hcju/isabelleps huggingface dataset:
measurable_lfp_coinduct: fixes P :: "'c measure \<Rightarrow> bool"   and M :: "'c measure"   and F :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "P M"   and "sup_continuous F"   and "\<And>M A. \<lbrakk>P M; \<And>N. P N \<Longrightarrow> A \<in> N \<rightarrow>\<^sub>M count_space UNIV\<rbrakk> \<Longrightarrow> F A \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "lfp F \<in> M \<rightarrow>\<^sub>M count_space UNIV"
trivial_RModule: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Ring1 R"   and "\<forall>r\<in>R. smult r (0::'b) = (0::'b)" shows "RModule R smult 0"
Step1: shows "(\<lambda>g. \<lambda>H\<in>{H. subgroup H G}. g <# H #> inv g) \<in> carrier G \<rightarrow> carrier (BijGroup {H. subgroup H G})"
neg_eq_simps_1: fixes x :: "tr" assumes "neg\<cdot>x = TT" shows "x = FF"
ennreal_INF_const_minus: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "(\<Squnion>x\<in>I. c - f x) = c - \<Sqinter> (f ` I)"
that: fixes p1 :: "('v \<times> 'w \<times> 'v) list"   and p2 :: "('v \<times> 'w \<times> 'v) list" assumes "p = p1 @ p2"   and "is_path v p1 u"   and "u \<notin> int_vertices p1" shows "thesis"
elements_xs: shows "\<forall>x. x \<in> set (j # xs) \<longrightarrow> x < min (nrows A) (ncols A)"
incseq': shows "incseq (\<lambda>i x. g x i)"
three_div_general: fixes D :: "nat \<Rightarrow> nat"   and nd :: "nat" shows "(3 dvd (\<Sum>x<nd. D x * 10 ^ x)) = (3 dvd sum D {..<nd})"
le_less_trans: fixes x :: "'b list"   and y :: "'b list"   and z :: "'b list" assumes "x \<le>p y"   and "y <p z" shows "x <p z"
sums_emeasure: fixes F :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "disjoint_family F"   and "\<And>i. F i \<in> sets M" shows "(\<lambda>i. emeasure M (F i)) sums emeasure M (\<Union> (range F))"
array_shrink'_def: fixes a :: "'a array" shows "array_shrink' a \<equiv> array_shrink a \<circ> nat_of_integer"
degree_normalize: fixes f :: "'b poly" shows "degree (normalize f) = degree f"
isodeflI: fixes t :: "udom defl"   and d :: "'a \<rightarrow> 'a" assumes "\<And>x. cast\<cdot>t\<cdot>x = EMB('a)\<cdot>(d\<cdot>(PRJ('a)\<cdot>x))" shows "isodefl d t"
HTsetI: assumes "wf (inv_image R (f \<circ> snd))"   and "\<And>Pa c Qa s\<^sub>0. \<lbrakk>HTset_r (\<lambda>c' s'. ((c', s'), c, s\<^sub>0) \<in> inv_image R (f \<circ> snd)) \<pi> {(P, PCall p, Q)}; (Pa, c, Qa) \<in> {(P, PCall p, Q)}; Pa s\<^sub>0\<rbrakk> \<Longrightarrow> wp \<pi> c (Qa s\<^sub>0) s\<^sub>0" shows "HTset \<pi> {(P, PCall p, Q)}"
add: fixes i :: "'a"   and b1 :: "'b"   and b2 :: "'b" shows "f i (b1 + b2) = f i b1 + f i b2"
deliver_locally: fixes m :: "'a"   and i :: "nat" assumes "Broadcast m \<in> set (history i)" shows "Broadcast m \<sqsubset>\<^sup>i Deliver m"
distinct_13: fixes x1 :: "int \<times> int"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "C x1 \<noteq> CN x81 x82 x83"
simps: fixes x :: "real"   and y :: "real" shows "(Rmin x < Rmin y) = (x < y)"
sub_monom_2: fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "g \<in> carrier (UP R)"   and "a \<in> carrier R" shows "Cring_Poly.compose R (monom (UP R) a n) g = a \<odot>\<^bsub>UP R\<^esub> g [^]\<^bsub>UP R\<^esub> n"
B: shows "ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd(d := Pd')) Ch"
simps_1: fixes thy :: "exetheory"   and uu :: "(variable \<times> typ) list"   and uv :: "variable set"   and Hs :: "Core.term list"   and t :: "Core.term"   and Tis :: "((variable \<times> String.literal set) \<times> typ) list" shows "exereplay' thy uu uv Hs (PAxm t Tis) = (if exeinst_ok thy Tis \<and> exeterm_ok thy t then if t \<in> set (exeaxioms_of thy) then Some (forall_intro_vars (subst_typ' Tis t) []) else None else None)"
