hcju/isabelleps huggingface dataset:
sublinearD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and a :: "real"   and b :: "real"   and c :: "real"   and s :: "'a" assumes "sublinear t"   and "sound P"   and "sound Q"   and "0 \<le> a"   and "0 \<le> b"   and "0 \<le> c" shows "a * t P s + b * t Q s \<ominus> c \<le> t (\<lambda>s'. a * P s' + b * Q s' \<ominus> c) s"
Abs_inject: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "type_definition Rep Abs A"   and "x \<in> A"   and "y \<in> A" shows "(Abs x = Abs y) = (x = y)"
nextmem_exists_and_unique: fixes c :: "('e, 'c) MWLsCom"   and m :: "'c \<Rightarrow> 'd"   and E :: "'e \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and BMap :: "'d \<Rightarrow> bool" shows "\<exists>m' p \<alpha>. ((c, m), \<alpha>, p, m') \<in> MWLs_semantics.MWLsSteps_det E BMap \<and> (\<forall>m''. (\<exists>p \<alpha>. ((c, m), \<alpha>, p, m'') \<in> MWLs_semantics.MWLsSteps_det E BMap) \<longrightarrow> m'' = m')"
assms_1: shows "integrable M (\<lambda>x. f (T x) * g x)"
compl_fam_iff_relations_triv: fixes gs :: "'a set" assumes "finite gs"   and "gs \<subseteq> carrier G"   and "\<forall>g\<in>gs. ord g = 0" shows "(relations gs = {\<lambda>_\<in>gs. 0}) = compl_fam (\<lambda>g. generate G {g}) gs"
Rep_ttree_inject: fixes x :: "'b ttree"   and y :: "'b ttree" shows "(Rep_ttree x = Rep_ttree y) = (x = y)"
that: fixes A\<^sub>P' :: "name list"   and \<Psi>\<^sub>P' :: "'b"   and \<Psi>' :: "'b" assumes "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "\<Psi>\<^sub>P \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'"   and "A\<^sub>P' \<sharp>* P'"   and "A\<^sub>P' \<sharp>* C"   and "distinct A\<^sub>P'" shows "thesis"
ext: fixes g :: "'s"   and f :: "'s" assumes "g \<cdot> f \<noteq> S.null" shows "S.seq g f"
image_mset_subseteq_mono: fixes A :: "'a multiset"   and B :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "A \<subseteq># B" shows "f `# A \<subseteq># f `# B"
cont: fixes x :: "'a" assumes "x \<in> X" shows "continuous_on UNIV (\<lambda>t. f x)"
fin: fixes F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a"   and fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"   and a :: "int"   and N :: "nat"   and C :: "'a"   and Y :: "real set" assumes "euler_maclaurin_int F f fs a N C Y" shows "finite Y"
simps_2: fixes S :: "'b fm set"   and C :: "'b fm set set"   and f :: "nat \<Rightarrow> 'b fm"   and n :: "nat" shows "extend S C f (Suc n) = (if extend S C f n \<union> {f n} \<in> C then extend S C f n \<union> {f n} else extend S C f n)"
simps_2: fixes G :: "prog"   and tn :: "qtname" shows "isrtype G (IfaceT tn) = is_iface G tn"
finite_abs_int_segment: fixes a :: "'c" shows "finite {k \<in> \<int>. \<bar>k\<bar> \<le> a}"
\<alpha>n_distinct: fixes g :: "'g" assumes "invar g" shows "distinct (\<alpha>n g)"
take_cols_def: fixes A :: "'b mat"   and inds :: "nat list" shows "take_cols A inds = mat_of_cols (dim_row A) (map ((!) (cols A)) (filter (\<lambda>y. y < dim_col A) inds))"
rtip: shows "rt ip = Some (dsn, dsk, f, hops, nhop, pre)"
deriv_closed: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "deriv f a \<in> carrier R"
test_bit_of_bl: fixes bl :: "bool list"   and n :: "nat" shows "of_bl bl !! n = (rev bl ! n \<and> n < LENGTH('a) \<and> n < length bl)"
zero_fun_apply: fixes x :: "'d" shows "0 x = (0::'c)"
