hcju/isabelleps huggingface dataset:
cases: fixes x :: "'b val \<times> 'b val" obtains i1 :: "32 word" and i2 :: "32 word" where "x = (Intg i1, Intg i2)"   | b1 :: "bool" and b2 :: "bool" where "x = (Bool b1, Bool b2)"   | v2 :: "'b val" where "x = (Unit, v2__)"   | v2 :: "'b val" where "x = (Null, v2a__)"   | v :: "bool" where "x = (Bool v__, Unit)"   | v :: "bool" where "x = (Bool va__, Null)"   | v :: "bool" and va :: "32 word" where "x = (Bool vb__, Intg vaa__)"   | v :: "bool" and va :: "'b" where "x = (Bool vc__, Addr vab__)"   | v :: "'b" and v2 :: "'b val" where "x = (Addr vd__, v2)"   | v1 :: "'b val" where "x = (v1__, Unit)"   | v1 :: "'b val" where "x = (v1a__, Null)"   | v :: "bool" where "x = (Unit, Bool ve__)"   | v :: "bool" where "x = (Null, Bool vf__)"   | va :: "32 word" and v :: "bool" where "x = (Intg vac__, Bool vg__)"   | va :: "'b" and v :: "bool" where "x = (Addr va, Bool vh__)"   | v1 :: "'b val" and v :: "'b" where "x = (v1, Addr v)"
simps_3: fixes f1 :: "'c"   and f2 :: "'c" shows "(case True of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f1"
lb_minimal: fixes y :: "'a"   and i :: "nat" assumes "y \<in> A"   and "size y < size (lb i)" shows "y \<notin> ith (eq_upto (BAD P) lb i) i"
False: shows "\<not> (p = 0 \<or> q = 0 \<or> p mod q = 0)"
simps_6: fixes x :: "nat"   and e :: "cexpr"   and e' :: "cexpr"   and t :: "pdf_type" shows "cexpr_subst x e (\<integral>\<^sub>c e' \<partial>t) = \<integral>\<^sub>c cexpr_subst (Suc x) (map_vars Suc e) e' \<partial>t"
iT_finite_2: fixes n :: "nat"   and d :: "nat" shows "finite [n\<dots>,d]"
clm_consec_res: fixes ts :: "traffic"   and c :: "cars" shows "clm ts c \<noteq> bot \<longrightarrow> consec (clm ts c) (res ts c) \<or> consec (res ts c) (clm ts c)"
f': shows "f = fls_shift (int n) (fps_to_fls f')"
pick_middlep: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'d"   and c :: "'f" assumes "(P OO Q) a c" shows "P a (BNF_Def.pick_middlep P Q a c) \<and> Q (BNF_Def.pick_middlep P Q a c) c"
swap_triple: fixes a :: "'b"   and c :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> c"   and "b \<noteq> c" shows "Fun.swap a b (Fun.swap b c (Fun.swap a b f)) = Fun.swap a c f"
fin_nrename2: fixes M :: "trm"   and x :: "name"   and y :: "name"   and z :: "name" assumes "fin (M[x\<turnstile>n>y]) z"   and "z \<sharp> (x, y)" shows "fin M z"
simps_1: fixes X :: "char list set" shows "L SKIP X = X"
get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes element_ptr :: "(_) element_ptr"   and uu :: "unit" shows "CD.get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr uu = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr RElement.child_nodes"
steps_imp_not_SN: fixes t :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'a"   and R :: "('b \<times> 'b) set" assumes "\<And>x. (t x, t (f x)) \<in> R" shows "\<not> SN R"
imp_graph_alt: fixes cnf :: "lit uprod set" shows "imp_graph cnf = {(negate l1, l2) |l1 l2. Upair l1 l2 \<in> cnf}"
set_to_map_inverse: fixes S :: "('g \<times> 'h) set" assumes "inj_on fst S" shows "map_to_set (set_to_map S) = S"
pre_mono: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and C :: "acom"   and s :: "char list \<Rightarrow> int" assumes "\<And>s. P' s \<le> P s" shows "pre C P' s \<le> pre C P s"
delete_edges_set_edges: fixes G :: "'b list_graph"   and E :: "('b \<times> 'b) list" shows "set (edgesL (FiniteListGraph.delete_edges G E)) = set (edgesL G) - set E"
eq: fixes a :: "nat list"   and ba :: "nat"   and a2 :: "nat list"   and a1 :: "nat" assumes "(a, ba) \<in> set (c1.gen_check a2)" shows "c1_incs b ys a1 0 (a, ba) = c1.incs a1 0 (a, ba)"
fmrestrict_set_dom: fixes m :: "('c, 'd) fmap" shows "fmrestrict_set (fmdom' m) m = m"
thetaSeqZO_sym: shows "sym thetaSeqZO"
small_vint_ls: shows "small {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> int_of_vint a < int_of_vint b}"