runit_char_1: fixes f :: "'a" assumes "Right.ide f" shows "Right.in_hom (Right.runit f) (Right.R f) f"
path2_2: shows "s2 = sum_weights (map (\<lambda>(s, l, t). l) path2)"
abs_gcd_int: fixes x :: "int"   and y :: "int" shows "\<bar>gcd x y\<bar> = gcd x y"
iteration_step_ex: fixes ES :: "('b list set \<times> 'b trm set) set"   and X :: "'b list set"   and xrhs :: "'b trm set" assumes "invariant ES"   and "(X, xrhs) \<in> ES"   and "Cond ES" shows "\<exists>xrhs'. (X, xrhs') \<in> Iter X ES"
\<I>_eqI: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('h, 'i) \<I>" assumes "outs_\<I> \<I> = outs_\<I> \<I>'"   and "\<And>x. x \<in> outs_\<I> \<I>' \<Longrightarrow> responses_\<I> \<I> x = responses_\<I> \<I>' x" shows "\<I> = \<I>'"
ide_iso_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>a : a \<cong> a\<guillemotright>"
returns_to_def: fixes P :: "'a set"   and x :: "'a" shows "returns_to P x = ((\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> P) \<and> (\<exists>t>0. t \<in> existence_ivl0 x \<and> flow0 x t \<in> P))"
OF_priority_match_ana_def: fixes \<gamma> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and flow_entries :: "('c, 'd) flow_entry_match list"   and packet :: "'e" shows "OF_priority_match_ana \<gamma> flow_entries packet \<equiv> let m = filter (\<lambda>f. \<gamma> (ofe_fields f) packet) flow_entries; m' = filter (\<lambda>f. \<forall>fo\<in>set m. ofe_prio fo \<le> ofe_prio f) m in case m' of [] \<Rightarrow> NoAction | [s] \<Rightarrow> Action s | s # aa # lista \<Rightarrow> Undefined"
prefix_code_3: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "prefix (x # xs) (y # ys) = (x = y \<and> prefix xs ys)"
prems_4: shows "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> q"
in_sets_embed_measure: fixes A :: "'a set"   and M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "A \<in> sets M" shows "f ` A \<in> sets (embed_measure M f)"
1: shows "rev (map (\<lambda>j. nth_default 0 (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) (p - j)) [0..<p + 1]) = map (nth_default 0 (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]))) [0..<p + 1]"
assms_2: shows "deadend v \<or> v \<notin> VV p"
vec_tendstoI: fixes f :: "'b \<Rightarrow> ('c, 'a) vec"   and a :: "('c, 'a) vec"   and net :: "'b filter" assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net" shows "(f \<longlongrightarrow> a) net"
gen_subgroup_is_subgroup: fixes G :: "('e, 'f) monoid_scheme"   and gens :: "'e set" assumes "Group.group G"   and "gens \<subseteq> carrier G" shows "\<langle>gens\<rangle>\<^bsub>G\<^esub> \<le> G"
has_as_product_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> nat"   and a :: "nat" shows "J.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J J.comp D a \<pi>"
strong_postdominate_path_branch: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and n' :: "'node" assumes "n -as\<rightarrow>* n''"   and "n' strongly-postdominates n''"   and "\<not> n' strongly-postdominates n" obtains a :: "'edge" and as' :: "'edge list" and as'' :: "'edge list" where "as = as' @ a # as''" and "valid_edge a" and "\<not> n' strongly-postdominates sourcenode a" and "n' strongly-postdominates targetnode a"
implode_def: shows "implode \<equiv> map_fun id Abs_literal (map ascii_of)"
D_LUB: fixes S :: "nat \<Rightarrow> 'c process" assumes "chain S" shows "D (lim_proc (range S)) = \<Inter> (D ` range S)"
