hcju/isabelleps huggingface dataset:
above_def: shows "above = case_option UNIV (Collect \<circ> (<))"
h'_dom: fixes x :: "'v"   and y :: "'v" assumes "0 < h' x y" shows "(x, y) \<in> E'"
ivl_limpt_diff: fixes s :: "real set"   and a :: "real"   and b :: "real"   and x :: "real" assumes "finite s"   and "a < b"   and "x \<in> {a..b} - s" shows "x islimpt {a..b} - s"
one_chain_line_integral_eq_line_integral_on_sudivision'_2: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "chain_subdiv_chain one_chain1 subdiv"   and "boundary_chain one_chain1"   and "boundary_chain subdiv"   and "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"   and "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>"   and "finite one_chain1"   and "finite basis" shows "\<forall>(k, \<gamma>)\<in>subdiv. line_integral_exists F basis \<gamma>"
Arden_preserves_soundness: fixes X :: "'b list set"   and rhs :: "'b trm set" assumes "X = lang_rhs rhs"   and "ardenable rhs"   and "finite rhs" shows "X = lang_rhs (Arden X rhs)"
ex_moebius_01inf: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and z3 :: "complex_homo" assumes "z1 \<noteq> z2"   and "z1 \<noteq> z3"   and "z2 \<noteq> z3" shows "\<exists>M. moebius_pt M z1 = 0\<^sub>h \<and> moebius_pt M z2 = 1\<^sub>h \<and> moebius_pt M z3 = \<infinity>\<^sub>h"
S1_le_S2: fixes n :: "nat" shows "{..<n div 2} \<times> {..<n div 2} \<subseteq> {(i, j). i + j < n}"
card_finite_linorders: fixes A :: "'b set" assumes "finite A" shows "card {R. linorder_on A R} = fact (card A)"
map_to_set_def: fixes m :: "'c \<Rightarrow> 'd option" shows "map_to_set m = {(k, v). m k = Some v}"
IH: assumes "wf_ruleset \<gamma> p rs" shows "wf_ruleset \<gamma> p (normalize_rules_dnf rs)"
map_id_on_in: fixes x :: "'c"   and xs :: "'c set" assumes "x \<in> xs" shows "map_id_on xs x = Some x"
distinct_conv_nth: fixes xs :: "'a list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
swaprows_carrier: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat"   and n :: "nat"   and nc :: "nat" shows "(swaprows k l A \<in> carrier_mat n nc) = (A \<in> carrier_mat n nc)"
termFSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "termFSbImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSubstAll h hA MOD"
vconn_implies_hpar_1: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>" shows "src \<mu> = src f"
map_ide: fixes a :: "'c" assumes "C.ide a" shows "local.map a = S.mkIde (\<iota> ` D.cones a)"
sig_trd_aux_alt_spp: fixes args :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list" assumes "fst args \<in> keys (rep_list (snd args))" shows "rep_list (sig_trd_aux bs args) = sig_trd_spp_aux (map spp_of bs) (lt (snd args)) (rep_list (snd args) - punit.higher (rep_list (snd args)) (fst args), punit.higher (rep_list (snd args)) (fst args))"
coinduct: fixes X :: "nat \<Rightarrow> bool"   and x :: "nat" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>n. x = n \<and> (X n \<or> q2 n)" shows "q2 x"
yD_4: shows "\<exists>!x. root_cond (p2, l2, r2) x"
LtK_Sign: fixes X :: "ltkey"   and X' :: "msg"   and Y' :: "msg" shows "LtK X \<noteq> Sign X' Y'"
pred_spmf_calleeI: fixes s :: "'a"   and x :: "'c" assumes "I1 s"   and "x \<in> outs_\<I> \<I>_full" shows "pred_spmf (pred_prod (\<lambda>_. True) I1) (callee1 s x)"
1: shows "\<forall>m n. m \<noteq> n \<longrightarrow> A m \<inter> A n = {}"
sigma_finite_measure_axioms: shows "sigma_finite_measure (Pi\<^sub>M J M)"
check_def: fixes eX :: "'b withERR" shows "check eX \<equiv> THE X. eX = OK X"
fps_mult_equiv_coeff: fixes f :: "'b fps"   and g :: "'b fps"   and p :: "'b"   and h :: "'b fps"   and n :: "nat" assumes "(f, g) \<in> fpsmodrel p" shows "(f * h) $ n mod p = (g * h) $ n mod p"