axioms_4: fixes step1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and step2 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and final1 :: "'d \<Rightarrow> bool"   and final2 :: "'e \<Rightarrow> bool"   and load1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and load2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and order :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and match :: "'h \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and compile :: "'f \<Rightarrow> 'g option" assumes "compiler step1 step2 final1 final2 load1 load2 order match compile" shows "compiler_axioms load1 load2 match compile"
bool_list_fin: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and m :: "nat" assumes "source_code M b fi X enc dec" shows "finite {bl. length bl = m}"
H_def: shows "H = {x. coprime x p \<and> euler_liar (int x) p \<and> x \<in> {1..<p}}"
gC_def: fixes c :: "('fsym, 'psym) lit list" shows "gC c \<equiv> map gL c @ list (glitOfC c)"
mult_zero_left: fixes a :: "'a" shows "bot \<sqinter> a = bot"
fminus_finsert: fixes A :: "'b fset"   and a :: "'b"   and B :: "'b fset" shows "A |-| finsert a B = A |-| B |-| {|a|}"
det_def: fixes A :: "(('a, 'b) vec, 'b) vec" shows "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A $ i $ p i))"
Union_connected_components_of: fixes X :: "'a topology" shows "\<Union> (connected_components_of X) = topspace X"
natural_transformation_axioms: shows "natural_transformation A B F H \<tau>'\<tau>.map"
mult_is_0: fixes m :: "nat"   and n :: "nat" shows "(m * n = 0) = (m = 0 \<or> n = 0)"
map_sel_1: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c stream" shows "shd (smap f a) = f (shd a)"
some_distinguishes: fixes p :: "perm" shows "(SOME x. weak_formula x \<and> supp x \<subseteq> supp (p \<bullet> P) \<and> x distinguishes (p \<bullet> P) from (p \<bullet> Q)) distinguishes (p \<bullet> P) from (p \<bullet> Q)"
whole_ideal: fixes R :: "('h, 'i) Ring_scheme" assumes "Ring R" shows "ideal R (carrier R)"
hn_refine_preI: fixes \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and a :: "'b nres" assumes "\<And>h. h \<Turnstile> \<Gamma> \<Longrightarrow> hn_refine \<Gamma> c \<Gamma>' R a" shows "hn_refine \<Gamma> c \<Gamma>' R a"
simps_6: fixes I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "satisfies I1 I2 (ws1s.Ex1 x \<phi>) = (\<exists>n. satisfies (I1(x := n)) I2 \<phi>)"
braun_ssa_CFG_SSA_wf: shows "CFG_SSA_wf \<alpha>e \<alpha>n invar inEdges' Entry defs' uses' phis'"
ih1: assumes "s'' = 0" shows "steps0 (Suc (length A div 2), l, r) (A |+| B) stp = (0, l'', r'')"
a_pos: fixes i :: "'b" assumes "i \<in> S" shows "0 \<le> a i"
vdisjnt_vemptyI: fixes A :: "V"   and B :: "V" assumes "\<And>x. \<lbrakk>x \<in>\<^sub>\<circ> A; x \<in>\<^sub>\<circ> B\<rbrakk> \<Longrightarrow> False" shows "vdisjnt A B"
2: shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (unlabel Di) \<subseteq> X"
mult_left_one: fixes x :: "'c" shows "bounded_distrib_lattice_class.comp_inf.covector x"
not_less_eq_eq: fixes m :: "nat"   and n :: "nat" shows "(\<not> order.greater_eq n m) = order.greater_eq m (Suc n)"
hyps_2: shows "s = 0 \<or> Re (g s) = Re z"
union_fold_insert: fixes A :: "'a set"   and B :: "'a set" assumes "finite A" shows "A \<union> B = Finite_Set.fold insert B A"
circ_plus_sub: fixes x :: "'a" shows "x\<^sup>\<star> * x \<le> x\<^sup>+"
assms_1: shows "M \<turnstile>\<^sub>c Fun c []"
prop_make_vert_equiv_1: fixes n :: "nat" shows "rat_poly.row_length (make_vert_equiv n) = 2 ^ n"
continuity_lem_eq0: fixes r :: "real"   and p :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "r < p"   and "\<forall>x\<in>{r<..p}. a * x\<^sup>2 + b * x + c = 0" shows "a = 0 \<and> b = 0 \<and> c = 0"
