hcju/isabelleps huggingface dataset:
tkt_thread_eq: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and k :: "nat" assumes "A.reachable (c, n, ts)"   and "t < N"   and "has_ticket (ts t) k" shows "tkt_thread ts k = t"
open_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "open S" shows "openin (top_of_set T) S"
supp_finite_set_at_base: fixes S :: "'b set" assumes "finite S" shows "supp S = atom ` S"
sum_eq_sum_append: fixes i :: "nat"   and j :: "nat"   and as :: "'b list"   and as' :: "'b list" assumes "0 < i"   and "i \<le> j"   and "j \<le> length as" shows "SWA.sum as i j = SWA.sum (as @ as') i j"
mult_nonneg_nonneg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
is_infinity_uminus: fixes a :: "('a, 'b) IEEE.float" shows "is_infinity (- a) = is_infinity a"
complex_norm_eq_1_exp: fixes z :: "complex" shows "(cmod z = 1) = (exp (\<i> * complex_of_real (Arg2pi z)) = z)"
Arefl_def: fixes R :: "('b \<times> 'b) set" shows "Arefl R = {e. (e, e) \<in> R}"
lcf_closed: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "f (deg R f) \<in> carrier R"
in_g: fixes a :: "'b"   and b :: "nat" assumes "(a, b) \<in> g" shows "b < max n (nextMax (Range g))"
infs_coinduct_shift: fixes R :: "'b stream \<Rightarrow> bool"   and w :: "'b stream"   and P :: "'b \<Rightarrow> bool" assumes "R w"   and "\<And>w. R w \<Longrightarrow> \<exists>u v. w = u @- v \<and> Bex (set u) P \<and> R v" shows "infs P w"
lm038: fixes P :: "('c \<times> 'd) set"   and X :: "'c set"   and Q :: "('c \<times> 'e) set" shows "Domain (P outside X) \<inter> Domain (Q || X) = {}"
reachable_nodes_steps'_2: fixes u :: "'b"   and g :: "'b ugraph"   and r :: "'b"   and v :: "'b" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> (edges g)\<^sup>*" shows "(u, v) \<in> (edges (component_of g r))\<^sup>*"
restr_delete: fixes x :: "'a"   and D :: "'a set"   and al :: "('a \<times> 'b) list" shows "delete x (restrictA D al) = (if x \<in> D then restrictA (D - {x}) al else restrictA D al)"
rel_option_restrict_relpI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "rel_option R x y"   and "pred_option P x"   and "pred_option Q y" shows "rel_option (R \<upharpoonleft> P \<otimes> Q) x y"
functions_of_def: fixes P :: "'c set set"   and A :: "'c set"   and B :: "'d set" shows "Twelvefold_Way_Entry7.functions_of P A B = {f \<in> A \<rightarrow>\<^sub>E B. (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P}"
offending_not_empty: fixes F :: "('c \<times> 'c) set"   and sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "F \<in> SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP" shows "F \<noteq> {}"
ih: fixes tt :: "('s, 'v) tm"   and ss :: "('s, 'v) tm" assumes "hsize tt + hsize ss < hsize t + hsize s"   and "tt \<rightarrow>\<^sub>e\<^sub>m\<^sub>b ss" shows "tt >\<^sub>t ss"
abs_eq_iff: fixes x :: "('a, 'b, 'c) Tree"   and y :: "('a, 'b, 'c) Tree" shows "(abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
harmonic_aux2: fixes M :: "nat" assumes "0 < M" shows "(\<Sum>n = 1..2 ^ M. 1 / real n) = 1 + (\<Sum>m = 1..M. \<Sum>n = 2 ^ (m - 1) + 1..2 ^ m. 1 / real n)"
pHom_X_Y: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y" shows "f X = Y"
that: fixes ns\<^sub>1 :: "'node list"   and ns\<^sub>3 :: "'node list"   and ns\<^sub>2 :: "'node list" assumes "ns = ns\<^sub>1 @ ns\<^sub>3 @ ns\<^sub>2"   and "prefix (ns\<^sub>1 @ [x]) ns"   and "suffix (x # ns\<^sub>2) ns"   and "g \<turnstile> n-ns\<^sub>1 @ [x]\<rightarrow>x"   and "x \<notin> set ns\<^sub>1"   and "g \<turnstile> x-ns\<^sub>3\<rightarrow>x"   and "g \<turnstile> x-x # ns\<^sub>2\<rightarrow>m"   and "x \<notin> set ns\<^sub>2" shows "thesis"
