hcju/isabelleps huggingface dataset:
simps_7: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Inverse f) env = compute_tm_by_comp prec ord I a (Inverse (Var 0)) (approx_tm prec ord I a f env) (\<lambda>x. 0 < lower x \<or> upper x < 0)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) G' G' G'"
between_midpoint_2: fixes b :: "'b"   and a :: "'b" shows "between (b, a) (midpoint a b)"
simple_fw_valid_def: shows "simple_fw_valid \<equiv> list_all (simple_match_valid \<circ> match_sel)"
less_cSupD: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and z :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "less z (Sup X)" shows "\<exists>x\<in>X. less z x"
set_integrableI_nonneg: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "set_borel_measurable M A f"   and "AE x\<in>A in M. 0 \<le> f x"   and "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M < \<infinity>" shows "set_integrable M A f"
parts_increasing: fixes H :: "msg set" shows "H \<subseteq> parts H"
vector_export_comp_unit: fixes x :: "'a"   and y :: "'a" shows "(x * top \<sqinter> (1::'a)) * y = x * top \<sqinter> y"
leq_path_rep: fixes 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 C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and C'' :: "char list" assumes "Subobjs P C (Cs @ [C'])"   and "Subobjs P C (Cs @ [C', C''])" shows "P,C \<turnstile> (Cs @ [C']) \<sqsubset>\<^sup>1 Cs @ [C', C'']"
op_norm_matrix_matrix_mult_le: fixes A :: "(('a, 'd) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'd) vec" shows "\<parallel>A ** B\<parallel>\<^sub>o\<^sub>p \<le> (\<parallel>A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>B\<parallel>\<^sub>o\<^sub>p)"
mult_1_left: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.monoid_mult one times" shows "times one a = a"
dg_Set_is_arrE: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V"   and S :: "V" assumes "S : A \<mapsto>\<^bsub>dg_Set \<alpha>\<^esub> B" obtains "arr_Set \<alpha> S" and "S\<lparr>ArrDom\<rparr> = A" and "S\<lparr>ArrCod\<rparr> = B"
alpha_qVar_iff: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and xs' :: "'f"   and x' :: "'g" shows "(X #= qVar xs' x') = (X = qVar xs' x')"
convexD: fixes s :: "'c set"   and x :: "'c"   and y :: "'c"   and u :: "real"   and v :: "real" assumes "convex s"   and "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
locally_finite_on_subset: fixes X :: "'c set"   and J :: "'d set"   and U :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and V :: "'d \<Rightarrow> 'c set" assumes "locally_finite_on X J U"   and "\<And>i. i \<in> I \<Longrightarrow> V i \<subseteq> U i"   and "I \<subseteq> J" shows "locally_finite_on X I V"
lsu_Inf_closed: shows "Inf_closed_set (range \<nu>\<^sup>\<natural>)"
degen_run_complete: fixes r :: "nat \<Rightarrow> 'Q"   and m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b" assumes "num_acc \<noteq> 0"   and "is_run r" shows "\<exists>r'. graph_defs.is_run (degeneralize_ext m) r' \<and> r = fst \<circ> r'"
below_def: shows "below \<equiv> map_fun rep_ivl abs_ivl below_rep"
all_after_unl_7: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'a" shows "w \<Turnstile> (\<exists>x. P x)$ = (\<exists>x. P x$)"
exec_skip: fixes \<sigma> :: "'a"   and M :: "'a \<Rightarrow> (bool \<times> 'a) option" shows "(\<sigma> \<Turnstile> skip\<^sub>S\<^sub>E ;- M) = (\<sigma> \<Turnstile> M)"
P: shows "P \<in> carrier_mat CARD('nr) CARD('nr)"
invocation_class_def: fixes m :: "inv_mode"   and s :: "st"   and a' :: "val"   and statT :: "ref_ty" shows "invocation_class m s a' statT = (case m of Static \<Rightarrow> if \<exists>statC. statT = ClassT statC then the_Class (RefT statT) else Object | SuperM \<Rightarrow> the_Class (RefT statT) | IntVir \<Rightarrow> obj_class (lookup_obj s a'))"
