hcju/isabelleps huggingface dataset:
domID: fixes p :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "p \<noteq> [] \<and> x \<in> dom (C (list2FWpolicy p))" shows "x \<in> dom (C (list2FWpolicy (insertDenies p)))"
invariantI: fixes ES :: "('b list set \<times> 'b trm set) set" assumes "soundness ES"   and "finite ES"   and "distinctness ES"   and "ardenable_all ES"   and "finite_rhs ES"   and "validity ES" shows "invariant ES"
sums_def': fixes f :: "nat \<Rightarrow> 'c"   and s :: "'c" shows "f sums s = (\<lambda>n. sum f {0..n}) \<longlonglongrightarrow> s"
Int_left_commute: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<inter> (B \<inter> C) = B \<inter> (A \<inter> C)"
tdev_msum_pdevs: fixes f :: "'b pdevs"   and n :: "nat"   and g :: "'b pdevs" assumes "degree f \<le> n" shows "tdev (msum_pdevs n f g) = tdev f + tdev g"
ground_elig: shows "ground_resolution_with_selection.eligible S_G As (D + negs (mset As))"
has_integral_affinity_iff: fixes m :: "real"   and f :: "'a \<Rightarrow> 'b"   and c :: "'a"   and I :: "'b"   and a :: "'a"   and b :: "'a" assumes "0 < m" shows "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral I /\<^sub>R m ^ DIM('a)) (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) = (f has_integral I) (cbox a b)"
LID\<^sub>SI_2: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and l' :: "'b"   and v :: "('a, 'b, 'c) val"   and l :: "'b" assumes "\<sigma> l' = Some v"   and "l \<in> LID\<^sub>V v" shows "l \<in> LID\<^sub>S \<sigma>"
vec_of_basis_enum_times: fixes \<psi> :: "'a"   and \<phi> :: "'a" shows "vec_of_basis_enum (\<psi> * \<phi>) = vec_of_list [vec_of_basis_enum \<psi> $ 0 * vec_of_basis_enum \<phi> $ 0]"
of_int_of_nat_eq: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
c_3: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Equality stateful_strand_step" shows "\<langle>the_check stateful_strand_step: the_lhs stateful_strand_step \<doteq> the_rhs stateful_strand_step\<rangle> = stateful_strand_step"
extensional'I: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 's" assumes "\<And>x. x \<notin> A \<Longrightarrow> f x = unity" shows "f \<in> extensional' A"
first_countable_basisE: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.first_countable_topology open" obtains \<A> :: "'b set set" where "countable \<A>" and "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" and "\<And>A. A \<in> \<A> \<Longrightarrow> open A" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
in_sstart: fixes s :: "'a stream"   and S :: "'a set"   and x :: "'a"   and xs :: "'a list" shows "(s \<in> sstart S (x # xs)) = (shd s = x \<and> stl s \<in> sstart S xs)"
lleq_antisym: fixes x :: "char list"   and y :: "char list" assumes "lleq_charlist x y"   and "lleq_charlist y x" shows "x = y"
seq_char: fixes g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2"   and f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" shows "A2_BxA2.seq g f = (A2_B.seq (fst g) (fst f) \<and> A2.seq (snd g) (snd f))"
reduced_GB_is_auto_reduced_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m" shows "punit.is_auto_reduced (punit.reduced_GB F)"
a: shows "\<forall>j. mod_type_class.to_nat j < CARD('a) \<longrightarrow> P j"
that: fixes p' :: "pr_operation' list" assumes "((f, l), p', f', l') \<in> trcl pr_algo_lts'"   and "p = map project_operation p'" shows "thesis"
mreq_end: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and x :: "'e"   and b :: "'d list"   and r :: "'d"   and c :: "'d list"   and a :: "'d" assumes "applied_rule_rev C x b = \<lfloor>r\<rfloor>"   and "applied_rule_rev C x c = \<lfloor>r\<rfloor>" shows "applied_rule_rev C x (a # b) = applied_rule_rev C x (a # c)"
pirreducibleE_2: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" assumes "domain R"   and "subring K R"   and "p \<in> carrier (K [X]\<^bsub>R\<^esub>)"   and "pirreducible\<^bsub>R\<^esub> K p" shows "p \<notin> Units (K [X]\<^bsub>R\<^esub>)"
NSDERIV_inverse: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "NSDERIV inverse x :> - (inverse x ^ Suc (Suc 0))"