assms_2: shows "A \<inter> {a. en a s} \<noteq> {}"
simps_3: fixes ob :: "'c set"   and ar :: "'b set"   and dom :: "'b \<Rightarrow> 'c"   and cod :: "'b \<Rightarrow> 'c"   and id :: "'c \<Rightarrow> 'b"   and comp :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'d" shows "category.dom \<lparr>ob = ob, ar = ar, dom = dom, cod = cod, id = id, comp = comp, \<dots> = more\<rparr> = dom"
emeasure_INT_decseq_subset: fixes I :: "nat set"   and F :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "I \<noteq> {}"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; i \<le> j\<rbrakk> \<Longrightarrow> F j \<subseteq> F i"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"   and "\<And>i. i \<in> I \<Longrightarrow> emeasure M (F i) \<noteq> \<infinity>" shows "emeasure M (\<Inter> (F ` I)) = (INF i\<in>I. emeasure M (F i))"
le_boolE: fixes P :: "bool"   and Q :: "bool" assumes "P \<le> Q"   and "P" obtains "Q"
set_def: shows "local.set = list.set \<circ> inorder"
WSS_DR_iff_source_target_rel: fixes success :: "'barbs" shows "(((\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)))) \<and> enc_reflects_divergence) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_weakly_respects_barb_set Rel (STCalWB SWB TWB) {success} \<and> rel_reflects_divergence Rel (STCal Source Target))"
regular: shows "emeasure lborel (closure S) = emeasure lborel S"
alpha_res: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>res R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y)"
gcd_div_unit2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "gcd b (c div a) = gcd b c"
assms: shows "CT;\<Gamma> \<turnstile>+ es : Cs"
assms_5: shows "f (- 1) $ 1 = - 1"
tpOf_tNN: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and T :: "'a trm" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "Signature.tpOf (ProblemIkTpart.TE_resOf resOf) (ProblemIkTpart.tNN resOf prot protFw T) = Signature.tpOf resOf T"
sorted_wrt_quicksort: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "transp R"   and "Restricted_Predicates.total_on R (set xs)"   and "reflp_on R (set xs)" shows "sorted_wrt R (quicksort R xs)"
pos0: fixes i :: "nat"   and x :: "'a" shows "0 \<le> f ((T ^^ i) x)"
invertible_inverse_invertible: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> H" shows "sub.invertible (sub.inverse u)"
span: shows "carrier_vec n \<subseteq> local.span (set fs)"
cod_simps_3: shows "g.cod.leg1 = g.leg1"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + b + c = a + (b + c)"
Rep_cases: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and y :: "'c" assumes "type_definition Rep Abs A"   and "y \<in> A" obtains x :: "'b" where "y = Rep x"
assms: shows "\<Psi> \<rhd> P \<doteq> Q"
runit_commutes_with_R: fixes f :: "'a" assumes "ide f" shows "PB.runit (R f) = R (PB.runit f)"
card_eq_sum: fixes A :: "'a set" shows "card A = (\<Sum>x\<in>A. 1)"
n_valTr: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(THE l. l * Lv K v = v x) * Lv K v = v x"
h11: shows "[(ainv * b') ^ (4 * k + 2) = 1] (mod 4 * k + 3)"
bind_fv_unchanged: fixes x :: "variable"   and \<tau> :: "typ"   and t :: "Core.term" assumes "(x, \<tau>) \<notin> fv t" shows "bind_fv (x, \<tau>) t = t"
one_eq_divide_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(one = divide a b) = (b \<noteq> zero \<and> a = b)"