special_subgroup_special_coset: fixes S :: "'b set"   and T :: "'b set" assumes "PreCoxeterSystem S"   and "T \<in> Pow S" shows "\<langle>T\<rangle> \<in> PreCoxeterSystem.special_cosets S"
Vs_are_values: fixes e :: "trm"   and T :: "ty" assumes "e \<in> V T" shows "val e"
BinOpRed1: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and bop :: "bop"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s'\<rangle>"
simps_2: fixes z :: "quat" shows "Im1 (Quaternions.cnj z) = - Im1 z"
return_time_exivl: fixes f :: "'c \<Rightarrow> 'c"   and X :: "'c set"   and P :: "'c set"   and x :: "'c" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "auto_ll_on_open.return_time f X P x \<in> auto_ll_on_open.existence_ivl0 f X x"
FAddSA_States_neq: fixes S' :: "'d"   and SA :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and A :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "S' \<notin> States SA"   and "S \<noteq> S'" shows "(A [f+] (S, SA)) S' = A S'"
elims_3: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'h"   and xa :: "('e, 'f) lit" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP"   and "\<not> Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and a :: "('e, 'f) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> Struct.satA intF intP \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'h" and a :: "('e, 'f) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> Struct.satA intF intP \<xi> a"
MakeFaceFinal_face_face_op: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "face_face_op (makeFaceFinal f g)"
oa_facts_7: fixes v :: "i"   and x :: "\<nu>" shows "[\<lparr>O!,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<A>\<lparr>O!,x\<^sup>P\<rparr> in v]"
map_cong: fixes x :: "'a multiset"   and ya :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in># ya \<Longrightarrow> f z = g z" shows "image_mset f x = image_mset g ya"
inner_is_valid: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and n :: "'b" assumes "CFGExit sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit n" shows "CFG.valid_node sourcenode targetnode valid_edge n"
paths_nxt: fixes xs :: "'b list"   and t :: "'b ttree"   and x :: "'b" assumes "xs \<in> paths (nxt t x)" obtains "x # xs \<in> paths t"   | "xs = []"
atLeastLessThan_iff: fixes i :: "'a Word.word"   and l :: "'a Word.word"   and u :: "'a Word.word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
distinct_280: fixes x15 :: "Syntax.term"   and x13 :: "Syntax.term" shows "Auth x15 \<noteq> Roll x13"
check_tableau_equiv: fixes s :: "('i, 'a) state"   and v :: "nat \<Rightarrow> 'a" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> (check s)"
final_threads_def: fixes s :: "('b, 'a) locks \<times> (('a \<Rightarrow> ('x1 \<times> 'b \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'd wait_set_status option) \<times> 'a set" shows "r1.final_threads s \<equiv> {t. r1.final_thread s t}"
elims: fixes x :: "'a \<Rightarrow> bool option"   and xa :: "'a strips_operator list list"   and y :: "'a \<Rightarrow> bool option" assumes "execute_parallel_plan x xa = y" obtains s :: "'a \<Rightarrow> bool option" where "x = s__" and "xa = []" and "y = s__"   | s :: "'a \<Rightarrow> bool option" and ops :: "'a strips_operator list" and opss :: "'a strips_operator list list" where "x = s" and "xa = ops # opss" and "y = (if are_all_operators_applicable s ops \<and> are_all_operator_effects_consistent ops then execute_parallel_plan (execute_parallel_operator s ops) opss else s)"
mcont_mono: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "mcont luba orda lubb ordb f" shows "monotone orda ordb f"
