hcju/isabelleps huggingface dataset:
leq_lit_match: fixes L :: "('b, 'a) Term.term literal"   and K :: "('b, 'a) Term.term literal"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term option" assumes "match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>"   and "is_pos L = is_pos K" shows "leq_lit L K"
intro: fixes A :: "'c set" assumes "csubspace A"   and "closed A" shows "closed_csubspace A"
deriv_c: shows "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
k_memb_def: fixes s :: "'s odlist"   and A :: "(('s, 'aAct list) trie, ('s, ('obs, 's odlist) mapping) trie) AlgState" shows "SPRsingle.KBP.k_memb s A \<equiv> isSome (MapOps.lookup trie_odlist_MapOps (aActs A) s)"
pelims: fixes x :: "(int \<times> int) list"   and y :: "(int \<times> int) \<times> int \<times> int" assumes "closest_pair_bf x = y"   and "closest_pair_bf_dom x" obtains "x = []" and "y = undefined" and "closest_pair_bf_dom []"   | uu_ :: "int \<times> int" where "x = [uu_]" and "y = undefined" and "closest_pair_bf_dom [uu_]"   | p\<^sub>0 :: "int \<times> int" and p\<^sub>1 :: "int \<times> int" where "x = [p\<^sub>0'__, p\<^sub>1]" and "y = (p\<^sub>0'__, p\<^sub>1)" and "closest_pair_bf_dom [p\<^sub>0'__, p\<^sub>1]"   | p\<^sub>0 :: "int \<times> int" and v :: "int \<times> int" and vb :: "int \<times> int" and vc :: "(int \<times> int) list" where "x = p\<^sub>0 # v # vb # vc" and "y = (let (c\<^sub>0, c\<^sub>1) = closest_pair_bf (v # vb # vc); p\<^sub>1 = find_closest_bf p\<^sub>0 (v # vb # vc) in if dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (c\<^sub>0, c\<^sub>1) else (p\<^sub>0, p\<^sub>1))" and "closest_pair_bf_dom (p\<^sub>0 # v # vb # vc)"
synth_Un: fixes G :: "msg set"   and H :: "msg set" shows "synth G \<union> synth H \<subseteq> synth (G \<union> H)"
pref_concat_pref: fixes us :: "'b list list"   and ws :: "'b list list" assumes "us \<le>p ws" shows "concat us \<le>p concat ws"
urD_1: shows "real_of_rat l \<le> real_of_1 (p, l, r)"
mono_bis: fixes Retr1 :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"   and Retr2 :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "Bisim.mono Retr1"   and "Bisim.mono Retr2"   and "\<And>theta. Retr1 theta \<subseteq> Retr2 theta" shows "bis Retr1 \<subseteq> bis Retr2"
cases: fixes x :: "nat \<times> ('vkey \<times> 'sigkey \<times> ('message \<times> 'signature) list) option \<times> unit" obtains \<eta> :: "nat" and uu_ :: "unit" where "x = (\<eta>'__, None, uu_)"   | \<eta> :: "nat" and vkey :: "'vkey" and sigkey :: "'sigkey" and log :: "('message \<times> 'signature) list" and uv_ :: "unit" where "x = (\<eta>, Some (vkey, sigkey, log), uv_)"
measure_nonneg: fixes M :: "'d measure"   and A :: "'d set" shows "0 \<le> Sigma_Algebra.measure M A"
m_comm: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"
conga_preserves_lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p" assumes "A B C CongA A' B' C'"   and "D E F CongA D' E' F'"   and "A B C LtA D E F" shows "A' B' C' LtA D' E' F'"
cases: fixes x :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr" obtains \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" and e :: "('b, 'a, 'c) expr" where "x = (\<sigma>, \<tau>, e)"
IH: shows "make_vert_equiv k = mat1 (2 ^ k)"
bsok_simps_10: fixes a :: "'a expr1"   and i :: "'a expr1"   and e :: "'a expr1"   and n :: "nat" shows "bsok (a\<lfloor>i\<rceil> := e) n = (bsok a n \<and> bsok i n \<and> bsok e n)"