apply_cm_def: fixes c :: "('a, 't, 'b) Propagate.configuration_scheme"   and loc :: "'a"   and t :: "'t"   and n :: "int" shows "apply_cm c loc t n = (let new_pointstamps = \<lambda>loc'. if loc' = loc then update_zmultiset (c_pts c loc') t n else c_pts c loc' in c\<lparr>c_pts := new_pointstamps, c_work := \<lambda>loc'. c_work c loc' + frontier_changes (new_pointstamps loc') (c_pts c loc')\<rparr>)"
bind_fv2_preserves_type: fixes Ts :: "typ list"   and t :: "Core.term"   and ty :: "typ"   and T :: "typ"   and v :: "variable" assumes "typ_of1 Ts t = Some ty" shows "typ_of1 (Ts @ [T]) (bind_fv2 (v, T) (length Ts) t) = Some ty"
sign_tjk: shows "sign (Fun.swap j k id) = - 1"
isomorphic_sym: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and W :: "'c set" assumes "isomorphic smult' W" shows "\<exists>T. FGModuleIso G smult' W (\<cdot>) T V"
worklists_vacant_to_trans: fixes c :: "('loc, 't) configuration"   and t :: "'t"   and t' :: "'t" assumes "worklists_vacant_to c t"   and "t' \<le> t" shows "worklists_vacant_to c t'"
remove_Min_inext_nth_Suc_conv: fixes I :: "nat set"   and n :: "nat" assumes "Suc 0 < card I \<or> infinite I" shows "(I - {iMin I}) \<rightarrow> n = I \<rightarrow> Suc n"
inj_take: fixes n1 :: "nat"   and al :: "'a list"   and n2 :: "nat" assumes "n1 \<le> length al"   and "n2 \<le> length al" shows "(take n1 al = take n2 al) = (n1 = n2)"
shift2_mrexps2: fixes r :: "'b mrexp2"   and s :: "'b rexp"   and x :: "bool"   and a :: "'b" assumes "r \<in> mrexps2 s" shows "shift2 x r a \<in> mrexps2 s"
large_Gromov_product_approx: fixes M :: "ereal" assumes "M < \<infinity>" shows "\<exists>e D. 0 < e \<and> D < \<infinity> \<and> (\<forall>x y. dist x y \<le> e \<longrightarrow> D \<le> extended_Gromov_distance x (to_Gromov_completion basepoint) \<longrightarrow> M \<le> extended_Gromov_product_at basepoint x y)"
diff_le_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) c = less_eq a (plus c b)"
sparsegrid'I: fixes b :: "(nat \<times> int) list"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and ds :: "nat set" assumes "b \<in> sparsegrid' dm"   and "p \<in> grid b ds" shows "p \<in> sparsegrid' dm"
le_ASSUMEI: fixes \<Phi> :: "bool"   and M :: "'b nres"   and M' :: "'b nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> ASSUME \<Phi> \<bind> (\<lambda>_. M')"
sharing_consistent_share_all_until_volatile_write: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S> :: "nat \<Rightarrow> bool option"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "ownership_distinct ts"   and "sharing_consis \<S> ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent (share_all_until_volatile_write ts \<S>) (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
Block1Throw: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and T :: "ty"   and a :: "'addr"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>{V:T=None; Throw a},s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
leadsTo_induct: fixes F :: "'c program"   and za :: "'c set"   and zb :: "'c set"   and P :: "'c set \<Rightarrow> 'c set \<Rightarrow> bool" assumes "F \<in> za \<longmapsto> zb"   and "\<And>A B. F \<in> A ensures B \<Longrightarrow> P A B"   and "\<And>A B C. \<lbrakk>F \<in> A \<longmapsto> B; P A B; F \<in> B \<longmapsto> C; P B C\<rbrakk> \<Longrightarrow> P A C"   and "\<And>B S. \<forall>A\<in>S. F \<in> A \<longmapsto> B \<and> P A B \<Longrightarrow> P (\<Union> S) B" shows "P za zb"
assms_1: shows "P \<in> carrier_mat (dim_row A) (dim_row A)"
