hcju/isabelleps huggingface dataset:
simps_1: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "minus_poly_rev_list (x # xs) (y # ys) = (x - y) # minus_poly_rev_list xs ys"
language_entails: fixes w :: "'a stream"   and \<phi> :: "'a pltl" shows "(w \<in> formula.language \<phi>) = (!!) w \<Turnstile>\<^sub>p \<phi>"
jkbpC_represents: shows "SPR.represents SPR.jkbpC"
order_less_subst2: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and c :: "'b" assumes "a < b"   and "f b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "f a < c"
host_apply_respect_type: fixes t1s :: "t list"   and vs :: "v list"   and s :: "s"   and t2s :: "t list"   and f :: "host"   and hs :: "host_state"   and s' :: "s"   and vs' :: "v list" assumes "list_all2 types_agree t1s vs"   and "host_apply s (t1s _> t2s) f vs hs = Some (s', vs')" shows "list_all2 types_agree t2s vs'"
compact_imp_bounded: fixes U :: "'b set" assumes "compact U" shows "bounded U"
prerun_valid_def: fixes sys :: "('g, 'h, 'i, 'j, 'k, 'l) pre_system_scheme"   and \<phi> :: "('g, 'h, 'i, 'j, 'k) system_state \<Rightarrow> bool" shows "(sys \<Turnstile>\<^bsub>pre\<^esub> \<phi>) = (\<forall>\<sigma>. prerun sys \<sigma> \<longrightarrow> (\<box>\<lceil>\<phi>\<rceil>) \<sigma>)"
factors_equal: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a = b"   and "c = d" shows "a * c = b * d"
ord_not_coprime: fixes n :: "'a"   and a :: "'a" assumes "\<not> coprime n a" shows "Pocklington.ord n a = 0"
Maschke_proper: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V"   and "U \<noteq> 0"   and "U \<noteq> V" shows "\<exists>W. (RModule FG (\<cdot>) W \<and> W \<subseteq> V) \<and> W \<noteq> 0 \<and> W \<noteq> V \<and> V = W \<oplus> U"
fill_sort_inv: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and u :: "nat"   and n :: "nat" assumes "index_less index key"   and "index_mono index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "sort_inv key (u, enum xs index key n mi ma, map the (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma))"
vec_component: fixes x :: "'b"   and i :: "'c" shows "vec x $ i = x"
disjoint_family_indicator_le_1: fixes A :: "'b \<Rightarrow> 'c set"   and I :: "'b set"   and x :: "'c" assumes "disjoint_family_on A I" shows "(\<Sum>i\<in>I. indicator (A i) x) \<le> (1::'d)"
bind_det_aux: fixes x :: "'a"   and m :: "'a nres"   and y :: "'b"   and f :: "'a \<Rightarrow> 'b nres" assumes "RETURN x \<le> m"   and "RETURN y \<le> f x" shows "RETURN y \<le> m \<bind> f"
inv: shows "matrix_change_of_basis Y Y' = matrix_inv (matrix_change_of_basis Y' Y)"
ipPartitionSingleSet: fixes t :: "'b set"   and Bs :: "'b set set"   and ts :: "'b set list" assumes "ipPartition {t} (addSubsetSet t Bs)" shows "ipPartition {t} (partitioning ts (addSubsetSet t Bs))"
agsum_add3pm: fixes a :: "'a"   and n :: "int"   and m :: "int" assumes "a \<in> carrier A"   and "0 < n"   and "m < 0" shows "(n + m)\<triangleright>a\<^bsub>A\<^esub> = n\<triangleright>a\<^bsub>A\<^esub> \<plusminus> m\<triangleright>a\<^bsub>A\<^esub>"
Vset_succ: fixes k :: "hf" assumes "Ord k" shows "Vset (succ k) = HPow (Vset k)"
sorted_wrt_take_drop: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and n :: "nat" assumes "sorted_wrt f xs" shows "\<forall>x\<in>set (take n xs). \<forall>y\<in>set (drop n xs). f x y"
g_restrict_def: fixes P :: "'k \<times> 'v \<Rightarrow> bool"   and m :: "'s" shows "g_restrict P m \<equiv> iterate m (\<lambda>(k, v) \<sigma>. if P (k, v) then update_dj k v \<sigma> else \<sigma>) (local.empty ())"