knows2: fixes e2 :: "Expression list"   and e1 :: "Expression list"   and e :: "Expression list"   and A :: "specID" assumes "e2 = e1 @ e \<or> e2 = e @ e1"   and "knows A e2" shows "knows A e"
is_none_code_2: fixes x :: "'b" shows "Option.is_none (Some x) = False"
assoc_is_natural_2: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>[\<mu>, \<nu>, \<tau>] = \<a>[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> ((\<mu> \<star> \<nu>) \<star> \<tau>)"
distinct_entries: fixes t :: "('a, 'b) RBT.rbt" shows "distinct (list.map fst (entries t))"
ferrack: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "qfree (ferrack p) \<and> Ifm bs (ferrack p) = (\<exists>x. Ifm (x # bs) p)"
kripke_rels_trc_worlds: fixes w :: "'b"   and w' :: "'b"   and M :: "('c, 'd, 'b) KripkeStructure"   and W :: "'b set" assumes "(w, w') \<in> (\<Union> (range (relations M)))\<^sup>*"   and "w \<in> worlds M"   and "kripke M"   and "W = worlds M" shows "w' \<in> W"
list_rel_imp_same_length: fixes l :: "'c list"   and l' :: "'d list"   and R :: "('c \<times> 'd) set" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel" shows "length l = length l'"
m_closed: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" assumes "x \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)" shows "x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"
finite_vars_of: fixes t :: "'c trm" shows "finite (vars_of t)"
simps_24: fixes x1 :: "'a"   and x91 :: "'a local.term"   and x92 :: "'a local.term"   and x93 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<a>\<^bold>[x91, x92, x93\<^bold>]"
assms_1: shows "B \<in> f \<turnstile> folding_g.\<C>"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b" shows "x # xs \<Squnion>\<^bsub>f\<^esub> y = xs \<Squnion>\<^bsub>f\<^esub> (x \<squnion>\<^bsub>f\<^esub> y)"
fold_Option_bind_eq_None_start_None: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a option"   and list :: "('b \<times> 'c) list" shows "fold (\<lambda>(x, y) option. Option.bind option (f x y)) list None = None"
is_process7: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set" assumes "is_process P" shows "\<forall>s t. s \<notin> DIVERGENCES P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> DIVERGENCES P"
f_simps_7: fixes n :: "nat"   and xo :: "'v vertex" shows "f n (xo, SOURCE) = 0"
enc_and_TRel_impl_indRelRTPO_weakly_respects_success: fixes success :: "'barbs"   and TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<Down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<Down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<Down><TWB>x)))"   and "rel_weakly_preserves_barb_set TRel TWB {success}"   and "rel_weakly_reflects_barb_set TRel TWB {success}" shows "rel_weakly_respects_barb_set (indRelRTPO TRel) (STCalWB SWB TWB) {success}"
simps_1: shows "\<lparr>Unit\<rparr> = Unit"
part1_Un_part2: fixes V :: "'a set" assumes "infinite V" shows "part1 V \<union> part2 V = V"
mult_less_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(k * m < k * n) = (0 < k \<and> m < n)"
abs_eq: shows "{}\<^sub>p = Abs_pfun Map.empty"
\<alpha>sON: shows "list.set \<alpha>s \<subseteq> ON"
residue_times_one_l: fixes k :: "nat"   and y :: "int" shows "y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> 1 = y mod p ^ k"
csucc_def: fixes \<kappa> :: "V" shows "csucc \<kappa> \<equiv> LEAST \<kappa>'. Ord \<kappa>' \<and> Card \<kappa>' \<and> \<kappa> < \<kappa>'"
anti_mono': fixes l :: "nat"   and c :: "com"   and l' :: "nat" assumes "l \<turnstile>' c"   and "l' \<le> l" shows "l' \<turnstile>' c"
in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "B.VV.arr f"   and "B.VV.dom f = a"   and "B.VV.cod f = b" shows "B.VV.in_hom f a b"
H_2: shows "g \<in> space\<^sub>N N2"
tab_simps_4: shows "local.dom \<rho> = r\<^sub>1"
approximation_preproc_nat_4: fixes i :: "nat"   and j :: "nat" shows "real (i - j) = max (real i - real j) 0"
alen: shows "\<forall>as. set as \<subseteq> carrier G \<and> wfactors G as a \<longrightarrow> ac = length as"