Lcm_0_iff': fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "(Lcm A = zero) = (\<nexists>l. l \<noteq> zero \<and> (\<forall>a\<in>A. dvd.dvd times a l))"
compose_assoc: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and h :: "'h \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "f \<in> A \<rightarrow> B" shows "compose A h (compose A g f) = compose A (compose B h g) f"
assms: fixes distinguisher :: "('a, 'b) distinguisher" assumes "\<I> \<turnstile>g distinguisher \<surd>" shows "connect distinguisher res = connect distinguisher res'"
\<C>_vars_subset_\<C>: fixes x :: "var\<^sub>C" shows "\<C>_vars\<^sub>C x \<subseteq> \<C>\<^sub>C"
dvd_lcm1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times a (lcm a b)"
sets_Collect_countable_Ex1: fixes M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. {x \<in> space M. P i x} \<in> sets M" shows "{x \<in> space M. \<exists>!i. P i x} \<in> sets M"
triangle_G': fixes a :: "'c" assumes "C.ide a" shows "D.inverse_arrows (\<eta> (G a)) (G (\<epsilon> a))"
inj_on_opp_chambers': fixes C :: "'a set"   and D :: "'a set" assumes "chamber C"   and "C \<notin> f \<turnstile> X"   and "chamber D"   and "D \<notin> f \<turnstile> X"   and "f ` C = f ` D" shows "C = D"
exterior_def: fixes X :: "'a set" shows "exterior X = - (interior X \<union> boundary X)"
iso_reduced_homology_by_contractible: fixes X :: "'b topology"   and S :: "'b set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "hom_induced p X {} X S id \<in> Group.iso (reduced_homology_group p X) (relative_homology_group p X S)"
elims: fixes x :: "tree"   and y :: "nat" assumes "treesize x = y" obtains "x = Leaf" and "y = 0"   | l :: "tree" and r :: "tree" where "x = Branching l r" and "y = 1 + treesize l + treesize r"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. Norm N (u n - u m) < e"
\<rho>\<^sub>1_def: shows "\<rho>\<^sub>1 \<equiv> run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w"
Inf_atLeastAtMost: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq x y" shows "Inf (ord.atLeastAtMost less_eq x y) = x"
LSC_BigAndL: fixes F :: "'b formula list"   and \<Gamma> :: "'b formula multiset" assumes "mset F + \<Gamma> \<Rightarrow>\<^sub>n" shows "\<^bold>\<And>F, \<Gamma> \<Rightarrow>\<^sub>n"
ith: shows "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
wf_mstate_minit_safe: fixes \<phi> :: "Formula.formula"   and R :: "event_data list set" assumes "mmonitorable \<phi>" shows "wf_mstate \<phi> pnil R (minit_safe \<phi>)"
wf_nFAnd: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" assumes "local.wf idx (FAnd \<phi> \<psi>)" shows "local.wf idx (nFAnd \<phi> \<psi>)"
derived_obtain_orig_block: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>D" obtains b2 :: "'a set" where "b = b2 \<inter> bl" and "b2 \<in># remove1_mset bl \<B>"
ode_d_fa_nth: fixes i :: "nat"   and n :: "nat"   and xs :: "floatarith list"   and ds :: "floatarith list" assumes "i < length ode_e" shows "ode_d_fa_nth n xs ds i = ode_d_fa n xs ds ! i"
R'_no_Q: fixes v :: "'a" assumes "v \<in> set R'"   and "v \<noteq> z" shows "\<not> Q'.hitting_paths v"
sym_conv_converse_eq: fixes r :: "('a \<times> 'a) set" shows "sym r = (r\<inverse> = r)"
imirror_def: fixes I :: "nat set" shows "imirror I \<equiv> (-) (iMin I + Max I) ` I"
inner_psi_l: fixes l :: "nat" shows "inner_prod (psi_l l) (psi_l l) = 1"
