hcju/isabelleps huggingface dataset:
height_gt_0_notin_alphabet_imp_sibling_right: fixes t\<^sub>2 :: "'a tree"   and a :: "'a"   and t\<^sub>1 :: "'a tree"   and w :: "nat" assumes "0 < height t\<^sub>2"   and "a \<notin> alphabet t\<^sub>1" shows "sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>2 a"
assoc: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" shows "a \<union># b \<union># c = a \<union># (b \<union># c)"
simps_3: fixes disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'b"   and a :: "'c" shows "primitive_extractor (disc, sel) (MatchNot (Match a)) = (if disc a then ([Neg (sel a)], MatchAny) else ([], MatchNot (Match a)))"
rep_eq: fixes x :: "'k word"   and xa :: "'k word" shows "(x < xa) = (take_bit LENGTH('k) (Word.rep x) < take_bit LENGTH('k) (Word.rep xa))"
Conf_InitStatus_InitConf: fixes A :: "('d, 'e, 'f) hierauto" shows "Conf (InitStatus A) = InitConf A"
a_objects_exist_3: fixes v :: "i" shows "[\<^bold>\<box>\<^bold>\<not>(\<^bold>\<forall>x. \<lparr>E!,x\<^sup>P\<rparr>) in v]"
Zorn_Lemma: fixes A :: "'b set set" assumes "\<forall>C\<in>chains A. \<Union> C \<in> A" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
ln_series': fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "(\<lambda>n. - ((- x) ^ n) / real n) sums ln (1 + x)"
hb_trans: fixes m1 :: "'a"   and m2 :: "'a"   and m3 :: "'a" assumes "hb m1 m2"   and "hb m2 m3" shows "hb m1 m3"
nsmulextp_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "'a multiset"   and n :: "'a multiset" shows "nsmulextp f m n = ((m, n) \<in> ns_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)})"
H_iso_cond2: fixes f :: "'d \<Rightarrow> 'e"   and y :: "'d"   and z :: "'d"   and x :: "'e" assumes "mono f"   and "y \<le> z"   and "x \<le> f y" shows "x \<le> f z"
assms_2: fixes a :: "'a"   and b :: "'b" assumes "a \<in> set1_FGf x"   and "b \<in> set1_FGf y"   and "L1 a b" shows "L1' a b"
FaultPropBlock: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and abrs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f" shows "\<Gamma>\<turnstile> ([], (nrms, abrs) # css, Fault f) \<rightarrow> (nrms, css, Fault f)"
weakBisimReflexive: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'" shows "(\<Psi>, P, P) \<in> env.weakBisim substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"
is_acc_def: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'c"   and m :: "'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat" shows "gb_graph.is_acc (b_graph.to_gbg_ext (degeneralize_ext ma) m) r \<equiv> \<forall>A\<in>gbg_F (b_graph.to_gbg_ext (degeneralize_ext ma) m). \<exists>\<^sub>\<infinity>i. r i \<in> A"
sheaf_of_rings_axioms: shows "sheaf_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
elims: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and x :: "'b set set"   and y :: "'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)" assumes "ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M x = y" obtains \<Sigma> :: "'b set set" where "x = \<Sigma>" and "y = \<delta> \<times> \<Delta>\<^sub>\<times> (semi_mojmir_def.step \<Sigma> \<delta>\<^sub>M \<circ> q\<^sub>0\<^sub>M \<circ> theG)"
reduce_poly_simple: fixes b :: "complex"   and n :: "nat" assumes "b \<noteq> 0"   and "n \<noteq> 0" shows "\<exists>z. cmod (1 + b * z ^ n) < 1"
absorb_iff2: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a \<sqinter> b = b)"
get_child_nodes_delete_shadow_root: fixes ptr' :: "(_) object_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr"   and "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr'" shows "r h h'"
i_drop_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Up> n \<equiv> \<lambda>x. f (n + x)"
pfa_blit_def: shows "pfa_blit \<equiv> \<lambda>(src, sn) si (dst, dn). blit src si dst"
ranking_index_eqI: fixes i :: "nat"   and x :: "'a" assumes "i < length (weak_ranking le)"   and "x \<in> weak_ranking le ! i" shows "weak_ranking_index x = i"
assms_5: shows "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"
