hcju/isabelleps huggingface dataset:
same_key2D: fixes s\<^sub>2 :: "event list"   and g :: "guest"   and r :: "room"   and k\<^sub>2 :: "key"   and k :: "key"   and s\<^sub>1 :: "event list"   and k\<^sub>1 :: "key"   and g' :: "guest" assumes "hotel (s\<^sub>2 @ Check_in g r (k\<^sub>2, k) # s\<^sub>1)"   and "(k\<^sub>1, k) \<in> Trace.cards (s\<^sub>2 @ Check_in g r (k\<^sub>2, k) # s\<^sub>1) g'" shows "g = g' \<and> k\<^sub>1 = k\<^sub>2"
filterlim_sequentially_iff_filterlim_real: fixes f :: "'b \<Rightarrow> nat"   and F :: "'b filter" shows "filterlim f sequentially F = (LIM x F. real (f x) :> at_top)"
ltln_weak_to_strong_1: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi>"   and "\<not> w \<Turnstile>\<^sub>n G\<^sub>n \<phi>" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi>"
to_int_mod_ring_rebase: fixes x :: "'a mod_ring" assumes "rebase_ge TYPE('a) TYPE('b)" shows "to_int_mod_ring (@x) = to_int_mod_ring x"
finiteExpand: fixes P :: "pi"   and Q :: "pi" shows "finite (expandSet P Q)"
atoms_def: shows "atoms \<equiv> \<lambda>H. {x. atomsp (\<lambda>x. x \<in> H) x}"
Op_preserves_wls: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "wls (stOf delta) (Op delta inp binp)"
adjuster_finsum: fixes us :: "'a vec list"   and w :: "'a vec" assumes "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us = (\<Oplus>\<^bsub>V\<^esub>u\<in>set us. - (w \<bullet>c u) / (u \<bullet>c u) \<cdot>\<^sub>v u)"
getMinTree_min_prio: fixes bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and y :: "'a \<times> 'b" assumes "BinomialHeapStruc.queue_invar bq"   and "y \<in># BinomialHeapStruc.queue_to_multiset bq" shows "BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree bq) \<le> snd y"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "h (inv x) \<in> carrier H"
update_convs_2: fixes channels' :: "channel list \<Rightarrow> channel list"   and vars :: "(String.literal, variable) assoc_list"   and channels :: "channel list"   and timeout :: "bool"   and procs :: "pState list"   and more :: "'b" shows "gState.channels_update channels' \<lparr>gState.vars = vars, channels = channels, timeout = timeout, procs = procs, \<dots> = more\<rparr> = \<lparr>gState.vars = vars, channels = channels' channels, timeout = timeout, procs = procs, \<dots> = more\<rparr>"
game_equiv_subst: fixes \<alpha> :: "game"   and \<beta> :: "game"   and P :: "(variable \<Rightarrow> real) set \<Rightarrow> bool"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "game_equiv \<alpha> \<beta>"   and "P (game_sem I \<alpha> X)" shows "P (game_sem I \<beta> X)"
intros_4: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_term \<Sigma> t"   and "wf_term \<Sigma> u" shows "wf_term \<Sigma> (t $ u)"
component_edges_subset: fixes g :: "'a ugraph"   and r :: "'a" shows "edges (component_of g r) \<subseteq> edges g"
liftE_length: fixes n :: "nat"   and k :: "nat"   and \<Gamma> :: "binding list" shows "\<parallel>\<up>\<^sub>e n k \<Gamma>\<parallel> = \<parallel>\<Gamma>\<parallel>"
SatN: fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>@ a p)) at b in' set ((ps, a) # branch)"   and "new (\<^bold>\<not> p) a ((ps, a) # branch)"   and "A, Suc n \<turnstile> ((\<^bold>\<not> p) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
sums_of_real: fixes X :: "nat \<Rightarrow> real"   and a :: "real" assumes "X sums a" shows "(\<lambda>n. of_real (X n)) sums of_real a"
times_divide_eq_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * (b / c) = a * b / c"
pmdl_induct: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and B :: "('b \<Rightarrow>\<^sub>0 'c) set"   and P :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool" assumes "p \<in> MPoly_Type_Class.punit.pmdl B"   and "P 0"   and "\<And>a p c t. \<lbrakk>a \<in> MPoly_Type_Class.punit.pmdl B; P a; p \<in> B; c \<noteq> (0::'c)\<rbrakk> \<Longrightarrow> P (a + MPoly_Type_Class.punit.monom_mult c t p)" shows "P p"