dsum_pOp_func: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_pOp I A \<in> carr_dsumag I A \<rightarrow> carr_dsumag I A \<rightarrow> carr_dsumag I A"
prems_2: shows "ideal F \<div> monomial (1::'a) t = ideal (monomial (1::'a) ` S)"
cr_view\<^sub>h_Grp: shows "cr_view\<^sub>h = Grp UNIV to_view\<^sub>h"
States_Int_not_mem: fixes F :: "('d, 'e, 'f) seqauto set"   and SA :: "('d, 'e, 'f) seqauto" assumes "\<Union> (States ` F) \<inter> States SA = {}" shows "SA \<notin> F"
shift_snth_less: fixes p :: "nat"   and xs :: "'a list"   and s :: "'a stream" assumes "p < length xs" shows "(xs @- s) !! p = xs ! p"
div_invert: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "0 < x"   and "0 < y"   and "y \<le> n div x" shows "x \<le> n div y"
sat_not_marked: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb"   and "rv \<in> red_vertices prb"   and "Conf.sat (confs prb rv)" shows "\<not> marked prb rv"
pos_divide_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq (divide b c) a = less_eq b (times a c)"
lemma3_5_d: fixes ars :: "('c \<times> 'd \<times> 'c) set"   and \<tau> :: "'c \<times> ('d \<times> 'c) list"   and \<sigma> :: "'c \<times> ('d \<times> 'c) list"   and \<sigma>' :: "'c \<times> ('d \<times> 'c) list"   and \<tau>' :: "'c \<times> ('d \<times> 'c) list"   and \<upsilon> :: "'c \<times> ('d \<times> 'c) list"   and \<sigma>'' :: "'c \<times> ('d \<times> 'c) list"   and \<upsilon>' :: "'c \<times> ('d \<times> 'c) list" assumes "diagram ars (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "diagram ars (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')" shows "diagram ars ((fst \<tau>, snd \<tau> @ snd \<upsilon>), \<sigma>, \<sigma>'', fst \<tau>', snd \<tau>' @ snd \<upsilon>')"
uncountable_infinite: fixes A :: "'f set" assumes "uncountable A" shows "infinite A"
Id_on_iff: fixes x :: "'a"   and y :: "'a"   and A :: "'a set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
sorted_nth_less: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and i :: "nat"   and j :: "nat" assumes "ccw'.sortedP x xs"   and "i < j"   and "j < length xs" shows "ccw' x (xs ! i) (xs ! j)"
d_cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "d\<^sub>y X \<tau> = d\<^sub>y (\<lambda>_. X \<tau>) \<tau>"
paths_either: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "paths (t \<oplus>\<oplus> t') = paths t \<union> paths t'"
assms_3: fixes i :: "'b" assumes "i \<in> S" shows "0 \<le> a i"
ord_less_eq_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "lt a b"   and "b = c" shows "lt a c"
that: fixes as :: "nat \<Rightarrow> 'a" assumes "A.is_fair_run as"   and "\<forall>i. R (as i) (cs i)" shows "thesis"
hyps_1: shows "P \<turnstile> D' \<prec>\<^sup>1 D"
supr_least: fixes B :: "'a set"   and a :: "'a" assumes "B \<subseteq> Field r"   and "a \<in> Field r"   and "\<And>b. b \<in> B \<Longrightarrow> (b, a) \<in> r" shows "(local.supr B, a) \<in> r"
assms_5: shows "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])"
linear_simps_3: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "f (0::'b) = (0::'c)"
imp_igWlsBinp: fixes MOD :: "('o, 'p, 'varSort, 'sort, 'q, 'r, 's, 't, 'u) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 't option"   and hA :: "'t \<Rightarrow> 'v"   and MOD' :: "('w, 'x, 'varSort, 'sort, 'y, 'z, 'aa, 'v, 'ab) model_scheme" assumes "igWlsBinp MOD delta binp"   and "ipresIGWlsAbs hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
