hcju/isabelleps huggingface dataset:
mbuf2t_take_add_induct': fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "event_data option list set list"   and ys :: "event_data option list set list"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and nts' :: "nat list"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and j' :: "nat"   and P' :: "String.literal \<Rightarrow> nat option"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "pred_mapping (\<lambda>x. x \<le> j) P"   and "pred_mapping (\<lambda>x. x \<le> j') P'"   and "rel_mapping (\<le>) P P'"   and "wf_mbuf2' \<sigma> P V j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>)) [Monitor.progress \<sigma> P \<phi> j..<Monitor.progress \<sigma> P' \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"   and "j \<le> j'"   and "U (min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)) z"   and "\<And>k X Y z. \<lbrakk>min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j) \<le> k; Suc k \<le> Monitor.progress \<sigma> P' \<phi> j'; Suc k \<le> Monitor.progress \<sigma> P' \<psi> j'; qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<phi>) X; qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<psi>) Y; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f X Y (\<tau> \<sigma> k) z)" shows "U (min (Monitor.progress \<sigma> P' \<phi> j') (Monitor.progress \<sigma> P' \<psi> j')) z'"
inv_dghm_ArrMap_vrange: shows "\<R>\<^sub>\<circ> (inv_dghm \<FF>\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
elims: fixes x :: "(Vars.var \<times> exp) list"   and y :: "assn" assumes "heapToAssn x = y" obtains "x = []" and "y = ANil"   | v :: "Vars.var" and e :: "exp" and \<Gamma> :: "(Vars.var \<times> exp) list" where "x = (v, e) # \<Gamma>" and "y = ACons v e (heapToAssn \<Gamma>)"
order_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "x \<le> y"
Limsup_const: fixes F :: "'a filter"   and c :: "'b" assumes "F \<noteq> bot" shows "Limsup F (\<lambda>x. c) = c"
valid_store_sops: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>store_sops is. valid_sop sop"
set_sorted_list_of_multiset: fixes M :: "'a multiset" shows "set (sorted_list_of_multiset M) = set_mset M"
vsubstInp_def2: fixes ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" shows "inp %[y1 // y2]_ys = lift (vsubst ys y1 y2) inp"
find_max_impl_refine: fixes S :: "('a, unit) RBT.rbt"   and P :: "'a \<Rightarrow> bool" assumes "rs.invar S" shows "RETURN (find_max_impl P S) \<le> find_max P (rs.\<alpha> S)"
fmeasurable_box: fixes a :: "'b"   and b :: "'b" shows "box a b \<in> fmeasurable lborel"
clo2: shows "closedin (top_of_set (S \<union> \<Union> G \<union> (S \<union> UF))) (S \<union> UF)"
4: shows "\<forall>a\<in>sset (smap f (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p)))). odd a"
simps: fixes s :: "'b + 'c"   and a :: "'b" shows "setlp s a = (\<exists>x. a = x \<and> s = Inl x)"
Osum_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" assumes "Field r \<inter> Field r' = {}"   and "Well_order r"   and "Well_order r'" shows "r \<le>o r \<union>o r'"
simps_5: fixes a :: "float" shows "max_Var_floatarith (Num a) = 0"
fail_cont_def: shows "local.fail_cont = ContT (\<lambda>_. fail)"
prod_filter_mono: fixes F :: "'b filter"   and F' :: "'b filter"   and G :: "'c filter"   and G' :: "'c filter" assumes "F \<le> F'"   and "G \<le> G'" shows "F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'"
invar_def: fixes \<phi> :: "'state \<Rightarrow> bool" shows "invar \<phi> \<equiv> \<forall>s a. reach s \<and> \<phi> s \<longrightarrow> \<phi> (snd (step s a))"
is_gcd_is_bezout_ext: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "class.bezout_ring times one plus zero minus uminus"   and "bezout_ring.is_bezout_ext times plus uminus bezout" shows "GCD_ring.is_gcd times (\<lambda>a b. case bezout a b of (x, xa, u, v, gcd') \<Rightarrow> gcd')"