length_filter_conv_card: fixes p :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (filter p xs) = card {i. i < length xs \<and> p (xs ! i)}"
line_integral_exists_subpath: fixes f :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and g :: "real \<Rightarrow> 'a"   and u :: "real"   and v :: "real" assumes "line_integral_exists f basis g"   and "valid_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "u \<le> v" shows "line_integral_exists f basis (subpath u v g)"
iIN_iT_Div_Int: fixes k :: "nat"   and n :: "nat"   and A :: "nat set"   and d :: "nat" assumes "0 < k"   and "n \<le> iMin A"   and "\<forall>x\<in>A. x div k \<le> (n + d) div k \<longrightarrow> x \<le> n + d" shows "A \<inter> [n\<dots>,d] \<oslash> k = (A \<oslash> k) \<inter> ([n\<dots>,d] \<oslash> k)"
exec_step_ind_Invoke: fixes ps :: "val list"   and n :: "nat"   and stk :: "val list"   and r :: "val"   and C :: "char list"   and h :: "heap"   and D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and M :: "char list"   and m :: "jvm_method"   and f' :: "frame"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "ps = take n stk"   and "r = stk ! n"   and "C = cname_of h (the_Addr r)"   and "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "r \<noteq> Null"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = m in D"   and "f' = ([], [r] @ rev ps @ replicate mxl\<^sub>0 undefined, D, M, 0, No_ics)" shows "exec_step_ind (StepI (Invoke M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, f' # (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
access_mbr_simp: fixes m :: "qtname \<times> memberdecl" shows "accmodi (mbr m) = accmodi m"
inj_on_fun_updI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and y :: "'d"   and x :: "'c" assumes "inj_on f A"   and "y \<notin> f ` A" shows "inj_on (f(x := y)) A"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Listn.le r = list_all2 (\<lambda>x. lesub x r)"
isomorphicE: fixes a :: "'b"   and a' :: "'b" assumes "B.isomorphic a a'" obtains f :: "'b" where "\<guillemotleft>f : a \<rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f"
subgroupE_2: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)" shows "H \<noteq> {}"
cart_eq_inner_axis: fixes a :: "(real, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "a $h i = a \<bullet> axis i 1"
agreeL: shows "Vagree \<nu> \<nu>' (\<Union> (primify ` FVT t1))"
is_trail_split': fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and u :: "'v"   and w :: "'w"   and u' :: "'v"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v (p1 @ (u, w, u') # p2) v'" shows "is_trail v p1 u \<and> (u, w, u') \<in> E \<and> is_trail u' p2 v'"
u2: shows "projection (space_as_set M) \<circ> projection (space_as_set M) = projection (space_as_set M)"
abs_eq: shows "(\<squnion>) = (\<lambda>f g. (f\<^sub>\<bullet> \<squnion> g\<^sub>\<bullet>)\<^sup>\<bullet>)"
assms_1: shows "\<forall>(k, \<gamma>)\<in>chain. point_path \<gamma>"
homotopic_paths_nearby_explicit: fixes g :: "real \<Rightarrow> 'b"   and h :: "real \<Rightarrow> 'b"   and S :: "'b set" assumes "path g"   and "path h"   and "pathstart h = pathstart g"   and "pathfinish h = pathfinish g"   and "\<And>t x. \<lbrakk>t \<in> {0..1}; x \<notin> S\<rbrakk> \<Longrightarrow> norm (h t - g t) < norm (g t - x)" shows "homotopic_paths S g h"
ignore_def: fixes \<Delta> :: "'state \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'e"   and vl1 :: "'f"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'d" shows "ignore \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<not> \<gamma> (Trans s a ou s') \<and> \<Delta> s' vl' s1 vl1"
assms_2: assumes "tree H \<and> spanning H G" shows "P"
observable_def: fixes ob :: "'e \<Rightarrow> 'f"   and P :: "'e set" shows "observable ob P \<equiv> \<forall>s s'. ob s = ob s' \<longrightarrow> s' \<in> P \<longrightarrow> s \<in> P"
not_fork: fixes \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e :: "('r, 'l, 'v) expr" shows "s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Rfork e])"