bij_betw_iff_bijections: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" shows "bij_betw f A B = (\<exists>g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
gb_schema_aux_term_def: fixes d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "gb_schema_aux_term d gs = gb_schema_aux_term1 \<inter> gb_schema_aux_term2 d gs"
\<psi>i_B: fixes i :: "nat" shows "\<psi>i i \<in> rangeB'"
comp_isotone: fixes x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "x \<le> y"   and "w \<le> z" shows "x * w \<le> y * z"
subseq_sum_weights_le: fixes xs :: "'sum list"   and ys :: "'sum list" assumes "xs \<preceq> ys" shows "sum_weights xs \<le> sum_weights ys"
subst_compose_diff: fixes \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm"   and x1 :: "'var"   and x2 :: "'var" assumes "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<in> var"   and "x2 \<in> var"   and "x1 \<noteq> x2"   and "x1 \<notin> FvarsT t2" shows "subst (subst \<phi> t1 x1) t2 x2 = subst (subst \<phi> t2 x2) (substT t1 t2 x2) x1"
refl: fixes x :: "'a" assumes "x \<in> set_spmf p" shows "R x x"
path_connected_space_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "path_connected_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> (\<forall>i\<in>I. path_connected_space (X i)))"
d: shows "x |\<in>| (Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> f)"
rank_removed: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and v :: "nat \<times> 'a" assumes "finite (nodes A)"   and "w \<notin> language A" shows "v \<notin> graph A w (Suc (rank A w v))"
SCallParamsThrow\<^sub>1: fixes P :: "nat exp prog"   and es :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and es' :: "nat exp list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and ex :: "nat exp"   and es\<^sub>2 :: "nat exp list"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   and "es' = map Val vs @ throw ex # es\<^sub>2" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
NSsummable_NSCauchy: fixes f :: "nat \<Rightarrow> real" shows "NSsummable f = (\<forall>M\<in>HNatInfinite. \<forall>N\<in>HNatInfinite. \<bar>sumhr (M, N, f)\<bar> \<approx> 0)"
pathfinish_pathstart_partcirclepath_simps_10: fixes z0 :: "complex"   and r :: "real"   and st :: "real" shows "pathfinish (part_circlepath z0 r st pi) = z0 - complex_of_real r"
unitary_twosided_isometry: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "unitary U = (isometry U \<and> isometry (U*))"
uniformly_Cauchy_on_def: fixes X :: "'c set"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'd" shows "uniformly_Cauchy_on X f = (\<forall>e>0. \<exists>M. \<forall>x\<in>X. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e)"
simps_12: fixes C :: "char list"   and M :: "char list"   and es :: "char list exp list" shows "fv (C\<bullet>\<^sub>sM(es)) = fvs es"
ns'_2: shows "\<forall>n\<in>set ns'. v \<notin> defs g n"
checkpoint_s_def: shows "checkpoint_s = Incremental_State_Ops_Simplex.checkpoint_s"
linear_map_axioms: shows "linear_map class_ring (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}) (module_vec TYPE('a mod_ring) n) (\<lambda>v. vec n (\<lambda>i. coeff (v mod m i) 0))"
a_Pi: shows "a \<in> B \<rightarrow>\<^sub>E carrier class_ring"
closure_affine_hull: fixes S :: "'b set" shows "closure S \<subseteq> affine hull S"
rMulZeroDiv: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x = \<zero> \<or> y = \<zero>" shows "x \<cdot>\<^sub>r y = \<zero>"
assms: fixes i :: "nat" assumes "i \<in> s" shows "k \<le> i"
prems: fixes i :: "nat" assumes "Suc i \<le> Suc n" shows "w !! i = x !! i"
A_7: shows "real (c + 1) \<le> (v \<oplus> t') x + (t - t')"
summable_LIMSEQ_zero: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "f \<longlonglongrightarrow> (0::'b)"
assms_6: shows "QQ = (\<lambda>tab. semiJoin tab (st, t)) ` Q"
independent_span_bound: fixes T :: "'c set"   and S :: "'c set" assumes "finite T"   and "vs2.independent S"   and "S \<subseteq> vs2.span T" shows "finite S \<and> card S \<le> card T"
assms_3: fixes \<phi> :: "'a fm"   and xs :: "'b list" assumes "normal \<phi>"   and "nqfree \<phi>" shows "I (qe \<phi>) xs = (\<exists>x. I \<phi> (x # xs))"