simps_6: fixes r1 :: "'a rexp" shows "awidth (Star r1) = awidth r1"
hd_construct: fixes q :: "('c, 'd) pq" assumes "\<not> is_empty q" shows "hd (alist_of q) = (PQ.min q, the (priority q (PQ.min q)))"
AXIOMS_preserved_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and M' :: "trm" assumes "<a>:M \<in> AXIOMSc B"   and "M \<longrightarrow>\<^sub>a* M'" shows "<a>:M' \<in> AXIOMSc B"
Un_insert_left: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "insert a B \<union> C = insert a (B \<union> C)"
d\<^sub>3_def: shows "d\<^sub>3 \<equiv> map_fun eval\<Pi>\<^sub>3 id Some"
assms_2: shows "openin (top_of_set (affine hull S)) T"
assms: shows "x \<in> open_segment (0::'a) b"
ge_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.non_strict_order less_eq"   and "less_eq y x"   and "less_eq z y" shows "less_eq z x"
nsqn_quality_increases_nsqn_lt_lt: fixes i :: "nat"   and \<xi> :: "state"   and \<xi>' :: "state"   and s :: "nat" assumes "i \<in> kD (rt \<xi>)"   and "quality_increases \<xi> \<xi>'"   and "s < nsqn (rt \<xi>) i" shows "s < nsqn (rt \<xi>') i"
notin_fset: fixes x :: "'f"   and S :: "'f fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
default_unique_IFS: fixes otherbot :: "'a" assumes "\<forall>G f nP i. wf_graph G \<and> f \<in> set_offending_flows G nP \<and> i \<in> snd ` f \<longrightarrow> \<not> sinvar G (nP(i := otherbot))" shows "otherbot = \<bottom>"
fcomp_assoc: fixes f :: "'a \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c"   and h :: "'c \<Rightarrow> 'b" shows "fcomp (fcomp f g) h = fcomp f (fcomp g h)"
hb: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) b = B"
arcs_union: fixes G :: "('d, 'c) pre_digraph"   and H :: "('d, 'c) pre_digraph" shows "arcs (Digraph_Component.union G H) = arcs G \<union> arcs H"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> src s\<^sub>0 \<Rightarrow> s\<^sub>0\<guillemotright>"
\<epsilon>_cod: shows "C.inverse_arrows (\<epsilon> (C.cod f)) (C.inv (\<epsilon> (C.cod f)))"
P6: shows "\<lfloor>\<lambda>w. \<forall>x. (Dependent x \<^bold>\<rightarrow> Explained x) w\<rfloor>"
insertI1: fixes a :: "'b"   and B :: "'b set" shows "a \<in> insert a B"
span_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "ideal A \<subseteq> ideal B"
emb_LConsE: fixes z :: "'b"   and zs :: "'b llist"   and ys :: "'b llist" assumes "emb (LCons z zs) ys" obtains ys :: "'b llist" and zsa :: "'b list" where "ysa__ = prepend zsa (LCons z ys)" and "emb zs ys"
remap_rta: fixes f :: "'Q \<Rightarrow> 'b" shows "ranked_tree_automaton (ta_remap f TA) A"
new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ag" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> new_element_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
pure_assn_eq_emp_iff: fixes P :: "bool" shows "(\<up> P = emp) = P"
sd_chain_strictly_preferred: fixes m :: "nat"   and n :: "nat"   and q :: "'alt pmf"   and s :: "'alt pmf" assumes "m < n"   and "sd_chain m = Some q"   and "sd_chain n = Some s" shows "q \<prec>[Pareto (SD \<circ> R)] s"
measurable_empty_iff: fixes N :: "'b measure"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "space N = {}" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (space M = {})"
inj_on_right_lang_PR: assumes "dfa.states M = accessible" shows "inj_on (dfa.right_lang (nfa.Power_dfa (Reverse_nfa M))) (dfa.states (nfa.Power_dfa (Reverse_nfa M)))"
\<pi>_inverse_exists: shows "\<exists>na. \<pi> (real na) = real n"
