"   and "\<parallel>Matrix.col v 0\<parallel> = 1" shows "state n v"
inf_distribD_1: fixes f :: "'g \<Rightarrow> 'h" assumes "inf_distrib f" shows "strict f"
select_convs_2: fixes store :: "'c \<Rightarrow> nat"   and pred :: "(('c \<times> nat \<Rightarrow> 'd) \<Rightarrow> bool) set"   and more :: "'e" shows "pred \<lparr>store = store, pred = pred, \<dots> = more\<rparr> = pred"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ts[i := (p\<^sub>2, is @ mis, \<theta>, sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis], \<D>', \<O>, \<R>')]) m = flush_all_until_volatile_write ts m"
alphaBinp_refl: fixes qbinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qAbs option" assumes "qGoodBinp qbinp" shows "(qbinp, qbinp) \<in> alphaBinp"
asIMOD_igAbsRen: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igAbsRen (asIMOD SEM)"
induct: fixes x1 :: "('a, 'b, 'c, 'd, 'e) qTerm"   and x2 :: "('a, 'b, 'c, 'd, 'e) qTerm"   and P :: "('a, 'b, 'c, 'd, 'e) qTerm \<Rightarrow> ('a, 'b, 'c, 'd, 'e) qTerm \<Rightarrow> bool" assumes "(x1, x2) \<in> qSwapped"   and "\<And>X. P X X"   and "\<And>X Y Z. \<lbrakk>(X, Y) \<in> qSwapped; P X Y; (Y, Z) \<in> qSwapped; P Y Z\<rbrakk> \<Longrightarrow> P X Z"   and "\<And>X Y x y zs. \<lbrakk>(X, Y) \<in> qSwapped; P X Y\<rbrakk> \<Longrightarrow> P X (Y #[[x \<and> y]]_zs)" shows "P x1 x2"
ppag: shows "path_image \<gamma> \<subseteq> - cball z (e / 2)"
not_all_zero: shows "\<not> (\<forall>a. is_zero_row_upt_k a (ncols (Gauss_Jordan A)) (Gauss_Jordan A))"
Map_comp: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "('a, 'b) concrete_category.arr"   and g :: "('a, 'b) concrete_category.arr" assumes "concrete_category Obj Hom Id Comp"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) f"   and "partial_magma.arr (concrete_category.COMP Obj Hom Comp) g"   and "concrete_category.Dom g = concrete_category.Cod f" shows "concrete_category.Map (concrete_category.COMP Obj Hom Comp g f) = Comp (concrete_category.Cod g) (concrete_category.Dom g) (concrete_category.Dom f) (concrete_category.Map g) (concrete_category.Map f)"
conj_blt: fixes n :: "nat"   and p :: "fm"   and q :: "fm" assumes "boundslt n p"   and "boundslt n q" shows "boundslt n (Parametric_Ferrante_Rackoff.conj p q)"
characteriseLast: fixes C :: "('a, 'b) sequent"   and m :: "nat"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set" assumes "(C, m + 1) \<in> derivable R" shows "\<exists>Ps. Ps \<noteq> [] \<and> (Ps, C) \<in> R \<and> (\<forall>p\<in>set Ps. \<exists>n\<le>m. (p, n) \<in> derivable R)"
arr_dom_iff_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.arr (J.dom f) = J.arr f"
equiv_imp_dvd_card: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and k :: "nat" assumes "finite A"   and "equiv A r"   and "\<And>X. X \<in> A // r \<Longrightarrow> k dvd card X" shows "k dvd card A"
seqSubst2: fixes x :: "name"   and \<sigma> :: "(name list \<times> 'a list) list"   and T :: "'a" assumes "x \<sharp> \<sigma>"   and "x \<sharp> T" shows "x \<sharp> T[<\<sigma>>]"
has_codomain_iff_arr: fixes f :: "'c" shows "(C.codomains f \<noteq> {}) = C.arr f"
inject_2: fixes x21 :: "variable"   and x22 :: "typ"   and y21 :: "variable"   and y22 :: "typ" shows "(Fv x21 x22 = Fv y21 y22) = (x21 = y21 \<and> x22 = y22)"
divide_strict_left_mono: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "(0::'a) < c"   and "(0::'a) < a * b" shows "c / a < c / b"