infty_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>infty_rel = {(Num a, Num a') |a a'. (a, a') \<in> R} \<union> {(infty.Infty, infty.Infty)}"
minGraphProps9: fixes g :: "graph"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "v \<in> \<V> f" shows "v \<in> \<V> g"
len_lexext_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "len_lexext gt [y] [x] = gt y x"
fun_add_zeroR: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F" shows "a \<oplus>\<^bsub>F\<^esub> \<zero>\<^bsub>F\<^esub> = a"
simps_2: fixes i :: "nat"   and j :: "nat"   and a :: "'a option"   and t :: "'a tree"   and u :: "'a tree" shows "local.well_valued0 (Node i j a t u) = (0 < i \<and> j \<le> length as \<and> (a \<noteq> None \<longrightarrow> a = Some (local.sum i j)) \<and> local.well_valued0 t \<and> local.well_valued0 u \<and> (u = Leaf \<or> val u \<noteq> None))"
simpeq_islin: fixes t :: "tm" shows "islin (simpeq t)"
lspasl_magicl_der: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "h1,h2\<triangleright>h0"   and "(A \<longrightarrow>* B) h2" shows "(h1,h2\<triangleright>h0) \<and> \<not> A h1 \<and> (A \<longrightarrow>* B) h2 \<and> magicl_applied h1 h2 h0 (A \<longrightarrow>* B) \<or> (h1,h2\<triangleright>h0) \<and> B h0 \<and> (A \<longrightarrow>* B) h2 \<and> magicl_applied h1 h2 h0 (A \<longrightarrow>* B)"
distinct_left_right: fixes l :: "'c tree"   and z :: "'c"   and b :: "bool"   and r :: "'c tree"   and x :: "'c"   and y :: "'c" assumes "all_distinct (Node l z b r)"   and "x \<in> set_of l"   and "y \<in> set_of r" shows "x \<noteq> y"
smap_streams: fixes s :: "'d stream"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "s \<in> streams A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "smap f s \<in> streams B"
hom_uminus: fixes x :: "'b" shows "to_fract (- x) = - to_fract x"
Union_insert: fixes a :: "'b set"   and B :: "'b set set" shows "\<Union> (insert a B) = a \<union> \<Union> B"
intros: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
assms_1: shows "\<forall>y. \<exists>n. y = f n"
assms: shows "\<forall>x\<in>set xs. sorted (map key x)"
all_zero_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(\<forall>u. inner x u = 0) = (x = zero)"
\<epsilon>\<^sub>h_gt_0: shows "0 < \<epsilon>\<^sub>h"
RM_2: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<^bold>\<diamond>\<psi> in v]"
card_of_UNION_Sigma: fixes A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "|\<Union> (A ` I)| \<le>o |Sigma I A|"
card_of_lists_mono: fixes A :: "'a set"   and B :: "'b set" assumes "|A| \<le>o |B|" shows "|lists A| \<le>o |lists B|"
cINF_greatest: fixes A :: "'d set"   and m :: "'e multiset"   and f :: "'d \<Rightarrow> 'e multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># \<Sqinter> (f ` A)"
prob1_def: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" shows "prob1 n v i \<equiv> \<Sum>j\<in>{k |k. select_index n i k}. (cmod (v $$ (j, 0)))\<^sup>2"
ND_caseDistinction: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> H"   and "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> H" shows "\<Gamma> \<turnstile> H"
invc_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree" assumes "invc l"   and "invc r" shows "(color l = Black \<and> color r = Black \<longrightarrow> invc (join l r)) \<and> invc2 (join l r)"
inf_sup_aci_5: fixes x :: "'a"   and y :: "'a" shows "sup x y = sup y x"
minus_sat_def: fixes x :: "'a sat"   and y :: "'a sat" shows "x - y = Abs_sat' (nat_of x - nat_of y)"
name_list_set_fresh: fixes xvec :: "name list"   and x :: "'a" shows "set xvec \<sharp>* x = xvec \<sharp>* x"
smallo_imp_asymp_equiv: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "(\<lambda>x. f x - g x) \<in> o[F](g)" shows "f \<sim>[F] g"