distinct_2: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x23 :: "('d, 'e) ctxt"   and x24 :: "('d, 'e) Term.term list" shows "More x21 x22 x23 x24 \<noteq> \<box>"
cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "O[F](f) = O[F](g)"
P: shows "P \<in> Units (ring_mat TYPE('a) n b)"
size_list_conv_sum_list: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> nat"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "size_list f xs = sum_list (map f xs) + length xs"
while_never: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and u :: "char list \<Rightarrow> int"   and c' :: "com" assumes "(c, s) \<Rightarrow> u" shows "c \<noteq> WHILE Bc True DO c'"
numsub_def: fixes s :: "MIR.num"   and t :: "MIR.num" shows "numsub s t \<equiv> if s = t then C 0 else numadd (s, numneg t)"
f21_def: shows "f21 = snd (split_face f' v a ws)"
acute_conga__acute: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.Acute Bet Cong D E F"
current_memory: fixes \<C> :: "t_context"   and n :: "nat" assumes "memory \<C> = Some n" shows "\<C> \<turnstile> [Current_memory] : [] _> [T_i32]"
set_cases: fixes e :: "'d"   and a :: "'d strand_label" assumes "e \<in> set_strand_label a" obtains "a = ln e"
IArray_sub: fixes as :: "'c iarray" shows "(!!) as = (!) (IArray.list_of as)"
ms_1: shows "G ` I \<subseteq> Pow \<Omega>"
scale_left_commute: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b"   and x :: "'c" assumes "Modules.module scale" shows "scale a (scale b x) = scale b (scale a x)"
alw_relatesD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a stream" assumes "alw (relates P) s" shows "P (shd s) (shd (stl s))"
gt_irrefl: fixes s :: "('s, 'v) tm" assumes "wary s" shows "\<not> s >\<^sub>t s"
into_from_WW: fixes ns :: "nat list" assumes "strict_sorted ns" shows "into_WW 0 (from_WW 0 ns) = ns"
assoc: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a + b + c = a + (b + c)"
get_map_map: fixes S :: "('a \<times> 'b) set"   and k :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "is_map S"   and "k \<in> fst ` S" shows "get_map ((\<lambda>(a, b). (a, f a b)) ` S) k = (case get_map S k of (a, b) \<Rightarrow> (a, f a b))"
assms_2: shows "zc \<noteq> \<infinity>\<^sub>h"
Int_Un_eq_4: fixes T :: "'a set"   and S :: "'a set" shows "T \<union> S \<inter> T = T"
simplex_real_1: fixes cs :: "constraint list"   and I :: "nat list" assumes "simplex cs = Inl I" shows "\<nexists>v. v \<Turnstile>\<^sub>r\<^sub>c\<^sub>s set cs"
ts_i: shows "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"
cases: fixes P :: "'a prog"   and a1 :: "String.literal"   and a2 :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" assumes "P \<turnstile> a1 sees_methods a2" obtains D :: "String.literal" and fs :: "fdecl list" and ms :: "'a option mdecl list" and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" where "a1 = Object" and "a2 = Mm__" and "class P Object = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "Mm__ = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms__"   | C :: "String.literal" and D :: "String.literal" and fs :: "fdecl list" and ms :: "'a option mdecl list" and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" and Mm' :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option" where "a1 = C" and "a2 = Mm'" and "class P C = \<lfloor>(D, fs, ms)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D sees_methods Mm" and "Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)"
Cons_path_4: shows "\<forall>x\<in>set (tl (m'' # ms)). x \<noteq> n"