meld_mset: fixes q :: "('c, 'd) BinomialTree list"   and q' :: "('c, 'd) BinomialTree list" assumes "queue_invar q"   and "queue_invar q'" shows "queue_to_multiset (meld q q') = queue_to_multiset q + queue_to_multiset q'"
subst_img_alt_def: fixes s :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subst_range s = {t. \<exists>v. s v = t \<and> t \<noteq> Var v}"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes " x1 \<longrightarrow>\<^sub>1\<^sup>* x2" shows " pi \<bullet> x1 \<longrightarrow>\<^sub>1\<^sup>* pi \<bullet> x2"
BIT_pairwise': fixes qs :: "'a list"   and init :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set qs \<subseteq> set init"   and "(x, y) \<in> {(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}"   and "x \<noteq> y"   and "distinct init" shows "Pbefore_in x y BIT qs init = Pbefore_in x y BIT (Lxy qs {x, y}) (Lxy init {x, y})"
sup_compl_top: fixes x :: "'a" shows "x \<squnion> - x = \<top>"
tm_comp_fetch_in_A: fixes A :: "(action \<times> nat) list"   and s :: "nat"   and x :: "cell"   and a :: "action"   and B :: "(action \<times> nat) list" assumes "fetch A s x = (a, 0)"   and "s \<le> length A div 2"   and "s \<noteq> 0" shows "fetch (A |+| B) s x = (a, Suc (length A div 2))"
unit_circle_fix_moebius_inv: fixes M :: "moebius" assumes "unit_circle_fix M" shows "unit_circle_fix (moebius_inv M)"
\<O>b_def: shows "\<O>b = (\<lambda>\<pp>. undefined)"
assms_1: shows "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright>"
convex_closure_rel_interior_inter: fixes I :: "'a set set" assumes "\<forall>S\<in>I. convex S"   and "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}" shows "\<Inter> {closure S |S. S \<in> I} \<subseteq> closure (\<Inter> {rel_interior S |S. S \<in> I})"
exL_exMap_lem: fixes f :: "Label \<Rightarrow> sterm option"   and lz :: "Label \<Rightarrow> sterm option"   and f' :: "Label \<Rightarrow> sterm option" assumes "dom f = dom lz"   and "dom f' = dom f" shows "\<forall>L1 L2. finite L1 \<longrightarrow> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L1 \<and> p \<notin> L1 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. (the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> (\<forall>z. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => z \<longrightarrow> (\<exists>u. t => u \<and> z => u))) \<and> the (f' l) = \<sigma>[s,p] t)) \<longrightarrow> finite L2 \<longrightarrow> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L2 \<and> p \<notin> L2 \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lz l) = \<sigma>[s,p] t)) \<longrightarrow> (\<exists>L'. finite L' \<and> (\<exists>lu. dom lu = dom f \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (f' l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom f. body (the (f' l))) \<and> (\<forall>l\<in>dom f. \<forall>s p. s \<notin> L' \<and> p \<notin> L' \<and> s \<noteq> p \<longrightarrow> (\<exists>t. the (lz l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> the (lu l) = \<sigma>[s,p] t)) \<and> (\<forall>l\<in>dom f. body (the (lz l)))))"
finite_acyclic_wf: fixes r :: "('b \<times> 'b) set" assumes "finite r"   and "acyclic r" shows "wf r"
all_eql: 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 l1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l2 :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TarskiLen Cong A B l1"   and "Tarski_neutral_dimensionless.TarskiLen Cong A B l2" shows "l1 = l2"
degree_Q: shows "Polynomial.degree (of_int_poly Q) = card Roots'"
iso_map\<^sub>1_ide: fixes f :: "'c" assumes "C.ide f" shows "C.iso (\<epsilon>'.map\<^sub>1 f)"
dvd_dvd_imp_unit_mult: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.idom times one plus zero minus uminus"   and "dvd.dvd times x y"   and "dvd.dvd times y x" shows "\<exists>z. dvd.dvd times z one \<and> y = times x z"
double_dual_iso_one: shows "G'.double_dual_iso \<one>\<^bsub>Characters G\<^esub> = principal_char (Characters (Characters G))"
jge: shows "\<forall>j'<k. k + n cd\<^bsup>\<pi>\<^esup>\<rightarrow> j' \<longrightarrow> j' \<le> j"