reaction_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'e"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and \<Delta> :: "'a \<Rightarrow> 'd list \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool"   and s :: "'a"   and vl :: "'d list"   and s1 :: "'a"   and vl1 :: "'d list" shows "BD_Security.reaction step \<phi> f \<gamma> g T \<Delta> s vl s1 vl1 \<equiv> \<forall>a ou s' vl'. let trn = Trans s a ou s' in IO_Automaton.validTrans step trn \<and> \<not> T trn \<and> BD_Security.consume \<phi> f trn vl vl' \<longrightarrow> BD_Security.match step \<phi> f \<gamma> g \<Delta> s s1 vl1 a ou s' vl' \<or> BD_Security.ignore \<gamma> \<Delta> s s1 vl1 a ou s' vl'"
abs_pdevs_val_le_tdev: fixes e :: "nat \<Rightarrow> real"   and x :: "'b pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "\<bar>pdevs_val e x\<bar> \<le> tdev x"
InvariantGetReasonIsReasonAfterApplyDecide: fixes Vbl :: "nat set"   and state :: "State" assumes "\<not> Vbl \<subseteq> vars (elements (getM state))"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "getQ state = []" shows "let state' = applyDecide state Vbl in InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
obtain_t_subset_points: obtains T :: "'a set" where "T \<subseteq> \<V>" and "int (card T) = \<t>" and "finite T"
simps_3: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and uw :: "'b"   and ux :: "'b list" shows "lex_ext_unbounded f [] (uw # ux) = (False, False)"
dom_char: fixes f :: "'c local.term set" shows "\<F>\<^sub>SC.dom f = (if \<F>\<^sub>SC.arr f then \<F>C.dom f else \<F>C.null)"
simps_1: fixes a :: "'b"   and m :: "nat" shows "FOV (FQ a m) = {m}"
ord_power: fixes m :: "nat"   and a :: "nat"   and k :: "nat" assumes "coprime m a" shows "ord m (a ^ k) = ord m a div gcd k (ord m a)"
simps_2: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "exhaustive_above x (y # ys) = (\<not> proper_interval (Some x) (Some y) \<and> exhaustive_above y ys)"
select_convs_1: fixes access :: "acc_modi"   and static :: "bool"   and more :: "'a" shows "static \<lparr>access = access, static = static, \<dots> = more\<rparr> = static"
reachable_stateE: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and sh :: "('g, 'h, 'i, 'j, 'k) system_state"   and P :: "('g, 'h, 'i, 'j, 'k) system_state \<Rightarrow> bool" assumes "reachable_state sys sh"   and "\<And>\<sigma> i. prerun sys \<sigma> \<Longrightarrow> P (\<sigma> i)" shows "P sh"
fin_mono: fixes A :: "'x fset"   and B :: "'x fset"   and x :: "'x" assumes "A |\<subseteq>| B" shows "x |\<in>| A \<longrightarrow> x |\<in>| B"
a: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "f (f x y) z = f x (f y z)"
extf_min_empty: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and ss :: "('s, 'v) tm list" shows "\<not> extf f gt [] ss"
remove_vertex_E'': fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and x :: "'a" assumes "Digraph G"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<noteq> x"   and "w \<noteq> x" shows "v \<rightarrow>\<^bsub>Digraph.remove_vertex G x\<^esub> w"
project_mem: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and j :: "'c"   and x :: "'c \<Rightarrow> 'd" assumes "\<forall>k\<in>I. aGroup (A k)"   and "j \<in> I"   and "x \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A)" shows "(\<pi>\<^bsub>I,A,j\<^esub>) x \<in> carrier (A j)"
VectorSpaceHomSet_def: fixes fsmult :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set"   and fsmult' :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and W :: "'e set" shows "VectorSpaceHomSet fsmult V fsmult' W \<equiv> {T. VectorSpaceHom fsmult V fsmult' T} \<inter> {T. T ` V \<subseteq> W}"
measurable_count_space_eq2_countable: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and A :: "'c set" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
reachable_refl: fixes v :: "'a" assumes "v \<in> verts (G \<restriction> {w. u \<rightarrow>\<^sup>* w})" shows "v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. u \<rightarrow>\<^sup>* w}\<^esub> v"
arr_dom_iff_arr: fixes f :: "'b" shows "C'.arr (C'.dom f) = C'.arr f"
not_HFinite_HInfinite: fixes x :: "'b star" assumes "x \<notin> HFinite" shows "x \<in> HInfinite"