set_category_def: fixes S :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and \<AA> :: "('ah \<times> 'ah) set" shows "set_category S \<AA> \<equiv> category S \<and> set_category_axioms S \<AA>"
list_augment_commute: fixes i :: "nat"   and j :: "nat"   and \<sigma> :: "'a list"   and v :: "'a"   and u :: "'a" assumes "i \<noteq> j" shows "list_augment (list_augment \<sigma> j v) i u = list_augment (list_augment \<sigma> i u) j v"
Fun_mkPoint: fixes a :: "'s"   and t :: "'s" assumes "ide a"   and "t \<in> local.set a" shows "Fun (mkPoint a t) = (\<lambda>_\<in>{unity}. t)"
size_Diff_singleton: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "size (remove1_mset x M) = size M - 1"
component_stable: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set" assumes "F \<le> G"   and "G \<in> stable A" shows "F \<in> stable A"
almost_all_suffixes_model_\<F>_generalized: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "closed \<G> w" shows "\<forall>\<^sub>\<infinity>j. \<forall>\<psi>. G \<psi> \<in> \<G> \<longrightarrow> suffix j w \<Turnstile> eval\<^sub>G \<G> (\<F> \<psi> w \<G> j)"
block_closed: fixes S :: "'c set"   and P :: "'c set set"   and a :: "'c"   and A :: "'c set" assumes "partition S P"   and "a \<in> A"   and "A \<in> P" shows "a \<in> S"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_def: fixes A :: "'a set" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<equiv> \<mu> x. SKIP |-| STOP |-| Mprefix A (\<lambda>xa. x)"
\<rho>s_def: shows "\<rho>s = tl (renamings_apart (DA0 # CAs0))"
os_reverse_aux_rule: fixes xs :: "'a list"   and p :: "'a node ref option"   and ys :: "'a list"   and q :: "'a node ref option" shows "<os_list xs p * os_list ys q> os_reverse_aux q p <os_list (rev xs @ ys)>"
p_dist_inf: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.stone_algebra inf less_eq less sup bot top uminus" shows "uminus (inf x y) = sup (uminus x) (uminus y)"
tagged_division_of_self_real: fixes x :: "real"   and a :: "real"   and b :: "real" assumes "x \<in> {a..b}" shows "{(x, {a..b})} tagged_division_of {a..b}"
exec_instr_CM_change: fixes ta :: "('a, 'b, 'c) jvm_thread_action"   and xcp' :: "'a option"   and h' :: "'c"   and stk' :: "'a val list"   and loc' :: "'a val list"   and C :: "String.literal"   and M :: "String.literal"   and pc' :: "nat"   and frs :: "'a frame list"   and addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ins :: "'a instr"   and P :: "(nat \<times> nat \<times> 'a instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'b"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and C' :: "String.literal"   and M' :: "String.literal" assumes "(ta, xcp', h', (stk', loc', C, M, pc') # frs) \<in> JVM_heap_base.exec_instr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ins P t h stk loc C M pc frs" shows "(ta, xcp', h', (stk', loc', C', M', pc') # frs) \<in> JVM_heap_base.exec_instr addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ins P t h stk loc C' M' pc frs"
assms_2: shows "(qbinp, qbinp') \<in> alphaBinp"
\<alpha>_ne_m: shows "\<alpha> \<noteq> m"
le_Inf_iff: fixes b :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set" shows "less_eq_prod b (Inf_prod A) = (\<forall>a\<in>A. less_eq_prod b a)"
cdijkstra_refines: shows "RETURN (cdijkstra g v0) \<le> \<Down> (br mr.\<alpha> mr.invar) mdijkstra"
fmdom_notD: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x |\<notin>| fmdom m" shows "fmlookup m x = None"
triangle': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "a \<otimes> \<l>[b] = (\<r>[a] \<otimes> b) \<cdot> \<a>\<^sup>-\<^sup>1[a, \<I>, b]"
obj_src: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "D.obj (src\<^sub>D \<mu>)"
isSurj_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'f, 'sort, 'g, 'h, 'c, 'i, 'j) Recursion.model_scheme" shows "isSurj h MOD \<equiv> \<forall>s X. gWls MOD s X \<longrightarrow> (\<exists>X'. wls s X' \<and> h X' = X)"
count_Mn_normal_form: shows "count_Mn r_normal_form = 1"