norm_ci: fixes l :: "'a list"   and Pe :: "polex" assumes "in_carrier l" shows "Ipolex l Pe = Ipol l (norm Pe)"
east_west_tangents_2: shows "polar west = proj2_line_abs (vector [1, 0, 1])"
through: shows "f (poincare_map {x \<in> R. x \<bullet> n = c} x) \<bullet> n \<noteq> 0"
Enabled_conj1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<and> G) \<longrightarrow> Enabled F"
w_lt_n: shows "\<forall>p\<in>set w. p < n"
lock_actions_ok'_iff: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and las :: "lock_action list" shows "lock_actions_ok' l t las = (lock_actions_ok l t las \<or> (\<exists>xs ys. las = xs @ Lock # ys \<and> lock_actions_ok l t xs \<and> \<not> may_lock (upd_locks l t xs) t))"
SpecAnno': fixes P :: "'d set"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and Q :: "'d set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> t \<in> Q) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> t \<in> A)}"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) c Z (Q' Z),(A' Z)"   and "\<forall>Z. c Z = c undefined" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P specAnno P' c Q' A' Q,A"
fls_integral_mult_const_left_comm: fixes c :: "'a"   and f :: "'a fls" shows "fls_integral (fls_const c * f) = fls_const c * fls_integral f"
sel_2: fixes x1 :: "'b"   and x2 :: "'a" shows "snd (x1, x2) = x2"
Joints_appendD2: fixes ys :: "'a list"   and xs :: "'a list"   and YS :: "('a \<times> 'a pdevs) list"   and XS :: "('a \<times> 'a pdevs) list" assumes "ys @ xs \<in> Joints (YS @ XS)"   and "length ys = length YS" shows "ys \<in> Joints YS"
connected_closedin: fixes S :: "'b set" shows "connected S = (\<nexists>E1 E2. closedin (top_of_set S) E1 \<and> closedin (top_of_set S) E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
simps_5: fixes op :: "'op" shows "norm_instr (Inca.instr.IOp op) = Std.instr.IOp op"
sqn_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat" shows "sqn rt dip \<equiv> case rt dip of None \<Rightarrow> 0 | Some r \<Rightarrow> \<pi>\<^sub>2 r"
axioms_2: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set" assumes "bijective_map \<alpha> S T" shows "bijective \<alpha> S T"
TC_psubst: fixes G :: "('a, 'b) form list"   and f :: "'a \<Rightarrow> 'a" assumes "infinite UNIV"   and "\<stileturn> G" shows "\<stileturn> map (psubst f) G"
col_cop2__cop: fixes U :: "'p"   and V :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" assumes "U \<noteq> V"   and "Coplanar A B C U"   and "Coplanar A B C V"   and "Col U V P" shows "Coplanar A B C P"
distinct_1: fixes x1 :: "(real, 'b) vec"   and x21 :: "'b"   and x22 :: "real"   and x23 :: "'b kdt"   and x24 :: "'b kdt" shows "Leaf x1 \<noteq> Node x21 x22 x23 x24"
power_mono: fixes a :: "'d"   and b :: "'d"   and n :: "nat" assumes "a \<le> b"   and "(0::'d) \<le> a" shows "a ^ n \<le> b ^ n"
ssubst_option_lem: fixes f :: "'a \<Rightarrow> sterm option"   and z :: "char list"   and u :: "sterm" shows "\<forall>l\<in>dom f. [z \<rightarrow> u] the (f l) = the (ssubst_option z u (f l))"
simps_2: fixes x :: "'a"   and y :: "'a"   and l :: "'a list" shows "x mem y # l = (x = y \<or> x mem l)"
Abs_poly_mapping_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'d)}}" shows "lookup (Abs_poly_mapping y) = y"
Inf_inners_def: fixes X :: "real list set"   and y :: "real list" shows "Inf_inners X y = SPEC (\<lambda>r. \<forall>x\<in>X. r \<le> inner_lv_rel x y)"
t1: shows "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"
fresh_18: fixes a :: "atom"   and hash :: "hash"   and "term" :: "Syntax.term" shows "a \<sharp> Hashed hash term = (a \<sharp> hash \<and> a \<sharp> term)"