castAbs_Abs_subst: fixes xs :: "'varSort"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var"   and x' :: "'var"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isInBar (xs, s)"   and "wls s X"   and "wls (asSort xs) Y"   and "castAbs xs s (Abs xs x X) = (x', X')" shows "X #[Y / x]_xs = X' #[Y / x']_xs"
NF_no_trancl_step: fixes a :: "'b"   and A :: "('b \<times> 'b) set" assumes "a \<in> NF A" shows "\<forall>b. (a, b) \<notin> A\<^sup>+"
common_root_len_gcd: fixes u :: "'b list"   and t :: "'b list"   and v :: "'b list" assumes "u \<in> t*"   and "v \<in> t*" shows "\<^bold>|t\<^bold>| dvd gcd \<^bold>|u\<^bold>| \<^bold>|v\<^bold>|"
AE_all_countable: fixes P :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and M :: "'d measure" shows "(AE x in M. \<forall>i. P i x) = (\<forall>i. almost_everywhere M (P i))"
dg_prod_3_is_arrE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and A :: "V"   and B :: "V"   and F :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "digraph \<alpha> \<CC>"   and "F : A \<mapsto>\<^bsub>\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>\<^esub> B" obtains f :: "V" and f' :: "V" and f'' :: "V" and a :: "V" and a' :: "V" and a'' :: "V" and b :: "V" and b' :: "V" and b'' :: "V" where "F = [f, f', f'']\<^sub>\<circ>" and "A = [a, a', a'']\<^sub>\<circ>" and "B = [b, b', b'']\<^sub>\<circ>" and "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" and "f' : a' \<mapsto>\<^bsub>\<BB>\<^esub> b'" and "f'' : a'' \<mapsto>\<^bsub>\<CC>\<^esub> b''"
abs_induct: fixes P :: "'b proj_space \<Rightarrow> bool"   and x :: "'b proj_space" assumes "\<And>y. P (Proj y)" shows "P x"
mmore_ext: fixes a :: "\<lparr>unit_sys :: 'a, \<dots> :: 'b\<rparr>" shows "\<lparr>unit_sys = unit, \<dots> = mmore a\<rparr> = a"
ilist_eq_iff: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "(f = g) = (\<forall>x. f x = g x)"
binding: fixes \<eta> :: "nat"   and \<A> :: "'grp \<Rightarrow> ('grp \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" shows "okamoto_\<Sigma>_commit.abstract_com.bind_advantage \<eta> \<A> \<le> local.dis_log.advantage \<eta> (local.dis_log_alt.adversary2 (okamoto_\<Sigma>_commit.adversary \<eta> \<A>))"
nodesI_2: fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> edges g" shows "v \<in> nodes g"
minus_ow_axioms: shows "minus_ow U (-\<^sub>o\<^sub>w)"
wf_trancl: fixes r :: "('d \<times> 'd) set" assumes "wf r" shows "wf (r\<^sup>+)"
finite_finite_subset_caract': fixes F :: "'b set"   and C :: "real" assumes "\<And>G. \<lbrakk>G \<subseteq> F; finite G\<rbrakk> \<Longrightarrow> real (card G) \<le> C" shows "finite F \<and> real (card F) \<le> C"
integral_indicator: fixes M :: "'a measure"   and A :: "'a set" shows "integral\<^sup>L M (indicat_real A) = Sigma_Algebra.measure M (A \<inter> space M)"
rk_A'C'PR: fixes P :: "Points"   and Q :: "Points"   and R :: "Points"   and A' :: "Points"   and C' :: "Points" assumes "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {A', C', P, Q} = 4" shows "rk {A', C', P, R} = 4"
proj_specific_subterm: fixes t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and l :: "'b"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set" assumes "t \<sqsubseteq> t'"   and "proj_specific l t' \<A> Sec" shows "proj_specific l t \<A> Sec \<or> t \<in> Sec \<or> {} \<turnstile>\<^sub>c t"
dc: assumes "det (mat_of_rows n G) \<noteq> (0::'a)" shows "carrier_vec n \<subseteq> local.span (set G)"
acdb: fixes i :: "'a" assumes "i \<in> Basis"   and "box c d \<subseteq> cbox a b"   and "\<And>i. i \<in> Basis \<Longrightarrow> c \<bullet> i < d \<bullet> i" shows "a \<bullet> i \<le> c \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i"