Units_r_inv_ex: fixes x :: "nat \<times> nat" assumes "x \<in> Units S" shows "\<exists>y\<in>carrier S. x \<otimes>\<^bsub>S\<^esub> y = \<one>\<^bsub>S\<^esub>"
dcache_val_def: fixes c :: "word20 \<times> word12"   and state :: "'b sparc_state" shows "dcache_val c state \<equiv> dcache (cache state) c"
eq_fst_iff: fixes a :: "'h"   and p :: "'h \<times> 'i" shows "(a = fst p) = (\<exists>b. p = (a, b))"
tab1: shows "table (args_n args) (args_L args) rel1"
assms: shows "b_e_type_checker \<C> es (t1s _> t2s)"
assms_1: fixes Y :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "\<And>x. countable (Y x)" shows "countable (F Y x)"
subl2: shows "{flow0 x t1--<flow0 x t2} \<subseteq> {a<--<flow0 x t2}"
FORK_refines_FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m: fixes i :: "nat" shows "FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m i 0 \<sqsubseteq> FORK i"
special_cosets_have_glbs: fixes X :: "'w set"   and Y :: "'w set" assumes "X \<in> \<P>"   and "Y \<in> \<P>" shows "\<exists>B. dual_order.glbound_in_of \<P> X Y B"
simps_21: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Skip \<noteq> x31@[x32]"
two_power_increasing: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m"   and "m < LENGTH('b)" shows "2 ^ n \<le> 2 ^ m"
mat_of_cblinfun_cblinfun_apply: fixes F :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and u :: "'b" shows "vec_of_basis_enum (F *\<^sub>V u) = mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u"
eqSm: shows "exH_k (Suc m) = stm2.pmat_extension (H_k m) * stm1.pmat_extension hadamard"
fresh_star_supp_conv: fixes x :: "'b"   and y :: "'a" assumes "supp x \<sharp>* y" shows "supp y \<sharp>* x"
multi_member_this: fixes x :: "'b"   and XS :: "'b multiset" shows "x \<in># \<LM> x  \<RM> + XS"
dependent_explicit: fixes s :: "'c set" shows "m2.dependent s = (\<exists>t u. finite t \<and> t \<subseteq> s \<and> (\<Sum>v\<in>t. u v *b v) = (0::'c) \<and> (\<exists>v\<in>t. u v \<noteq> (0::'a)))"
affine_basis_exists: fixes V :: "'c set" shows "\<exists>B\<subseteq>V. \<not> affine_dependent B \<and> affine hull V = affine hull B"
introR: fixes a :: "'e"   and a' :: "'f"   and R :: "('e \<times> 'f) set" assumes "(a, a') \<in> R" shows "(a, a') \<in> R"
call\<^sub>C_def: fixes M :: "'c \<Rightarrow> 'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and A\<^sub>1 :: "'a control_state_scheme \<Rightarrow> 'c" shows "call\<^sub>C M A\<^sub>1 = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else M (A\<^sub>1 \<sigma>) \<sigma>)"
SourcesOnlyDSources: fixes i :: "AbstrLevelsID"   and C :: "CSet" assumes "\<forall>X. X \<in> DSources i C \<longrightarrow> DSources i X = {}" shows "Sources i C = DSources i C"
is_Runit'_def: fixes "term" :: "'b local.term" shows "is_Runit' term = (\<exists>x8. term = \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[x8\<^bold>])"
Rep_upset_cases: fixes y :: "'a set" assumes "y \<in> range \<up>" obtains x :: "'a upset" where "y = Rep_upset x"
type_definition_num0: shows "type_definition Rep_num0 Abs_num0 UNIV"
UN_upper: fixes a :: "'d"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
ide_is_unary_product: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "ide a"   and "m \<noteq> n" shows "has_as_product (DiscreteCategory.discrete_category.comp {m} n) (\<lambda>i. if i = m then a else null) a"
assms_2: shows "w\<^sub>2 \<le> w\<^sub>3"
Compl_iff: fixes c :: "'f"   and A :: "'f set" shows "(c \<in> - A) = (c \<notin> A)"
o_apply: fixes f :: "'ah \<Rightarrow> 'ag"   and g :: "'ai \<Rightarrow> 'ah"   and x :: "'ai" shows "(f \<circ> g) x = f (g x)"
subsets_smash_property_R: fixes n :: "nat"   and subsets1 :: "(nat list \<times> nat list) list"   and subsets2 :: "(nat list \<times> nat list) list" shows "\<forall>elem. List.member (subsets_smash_R n subsets1 subsets2) elem \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<and> elem = (fst elem1 @ map ((+) n) (fst elem2), snd elem1 @ map ((+) n) (snd elem2)))"