le_dag_set_of: fixes x :: "dag"   and y :: "dag" assumes "x \<le> y" shows "set_of x \<subseteq> set_of y"
listall_conj2: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "listall (\<lambda>x. P x \<and> Q x) xs" shows "listall Q xs"
deriv: fixes x :: "real" assumes "a < x" shows "((\<lambda>x. - 1 / ln x) has_real_derivative 1 / (x * (ln x)\<^sup>2)) (at x)"
A: shows "Esigma (m ^ k) = sum ((^) m) {0..k}"
subset: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme" assumes "subgroup H G" shows "H \<subseteq> carrier G"
pt_pos: fixes t :: "'a" assumes "t \<in> S - U" shows "0 < p t"
M_perp_sym: fixes l :: "proj2_line"   and m :: "proj2_line" assumes "M_perp l m" shows "M_perp m l"
tendsto_sin: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. sin (f x)) \<longlongrightarrow> sin a) F"
prj_in_hom_6: shows "\<guillemotleft>\<nu>\<pi>\<rho>.Prj\<^sub>0\<^sub>0 : H\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<rho>.apex\<guillemotright>"
simps_12: fixes i :: "int"   and a :: "Cooper.num" shows "simpfm (Dvd i a) = (if i = 0 then simpfm (Eq a) else if \<bar>i\<bar> = 1 then T else let a' = simpnum a in case a' of C v \<Rightarrow> if i dvd v then T else F | _ \<Rightarrow> Dvd i a')"
prod_mset_const_poly: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "(\<Prod>x\<in>#A. [:f x:]) = [:\<Prod>\<^sub># (image_mset f A):]"
select_convs_1: fixes precondition_of :: "('c \<times> 'd) list"   and effect_of :: "('c \<times> 'd) list"   and more :: "'e" shows "sas_plus_operator.precondition_of \<lparr>sas_plus_operator.precondition_of = precondition_of, effect_of = effect_of, \<dots> = more\<rparr> = precondition_of"
le_any_fun_inv: fixes v :: "val"   and t :: "(val \<times> val) fset" assumes "v \<sqsubseteq> VFun t" obtains t1 :: "(val \<times> val) fset" where "v = VFun t1" and "fset t1 \<subseteq> fset t"
RecvMarker_implies_Marker_in_set: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "ev = RecvMarker cid p q" shows "Marker \<in> set (msgs c cid)"
lfp_wf_induction: fixes f :: "'a \<Rightarrow> 'a"   and p :: "'b \<Rightarrow> 'a" assumes "mono f"   and "\<forall>w. p w \<sqsubseteq> f (Sup_less p w)" shows "\<Squnion>range p \<sqsubseteq> lfp f"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<Pi>\<^sub>1 set \<Rightarrow> 'a"   and \<nu> :: "\<nu>" shows "P (case \<nu> of \<omega>\<nu> x \<Rightarrow> f1 x | \<alpha>\<nu> x \<Rightarrow> f2 x) = ((\<forall>x1. \<nu> = \<omega>\<nu> x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. \<nu> = \<alpha>\<nu> x2 \<longrightarrow> P (f2 x2)))"
elims: fixes x :: "game option"   and xa :: "game option"   and y :: "game option" assumes "Composeo x xa = y" obtains \<alpha> :: "game" and \<beta> :: "game" where "x = Agame \<alpha>__" and "xa = Agame \<beta>" and "y = Agame (\<alpha>__ ;; \<beta>)"   | \<alpha> :: "game option" where "x = \<alpha>" and "xa = undefg" and "y = undefg"   | v :: "game" where "x = undefg" and "xa = Agame v" and "y = undefg"
dghm_const_ArrMap_app: fixes x :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "x \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "dghm_const \<CC> \<DD> a f\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = f"
IH_1: fixes t :: "char list \<Rightarrow> int option" assumes "approx t s1" shows "approx (defs c t) s2"
ring_1_ow_def: fixes U :: "'b set"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "ring_1_ow U one times plus zero minus uminus \<equiv> ring_ow U plus zero minus uminus times \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
category_with_binary_products_axioms: shows "category_with_binary_products C"
induct: fixes \<delta> :: "real \<Rightarrow> real"   and x1 :: "real \<times> real"   and x2 :: "(real \<times> real \<times> real) list"   and P :: "real \<times> real \<Rightarrow> (real \<times> real \<times> real) list \<Rightarrow> bool" assumes "fine \<delta> x1 x2"   and "\<And>a. P (a, a) []"   and "\<And>b c D a x. \<lbrakk>fine \<delta> (b, c) D; P (b, c) D; a < b; a \<le> x; x \<le> b; b - a < \<delta> x\<rbrakk> \<Longrightarrow> P (a, c) ((a, x, b) # D)" shows "P x1 x2"