I_def: shows "I = {i. X i \<noteq> topspace euclidean}"
prems: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : ts _> ts'"
intro: fixes a :: "ordinal" assumes "0 < a"   and "\<And>b. b < a \<Longrightarrow> b + a = a" shows "additive_principal a"
szip_smap: fixes zs :: "('c \<times> 'd) stream" shows "smap fst zs ||| smap snd zs = zs"
assms_2: shows "G \<subseteq> dgrad_sig_set' (length fs) d"
dvd_times_right_cancel_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> (0::'b)" shows "(b * a dvd c * a) = (b dvd c)"
add_opp: fixes a :: "'b"   and b :: "'b"   and p :: "'b point" assumes "on_curve a b p" shows "ell_field_class.add a p (opp p) = Infinity"
adj_path_app': fixes v :: "'v"   and ps :: "'v list"   and q :: "'v" assumes "adj_path v (ps @ [q])"   and "ps \<noteq> []" shows "adjacent (last ps) q"
rQuot_rev_lQuot: fixes x :: "'b"   and A :: "'b list set" shows "rQuot x A = rev ` lQuot x (rev ` A)"
derivation_A: fixes P :: "'a set" assumes "derivation P" shows "P \<subseteq> A"
qSwapAbs_preserves_qGoodAbs2: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs (A $[[x \<and> y]]_zs)" shows "qGoodAbs A"
nat_of_ord_inject: fixes x :: "hf"   and y :: "hf" assumes "Ord x"   and "Ord y" shows "(nat_of_ord x = nat_of_ord y) = (x = y)"
bound_compare_defs_8: fixes lt :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'a"   and b :: "'b option" shows "\<lhd>\<^sub>l\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> False | Some b' \<Rightarrow> lt c b'"
AAss1Red2: 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"   and e :: "'addr expr1" 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> := e,s\<rangle> -ta\<rightarrow> \<langle>Val a\<lfloor>i'\<rceil> := e,s'\<rangle>"
generate_in_carrier: fixes H :: "nat set"   and h :: "nat" assumes "H \<subseteq> carrier G"   and "h \<in> generate G H" shows "h \<in> carrier G"
max_def: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'h"   and b :: "'h" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
Cons_3: fixes ucl :: "('U \<times> 'C) list"   and ss' :: "('St, 'U, 'C) state" assumes "list_all isState sl"   and "map getGMUserCom (tl sl) = ucl" shows "\<exists>sl'. hd sl' = ss' \<and> wffp sl' \<and> list_all2 f (tl sl) (tl sl') \<and> map getGMUserCom (purgeIdle (tl sl')) = purge GH ucl"
mono_Field: fixes r :: "('c \<times> 'c) set"   and s :: "('c \<times> 'c) set" assumes "r \<subseteq> s" shows "Field r \<subseteq> Field s"
Var_left_vars_mset_less: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "x \<notin> vars_term t" shows "vars_mset (subst_mset (subst x t) M) \<subset> vars_mset (add_mset (Var x, t) M)"
intro: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "incidence_system \<V> \<B>"   and "sub_set_system_axioms \<U> \<A> \<V> \<B>" shows "sub_set_system \<U> \<A> \<V> \<B>"
leg_simps_3: shows "C.arr \<rho>.cod.leg1"
R: fixes s :: "'a llist" assumes "s \<in> A\<^sup>\<star>"   and "s \<le> x" shows "s \<le> y"
set_integral_at_point: fixes M :: "real measure"   and a :: "real"   and f :: "real \<Rightarrow> real" assumes "set_integrable M {a} f"   and "{a} \<in> sets M"   and "emeasure M {a} \<noteq> \<infinity>" shows "set_lebesgue_integral M {a} f = f a * Sigma_Algebra.measure M {a}"
face_cycle_succ_uv: shows "GM.face_cycle_succ uv = proj_arcs_H (H.face_cycle_succ (rev_H vw))"
1117: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<squnion> y) \<sqinter> (x \<squnion> - z \<sqinter> y) = bot"
orbit_sym: fixes x :: "'c"   and y :: "'c" assumes "x \<in> E"   and "y \<in> E"   and "y \<in> orbit G \<phi> x" shows "x \<in> orbit G \<phi> y"