id_apply: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm" shows "Lam_ml.id \<Gamma><t> = t"
cont: fixes T :: "real set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and x :: "'a" assumes "ll_on_open T f X"   and "x \<in> X" shows "continuous_on T (\<lambda>t. f t x)"
appl_methsD: fixes mh :: "ref_ty \<times> mhead"   and pTs' :: "ty list"   and G :: "prog"   and S :: "qtname"   and T :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list" assumes "(mh, pTs') \<in> appl_methds G S T \<lparr>name = mn, parTs = pTs\<rparr>" shows "mh \<in> mheads G S T \<lparr>name = mn, parTs = pTs'\<rparr> \<and> G\<turnstile>pTs[\<preceq>]pTs'"
reduced_GB_unique: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite G"   and "punit.is_reduced_GB G"   and "local.punit.pmdl G = local.punit.pmdl F" shows "punit.reduced_GB F = G"
abs_update_length: fixes k :: "'a"   and v :: "'b"   and l :: "('a \<times> 'b) list list" shows "length (abs_update k v l) = length l"
coboundedI2: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "b \<le> c" shows "inf a b \<le> c"
total_onPD: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "total_onP A r"   and "x \<in> A"   and "y \<in> A" shows "r x y \<or> x = y \<or> r y x"
emb_only_below_vars: fixes \<rho> :: "'d \<Rightarrow> ('c, 'd) tm"   and s :: "('c, 'd) tm"   and u :: "('c, 'd) tm"   and w :: "('c, 'd) tm" assumes "subst \<rho> s \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "s \<unrhd>\<^sub>e\<^sub>m\<^sub>b w"   and "is_Sym (head w)"   and "subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "\<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u" obtains ws :: "('c, 'd) tm list" where "ws \<noteq> []" and "hd ws = subst \<rho> w" and "last ws = u" and "\<forall>i. Suc i < length ws \<longrightarrow> (\<exists>p d. emb_step_at p d (ws ! i) = ws ! Suc i \<and> \<not> list_all (\<lambda>x. x = dir.Left) p)" and "\<forall>i<length ws. head (ws ! i) = head w \<and> num_args (ws ! i) = num_args w" and "\<forall>i<length ws. \<forall>k<num_args w. args (subst \<rho> w) ! k \<unrhd>\<^sub>e\<^sub>m\<^sub>b args (ws ! i) ! k"
P_2: fixes n :: "nat"   and x :: "'a" assumes "P n x" shows "x \<in> S"
hyps_1: fixes j :: "nat"   and i :: "nat" assumes "k = j - i"   and "i \<le> j"   and "j < length \<GG>" shows "\<GG> ! i \<subseteq> \<GG> ! j"
circline_set_positive_x_axis_I: fixes z :: "complex" assumes "is_real z"   and "0 < Re z" shows "of_complex z \<in> positive_x_axis"
simps_1: fixes I :: "('e, 'f, 'd) interp"   and p :: "'d" shows "prog_sem I ($\<alpha> p) = Programs I p"
not_accp_down: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "\<not> Wellfounded.accp R x" obtains z :: "'a" where "R z x" and "\<not> Wellfounded.accp R z"
cSup_singleton: fixes x :: "'a multiset" shows "\<Squnion> {x} = x"
tendsto_add_left_ereal: fixes y :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'a filter" assumes "\<bar>y\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y) \<longlongrightarrow> x + y) F"
derivation_f_refl: shows "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"
BIT: shows "T\<^sub>p_on_rand' BIT (type0 init x y) qs = 75 / 10\<^sup>2 * real (length v) - 5 / 10"
sinks_empty: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" assumes "sinks I D u xs = {}" shows "ipurge_tr I D u xs = xs"
mult_divide_mult_cancel_left: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
sharing_correct_def: fixes input :: "bool" shows "share_correct.sharing_correct input \<equiv> share input \<bind> (\<lambda>(s1, s2). reconstruct (s1, s2)) = return_spmf input"
step: shows "S t i \<turnstile> t ! i \<mapsto> S t (i + 1)"
frechet_derivative_cos: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "f differentiable at x" shows "frechet_derivative (\<lambda>x. cos (f x)) (at x) = (\<lambda>xa. frechet_derivative f (at x) xa * - sin (f x))"
assms_13: shows "RETURN (Some PHI'') \<le> Picard_step X0 0 h PHI'"
one_more_paths_exists_trivial: assumes "new_last = v1" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"