word_plus_and_or_coroll: fixes x :: "'b word"   and y :: "'b word" assumes "x AND y = 0" shows "x + y = x OR y"
in_borel_measurable_borel: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "(f \<in> borel_measurable M) = (\<forall>S\<in>sets borel. f -` S \<inter> space M \<in> sets M)"
accept\<^sub>R_def: fixes R :: "('c \<Rightarrow> 'd \<Rightarrow> 'c) \<times> 'c \<times> (('c \<times> 'd \<times> 'c) set \<times> ('c \<times> 'd \<times> 'c) set) set"   and w :: "nat \<Rightarrow> 'd" shows "accept\<^sub>R R w \<equiv> \<exists>P\<in>snd (snd R). accepting_pair\<^sub>R (fst R) (fst (snd R)) P w"
Max_wrt_among_nonempty: fixes B :: "'alt set" assumes "B \<inter> alts \<noteq> {}"   and "finite (B \<inter> alts)" shows "Max_wrt_among (R i) B \<noteq> {}"
nn_integral_diff: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>"   and "AE x in M. g x \<le> f x" shows "\<integral>\<^sup>+ x. f x - g x \<partial>M = integral\<^sup>N M f - integral\<^sup>N M g"
LD_b1_c: shows "LeftDerivation b1 E (take j c)"
assms_1: shows "i \<in> {c \<bullet> x |x. x \<in> sat_primal A b}"
comp_def: fixes f :: "'i \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" shows "f \<circ> g = (\<lambda>x. f (g x))"
adjoint_transpose_left_3: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g = trg u"   and "\<guillemotleft>\<mu> : f \<star> v \<Rightarrow> u\<guillemotright>" shows "trnl\<^sub>\<epsilon> u (trnl\<^sub>\<eta> v \<mu>) = \<mu>"
simps_2: fixes s :: "char list"   and i :: "char list"   and xs :: "char list" shows "??.Parser_Monad.exactly_aux s i [] xs = Inr (s, trim xs)"
select_convs_8: fixes igWls :: "'f \<Rightarrow> 'c \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'h \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'h"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'c option) \<Rightarrow> ('e \<Rightarrow> 'h option) \<Rightarrow> 'c"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubst :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubstAbs :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "igSwap \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igSwap"
zero_natural_def: shows "0 \<equiv> natural_of_nat 0"
lphil0_state_update_def: fixes s :: "nat"   and e :: "dining_event" shows "Up\<^sub>l\<^sub>p s e = (if e = picks 0 (N - 1) then 1 else if e = picks 0 0 then 2 else if e = putsdown 0 0 then 3 else 0)"
lm086: fixes x :: "'c"   and X :: "'c set"   and Y :: "'d set" assumes "x \<notin> X" shows "(\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f}) \<subseteq> injections ({x} \<union> X) Y"
preservation1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> x \<cdot> y \<cdot> x" shows "x \<cdot> y\<^sup>\<star> \<le> (x \<cdot> y + z)\<^sup>\<star> \<cdot> x"
proj_Cons_4: fixes l :: "'d strand_label"   and n :: "'d"   and a :: "'e"   and A :: "('d strand_label \<times> 'e) list" assumes "l = ln n" shows "proj n ((l, a) # A) = (l, a) # proj n A"
12394: fixes x :: "'a"   and y :: "'a" shows "- - (- x \<sqinter> - y) = - x \<sqinter> - y"
char_one_nz: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "\<chi> \<one>\<^bsub>G\<^esub> \<noteq> 0"
embed_bothWays_iso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "Well_order r"   and "Well_order r'"   and "embed r r' f"   and "embed r' r g" shows "iso r r' f"
f_def: shows "f \<equiv> \<lambda>x. x powr - c * ln x"
antipar_2: shows "src d\<^sub>1 = trg e\<^sub>1"
expand_finfun_eq: fixes f :: "('a, 'b) finfun"   and g :: "('a, 'b) finfun" shows "(f = g) = (finfun_apply f = finfun_apply g)"
rem_cycles_removes_last: fixes j :: "'b"   and i :: "'b"   and xs :: "'b list" shows "j \<notin> set (rem_cycles i j xs)"
map_iterator_rule_P: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and I :: "'e set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "map_iterator it m"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