lborel_integral_real_affine: fixes c :: "real"   and f :: "real \<Rightarrow> 'c"   and t :: "real" assumes "c \<noteq> 0" shows "integral\<^sup>L lborel f = \<bar>c\<bar> *\<^sub>R (LBINT x. f (t + c * x))"
dsg3: fixes x :: "'a"   and y :: "'a" shows "ans_d (ans_d x \<cdot> y) = ans_d x \<cdot> ans_d y"
that: fixes a :: "'a"   and b :: "'a" assumes "x = Upair a b" shows "thesis"
assms_3: fixes x :: "'a" shows "f x \<sqsubseteq> g x"
TryReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>try e' catch(C V) e\<^sub>2,s'\<rangle>"
SD_preorderD_2: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'a pmf"   and p :: "'a pmf" assumes "Order_Predicates.preorder_on carrier le"   and "SD le q p" shows "q \<in> lotteries_on carrier"
converse_rtranclp_induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P b"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>*\<^sup>* z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
hyps_2: shows "P \<parallel> !P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'"
execute_parallel_operator_positive_effect_if_i: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and op :: "'a strips_operator"   and v :: "'a" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "op \<in> set ops"   and "v \<in> set (add_effects_of op)" shows "map_of (effect_to_assignments op) v = Some True"
FinalDenys_subseteq_in_doubt_allow: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs" shows "_Collect p ((\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny) \<subseteq> _Collect p (\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalDeny)"
insert: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<notin> A" shows "aggregation.sum_0 g (insert x A) = g x + aggregation.sum_0 g A"
\<Phi>_simp: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and yx :: "'d \<times> 'c" assumes "meta_adjunction C D F G \<phi> \<psi>"   and "partial_magma.ide (product_category.comp (dual_category.comp D) C) yx" shows "transformation_by_components.map (product_category.comp (dual_category.comp D) C) replete_setcat.comp (hom_functor.map C replete_setcat.comp (\<lambda>_. meta_adjunction.inC) \<circ> product_functor.map (dual_category.comp D) C (dual_category.comp C) C (dual_functor.map F) (identity_functor.map C)) (meta_adjunction.\<Phi>o C D F G \<phi>) yx = set_category_given_img.mkArr replete_setcat.comp (set_category.some_img replete_setcat.comp (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C (\<lambda>_. meta_adjunction.inC) (F (fst yx), snd yx)) (hom_functor.set D (\<lambda>_. meta_adjunction.inD) (fst yx, G (snd yx))) (meta_adjunction.inD \<circ> \<phi> (fst yx) \<circ> hom_functor.\<psi> C (\<lambda>_. meta_adjunction.inC) (F (fst yx), snd yx))"
alphaAbs_sym: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and A' :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "A $= A'" shows "A' $= A"
simps_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a" shows "(case \<bottom> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
h1: shows "M_mat_R (get_signs_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) (get_subsets_R (reduce_system_R p (qs, M_mat_R signs subsets, subsets, signs))) = M_mat_R (reduction_signs_R p qs signs subsets (M_mat_R signs subsets)) (reduction_subsets_R p qs signs subsets (M_mat_R signs subsets))"
not_fic_nrename_aux: fixes M :: "trm"   and c :: "coname"   and x :: "name"   and y :: "name" assumes "fic M c" shows "fic (M[x\<turnstile>n>y]) c"