Bseq_eq_bounded: fixes f :: "nat \<Rightarrow> 'a" shows "Bseq f = bounded (range f)"
assms_2: shows "y \<in> optimal_set (\<lambda>x y. x \<bullet> c \<le> y \<bullet> c) (sat_dual A c)"
cond_2r: assumes "other.s_stop \<le> u_max" shows "collision_react {0..}"
\<tau>mRedT_add_thread_inv: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and xln :: "'x2 \<times> 'l \<Rightarrow>f nat" assumes "\<tau>mRed2 s s'"   and "thr s t = None" shows "\<tau>mRed2 (locks s, (thr s(t \<mapsto> xln), shr s), wset s, interrupts s) (locks s', (thr s'(t \<mapsto> xln), shr s'), wset s', interrupts s')"
proves_beta_step: fixes \<Theta> :: "((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)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "t \<rightarrow>\<^sub>\<beta> u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
neqE: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
abs_fun_simps_2: fixes X :: "'b set"   and R :: "('b \<times> 'c) set" assumes "X \<subseteq> Domain R" shows "\<Up> R (RES X) = RES (R `` X)"
isoI: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and g :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "Cop_S.inverse_arrows f g" shows "Cop_S.iso f"
that: fixes f' :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)" shows "thesis"
conjugate_exponent_real_3: fixes p :: "real" assumes "1 < p" shows "conjugate_exponent (conjugate_exponent p) = p"
eq_imp_eq_mod_int: fixes a :: "int"   and b :: "int"   and m :: "int" assumes "a = b" shows "[a = b] (mod m)"
wfrestrictedvars\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfrestrictedvars\<^sub>s\<^sub>t\<^sub>p S))"
gt_db_db: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "db j >\<^sub>s db i"
ssubst_openz_distrib: fixes t' :: "sterm"   and x :: "char list"   and t :: "sterm"   and s :: "sterm"   and p :: "sterm" assumes "lc t'" shows "[x \<rightarrow> t'] (t\<^bsup>[s,p]\<^esup>) = ([x \<rightarrow> t'] t\<^bsup>[[x \<rightarrow> t'] s,[x \<rightarrow> t'] p]\<^esup>)"
fdia_star_induct: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "d z +  |x\<rangle> y \<le> d y" shows " |x\<^sup>\<star>\<rangle> z \<le> d y"
interp_ops_vals_domain: fixes ops :: "('c \<times> ('c, 'd) list_op) list" assumes "list_ops ops" shows "dom (snd (interp_ops ops)) = set (inserted_ids ops) - set (deleted_ids ops)"
tauChainRes: fixes P :: "ccs"   and P' :: "ccs"   and x :: "name" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "\<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<^sub>\<tau> \<lparr>\<nu>x\<rparr>P'"
diagonal_mat_def: fixes A :: "'a mat" shows "diagonal_mat A \<equiv> \<forall>i<dim_row A. \<forall>j<dim_col A. i \<noteq> j \<longrightarrow> A $$ (i, j) = (0::'a)"
DBM_val_bounded_neg_cycle1: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat"   and i :: "nat"   and xs :: "nat list" assumes "u \<turnstile>\<^bsub>v,n\<^esub> M"   and "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "len M i i xs < \<one>"   and "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "i \<noteq> 0"   and "cnt 0 xs \<le> 1" shows "False"
simplex_like_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ordering less_eq less" shows "ordering.simplex_like less_eq less P \<equiv> finite P \<and> (\<exists>f A. OrderingSetIso less_eq less dual_order.greater_eq dual_order.greater P f \<and> f ` P = Pow A)"
igSubstAllIPresIGWlsAll_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstAllIPresIGWlsAll MOD \<equiv> igSubstIPresIGWls MOD \<and> igSubstAbsIPresIGWlsAbs MOD"
lt_q'': shows "punit.lt (rep_list q') = punit.lt (rep_list q)"