aval_plus_symmetry: fixes x :: "'a aexp"   and y :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "aval (Plus x y) s = aval (Plus y x) s"
preserves_comp_2: fixes A1 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'c \<times> 'd \<Rightarrow> 'e"   and f2' :: "'d"   and f2 :: "'d" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
inres_def: fixes S :: "'b nres"   and x :: "'b" shows "inres S x \<equiv> RETURN x \<le> S"
counit_simps_1: shows "D.seq (D.inv (unit (trg\<^sub>C f))) (F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
span_eq: fixes S :: "'a set"   and T :: "'a set" shows "(More_Modules.ideal S = More_Modules.ideal T) = (S \<subseteq> More_Modules.ideal T \<and> T \<subseteq> More_Modules.ideal S)"
ope: fixes x :: "'a" assumes "x \<in> S" shows "openin (top_of_set S) (U x)"
list_ex_simps_2: fixes P :: "'c \<Rightarrow> bool" shows "list_ex P [] = False"
asymp_equivD_strong: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g"   and "\<forall>\<^sub>F x in F. f x \<noteq> (0::'b) \<or> g x \<noteq> (0::'b)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> (1::'b)) F"
\<psi>_def: fixes x :: "'c"   and h :: "'d" shows "\<psi> x h = \<epsilon> x \<cdot>\<^sub>C F h"
le_fin_word: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and p :: "'b" assumes "transition_system_traces ex en ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "transition_system.path ex en w\<^sub>2 p" shows "transition_system.path ex en w\<^sub>1 p"
ws_progI: fixes G :: "prog" assumes "\<forall>(I, i)\<in>set (ifaces G). \<forall>J\<in>set (isuperIfs i). is_iface G J \<and> (J, I) \<notin> (subint1 G)\<^sup>+"   and "\<forall>(C, c)\<in>set (classes G). C \<noteq> Object \<longrightarrow> is_class G (super c) \<and> (super c, C) \<notin> (subcls1 G)\<^sup>+" shows "ws_prog G"
scale_add3: fixes a :: "real \<times> real"   and x :: "real"   and b :: "real \<times> real" shows "ccw' 0 a (x *\<^sub>R a + b) = ccw' 0 a b"
maxsimpI: fixes X :: "'b set set"   and x :: "'b set" assumes "SimplicialComplex X"   and "x \<in> X"   and "\<And>z. \<lbrakk>z \<in> X; order.greater_eq z x\<rbrakk> \<Longrightarrow> z = x" shows "SimplicialComplex.maxsimp X x"
NatInterval_not_Empty: fixes n :: "nat"   and m :: "nat" shows "NatInterval n m \<noteq> Empty"
F: shows "Ps \<Longrightarrow>\<^sub>A P * F"
partition_on_size1: fixes A :: "'b set" assumes "finite A" shows "{P. partition_on A P \<and> (\<forall>X\<in>P. card X = 1)} = {(\<lambda>a. {a}) ` A}"
transitive_axioms: shows "transitive A (\<lambda>x y. y \<sqsubseteq> x)"
zeq: shows "subgroup_generated (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) {z} = reduced_homology_group (int n - 1) (nsphere (n - Suc 0))"
iterate_Suc: fixes n :: "nat"   and F :: "'d \<rightarrow> 'd"   and x :: "'d" shows "iterate (Suc n)\<cdot>F\<cdot>x = F\<cdot>(iterate n\<cdot>F\<cdot>x)"
stake_all_triv: fixes xs :: "[:'a:]" shows "stake\<cdot>(slength\<cdot>xs)\<cdot>xs = xs"
greaterThanLessThan_eq_ball: fixes a :: "real"   and b :: "real" shows "{a<..<b} = ball ((a + b) / 2) ((b - a) / 2)"
mgu_free: fixes t1 :: "typ"   and t2 :: "typ"   and U :: "nat \<Rightarrow> typ" assumes "mgu t1 t2 = Some U" shows "free_tv U \<subseteq> free_tv t1 \<union> free_tv t2"
semiring_norm_88: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m = num.Bit1 n) = False"
image_eq_UN: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "f ` A = (\<Union>x\<in>A. {f x})"
fully_faithful_functor_axioms: shows "fully_faithful_functor (\<cdot>) (\<cdot>) (\<lambda>f. f \<otimes> cod \<iota>)"
AAcc1Red2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and i :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and i' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and a :: "'addr val" assumes "uf,P,t \<turnstile>1 \<langle>i,s\<rangle> -ta\<rightarrow> \<langle>i',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val a\<lfloor>i\<rceil>,s\<rangle> -ta\<rightarrow> \<langle>Val a\<lfloor>i'\<rceil>,s'\<rangle>"