assms_1: shows "F \<hookrightarrow>\<^sub>F G"
abs_eq: fixes xa :: "'c ltl"   and x :: "'c ltl" shows "Abs xa \<up>and Abs x = Abs (xa and x)"
norm_trunc_ell2_finite: fixes S :: "'c set"   and x :: "'c ell2" assumes "finite S" shows "norm (trunc_ell2 S x) = sqrt (\<Sum>i\<in>S. (cmod (Rep_ell2 x i))\<^sup>2)"
pair_measure_count_space: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "count_space A \<Otimes>\<^sub>M count_space B = count_space (A \<times> B)"
lb: fixes v :: "'a" assumes "v \<noteq> (0::'a)" shows "\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0::'a} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>"
variety_of_def: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "\<V> F = {a. \<forall>f\<in>F. poly_eval a f = (0::'d)}"
of_rat_floor: fixes r :: "rat" shows "\<lfloor>of_rat r\<rfloor> = \<lfloor>r\<rfloor>"
mangoldt_odd_pos: fixes d :: "nat" shows "0 \<le> mangoldt_odd d"
one_is_add: fixes m :: "nat"   and n :: "nat" shows "(Suc 0 = m + n) = (m = Suc 0 \<and> n = 0 \<or> m = 0 \<and> n = Suc 0)"
powr_le_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(b powr y \<le> x) = (y \<le> log b x)"
subST_1: shows "S \<subseteq> topspace (Euclidean_space n)"
distinct_set_remove_last: fixes xs :: "'a list"   and x :: "'a" assumes "distinct (xs @ [x])" shows "set xs = set (xs @ [x]) - {x}"
simps_4: fixes uu :: "int \<Rightarrow> int \<Rightarrow> bool"   and uv :: "value option" shows "MaybeBoolInt uu uv None = invalid"
ge2_events_lax: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set" assumes "MinkowskiSpacetime \<E> \<P> betw"   and "Q \<in> \<P>" shows "\<exists>a\<in>Q. \<exists>b\<in>Q. a \<noteq> b"
partList4_subset: fixes a :: "'b set"   and ts :: "'b set list"   and b :: "'b set" assumes "a \<subseteq> \<Union> (set ts)" shows "a \<subseteq> \<Union> (set (partList4 b ts))"
assms_2: shows "der \<in> borel_measurable (G matur)"
\<sigma>_nb: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and t :: "MIR.num"   and k :: "int" assumes "iszlfm p (a # bs)"   and "numbound0 t" shows "bound0 (\<sigma> p k t)"
pivotandupdate_unsat_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<U> (pivot_and_update x\<^sub>i x\<^sub>j c s) = \<U> s"
fimage_cong: fixes M :: "'a fset"   and N :: "'a fset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. x |\<in>| N \<Longrightarrow> f x = g x" shows "f |`| M = g |`| N"
sets_uniform_count_measure_eq_UNIV_1: shows "(sets (uniform_count_measure UNIV) = UNIV) = True"
trivial_hom_iff: shows "(h ` carrier R = {\<zero>\<^bsub>S\<^esub>}) = (a_kernel R S h = carrier R)"
abelian_groupE_5: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
arcs_subdivide: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b"   and w :: "'b" shows "parcs (subdivide G (u, v) w) = parcs G - {(u, v), (v, u)} \<union> {(u, w), (w, u), (w, v), (v, w)}"
prems: fixes x :: "'b"   and a\<^sub>1 :: "'a"   and a\<^sub>2 :: "'a" assumes "x \<in> S'"   and "f x a\<^sub>1 a\<^sub>2" shows "r a\<^sub>1 a\<^sub>2"
IH: fixes x :: "'a"   and y :: "'a" assumes "x \<in> M"   and "y \<in> iterates_above a" shows "x \<in> iterates_above y \<or> y \<in> iterates_above x"
rel_simps_48: fixes m :: "num"   and n :: "num" shows "(numeral m < numeral n) = (m < n)"
fmdomI: fixes m :: "('c, 'b) fmap"   and x :: "'c"   and y :: "'b" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
icheck_curr_init: fixes P :: "'a prog"   and C :: "char list"   and e' :: "'b exp"   and e :: "'b exp" assumes "icheck P C e'"   and "ss_exp e = e'" shows "curr_init P e = \<lfloor>C\<rfloor>"