G_monotonic: shows "MonotProblem GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf gPB"
Mini_def: fixes i :: "'a"   and js :: "'a list" shows "Mini i js = (if js = [] then i else Min (set js))"
mult2_alt_s_implies_mult2_alt_ns: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> mult2_alt_s ns s" shows "(X, Y) \<in> mult2_alt_ns ns s"
rel_compile_fundef: fixes get_arity :: "'fun \<Rightarrow> nat option"   and load_oracle :: "nat \<Rightarrow> Unboxed.type option"   and fd1 :: "('b, 'l, 'fun, 'd, 'e) Inca.instr fundef"   and fd2 :: "('b, 'l, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef" assumes "compile_fundef get_arity load_oracle fd1 = Ok fd2" shows "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"
is_expansion_aux_coinduct_upto: fixes X :: "('a \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "X xs f basis"   and "basis_wf basis"   and "\<And>xs f basis. X xs f basis \<Longrightarrow> xs = MSLNil \<and> (\<forall>\<^sub>F x in at_top. f x = 0) \<and> length basis = Suc (expansion_level TYPE('a)) \<or> (\<exists>xs' b e basis' C. xs = MSLCons (C, e) xs' \<and> basis = b # basis' \<and> ms_closure X xs' (\<lambda>x. f x - eval C x * b x powr e) (b # basis') \<and> is_expansion C basis' \<and> (\<forall>e'>e. f \<in> o(\<lambda>x. b x powr e')) \<and> ms_exp_gt e (ms_aux_hd_exp xs'))" shows "is_expansion_aux xs f basis"
is_scc_pointwise: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" shows "is_scc E U = (U \<noteq> {} \<and> (\<forall>u\<in>U. \<forall>v\<in>U. (u, v) \<in> (Restr E U)\<^sup>*) \<and> (\<forall>u\<in>U. \<forall>v. v \<notin> U \<and> (u, v) \<in> E\<^sup>* \<longrightarrow> (\<forall>u'\<in>U. (v, u') \<notin> E\<^sup>*)))"
counit_in_hom_2: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<sigma>.counit a : \<tau>\<^sub>0 a \<star>\<^sub>D \<sigma>.map\<^sub>0 a \<Rightarrow>\<^sub>D G.map\<^sub>0 a\<guillemotright>"
Forall: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf n (Forall p) = dfa_of_pf n (Neg (Exist (Neg p)))"
real_sqrt_eq_0_iff: fixes x :: "real" shows "(sqrt x = 0) = (x = 0)"
spn_ide: fixes r :: "'a" assumes "ide r" shows "spn r \<cong> src (tab\<^sub>0 r)"
closed_Un: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<union> T)"
is_subseq_of_def: fixes s :: "nat \<Rightarrow> 'a"   and s' :: "nat \<Rightarrow> 'a" shows "is_subseq_of s s' = (\<exists>f. is_increasing f \<and> s' = take_subseq s f)"
\<phi>_simps_5: shows "cod f\<^sub>0g\<^sub>1.\<phi> = tab\<^sub>1 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0"
alw_ev_scons: fixes P :: "'a stream \<Rightarrow> bool"   and a :: "'a"   and w :: "'a stream" shows "alw (ev P) (a ## w) = alw (ev P) w"
A: shows "\<forall>x. \<exists>y. (y, x) \<in> R"
rel_of_succ_def: fixes s :: "'d \<Rightarrow> 'e set" shows "rel_of_succ s \<equiv> {(a, b). b \<in> s a}"
E: fixes i :: "'a" assumes "i \<in> is" shows "equiv A (f i)"
order_linear_power': fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "order a ([:b, 1::'b:] ^ Suc n) = (if b = - a then Suc n else 0)"
that_2: assumes "ord a b = EQUAL"   and "a = b" shows "thesis"
nths_Suc_n_dims: shows "nths dims {0..<Suc n} = dims"
power_0: fixes a :: "'a" shows "a +^ 0 = (0::'a)"
zero_le_Fract_iff: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(0 \<le> Fract a b) = ((0::'b) \<le> a)"
angle_collinear: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "angle a b c \<in> {0, pi}" shows "collinear {a, b, c}"
acyclic_1b_def: fixes x :: "'a" shows "acyclic_1b x \<equiv> \<forall>y. orientation x y \<longrightarrow> y\<^sup>\<star> \<sqinter> y\<^sup>T\<^sup>\<star> = (1::'a)"
zero_order_1: fixes x :: "'b multiset" shows "{#} \<subseteq># x"
proper_signed_list_map_uniform_snd: fixes b :: "bool"   and as :: "'b list" shows "proper_signed_list (map (\<lambda>s. (s, b)) as)"