fr_equsubst: fixes p :: "fm"   and bs :: "real list"   and x :: "real" assumes "isrlfm p" shows "(\<exists>x. Ifm (x # bs) p) = (Ifm (x # bs) (minusinf p) \<or> Ifm (x # bs) (plusinf p) \<or> (\<exists>(t, k)\<in>set (uset p). \<exists>(s, l)\<in>set (uset p). Ifm (x # bs) (usubst p (Add (Mul l t) (Mul k s), 2 * k * l))))"
distrib: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a + b) * c = a * c + b * c"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_hd_or_tl_axioms ext" shows "ext_hd_or_tl ext"
unit_factor_field: fixes x :: "'b" shows "unit_factor x = x"
not_Low_High: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and H :: "'c set"   and L :: "'c set"   and High :: "'b"   and Low :: "'b"   and V :: "'c" assumes "NonInterferenceInterGraph sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses H L High Low"   and "V \<notin> L" shows "V \<in> H"
concurrent_set_appendI: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "happens_before hb_weak hb"   and "happens_before.concurrent_set hb a xs"   and "happens_before.concurrent_set hb a ys" shows "happens_before.concurrent_set hb a (xs @ ys)"
thin_locs_pre_keep_atSE: fixes p :: "'e"   and ls :: "'f set"   and s :: "('g, 'f, 'e, 'h, 'i) system_state"   and P :: "bool"   and l :: "'f" assumes "atS p ls s \<longrightarrow> P"   and "at p l s"   and "l \<in> ls" obtains "P"
scaleR_nonneg_nonneg: fixes a :: "real"   and x :: "'c" assumes "0 \<le> a"   and "(0::'c) \<le> x" shows "(0::'c) \<le> a *\<^sub>R x"
a_set_tag_name_def: fixes ptr :: "(_) element_ptr"   and tag :: "char list" shows "a_set_tag_name ptr tag = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr tag_name_update tag)"
split: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
not_no_matching_Goto_singleton_cases: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and a :: "action" shows "(\<not> no_matching_Goto \<gamma> p [Rule m a]) = ((\<exists>chain. a = Goto chain) \<and> matches \<gamma> m p)"
prime_sum_upto_eqI: fixes a :: "real"   and b :: "real"   and f :: "nat \<Rightarrow> 'b" assumes "a \<le> b"   and "\<And>k. k \<in> {nat \<lfloor>a\<rfloor><..nat \<lfloor>b\<rfloor>} \<Longrightarrow> \<not> prime k" shows "prime_sum_upto f a = prime_sum_upto f b"
that: fixes s :: "'a llist" assumes "s \<in> A\<^sup>\<omega>"   and "t = r @@ s" shows "thesis"
mult_nonneg_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "Rings.class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq zero a"   and "less_eq zero b" shows "less_eq zero (times a b)"
row_greater_greatest_is_zero: fixes k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec"   and i :: "'d" assumes "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i" shows "is_zero_row_upt_k i k A"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "A.ide a"   and "A.seq f a" shows "f \<cdot>\<^sub>A a = f"
1: shows "f \<longlonglongrightarrow> lim f"
vsv_vimageI2: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
arity_2_result_def: fixes op2 :: "b_e" shows "arity_2_result op2 = (case op2 of Binop_i t x \<Rightarrow> t | Binop_f t x \<Rightarrow> t | Relop_i t x \<Rightarrow> T_i32 | Relop_f t x \<Rightarrow> T_i32)"
linear_exists_right_inverse_on: fixes f :: "'b \<Rightarrow> 'c"   and V :: "'b set" assumes "linear (*a) (*b) f"   and "vs1.subspace V" shows "\<exists>g. range g \<subseteq> V \<and> linear (*b) (*a) g \<and> (\<forall>v\<in>f ` V. f (g v) = v)"
rows_carrier: fixes A :: "'b mat" shows "set (rows A) \<subseteq> carrier_vec (dim_col A)"
Image_closed_trancl: fixes r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "r `` X \<subseteq> X" shows "r\<^sup>* `` X = X"
sup_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "sup f g = (\<lambda>x. sup (f x) (g x))"