invertibleI: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> M"   and "v \<in> M" shows "invertible u"
prime_sum_upto_altdef1: fixes f :: "nat \<Rightarrow> 'b"   and x :: "real" shows "prime_sum_upto f x = sum_upto (\<lambda>p. ind prime p * f p) x"
t2_def: shows "t2 \<equiv> punit.lt (rep_list q)"
word_lsb_alt: fixes w :: "'a word" shows "lsb w = w !! 0"
sat_Since_rec: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Since \<phi> I \<psi>) = (mem 0 I \<and> MFOTL.sat \<sigma> v i \<psi> \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> MFOTL.sat \<sigma> v i \<phi> \<and> MFOTL.sat \<sigma> v (i - 1) (formula.Since \<phi> (subtract (\<Delta> \<sigma> i) I) \<psi>))"
D_SKIP: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "(G, Skip, Sec (\<lambda>(s, t, \<beta>). s \<equiv>\<^sub>\<beta> t)) \<in> Deriv"
prod_topology_subtopology_2: fixes X :: "'b topology"   and Y :: "'c topology"   and T :: "'c set" shows "prod_topology X (subtopology Y T) = subtopology (prod_topology X Y) (topspace X \<times> T)"
assms: shows "is_pr_Hom \<phi> Fs\<^sub>1 Fs\<^sub>2"
from_nat_to_nat_id: fixes a :: "'b" shows "Bij_Nat.from_nat (Bij_Nat.to_nat a) = a"
conversep_Grp_id: fixes A :: "'b set" shows "(Grp A id)\<inverse>\<inverse> = Grp A id"
ODnum_segmentTr1: fixes D :: "'c Order"   and x :: "'c Order set"   and y :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D"   and "y \<in> ODnums"   and "y \<sqsubset> x" shows "\<exists>c. c \<in> carrier D \<and> y = ordinal_number (Iod D (segment D c))"
simps_8: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Min a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (min_aform_err p a1 a2)))"
exists_l_desc: fixes ct :: "'s \<Rightarrow> 's pmf" assumes "proper ct" shows "\<exists>l\<in>S\<^sub>r \<rightarrow> S\<^sub>r \<union> S2. \<forall>s\<in>S\<^sub>r. l_desc S\<^sub>r ct l s"
isMinCut_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and k :: "nat set" shows "isMinCut c s t k \<equiv> NCut c s t k \<and> (\<forall>k'. NCut c s t k' \<longrightarrow> NCut.cap c k \<le> NCut.cap c k')"
assms: shows "P \<equiv> \<lambda>l u x. f x \<in> {l1 x..u1 x} \<longrightarrow> g x \<in> {l2 x..u2 x} \<longrightarrow> f x * g x \<in> {l..u}"
lta_right_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LtA Bet Cong A B C F E D"
fold2_cong: fixes init1 :: "'a"   and init2 :: "'a"   and err1 :: "'a"   and err2 :: "'a"   and xs1 :: "'b list"   and xs2 :: "'b list"   and ys1 :: "'c list"   and ys2 :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "init1 = init2"   and "err1 = err2"   and "xs1 = xs2"   and "ys1 = ys2"   and "\<And>init x y. \<lbrakk>x \<in> set xs1; y \<in> set ys1\<rbrakk> \<Longrightarrow> f x y init = g x y init" shows "Matching.fold2 f err1 xs1 ys1 init1 = Matching.fold2 g err2 xs2 ys2 init2"
Plus: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and A :: "'d set"   and B :: "'e set"   and g :: "'d + 'e \<Rightarrow> 'c" assumes "comm_monoid_set f z"   and "finite A"   and "finite B" shows "comm_monoid_set.F f z g (A <+> B) = f (comm_monoid_set.F f z (g \<circ> Inl) A) (comm_monoid_set.F f z (g \<circ> Inr) B)"
fresh_list_nil: fixes a :: "'v" shows "a \<sharp> []"
double_orthogonal_complement_increasing: fixes M :: "'b set" shows "M \<subseteq> orthogonal_complement (orthogonal_complement M)"
skip_im_Tr2: fixes i :: "nat"   and n :: "nat"   and x :: "nat" assumes "0 < i"   and "i \<in> {i. i \<le> Suc n}"   and "i \<le> x" shows "skip i x = Suc x"