C2_def: shows "C2 = \<lparr>carrier = UNIV, mult = (=), one = True\<rparr>"
dropWhile_nth: fixes j :: "nat"   and P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "j < length (dropWhile P xs)" shows "dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
eint_enumeration_def: shows "eint_enumeration = int_option_enumeration \<circ> Rep_eint"
leq_imp_subseteq: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "M \<subseteq># N" shows "set_mset M \<subseteq> set_mset N"
R\<^sub>N_alt_def: fixes p :: "real"   and n :: "nat"   and u :: "nat"   and l :: "nat" shows "random_skip_list.R\<^sub>N p n u l = map_pmf (\<lambda>f. steps {..<n} f 0 u l) (random_skip_list.SL\<^sub>N p n)"
SUP_upper: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A" shows "f i \<le> \<Squnion> (f ` A)"
wf_tm_from_abacus: fixes tp :: "(action \<times> nat) list"   and ap :: "abc_inst list"   and n :: "nat" assumes "tp = tm_of ap" shows "tm_wf0 (tp @ shift (mopup n) (length tp div 2))"
Lang_rtrancl_eq: fixes w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" shows "(w \<in> Lang G S) = [Inl S] -G\<rightarrow>\<^sup>* map Inr w"
stutter_equiv_element_left: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<sigma> \<approx> \<tau>" obtains m :: "nat" where "\<sigma> m = \<tau> n" and "\<forall>j<m. \<exists>k<n. \<sigma> j = \<tau> k"
app_sterm_def: fixes t :: "sterm"   and u :: "sterm" shows "app t u = t $\<^sub>s u"
eq_box: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(box a b = box c d) = (box a b = {} \<and> box c d = {} \<or> a = c \<and> b = d)"
le: shows "\<forall>\<^sub>F n in F. X n \<le> C"
vcg_all_lift: fixes coms :: "'b \<Rightarrow> ('c, 'd, 'e, 'f) com"   and p :: "'b"   and aft :: "'f \<Rightarrow> 'd set"   and P :: "'a \<Rightarrow> ('c, 'd, 'b, 'e, 'f) system_state \<Rightarrow> bool"   and c :: "('c, 'd, 'e, 'f) com"   and Q :: "'a \<Rightarrow> ('c, 'd, 'b, 'e, 'f) system_state \<Rightarrow> bool" assumes "\<And>x. valid_syn coms p aft (P x) c (Q x)" shows "valid_syn coms p aft (\<lambda>s. \<forall>x. P x s) c (\<lambda>s. \<forall>x. Q x s)"
simple_function_def: fixes M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" shows "simple_function M g = (finite (g ` space M) \<and> (\<forall>x\<in>g ` space M. g -` {x} \<inter> space M \<in> sets M))"
sum_of_squares_nat_bound: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x\<^sup>2 + y\<^sup>2 = n" shows "x \<le> n"
uint_word_of_int_eq: fixes k :: "int" shows "uint (word_of_int k) = take_bit LENGTH('c) k"
new_character_data_ptr: shows "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"
ostep_invariant_def: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool" shows "(A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. \<forall>a s'. (s, a, s') \<in> automaton.trans A \<and> S (fst s) (fst s') a \<longrightarrow> P (s, a, s'))"
point_injective: fixes x :: "'a" assumes "arc x" shows "coreflexive (x\<^sup>T * top * x)"
bind_assoc: fixes x :: "'m"   and f :: "int \<Rightarrow> 'm"   and g :: "int \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
med23s_def: fixes t :: "l3_state" shows "med23s t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t, signalsInit2 = signalsInit2 t, signalsResp2 = signalsResp2 t, chan = local.abs (ik t), bad = l3_state.bad t\<rparr>"
test_bit_bin': fixes w :: "'b word"   and n :: "nat" shows "w !! n = (n < size w \<and> bit (uint w) n)"