inf_not_le_Lt: fixes x :: "'c" shows "(\<infinity> \<le> Lt x) = False"
restrict_size_less: fixes k :: "'a ifex"   and var :: "'a"   and val :: "bool" assumes "ifex_top_var k = Some var" shows "size (restrict_top k var val) < size k"
comp_arr_inv': fixes f :: "'c" assumes "C.iso f" shows "f \<cdot>\<^sub>C C.inv f = C.cod f"
Input: fixes x :: "name"   and a :: "name"   and P :: "pi" assumes "x \<noteq> a" shows "a<x>.P \<longmapsto> a<x> \<prec> P"
vc: shows "vc C (\<lambda>a. I a + 1)"
assms_1: shows "B P TS A C \<and> B' P' TS A' C' \<or> B P OS A C \<and> B' P' OS A' C'"
rev_cases: fixes xs :: "'a list" obtains "xs = []"   | ys :: "'a list" and y :: "'a" where "xs = ys @ [y]"
inv_in_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<phi>.B.iso f"   and "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>\<phi>.B.inv f : b \<rightarrow>\<^sub>B a\<guillemotright>"
comp_inv_arr': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "C (category.inv C f) f = partial_magma.dom C f"
T3_def: shows "T3 \<equiv> \<lparr>verts = {u0, v0, w0}, arcs = {(u0, v0), (v0, u0), (v0, w0), (w0, v0), (w0, u0), (u0, w0)}\<rparr>"
assms: shows "(Set{\<two>} \<doteq> null) = false"
symmetric_poly_of_roots_in_subring_monic: fixes C :: "'a set"   and q :: "'a mpoly"   and l :: "'b \<Rightarrow> 'a"   and A :: "nat set"   and p :: "'b poly"   and root :: "nat \<Rightarrow> 'b" assumes "ring_closed C"   and "\<forall>m. MPoly_Type.coeff q m \<in> C"   and "ring_homomorphism l"   and "finite A"   and "symmetric_mpoly A q"   and "vars q \<subseteq> A"   and "p = (\<Prod>i\<in>A. [:- root i, 1::'b:])"   and "\<forall>i. l (poly.coeff p i) \<in> C" shows "insertion (\<lambda>x. l (root x)) q \<in> C"
add_mset_add_single: fixes a :: "'a"   and A :: "'a multiset" shows "add_mset a A = A + \<LM> a  \<RM>"
vec_eq: shows "(\<chi>i. if i = x then 1::'a else (0::'a)) = (\<chi>i. if x = i then 1::'a else (0::'a))"
br_chain: fixes \<beta> :: "'a \<Rightarrow> 'c"   and J :: "'a \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'b"   and I :: "'c \<Rightarrow> bool" shows "br \<beta> J O br \<alpha> I = br (\<alpha> \<circ> \<beta>) (\<lambda>s. J s \<and> I (\<beta> s))"
id_FSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'e, 'f, 'h) model_scheme" shows "FSbImorph id id MOD MOD"
rep_eq: fixes x :: "'d \<Rightarrow> bool"   and xa :: "('d, 'c) lgenerator'" shows "lgenerator (ltakeWhile_trans' x xa) = takeWhile_raw x (lgenerator xa)"
maps_to_ex_def: fixes x :: "char list" shows "x \<hookrightarrow> - \<equiv> \<lambda>s. \<exists>x. (x \<hookrightarrow> x) s"
S_alt_def: shows "S (A Q \<pi>) = {r} \<union> fst ` A Q \<pi>"
Class_subset: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E" shows "Class a \<subseteq> Class b"
prv_lcnj_mono: fixes \<phi>1s :: "'fmla list"   and \<phi>2s :: "'fmla list" assumes "set \<phi>1s \<subseteq> fmla"   and "set \<phi>2s \<subseteq> set \<phi>1s" shows "prv (imp (lcnj \<phi>1s) (lcnj \<phi>2s))"
snoc_1: fixes m :: "'a"   and l' :: "'a list"   and m' :: "'a"   and t' :: "'a list" assumes "(m, l') = foldl f (m', t') xs" shows "mset (m # l') = mset (m' # t' @ xs)"
Min_ge_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less_eq x (linorder.Min less_eq A) = (\<forall>a\<in>A. less_eq x a)"
apartment_min_gallery_least_length: fixes X :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "X \<in> \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D" shows "SimplicialComplex.min_maxsimpchain X (C # (ARG_MIN length Cs. SimplicialComplex.maxsimpchain X (C # Cs @ [D])) @ [D])"