comp_in_homI': fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "arr g"   and "local.dom f = a"   and "cod g = c"   and "local.dom g = cod f" shows "\<guillemotleft>local.comp g f : a \<rightarrow> c\<guillemotright>"
homeomorphic_translation: fixes S :: "'c set"   and a :: "'c" shows "S homeomorphic (+) a ` S"
is_le_rel_le: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and x :: "'t"   and y :: "'t" assumes "ord_term_lin.is_le_rel rel"   and "rel x y" shows "x \<preceq>\<^sub>t y"
elem_is_gen_pow: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c"   and x :: "'c" assumes "Finite_And_Cyclic_Groups.cyclic_group G gen"   and "x \<in> carrier G" shows "\<exists>n. x = gen [^]\<^bsub>G\<^esub> n"
path_valid_node_1: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and n :: "'g"   and as :: "'f list"   and n' :: "'g" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "CFG.valid_node sourcenode targetnode valid_edge n"
invert_opposite_sides_of_square: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr"   and h :: "('a S.E.term, 'a) S.arr"   and k :: "('a S.E.term, 'a) S.arr" assumes "S.seq f g"   and "f \<cdot>\<^sub>S g = h \<cdot>\<^sub>S k"   and "S.iso f"   and "S.iso k" shows "S.seq g (S.inv k) \<and> S.seq (S.inv f) h \<and> g \<cdot>\<^sub>S S.inv k = S.inv f \<cdot>\<^sub>S h"
i: shows "i < IArray.length (matrix_to_iarray (divide_by_norm A))"
append_eq_append_conv2: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and ts :: "'b list" shows "(xs @ ys = zs @ ts) = (\<exists>us. xs = zs @ us \<and> us @ ys = ts \<or> xs @ us = zs \<and> ys = us @ ts)"
vars_st_snd_map: fixes X :: "('b, 'a) Term.term list"   and f :: "'b" shows "vars\<^sub>s\<^sub>t (map Send X) = fv (Fun f X)"
right_impl_one: fixes a :: "'b" shows "a l\<rightarrow> a = (1::'b)"
circle_diam_rescale: fixes l :: "real"   and r :: "real" assumes "l < r" shows "circle_diam l r = (\<lambda>x. x * (complex_of_real r - complex_of_real l) + complex_of_real l) ` circle_01_diam"
simps_11: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>n \<phi> \<and> (\<forall>j\<le>i. suffix j \<xi> \<Turnstile>\<^sub>n \<psi>))"
card_pick_inf: fixes S :: "nat set"   and n :: "nat" assumes "infinite S" shows "card {a \<in> S. a < pick S n} = n"
respection: shows "rel_weakly_respects_barbs (indRelSTEQ SRel TRel) (STCalWB SWB TWB)"
Predict_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.Predict \<RR> k I = I \<union> {uu_. \<exists>r x. uu_ = init_item r k \<and> r \<in> \<RR> \<and> x \<in> bin I k \<and> next_symbol x = Some (fst r)}"
abs_ge_zero: fixes abs :: "'c \<Rightarrow> '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 a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq zero (abs a)"
Equiv_Exec_output_eqI: fixes equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and m :: "'h"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and k1 :: "nat"   and k2 :: "nat" assumes "equiv_states (localState1 c1) (localState2 c2)"   and "Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2" shows "last_message (map output_fun1 (f_Exec_Comp_Stream trans_fun1 (input_fun1 m # NoMsg\<^bsup>k1 - Suc 0\<^esup>) c1)) = last_message (map output_fun2 (f_Exec_Comp_Stream trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2))"
properties_distinct: fixes j :: "nat"   and i :: "nat" assumes "j \<in> {0<..<lambda}" shows "seq (i + j) \<noteq> seq i"
2: shows "\<l>[f \<star> g] = (\<l>[f] \<star> g) \<cdot> \<a>\<^sup>-\<^sup>1[trg f, f, g]"
last_length: fixes a :: "'b"   and xs :: "'b list" shows "(a # xs) ! length xs = last (a # xs)"