cases: fixes a1 :: "'d \<Rightarrow> 'e tree option"   and a2 :: "'d set"   and a3 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and a4 :: "'d \<Rightarrow> 'e tree option"   and a5 :: "'d set"   and a6 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "((a1, a2, a3), a4, a5, a6) \<in> brw_step \<delta>" obtains q :: "'d" and W :: "'d set" and dsqr :: "('d, 'e) ta_rule set" and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" and Q' :: "'d \<Rightarrow> 'e tree option" and Q :: "'d \<Rightarrow> 'e tree option" and W' :: "'d set" and rcm' :: "('d, 'e) ta_rule \<Rightarrow> nat option" where "a1 = Q" and "a2 = W" and "a3 = rcm" and "a4 = Q'" and "a5 = W'" and "a6 = rcm'" and "q \<in> W" and "dsqr = {r \<in> \<delta>. q \<in> set (rhsq r) \<and> the (rcm r) \<le> 1}" and "dom Q' = dom Q \<union> lhs ` dsqr" and "\<And>q t. Q' q = Some t \<Longrightarrow> Q q = Some t \<or> (\<exists>r\<in>dsqr. q = lhs r \<and> t = construct_witness Q r)" and "W' = W - {q} \<union> (lhs ` dsqr - dom Q)" and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm' r = (if q \<in> set (rhsq r) then Some (the (rcm r) - 1) else rcm r)"
uniqueness_of_interpolation: fixes S :: "'a set"   and n :: "nat"   and p :: "'a poly"   and q :: "'a poly" assumes "card S = Suc n"   and "degree p \<le> n"   and "degree q \<le> n"   and "\<And>x. x \<in> S \<Longrightarrow> poly p x = poly q x" shows "p = q"
CARD_1_UNIV: fixes x :: "'c" assumes "class.CARD_1 TYPE('c)" shows "UNIV = {x}"
Nml_implies_Arr: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t" shows "bicategorical_language.Arr V src trg t"
Call: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn"   and args :: "expr list"   and R :: "val \<Rightarrow> 'b assn"   and mode :: "inv_mode"   and statT :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and S :: "'b assn"   and accC :: "qtname" assumes "G,A\<turnstile>{Normal P} e-\<succ> {Q}"   and "\<forall>a. G,A\<turnstile>{Q\<leftarrow>In1 a} args\<doteq>\<succ> {R a}"   and "\<forall>a vs invC declC l. G,A\<turnstile>{R a\<leftarrow>In3 vs \<and>. (\<lambda>s. declC = invocation_declclass G mode (snd s) a statT \<lparr>name = mn, parTs = pTs\<rparr> \<and> invC = invocation_class mode (snd s) a statT \<and> l = locals (snd s)) ;. init_lvars G declC \<lparr>name = mn, parTs = pTs\<rparr> mode a vs \<and>. (\<lambda>s. normal s \<longrightarrow> G\<turnstile>mode\<rightarrow>invC\<preceq>statT)} Methd declC \<lparr>name = mn, parTs = pTs\<rparr>-\<succ> {set_lvars l .; S}" shows "G,A\<turnstile>{Normal P} {accC,statT,mode}e\<cdot>mn( {pTs}args)-\<succ> {S}"
polyhedral_cone_carrier: fixes A :: "'a mat"   and nr :: "nat" assumes "A \<in> carrier_mat nr n" shows "polyhedral_cone A \<subseteq> carrier_vec n"
prime_dvd_mult_int: fixes p :: "int"   and a :: "int"   and b :: "int" assumes "prime p" shows "(p dvd a * b) = (p dvd a \<or> p dvd b)"
fin: fixes s :: "'c" assumes "wellformed s" shows "finite {fold delta w s |w. w \<in> lists (set alphabet)}"
preserves_hom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "C.VV.in_hom f a b" shows "\<guillemotleft>fst f \<star>\<^sub>C snd f : fst a \<star>\<^sub>C snd a \<Rightarrow>\<^sub>C fst b \<star>\<^sub>C snd b\<guillemotright>"
cases: fixes x :: "symbol list" obtains "xb__ = []"   | x :: "symbol" and xs :: "symbol list" where "xb__ = x # xs"
UnCI: fixes c :: "'c"   and B :: "'c set"   and A :: "'c set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