transp_ln_smallo_ln: shows "transp (\<lambda>f g. (\<lambda>x. ln (g x)) \<in> o(\<lambda>x. ln (f x)))"
ide_implies_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "free_monoidal_category.can C a"
iNextWeak_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextWeak t0 I P \<equiv> \<box> t {inext t0 I} \<down>> t0. P t"
differentiable_imp_continuous_within: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "f differentiable at x within s" shows "continuous (at x within s) f"
p_and_not_p: fixes P :: "'b upred" shows "(P \<and> \<not> P) = false"
less_imp_le: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subset># y" shows "x \<subseteq># y"
max_min_distrib1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.linorder less_eq less" shows "ord.max less_eq (ord.min less_eq b c) a = ord.min less_eq (ord.max less_eq b a) (ord.max less_eq c a)"
cases: fixes x :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list \<times> 't list \<times> (('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" obtains bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list" and ss :: "'t list" and z :: "nat" where "x = ((bsa__, ssa__, []), za__)"   | bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list" and ss :: "'t list" and p :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat" and ps :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list" and z :: "nat" where "x = ((bs, ss, p # ps), z)"
measurable_sets_borel: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'c measure"   and A :: "'c set" assumes "f \<in> borel \<rightarrow>\<^sub>M M"   and "A \<in> sets M" shows "f -` A \<in> sets borel"
split_B: shows "B = {i \<in> B. i < j} \<union> {i \<in> B. j \<le> i}"
set_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>set_rel \<equiv> {(A, B). (\<forall>x\<in>A. \<exists>y\<in>B. (x, y) \<in> R) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. (x, y) \<in> R)}"
converse_inv_image: fixes R :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c" shows "(inv_image R f)\<inverse> = inv_image (R\<inverse>) f"
numeral_octo_conv_of_nat: fixes x :: "num" shows "numeral x = octo_of_nat (numeral x)"
borel_measurable_nn_integral: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> ennreal"   and N :: "'c measure" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. integral\<^sup>N M2 (f x)) \<in> borel_measurable N"
hext_arrD: fixes h :: "'heap"   and h' :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat" assumes "h \<unlhd> h'"   and "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>" shows "typeof_addr h' a = \<lfloor>Array_type T n\<rfloor>"
var_renaming_preserves_size_2: fixes \<alpha> :: "'d \<Rightarrow> 'a"   and \<beta> :: "'e \<Rightarrow> 'b"   and \<zeta> :: "'f \<Rightarrow> 'c"   and e :: "('d, 'e, 'f) expr" shows "size (map_expr \<alpha> \<beta> \<zeta> e) = size e"
group_l_invI: assumes "\<And>x. x \<in> carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)) \<Longrightarrow> \<exists>y\<in>carrier (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)). y \<otimes>\<^bsub>DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)\<^esub> x = \<one>\<^bsub>DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n)\<^esub>" shows "Group.group (DirProds (\<lambda>p. Z (p ^ multiplicity p n)) (prime_factors n))"
left_neutral: fixes a :: "'g" shows "(1::'g) * a = a"