infinite_Collection: fixes X :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "infinite X"   and "\<forall>b\<in>X. P b" shows "infinite {b \<in> X. P b}"
eq: shows "x = 2 * u * v \<and> y = u\<^sup>2 - v\<^sup>2 \<or> x = u\<^sup>2 - v\<^sup>2 \<and> y = 2 * u * v"
vars_setsum: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c mpoly" assumes "finite S" shows "vars (sum f S) \<subseteq> (\<Union>m\<in>S. vars (f m))"
set_relator_eq_onp: fixes P :: "'d \<Rightarrow> bool" shows "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
trm_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and trm :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "elementary_category_with_terminal_object C one trm"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (trm a) a one"
singleton_orbit_eq_fixed_point: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set"   and m :: "'e" assumes "group_action G \<phi> M"   and "m \<in> M" shows "(card (group_action.orbit G \<phi> M m) = 1) = (m \<in> group_action.fixed_points G \<phi> M)"
iterator_to_list_unfold: fixes x :: "'a" shows "iterator_to_list x = (if cnt x then x # iterator_to_list (nxt x) else [])"
range_oDeriv: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "range (oDeriv F) = {x. F x = x}"
zero_less_of_rat_iff: fixes r :: "rat" shows "((0::'b) < of_rat r) = (0 < r)"
succsr_def: shows "succsr \<equiv> {(x, y). y \<in> set (map (\<delta> x) \<Sigma>)}"
Min_le: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "x \<in> A" shows "linordered_monoid.Min A \<preceq> x"
l2_refines_l1: shows "refines (R12s \<inter> reach l1 \<times> (l2_inv1 \<inter> l2_inv2 \<inter> l2_inv3 \<inter> l2_inv4 \<inter> l2_inv5 \<inter> l2_inv6 \<inter> l2_inv7 \<inter> l2_inv8)) med12s l1 l2"
integrable_cong: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integrable M f = integrable N g"
gallery_betw_map: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "domain.gallery (C # Cs @ [D])" shows "codomain.gallery (f ` C # f \<Turnstile> Cs @ [f ` D])"
comp_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<rightarrow>\<^sub>D c\<guillemotright>"
proper_dia: fixes p :: "('b, 'c) fm"   and q :: "('b, 'c) fm" assumes "proper_dia p = Some q" shows "p = (\<^bold>\<diamond> q) \<and> (\<nexists>a. q = Nom a)"
alphaAbs_preserves_qGoodAbs1: fixes A :: "('c, 'd, 'varSort, 'var, 'e) qAbs"   and A' :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "A $= A'"   and "qGoodAbs A" shows "qGoodAbs A'"
compat_def: fixes I :: "'c set"   and theta :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "Resumption_Based.compat I theta f \<equiv> \<forall>i j. {i, j} \<subseteq> I \<and> i \<noteq> j \<longrightarrow> (f i, f j) \<in> theta"
leg_simps_1: shows "C.arr Dom_\<rho>.leg0"
idNMT: fixes p :: "('b, 'c) Combinators list" assumes "p \<noteq> []" shows "insertDenies p \<noteq> []"
set_homI: fixes A :: "'b set"   and U :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "A \<subseteq> U"   and "B \<subseteq> U"   and "f \<in> A \<rightarrow> B"   and "f \<in> extensional A" shows "\<lparr>set_dom = A, set_func = f, set_cod = B\<rparr> \<in> Hom\<^bsub>set_cat U\<^esub> A B"
zg_2: shows "z = (interval_bij (a, b) (- 1, 1) \<circ> g) zg"
length_chop: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n" shows "length (chop n xs) = nat \<lceil>real (length xs) / real n\<rceil>"
abs_update_rehash_map_of: fixes l :: "('a \<times> 'b) list list"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l" shows "map_of (concat (abs_update k v (ls_rehash l))) = map_of (concat l)(k \<mapsto> v)"
cfcomp1: fixes f :: "'g \<rightarrow> 'f"   and g :: "'e \<rightarrow> 'g" shows "f oo g = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
in_ofail: fixes s :: "'e"   and v :: "'d" shows "ofail s \<noteq> Some v"
Ghyp: shows "\<forall>\<tau>. \<tau> \<in> U (X t\<^sub>0) \<and> \<tau> \<le> t \<longrightarrow> G (X \<tau>)"
Domain_mono: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "r \<subseteq> s" shows "Domain r \<subseteq> Domain s"