bi_unique_union_l: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e set"   and a' :: "'f set"   and b :: "'e set"   and b' :: "'f set"   and xl :: "'e set" assumes "bi_unique T"   and "rel_set T a a'"   and "rel_set T b b'"   and "rel_set T xl (a' \<union> b')" shows "a \<union> b = xl"
aux: fixes a :: "real"   and a' :: "real"   and b :: "real"   and b' :: "real" assumes "a \<le> a'"   and "b \<le> b'" shows "a + b \<le> a' + b'"
int2: shows "integrable (measure_spmf coin_spmf) (\<lambda>x. spmf (R1_14_OT (x \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), x \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), x \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), x \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) (b1, b2) \<bind> (\<lambda>view. A view x)) True)"
not_in_vars_coeff0: fixes x :: "nat"   and p :: "'a mpoly" assumes "x \<notin> vars p" shows "MPoly_Type.coeff p (monomial (Suc 0) x) = (0::'a)"
assms_1: shows "AE x in M. f (T x) \<le> f x"
simps_3: fixes p :: "perm"   and \<phi> :: "'f" shows "p \<bullet> tPred \<phi> = tPred (p \<bullet> \<phi>)"
fbox_star_unfold: fixes z :: "'a"   and x :: "'a" shows " |(1::'a)] z \<cdot>  |x]  |x\<^sup>\<star>] z =  |x\<^sup>\<star>] z"
simps_2: fixes p :: "('a \<times> 'b \<times> 'a) list"   and w :: "'b" shows "mpath' (Some (p, w)) = Some p"
nextVertex_in_face': fixes f :: "face"   and v :: "nat" assumes "vertices f \<noteq> []" shows "f \<bullet> v \<in> \<V> f"
simps_14: fixes \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula" shows "Formula.future_bounded (formula.Since \<phi> I \<psi>) = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi>)"
get_ancestors_remains_not_in_ancestors: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ancestors' :: "(_) object_ptr list"   and node :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "Shadow_DOM.heap_is_wellformed h'"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "h' \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors'"   and "\<And>p children children'. \<lbrakk>h \<turnstile> Shadow_DOM.get_child_nodes p \<rightarrow>\<^sub>r children; h' \<turnstile> Shadow_DOM.get_child_nodes p \<rightarrow>\<^sub>r children'\<rbrakk> \<Longrightarrow> set children' \<subseteq> set children"   and "node \<notin> set ancestors"   and "object_ptr_kinds h = object_ptr_kinds h'"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.type_wf h'" shows "node \<notin> set ancestors'"
imp_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b"   and A :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "(imp_PL p q \<in> Cn A) = (q \<in> Cn (A \<union> {p}))"
a0i_def: shows "a0i \<equiv> \<lparr>init = a0i_init, trans = a0i_trans, obs = id\<rparr>"
arcs_ends_def: fixes G :: "('a, 'b) pre_digraph" shows "arcs_ends G \<equiv> arc_to_ends G ` arcs G"
simps_6: fixes vs :: "'b list"   and bs :: "'b list"   and a :: "tm" shows "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> (0::'b))"
sel_15: fixes x71 :: "'a expr"   and x72 :: "call_kind"   and x73 :: "'a call_expr" shows "source (Call x71 x72 x73) = x71"
comp_arr_inv': fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.comp f (VoV.inv f) = VoV.cod f"
permutes_bij': fixes A :: "'d set"   and a_to_b :: "'d \<Rightarrow> 'e"   and B :: "'e set"   and b_to_a :: "'e \<Rightarrow> 'd" assumes "\<And>a. a \<in> A \<Longrightarrow> a_to_b a \<in> B"   and "\<And>b. b \<in> B \<Longrightarrow> b_to_a b \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> b_to_a (a_to_b a) = a"   and "\<And>b. b \<in> B \<Longrightarrow> a_to_b (b_to_a b) = b" shows "{p. p permutes A} = (\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a) ` {p. p permutes B}"