induct: fixes P :: "('c \<Rightarrow> bool) \<Rightarrow> ('d, 'c) trie \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> bool"   and a1 :: "('d, 'c) trie" assumes "\<And>p v ps. (\<And>x xa y. \<lbrakk>x \<in> set ps; (xa, y) = x\<rbrakk> \<Longrightarrow> P p y) \<Longrightarrow> P p (Trie v ps)" shows "P a0 a1"
segment_unique: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "ord_equiv (Iod D (segment D a)) (Iod D (segment D b))" shows "a = b"
a0: shows "0 < m div 2 ^ multiplicity 2 m"
takeWhile_dropWhile_id: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "takeWhile P xs @ dropWhile P xs = xs"
delete_def: fixes x :: "'b"   and t :: "('b \<times> color) tree" shows "RBT_Set2.delete x t = paint Black (RBT_Set2.del x t)"
cases: fixes a :: "('a, 'b) obs_event action" assumes "is_new_action a" obtains aa :: "'a" and hT :: "htype" where "a = NormalAction (NewHeapElem aa hT)"
Sksplit: shows "{Suc k} \<union> {Suc (Suc k)..<n} = {Suc k..<n}"
sup_dual: fixes x :: "'a"   and y :: "'a" shows "\<partial> (x \<squnion> y) = \<partial> x \<sqinter> \<partial> y"
index_of_id_main: shows "nat_to_pr index_of_id = (\<lambda>x. x)"
Seq_le: fixes c1 :: "'b acom"   and c2 :: "'b acom"   and c :: "'b acom" shows "(c1;; c2 \<le> c) = (\<exists>c1' c2'. c = c1';; c2' \<and> c1 \<le> c1' \<and> c2 \<le> c2')"
less: fixes y :: "nat" assumes "y < m" shows "(f ^^ (y mod n)) x = (f ^^ y) x"
canI_associator_hcomp_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "src f = trg g"   and "src g = trg h"   and "src h = trg k" shows "\<a>[f, g \<star> h, k] = can (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> (\<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>) ((\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>g\<^bold>\<rangle> \<^bold>\<star> \<^bold>\<langle>h\<^bold>\<rangle>) \<^bold>\<star> \<^bold>\<langle>k\<^bold>\<rangle>)"
f_def: fixes A :: "'a" shows "f A \<equiv> THE B. (A, B) \<in> F"
K_record_comp: fixes c :: "'l"   and f :: "'k \<Rightarrow> 'm" shows "(\<lambda>x. c) \<circ> f = (\<lambda>x. c)"
seq_cons_Goto_t: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and chain :: "char list"   and t :: "state"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Goto chain)], Undecided\<rangle> \<Rightarrow> t"   and "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>Rule m (Goto chain) # rs, Undecided\<rangle> \<Rightarrow> t"
case_optionE: fixes P :: "bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b option" assumes "case x of None \<Rightarrow> P | Some y \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'b" where "x = Some y" and "Q y"
Ord_TC_less_iff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "(\<beta> \<sqsubset> \<alpha>) = (\<beta> \<subset>\<^sub>\<circ> \<alpha>)"
tail_events_sets: fixes A :: "nat \<Rightarrow> 'a set set" assumes "\<And>i. A i \<subseteq> events" shows "tail_events A \<subseteq> events"
node\<^sub>i_bal: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and ts :: "('b btree \<times> 'b) list"   and t :: "'b btree"   and k :: "nat" assumes "split split"   and "bal (Node ts t)" shows "bal_up\<^sub>i (split.node\<^sub>i k ts t)"
countable_dense_setE: fixes p :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "countable_basis p B" obtains D :: "'c set" where "countable D" and "\<And>X. \<lbrakk>p X; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>d\<in>D. d \<in> X"
image_mset_empty: fixes f :: "'d \<Rightarrow> 'c" shows "image_mset f {#} = {#}"
inner_dirsum_doubleI: fixes M :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "M = A + B"   and "add_independentS [A, B]" shows "M = A \<oplus> B"