asym: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<prec> a"   and "a \<prec> b" shows "False"
det_addcol: fixes l :: "nat"   and n :: "nat"   and k :: "nat"   and A :: "'b mat"   and a :: "'b" assumes "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (addcol a k l A) = det A"
convex_on_dist: fixes S :: "'a set"   and a :: "'a" shows "convex_on S (dist a)"
basis_def: fixes X :: "'a set" shows "\<E>.basis X = (indep_in carrier X \<and> (\<forall>x\<in>carrier - X. \<not> indep_in carrier (insert x X)))"
neval_0: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and x :: "('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)"   and y :: "('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)" assumes "sifum_security_init dma \<C>_vars eval"   and "x = y" shows "sifum_security_init.neval eval x 0 y"
cut_same_shift_sconst: fixes w :: "'b list"   and x :: "'b" shows "\<exists>n. w = cut_same x (w @- sconst x) @ replicate n x"
hd_le_sum_list: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "\<forall>i<length xs. (0::'a) \<le> xs ! i" shows "hd xs \<le> sum_list xs"
cring_lcf_of_sub_in_ltrm: fixes R :: "('c, 'd) ring_scheme"   and g :: "nat \<Rightarrow> 'c"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_cring R"   and "g \<in> carrier (UP R)"   and "f \<in> carrier (UP R)"   and "deg R f = n"   and "0 < deg R g"   and "f (deg R f) \<otimes>\<^bsub>R\<^esub> g (deg R g) [^]\<^bsub>R\<^esub> n \<noteq> \<zero>\<^bsub>R\<^esub>" shows "Cring_Poly.compose R (monom (UP R) (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom (UP R) (f (deg R f)) (deg R f)) g)) = f (deg R f) \<otimes>\<^bsub>R\<^esub> g (deg R g) [^]\<^bsub>R\<^esub> n"
other: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and t :: "'b net_state"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U"   and "U \<sigma> \<sigma>'"   and "P \<sigma> s t" shows "P \<sigma>' s t"
set_bounded_supp: fixes S :: "atom set"   and X :: "'a set" assumes "finite S"   and "\<And>x. x \<in> X \<Longrightarrow> supp x \<subseteq> S" shows "supp X \<subseteq> S"
Atom_inject: fixes x :: "'a + nat"   and y :: "'a + nat" assumes "Atom x = Atom y" shows "x = y"
mHom_compos: fixes L :: "('e, 'b, 'f) Module_scheme"   and N :: "('g, 'b, 'h) Module_scheme"   and f :: "'e \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'g" assumes "R module L"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N" shows "compos L g f \<in> mHom R L N"
iso_\<i>: fixes a :: "'c" assumes "C.obj a" shows "D.iso (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a))"
embedding_map_into_euclideanreal: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> real" assumes "path_connected_space X" shows "embedding_map X euclideanreal f = (continuous_map X euclideanreal f \<and> inj_on f (topspace X))"
has_sort_Tv_imp_sort_leq: fixes oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and idn :: "variable"   and S :: "String.literal set"   and S' :: "String.literal set" assumes "has_sort oss (Tv idn S) S'" shows "sort_leq (subclass oss) S S'"
q01_1: shows "q 0 = a - complex_of_real d"
dim_image_eq: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and S :: "'a set" assumes "\<forall>x. x \<in> S \<longrightarrow> f x \<in> dual_space S"   and "S \<subseteq> S"   and "linear_on S (dual_space S) (*\<^sub>R) (*\<^sub>R) f"   and "inj_on f (span S)" shows "dual.dim (f ` S) = S.dim S"
l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog" shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr set_shadow_root set_mode attach_shadow_root get_tag_name get_shadow_root \<equiv> known_ptr = ShadowRootClass.known_ptr \<and> attach_shadow_root = l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_attach_shadow_root set_shadow_root set_mode get_tag_name get_shadow_root"
r0: shows "r (Suc k) ((a $ 0) ^ Suc k) = a $ 0"
openin_open_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin X S" shows "openin (subtopology X S) T = (openin X T \<and> T \<subseteq> S)"
all_info_hyp: shows "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set (snd (snd (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))))"