divide_minus_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (uminus a) b = uminus (divide a b)"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
bound2: fixes K :: "real"   and e :: "real" assumes "0 < K"   and "0 < e" shows "AE x in M. limsup (\<lambda>xa. ereal (u xa x / real xa)) \<le> ereal (F K e x)"
split_option_all: fixes P :: "'b option \<Rightarrow> bool" shows "(\<forall>x. P x) = (P None \<and> (\<forall>x. P (Some x)))"
closure_def: fixes as :: "'a list" shows "closure as = rtrancl_while (\<lambda>(r, s). nullable r = nullable s) (\<lambda>(r, s). map (\<lambda>a. (nderiv a r, nderiv a s)) as)"
Zfun_left: fixes f :: "'e \<Rightarrow> 'a"   and F :: "'e filter"   and a :: "'b" assumes "Zfun f F" shows "Zfun (\<lambda>x. f x ** a) F"
elims: fixes x :: "trm option"   and xa :: "trm option"   and y :: "fml option" assumes "Geqo x xa = y" obtains \<theta> :: "trm" and \<eta> :: "trm" where "x = Aterm \<theta>" and "xa = Aterm \<eta>__" and "y = Afml (Geq \<theta> \<eta>__)"   | \<eta> :: "trm option" where "x = undeft" and "xa = \<eta>" and "y = undeff"   | v :: "trm" where "x = Aterm v" and "xa = undeft" and "y = undeff"
app_not_pair: fixes A :: "'b trm"   and B :: "'b trm"   and C :: "'b trm"   and D :: "'b trm" shows "App A B \<noteq> SimplyTyped.Pair C D"
null_sets_def: fixes M :: "'a measure" shows "null_sets M = {N \<in> sets M. emeasure M N = 0}"
smooth_on_inverse: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "(0::'c) \<notin> f ` S"   and "open S" shows "k-smooth_on S (\<lambda>x. inverse (f x))"
proj_arcs_G_in_arcs_H: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and a :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"   and "a \<in> arcs G" shows "subdiv1_contr.proj_arcs_G rev_G uv uw vw a \<in> arcs H"
le_a_ln: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "a m + ln (real m) \<le> a n + ln (real n)"
existence_ivl_reverse: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> rev.existence_ivl0 x0" shows "0 - t \<in> rev.existence_ivl0 (rev.flow0 x0 t)"
lin_part_id: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "lin_part f = monom P (f 0) 0 \<oplus>\<^bsub>P\<^esub> monom P (f 1) 1"
vsv_vimage_vsingleton: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> F" shows "F `\<^sub>\<circ> ZFC_in_HOL.set {a} = ZFC_in_HOL.set {F\<lparr>a\<rparr>}"
subterm_inj_on_def: fixes f :: "'a \<Rightarrow> ('b, 'c) Term.term"   and A :: "'a set" shows "subterm_inj_on f A \<equiv> \<forall>x\<in>A. \<forall>y\<in>A. (\<exists>v. v \<sqsubseteq> f x \<and> v \<sqsubseteq> f y) \<longrightarrow> x = y"
simps_2: fixes uv :: "'b"   and uw :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "KD_Tree.size_kdt (Node uv uw l r) = KD_Tree.size_kdt l + KD_Tree.size_kdt r"
partial_function_definitions_axioms: shows "partial_function_definitions option_ord (flat_lub None)"
dord_pm_antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord :: "('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "class.linorder less_eq less"   and "\<lbrakk>ord s t; ord t s\<rbrakk> \<Longrightarrow> s = t"   and "linorder.dord_pm ord s t"   and "linorder.dord_pm ord t s" shows "s = t"
RETURN_SPEC_refine: fixes x :: "'b"   and R :: "('b \<times> 'a) set"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'" shows "RETURN x \<le> \<Down> R (SPEC \<Phi>)"
iso: shows "\<exists>F G. inverse_functors J' J.comp F G"
ile_eSuc: fixes n :: "eint" shows "n \<le> n + 1"
max_list: fixes x :: "nat"   and xs :: "nat list" assumes "x \<in> set xs" shows "x \<le> max_list xs"
assms: fixes Xs :: "'a set list" assumes "1 < length Xs"   and "distinct (map f Xs)"   and "\<forall>i. f (Xs !% i) \<subset> Xs !% i \<inter> Xs !% (i + 1)" shows "False"
pivot'pivot: fixes s :: "('c, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)" shows "pivot' x\<^sub>i x\<^sub>j s = pivot x\<^sub>i x\<^sub>j s"