holds_Stream: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and s :: "'b stream" shows "holds P (x ## s) = P x"
assms: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "y \<in>\<^sub>\<circ> B" shows "x \<noteq> y"
Inf_greatest: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e"   and A :: "'e set"   and z :: "'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
sv01_consistent_V01: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1" shows "\<forall>x\<le>n. \<phi> (the (s\<^bsub>01\<^esub> (f \<triangleright> n))) x = f x"
sel: fixes f :: "'j \<Rightarrow> 'g"   and g :: "'k \<Rightarrow> 'h"   and h :: "'i \<Rightarrow> 'l"   and gpv :: "('j, 'k, 'l) gpv" shows "the_gpv (map_gpv' f g h gpv) = map_spmf (map_generat f g ((\<circ>) (map_gpv' f g h))) (map_spmf (map_generat id id (h ---> id)) (the_gpv gpv))"
map_1: fixes f :: "'b \<Rightarrow> 'a" shows "map_option f None = None"
le_TC_succ: fixes x :: "V"   and \<beta> :: "V" shows "(x \<sqsubseteq> ZFC_in_HOL.succ \<beta>) = (x \<sqsubseteq> \<beta> \<or> x = ZFC_in_HOL.succ \<beta>)"
filter_M_filter: fixes h :: "'a"   and P :: "'c \<Rightarrow> ('a, 'b, bool) prog"   and xs :: "'c list"   and ys :: "'c list" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "(\<forall>x\<in>set xs. h \<turnstile> ok P x) \<and> ys = filter (\<lambda>x. |h \<turnstile> P x|\<^sub>r) xs"
matchT_MT_def2: fixes d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchT_MT d c = (\<forall>s t t'. s \<approx> t \<and> (d, t) \<rightarrow>t t' \<longrightarrow> (\<exists>s'. (c, s) \<rightarrow>*t s' \<and> s' \<approx> t'))"
primeidealI: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "cring R"   and "carrier R \<noteq> I"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes>\<^bsub>R\<^esub> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" shows "primeideal I R"
path_component_of_disjoint: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" shows "disjnt (path_component_of_set X x) (path_component_of_set X y) = (\<not> path_component_of X x y)"
set_less_eq_aux'_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubseteq>'' B"   and "B \<sqsubseteq>'' C" shows "A \<sqsubseteq>'' C"
SetMark_def: shows "SetMark \<equiv> \<lambda>(i, j). case (i, j) of (init, loop) \<Rightarrow> Q1_a \<sqinter> Q2_a | (init, _) \<Rightarrow> \<top> | (loop, init) \<Rightarrow> \<top> | (loop, loop) \<Rightarrow> Q3_a \<sqinter> Q4_a | (loop, final) \<Rightarrow> Q5_a | (final, b) \<Rightarrow> \<top>"
eval_fds_at_nat: fixes f :: "'b fds"   and k :: "nat" shows "eval_fds f (of_nat k) = (\<Sum>n. fds_nth f n / of_nat n ^ k)"
op_ivl_Suc: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {m..<Suc n} = (if n < m then 0::'a else sum g {m..<n} + g n)"
assms_2: fixes y :: "'a" assumes "m2 = Inr y" shows "f1 y = f2 y"
for_rec3'_def: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "for_rec3' f a n k i j = (if k = 0 then RETURN a else for_rec (\<lambda>a k. for_rec2' (\<lambda>a. f a k) a n n n) a (k - 1)) \<bind> (\<lambda>a. for_rec2' (\<lambda>a. f a k) a n i j)"
error_free_init_lvars: fixes s :: "state"   and G :: "prog"   and C :: "qtname"   and sig :: "sig"   and mode :: "inv_mode"   and a :: "val"   and pvs :: "val list" assumes "error_free s" shows "error_free (init_lvars G C sig mode a pvs s)"
perm_simps_4: fixes p :: "perm"   and var :: "Syntax.var"   and "term" :: "Syntax.term" shows "p \<bullet> Rec var term = Rec (p \<bullet> var) (p \<bullet> term)"