less_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "strict_prefix x y"   and "strict_prefix y z" shows "strict_prefix x z"
function_ring_car_eqI: fixes R :: "('a, 'b) ring_scheme"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)"   and "\<And>a. a \<in> S \<Longrightarrow> f a = g a" shows "f = g"
snd_of_exps_zero: fixes a :: "nat"   and b :: "nat" shows "snd (rep_nat_term (Term_Order.of_exps a b 0)) = 0"
in_set_butlast_appendI: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys)" shows "x \<in> set (butlast (xs @ ys))"
Lim_def: fixes f :: "'g \<Rightarrow> ('f, 'g) Old_Datatype.node set" shows "Old_Datatype.Lim f \<equiv> \<Union> {z. \<exists>x. z = Push_Node (Inl x) ` f x}"
bin_rest_numeral_simps_7: fixes w :: "num" shows "- numeral (num.Bit0 w) div 2 = - numeral w"
gen_tlength_def: fixes n :: "nat"   and xs :: "('a, 'b) tllist" shows "gen_tlength n xs = enat n + tlength xs"
prj_simps_5: shows "C.dom \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 = HH\<mu>H\<nu>\<pi>\<rho>.chine"
simps_1: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun [] c = []"
bit_c''2: fixes x :: "'a"   and y :: "'a"   and x0 :: "'a"   and y0 :: "'a"   and s :: "('a list \<times> bool list \<times> 'a list) pmf"   and qs :: "'a list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "BIT_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs - 1) - 5 / 10"
prems: shows "\<forall>f. \<forall>a\<in>set (P f). f = a"
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> zg" shows "X \<in> knows Spy evs"
mult_commute_abs: fixes c :: "'c" shows "(\<lambda>x. x * c) = (*) c"
tmod3: fixes ds :: "dec list"   and ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and mn :: "char list"   and specs :: "spec list option" assumes "\<not> no_dup_types ds" shows "evaluate_top ck env s1 (Tmod mn specs ds) (s1, Rerr (Rabort Rtype_error))"
simps_9: fixes x1 :: "'d" shows "Stuck \<noteq> Normal x1"
root_nonzero: fixes n :: "nat" shows "FFT.root n \<noteq> 0"
c'_def: shows "c' \<equiv> complex_of_real (root n (cmod c)) * cis (arg c / real n)"
simps_4: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and project :: "'b \<Rightarrow> 'b"   and lookup :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and b :: "'c" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n (Atom b) = {[x] |x. lookup b x \<and> x \<in> \<Sigma> n}"
bind_return_pmf: fixes x :: "'i"   and f :: "'i \<Rightarrow> 'h pmf" shows "return_pmf x \<bind> f = f x"
left_idem: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<inter># (a \<inter># b) = a \<inter># b"
fmupd_fmdrop_id: fixes k :: "'c"   and N' :: "('c, 'd) fmap" assumes "k |\<in>| fmdom N'" shows "fmupd k (the (fmlookup N' k)) (fmdrop k N') = N'"
dom_lm_\<alpha>_Assoc_List_set: fixes v :: "('b, 'c) assoc_list" shows "dom (lm.\<alpha> v) = fst ` Assoc_List.set v"
p_residue_padic_int: fixes x :: "nat \<Rightarrow> int"   and k :: "nat"   and m :: "nat" assumes "x \<in> carrier Zp"   and "k \<le> m" shows "p_residue k (x m) = x k"
closed_open: fixes S :: "'a set" shows "closed S = open (- S)"
\<psi>\<^sub>0_to_\<psi>\<^sub>1: shows "Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, 1 / sqrt 2]]) \<Otimes> Tensor.mat_of_cols_list 2 (map (map complex_of_real) [[1 / sqrt 2, - 1 / sqrt 2]]) = \<psi>\<^sub>1"
reduced_row_echelon_form_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "reduced_row_echelon_form A = reduced_row_echelon_form_upt_k A (ncols A)"
set_qsort: fixes xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "set (qsort xs l) = set xs"
keys_map_scale_subset: fixes k :: "'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (k \<cdot> t) \<subseteq> keys t"
Trans: shows "\<Psi> \<rhd> P \<longmapsto> Rs"
Q_\<alpha>_delete: fixes Q :: "'q"   and u :: "'v" assumes "Q_invar Q" shows "Q_\<alpha> (Q_delete u Q) = (Q_\<alpha> Q)(u := \<infinity>)"
assms_1: shows "ipPartition (set (map wordinterval_to_set (extract_IPSets_generic0 dst rs))) A"
SYM: fixes X :: "freeExp"   and Y :: "freeExp" assumes "X \<sim> Y" shows "Y \<sim> X"