is_least_def: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" shows "is_least P n = (P n \<and> (\<forall>n'<n. \<not> P n'))"
induct: fixes P :: "'a simple_rule list \<Rightarrow> ('a, 'b) simple_packet_scheme \<Rightarrow> bool"   and a0 :: "'a simple_rule list"   and a1 :: "('a, 'b) simple_packet_scheme" assumes "\<And>uu_. P [] uu_"   and "\<And>m rs p. (\<not> simple_matches m p \<Longrightarrow> P rs p) \<Longrightarrow> P (SimpleRule m Accept # rs) p"   and "\<And>m rs p. (\<not> simple_matches m p \<Longrightarrow> P rs p) \<Longrightarrow> P (SimpleRule m Drop # rs) p" shows "P a0 a1"
oV: fixes i :: "'a" shows "openin (Y i) (f i -` g -` V \<inter> topspace (Y i))"
factor_gauss_int_prime_nat_def: fixes p :: "nat" shows "factor_gauss_int_prime_nat p = (if p = 2 then [1 + \<i>\<^sub>\<int>] else if [p = 3] (mod 4) then [of_nat p] else case prime_square_sum_nat_decomp p of (x, y) \<Rightarrow> [of_nat x + \<i>\<^sub>\<int> * of_nat y, of_nat y + \<i>\<^sub>\<int> * of_nat x])"
lrev_is_LNil_conv: fixes xs :: "'b llist" assumes "xs \<in> UNIV\<^sup>\<star>" shows "(lrev xs = LNil) = (xs = LNil)"
wfactors_prod_exists: fixes as :: "'a list" assumes "\<forall>a\<in>set as. irreducible G a"   and "set as \<subseteq> carrier G" shows "\<exists>a. a \<in> carrier G \<and> wfactors G as a"
map_pmf_of_set_bij_betw: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set B"
A: shows "(yps @ [(y, p)], Y) \<in> c_failures step out s\<^sub>0"
simps_1: fixes shared :: "bool"   and M :: "nat"   and w :: "nat \<Rightarrow> real" shows "insert_weights shared (Input M) w = Input M"
abs_eq: fixes x :: "\<nu> option" shows "proper (make\<kappa> x) = (None \<noteq> x)"
vars: shows "free_vars b \<subseteq> set vs \<union> set vs'"
UN_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'c set"   and g :: "'a \<Rightarrow> 'c set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
smult_monom_mult: fixes a :: "'b"   and b :: "'b"   and n :: "nat"   and f :: "'b poly" shows "Polynomial.smult a (monom b n * f) = monom (a * b) n * f"
components_are_iso: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "B.iso (fst a \<star>\<^sub>B snd a)"
Arr_iff_cod_in_codomain: fixes f :: "'a" shows "Arr f = (C.cod f \<in> codomains f)"
Int_open: fixes x :: "'b set"   and y :: "'b set" assumes "x open\<^bsub>S\<^esub>"   and "y open\<^bsub>S\<^esub>" shows "x \<inter> y open\<^bsub>S\<^esub>"
rref_upt_condition2_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> is_zero_row_upt_k i k A" shows "A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)"
HomFtorContra_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "Hom\<^bsub>C\<^esub>[\<midarrow>,X] \<equiv> MakeFtor (HomP\<^bsub>C\<^esub>[\<midarrow>,X])"
Least_Min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "finite {a. P a}"   and "\<exists>a. P a" shows "ord.Least less_eq P = linorder.Min less_eq {a. P a}"
assms_4: shows "\<forall>n w. cls_val_process Mkt pf1 n w = cls_val_process Mkt pf2 n w"
unsigned_ucast_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and w :: "'e word" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.unsigned one plus zero (ucast w) = take_bit LENGTH('d) (semiring_1.unsigned one plus zero w)"
wf_rexp_of: fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of \<Sigma> n \<phi>)"
vec_module_col: fixes A :: "'b mat" shows "module class_ring (module_vec TYPE('b) (dim_row A)\<lparr>carrier := LinearCombinations.module.span class_ring (module_vec TYPE('b) (dim_row A)) (set (cols A))\<rparr>)"
fps_exp_eq_1_iff: fixes c :: "'a" shows "(fps_exp c = 1) = (c = (0::'a))"
lg_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_lg [x, y] = lg x y"
l2_inv3D: fixes x :: "l2_state" assumes "x \<in> l2_inv3" shows "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz (generators x)) \<and> chan x \<subseteq> faked_chan_msgs x \<union> chan_generators"
gen_ideal_subset: fixes x :: "'a" assumes "x \<in> R" shows "\<langle>x\<rangle> \<subseteq> R"