LSM_sub_ker: fixes M :: "('g, 'a, 'h) Module_scheme"   and N :: "('i, 'a, 'j) Module_scheme"   and T :: "'g set"   and f :: "'g \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "T \<subseteq> carrier M"   and "f \<in> mHom R M N"   and "T \<subseteq> ker\<^bsub>M,N\<^esub> f" shows "LSM\<^bsub>R\<^esub> M T \<subseteq> ker\<^bsub>M,N\<^esub> f"
count_filter_zmset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b zmultiset"   and a :: "'b" shows "zcount (filter_zmset P M) a = (if P a then zcount M a else 0)"
sort_oalist_id: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw x xs" shows "sort_oalist x xs = xs"
th: shows "a = Mul i (snd (zsplit0 t)) \<and> n = i * fst (zsplit0 t)"
trans_less: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "cmp a b = Less"   and "cmp b c = Less" shows "cmp a c = Less"
super_orientation_diagonal: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y \<squnion> y\<^sup>T"   and "y \<sqinter> y\<^sup>T \<le> x \<sqinter> (1::'a)" shows "y \<sqinter> y\<^sup>T = x \<sqinter> (1::'a)"
elims: fixes x :: "('c, 'd) tm"   and y :: "('c, 'd) tm list" assumes "args x = y" obtains uu_ :: "('c, 'd) hd" where "x = Hd uu_" and "y = []"   | s :: "('c, 'd) tm" and t :: "('c, 'd) tm" where "x = App s t" and "y = args s @ [t]"
semiring_normalization_rules_16: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and lx :: "'b"   and ly :: "'b"   and rx :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times (times lx ly) rx = times (times lx rx) ly"
assms_1: fixes x :: "nat" assumes "x < length vs" shows "g x \<down>= vs ! x"
checkVarValue_Var: fixes type :: "varType"   and val :: "integer" assumes "varType_inv type" shows "variable_inv (Var type (checkVarValue type val))"
induct: fixes P :: "Dom \<Rightarrow> bool"   and Dom :: "Dom" assumes "P low"   and "P high" shows "P Dom"
cut_cut_greater: fixes i :: "'a set"   and a :: "'a"   and b :: "'a" shows "i \<down>> a \<down>> b = i \<down>> max a b"
IVT2': fixes f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and y :: "'b"   and a :: "'c" assumes "f b \<le> y"   and "y \<le> f a"   and "a \<le> b"   and "continuous_on {a..b} f" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
image_lt_hom_deg_set: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and z :: "nat" assumes "homogeneous_set A" shows "lpp ` (hom_deg_set z A - {0}) = {t \<in> lpp ` (A - {0}). deg_pm t = z}"
image_subsetI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
mem_ODnums_nonempty: fixes X :: "'c Order set" assumes "X \<in> ODnums" shows "\<exists>x. x \<in> X"
enum_pos_inverse: fixes x :: "'c" shows "enum_class.enum ! enum_pos x = x"
\<xi>_3: shows "ln_Gamma y - ln_Gamma x = (y - x) * Digamma \<xi>"
gt_sym_def: fixes g :: "nat"   and f :: "nat" shows "local.gt_sym g f = (f < g)"
connectedin_continuous_map_image: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "continuous_map X Y f"   and "connectedin X S" shows "connectedin Y (f ` S)"
widen_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and T :: "ty" shows "subtype P T T"
mono_seq_DT_right: fixes S :: "'b process"   and S' :: "'b process"   and P :: "'b process" assumes "S \<sqsubseteq>\<^sub>D\<^sub>T S'" shows "(P `;` S) \<sqsubseteq>\<^sub>D\<^sub>T (P `;` S')"
RedALength: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>length,(h, l)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val (Intg (word_of_nat n)),(h, l)\<rangle>"
ARR_iff_has_codomain: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.ARR C f = (partial_magma.codomains (free_monoidal_category.comp C) f \<noteq> {})"
arccos_le_pi2: fixes y :: "real" assumes "0 \<le> y"   and "y \<le> 1" shows "arccos y \<le> pi / 2"