assms_1: shows "A\<^sub>F \<sharp>* \<Psi>\<^sub>G"
wiring: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and f :: "'i \<Rightarrow> 'k"   and g :: "'l \<Rightarrow> 'j" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
all_zero: shows "\<forall>m>mod_type_class.from_nat (nrows A). A $ m $ mod_type_class.from_nat k = (0::'b)"
comp_converter_map_converter1: fixes f :: "'a \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'b"   and h :: "'i \<Rightarrow> 'e"   and k :: "'f \<Rightarrow> 'j"   and conv :: "('g, 'h, 'i, 'j) converter"   and conv' :: "('e, 'f, 'c, 'd) converter" shows "map_converter f g h k conv \<odot> conv' = map_converter f g id id (conv \<odot> map_converter h k id id conv')"
welldefined_signs1: shows "well_def_signs (length qs) signs"
conc_lemma: fixes A :: "'b list set"   and B :: "'b list set"   and P :: "'b \<Rightarrow> bool" assumes "[] \<notin> A" shows "{w \<in> A @@ B. w \<noteq> [] \<and> P (hd w)} = {w \<in> A. w \<noteq> [] \<and> P (hd w)} @@ B"
appL: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "s \<degree> u \<rightarrow>\<^sub>\<eta> t \<degree> u"
join_related2: fixes y :: "nat"   and x :: "nat" assumes "y \<le> x" shows "nat.join x y = x"
hom_connected_3: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "(\<nu> \<star> \<mu> \<noteq> null) = (cod \<nu> \<star> \<mu> \<noteq> null)"
set_bound_1: fixes n :: "nat" shows "fst (A n) \<subseteq> S"
mea: fixes n :: "nat" shows "S - C n \<in> lmeasurable"
arrI: fixes f1 :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr"   and f2 :: "'a2" assumes "A2_B.arr f1"   and "A2.arr f2" shows "A2_BxA2.arr (f1, f2)"
M_U_completeness: shows "MModel (\<lambda>\<sigma>. D) intF intP"
asprod_1_x: fixes x :: "ant" shows "1 *\<^sub>a x = x"
llength_lmap: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f llist" shows "llength (lmap f xs) = llength xs"
ent_triv_1: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>A true"
P_def: shows "P = A o\<^sub>C\<^sub>L Proj S o\<^sub>C\<^sub>L A*"
dim_def: fixes K :: "'a set"   and E :: "'a set" shows "local.dim K E = (THE n. dimension n K E)"
u_Confidentiality_B: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "Key servK \<notin> analz (knows Spy evs)"
intro: fixes fs :: "'b vec list"   and n :: "nat" assumes "set fs \<subseteq> carrier_vec n" shows "gram_schmidt_fs_Rn n fs"
SwapElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some swap" shows "\<exists>v w ops. os = v # w # ops \<and> n = Suc 0 \<and> t = (w # v # ops, S, h) \<and> ll = l + 1"
assert_gpv_def: fixes b :: "bool" shows "assert_gpv b = (if b then Generative_Probabilistic_Value.Done () else Fail)"
get_mode_locs_def: shows "get_mode_locs = (\<lambda>shadow_root_ptr. {preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode)})"
outputEq: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" shows "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P' = P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"
countable_image_eq: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "countable (f ` S) = (\<exists>T. countable T \<and> T \<subseteq> S \<and> f ` S = f ` T)"
C: shows "sinks I (the \<circ> con_comp_map D E p q) (the u) xs = the ` sinks (con_comp_pol I) (con_comp_map D E p q) u xs"
tree_map_of_invar: fixes T :: "'m"   and pres :: "('a \<times> 'b) list" assumes "invar T" shows "invar (tree_map_of' T pres)"
list_digraph_simps_1: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" shows "pverts (list_digraph G_list) = set (fst G_list)"
proots_within_empty: fixes p :: "'a poly" shows "proots_within p {} = {}"
exp_ind: fixes x :: "nat"   and a :: "'b" shows "a \<up> Suc x = a \<up> x @ [a]"
simps_15: fixes a :: "floatarith"   and b :: "floatarith" shows "fold_const_fa (Powr a b) = Powr (fold_const_fa a) (fold_const_fa b)"