scaleC_right_mono: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "complex"   and b :: "complex"   and x :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "a \<le> b"   and "less_eq zero x" shows "less_eq (scaleC a x) (scaleC b x)"
knows_subset_knows_Gets: fixes A :: "agent"   and evs :: "event list"   and A' :: "agent"   and X :: "msg" shows "knows A evs \<subseteq> knows A (Gets A' X # evs)"
from_hma\<^sub>v_def: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v = Matrix.vec CARD('e) (\<lambda>i. v $h mod_type_class.from_nat i)"
set_mult_ring_times_closed: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a"   and r :: "'a" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "x \<in> set_mult R A B"   and "r \<in> carrier R" shows "r \<cdot>\<^sub>r x \<in> set_mult R A B"
not_bit_length: fixes w :: "'b word" shows "\<not> bit w LENGTH('b)"
Diagonalize_Diag: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Diagonalize C t = t"
cdf_nonneg: fixes M :: "real measure"   and x :: "real" assumes "finite_borel_measure M" shows "0 \<le> cdf M x"
g_def: shows "g \<equiv> fst (normalize_quot (c, b))"
bij_finite_const_subst_exists': fixes S :: "'a set"   and T :: "('b, 'a) Term.term set"   and U :: "'b set" assumes "finite S"   and "finite T"   and "infinite U" shows "\<exists>\<sigma>. subst_domain \<sigma> = S \<and> bij_betw \<sigma> (subst_domain \<sigma>) (subst_range \<sigma>) \<and> subst_range \<sigma> \<subseteq> (\<lambda>c. Fun c []) ` U - T"
distinct_43: shows "AtOp \<noteq> string_binop.GreaterEqOp"
relatedtail: shows "V \<approx>\<^bsub>d\<^esub> V'"
fix_def2: fixes F :: "'c \<rightarrow> 'c" shows "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
assms_5: fixes b :: "'c" assumes "b \<in> S" shows "g b \<in> carrier R"
R23sD: fixes s :: "l2_state"   and t :: "l3_state" assumes "(s, t) \<in> R23s" shows "ik s = ik t \<inter> payload \<and> secret s = secret t \<and> progress s = progress t \<and> signalsInit s = signalsInit t \<and> signalsResp s = signalsResp t \<and> chan s = local.abs (ik t) \<and> l2_state.bad s = l3_state.bad t"
Inverse_relE: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "Category C"   and "cinv\<^bsub>C\<^esub> f g" obtains "f \<approx>>\<^bsub>C\<^esub> g" and "f ;;\<^bsub>C\<^esub> g = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f)" and "g ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)"
bsmap_fs_depD_2: fixes e :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and a :: "'b"   and q :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" assumes "(e, a) \<in># SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)" shows "e \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map BsSkewBinomialHeapStruc.bsmapt q)) \<and> a = BsSkewBinomialHeapStruc.eprio e"
gen_subgroup_is_subgroup: fixes gens :: "'c set" assumes "gens \<subseteq> carrier H" shows "\<langle>gens\<rangle>\<^bsub>H\<^esub> \<le> H"
assms_5: shows "r \<cdot> q \<le> q \<cdot> r"
a1: shows "h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children'"
map_sum_sel_1: fixes s :: "'e + 'f"   and f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" assumes "isl s" shows "projl (map_sum f g s) = f (projl s)"
norm_mono: fixes v :: "'b \<Rightarrow> nat"   and u :: "'b \<Rightarrow> 'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat \<Rightarrow> 'c" assumes "\<forall>c. 0 < v c"   and "u \<in> [M]\<^bsub>v,n\<^esub>" shows "u \<in> [DBM_Normalization.norm M k n]\<^bsub>v,n\<^esub>"
setdist_Lipschitz: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" shows "\<bar>setdist {x} S - setdist {y} S\<bar> \<le> dist x y"
assms_9: fixes a :: "'a set" assumes "a \<in> A" shows "emeasure M a \<noteq> \<infinity>"