consistent_subset: fixes N :: "('a, 'b) message set"   and M :: "('a, 'b) message set" assumes "consistent N"   and "M \<subseteq> N"   and "\<Union> (deps ` M) \<subseteq> I ` insert_messages M" shows "consistent M"
monotone_TCons: fixes x :: "'c" shows "monotone local.tllist_ord local.tllist_ord (TCons x)"
Dynkin_system_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "Dynkin_system \<Omega> M \<equiv> subset_class \<Omega> M \<and> Dynkin_system_axioms \<Omega> M"
node_deliver_messages_Deliver: fixes m :: "'msg" shows "node_deliver_messages [Deliver m] = [m]"
Max_wrt_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Max_wrt R = Max_wrt_among R UNIV"
exhaust: fixes y :: "'b nmultiset" obtains x1 :: "'b" where "y = Elem x1"   | x2 :: "'b nmultiset multiset" where "y = MSet x2"
comp'_def: shows "comp' = vsplit (\<lambda>\<langle>a, b\<rangle>. vsplit (\<lambda>c. vpair a))"
g: shows "\<forall>x\<in>s. g (f x) = x"
valid_states_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.valid_states problem \<equiv> {s. dom s = ast_problem.Dom problem \<and> (\<forall>x\<in>ast_problem.Dom problem. the (s x) \<in> ast_problem.range_of_var problem x)}"
lt_map_scale: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "lt (c \<cdot> p) = lt p"
cont_custom_arccos: fixes S :: "real set" assumes "S \<subseteq> {- 1..1}" shows "continuous_on S (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x)"
vars_diff: fixes p1 :: "'a mpoly"   and p2 :: "'a mpoly" shows "vars (p1 - p2) \<subseteq> vars p1 \<union> vars p2"
DWaxiom_def: shows "DWaxiom = ([[EvolveODE (OVar vid1) (Pc pid1)]]Pc pid1)"
nsets_2_eq: fixes A :: "'b set" shows "[A]\<^bsup>2\<^esup> = (\<Union>x\<in>A. \<Union>y\<in>A - {x}. {{x, y}})"
closed_UN_well_founded: fixes XX :: "'c set set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>X\<in>XX. well_founded X r \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. r y x \<longrightarrow> y \<in> X)" shows "well_founded (\<Union> XX) r"
smult_distrib_left: fixes r :: "('f, 'g) aezfun"   and m :: "'v"   and n :: "'v" assumes "r \<in> FG"   and "m \<in> V"   and "n \<in> V" shows "r \<cdot> (m + n) = r \<cdot> m + r \<cdot> n"
assms_2: shows "interaction_bounded_by' isl \<A> q"
build_rel_def: fixes \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
alg_equiv_inv_auto_7: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "trm"   and T' :: "ty" assumes "\<Gamma> \<turnstile> Var x \<leftrightarrow> t : T'" obtains "t = Var x" and "valid \<Gamma>" and "(x, T') \<in> set \<Gamma>"
dvdm_imp_degree_le: fixes m :: "int"   and p :: "int poly"   and q :: "int poly" assumes "poly_mod.dvdm m p q"   and "monic p"   and "poly_mod.Mp m q \<noteq> 0"   and "1 < m" shows "degree p \<le> degree q"
span_singleton: fixes x :: "'b" shows "span {x} = range (\<lambda>k. k *s x)"
core_subset: fixes a :: "'a set" shows "core a \<subseteq> a"
finite_gpvD: fixes gpv :: "('c, 'out, 'in) gpv"   and out :: "'out"   and c :: "('c, 'out, 'in) rpv"   and input :: "'in" assumes "local.finite_gpv gpv"   and "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "local.finite_gpv (c input)"
red_stat_red_is_stat_red: shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) reduc_calc.Red_Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) reduc_calc.Red_Red_I Red_F"
ar_ar: fixes x :: "'a" shows "ar (r x) = ar x"
compos_aI_inj: fixes A :: "('c, 'd) aGroup_scheme"   and B :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c" assumes "aGroup A"   and "aGroup B"   and "f \<in> aHom A B"   and "g \<in> aHom B A"   and "compos A g f = aI\<^bsub>A\<^esub>" shows "injec\<^bsub>A,B\<^esub> f"
fds_truncate_mult: fixes m :: "nat"   and f :: "'a fds"   and g :: "'a fds" shows "fds_truncate m (fds_truncate m f * fds_truncate m g) = fds_truncate m (f * g)"
s_def: fixes x :: "'a" shows "s x = (x - a) \<bullet> rev.rot (b - a)"