redundant_inference_def: fixes C :: "'a eclause"   and S :: "'a eclause set"   and P :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "redundant_inference C S P \<sigma> = (\<exists>S'\<subseteq>instances S. set_entails_clause (clset_instances S') (cl_ecl C) \<and> (\<forall>x\<in>S'. subterms_inclusion (subst_set (trms_ecl (fst x)) (snd x)) (trms_ecl C)) \<and> (\<forall>x\<in>S'. \<exists>D'\<in>P. ((fst x, snd x), D', \<sigma>) \<in> ecl_ord))"
tm_comp_steps: fixes c :: "nat \<times> cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and n :: "nat"   and B :: "(action \<times> nat) list" assumes "\<not> is_final (steps0 c A n)" shows "steps0 c (A |+| B) n = steps0 c A n"
distinct_21: fixes x3 :: "'a formula"   and x51 :: "'a formula"   and x52 :: "'a formula" shows "\<^bold>\<not> x3 \<noteq> x51 \<^bold>\<or> x52"
null_set_Diff: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "B - A \<in> null_sets M"
bdd_aboveI: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x M" shows "preorder.bdd_above less_eq A"
obind_NF_wp: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and Q :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a option" assumes "\<And>r. ovalidNF (R r) (g r) Q"   and "ovalidNF P f R" shows "ovalidNF P (f |>> g) Q"
j_1: shows "psi j = (\<lambda>x. if x = Suc n\<^sub>0 then Some 1 else Some 0)"
r_div_eqn: fixes a :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<in> carrier G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<cdot> a = y \<cdot> a" shows "x = y"
H_edge_succ_tail_eqD: fixes a :: "'b"   and b :: "'b" assumes "edge_succ HM a = b" shows "tail H a = tail H b"
xs_eq: shows "xs = map (the \<circ> (\<lambda>b. bound_intersect_2d_ud p (inner2_aform Z n b) g)) Basis_list"
of_nat_closed: fixes n :: "nat" shows "of_nat n \<in> A"
pair_map_pmf2: fixes A :: "'d pmf"   and f :: "'f \<Rightarrow> 'e"   and B :: "'f pmf" shows "pair_pmf A (map_pmf f B) = map_pmf (apsnd f) (pair_pmf A B)"
lipschitz_on_Pair: fixes L :: "real"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and M :: "real"   and g :: "'b \<Rightarrow> 'd" assumes "L-lipschitz_on A f"   and "M-lipschitz_on A g" shows "(sqrt (L\<^sup>2 + M\<^sup>2))-lipschitz_on A (\<lambda>a. (f a, g a))"
sdom_not_mem: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd set" assumes "a \<notin> sdom m" shows "m a = {}"
maximal_Extend: fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b" assumes "infinite UNIV"   and "finite A"   and "consistent A S"   and "finite (\<Union> (block_nominals ` S))"   and "surj f" shows "maximal A (Extend A S f)"
h7: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set (get_signs_R (reduce_system_R p (qs, matrix, subsets, signs)))"
block_unique: fixes S :: "'c set"   and P :: "'c set set"   and a :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "partition S P"   and "a \<in> A"   and "A \<in> P"   and "a \<in> B"   and "B \<in> P" shows "A = B"
models_def: fixes h :: "pheap"   and P :: "assn" shows "(h \<Turnstile> P) = aseval (Rep_assn P) h"
swap: fixes p :: "perm"   and a :: "atom"   and b :: "atom" assumes "P p"   and "supp p \<subseteq> S"   and "a \<in> S"   and "b \<in> S"   and "a \<noteq> b"   and "sort_of a = sort_of b" shows "P ((a \<rightleftharpoons> b) + p)"
bisect_def: shows "bisect = rec_nat (a, b) (\<lambda>n (x, y). if P x ((x + y) / 2) then ((x + y) / 2, y) else (x, (x + y) / 2))"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"