scale: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and r :: "'a"   and b :: "'b" assumes "module_hom s1 s2 f" shows "f (s1 r b) = s2 r (f b)"
simple_match_none_def: shows "simple_match_none \<equiv> \<lparr>iiface = ifaceAny, oiface = ifaceAny, src = (1, 0), dst = (0, 0), proto = ProtoAny, sports = (1, 0), dports = (0, 65535)\<rparr>"
extremeI: fixes e :: "'c"   and X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "e \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> r x e" shows "extreme X r e"
nice_ind_inv_is_inductive_invariant: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr" assumes "substitution subst" shows "substitution.inductive_invariant subst (\<lambda>s. substitution.domains_subsume_globally s \<and> substitution.subsumes_accessible_globally s)"
exp_not_zero_imp_exp_diff_not_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero" shows "power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (n - m) \<noteq> zero"
h_def: shows "h = \<beta> \<ominus> \<alpha>"
theI: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
FAssNull: fixes 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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "expr"   and v :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>2\<rangle>"
set_mult_assoc: fixes G :: "('e, 'f) monoid_scheme"   and M :: "'e set"   and H :: "'e set"   and K :: "'e set" assumes "Group.group G"   and "M \<subseteq> carrier G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G" shows "M <#>\<^bsub>G\<^esub> H <#>\<^bsub>G\<^esub> K = M <#>\<^bsub>G\<^esub> (H <#>\<^bsub>G\<^esub> K)"
G_in_rell: fixes L1 :: "'n \<Rightarrow> 'u \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'v \<Rightarrow> bool" shows "rell_G L1 L2 = (\<lambda>x y. \<exists>z. (set1_G z \<subseteq> {(x, y). L1 x y} \<and> set2_G z \<subseteq> {(x, y). L2 x y}) \<and> mapl_G fst fst z = x \<and> mapl_G snd snd z = y)"
assms_1: shows "run (Array.new n x) \<sigma> \<sigma>' r"
distinct_1: fixes x2 :: "'b" shows "\<bottom> \<noteq> \<lfloor>x2\<rfloor>"
cong_ex_bigtheta: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f2 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> O[F](g1)) = (f2 \<in> O[F](g2))"
fpxs_shift_add_left: fixes m :: "rat"   and n :: "rat"   and f :: "'b fpxs" shows "fpxs_shift (m + n) f = fpxs_shift m (fpxs_shift n f)"
commE: fixes x :: "'b list"   and y :: "'b list" assumes "x \<cdot> y = y \<cdot> x" obtains t :: "'b list" and m :: "nat" and k :: "nat" where "x = t \<^sup>@ k" and "y = t \<^sup>@ m"
2: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" shows "lt (q' g \<odot> g) \<preceq>\<^sub>t lt p"
rel_set_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
elims: fixes x :: "nat set"   and xa :: "(nat set \<times> 'b option list set) set"   and xb :: "nat set \<times> 'b option list"   and y :: "(nat set \<times> 'b option list set) set" assumes "New_max.newQuery x xa xb = y" obtains V :: "nat set" and Q :: "(nat set \<times> 'b option list set) set" and st :: "nat set" and t :: "'b option list" where "x = V" and "xa = Q" and "xb = (st, t)" and "y = (\<lambda>tab. New_max.projectTable V (New_max.semiJoin tab (st, t))) ` Q"
span_is_subspace: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "subspace K (span S) V"
pf: shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
lrev_body_lrev_body1_eq: shows "lrev_body1 = unwrapH oo lrev_body oo wrapH"
l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs"
sum_list_simps_2: fixes x :: "'e"   and xs :: "'e list" shows "sum_list (x # xs) = x + sum_list xs"
nat_mod_lem: fixes n :: "nat"   and b :: "nat" assumes "0 < n" shows "(b < n) = (b mod n = b)"