smallest_ccdi_sets_Int: fixes b :: "'a set"   and a :: "'a set" assumes "b \<in> smallest_ccdi_sets \<Omega> M"   and "a \<in> smallest_ccdi_sets \<Omega> M" shows "a \<inter> b \<in> smallest_ccdi_sets \<Omega> M"
exhaust_sel: fixes state :: "'a state" obtains "state = make_state (clock state) (refs state) (ffi state) (defined_types state) (defined_mods state)"
isfresh: fixes S :: "'a set" assumes "finite S" shows "fresh S \<notin> S"
disconnected_nodes_eq_h2: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "owner_document \<noteq> doc_ptr" shows "h2 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h3 \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
a_get_ancestors_locs_def: shows "get_ancestors_locs = get_parent_locs"
remap_accs2: fixes \<delta>' :: "('a, 'b) ta_rule set"   and n :: "'b tree"   and q' :: "'a"   and f :: "'c \<Rightarrow> 'a"   and \<delta> :: "('c, 'b) ta_rule set"   and q :: "'c"   and Q :: "'c set" assumes "accs \<delta>' n q'"   and "\<delta>' = remap_rule f ` \<delta>"   and "q' = f q"   and "inj_on f Q"   and "q \<in> Q"   and "\<delta>_states \<delta> \<subseteq> Q" shows "accs \<delta> n q"
simps_2: fixes disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'b"   and a :: "'c" shows "primitive_extractor (disc, sel) (Match a) = (if disc a then ([Pos (sel a)], MatchAny) else ([], Match a))"
ack_add_bound2: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "i < ack k j" shows "i + j < ack (4 + k) j"
gamma_plus': fixes i1 :: "int"   and a1 :: "ivl"   and i2 :: "int"   and a2 :: "ivl" assumes "i1 \<in> \<gamma>_ivl a1"   and "i2 \<in> \<gamma>_ivl a2" shows "i1 + i2 \<in> \<gamma>_ivl (a1 + a2)"
flowstoE: fixes X0 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and CX :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and X1 :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and x0 :: "'a"   and d0 :: "'c \<Rightarrow>\<^sub>L 'a" assumes "flowsto X0 T CX X1"   and "(x0, d0) \<in> X0" obtains h :: "real" where "h \<in> T" and "h \<in> existence_ivl0 x0" and "(flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1" and "\<And>h'. h' \<in> {0<--<h} \<Longrightarrow> (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX"
that: fixes a :: "name"   and t :: "trm" assumes "u = Lam [a].t"   and "\<forall>v\<in>V T\<^sub>1. \<exists>v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2" shows "thesis"
language_coinduct_upto_Plus: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and L :: "'a language"   and K :: "'a language" assumes "R L K"   and "\<And>L K. R L K \<Longrightarrow> \<oo> L = \<oo> K \<and> (\<forall>x. Plus_cong R (\<dd> L x) (\<dd> K x))" shows "L = K"
transfer: fixes f :: "'b \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "cr_pmf (density (count_space UNIV) (ennreal \<circ> f)) (local.embed_pmf f)"
hyp2: assumes "distinct (map fst ps)" shows "distinct (ExcessNotAtRecList ps g)"
arr_tensor: fixes f :: "'a"   and g :: "'a" assumes "S.arr f"   and "S.arr g" shows "S.hseq (fst (f, g)) (snd (f, g))"
thm2_ind2b: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and X :: "'b set"   and k :: "nat"   and l :: "nat" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.long_ch_by_ord2 betw f X"   and "finite X"   and "0 < k \<and> k < l \<and> l < card X" shows "betw (f (k - 1)) (f k) (f l)"
continuous_on_pre_zeta: fixes a :: "real"   and A :: "complex set" assumes "0 < a" shows "continuous_on A (pre_zeta a)"
Fix_def: fixes f :: "'a \<Rightarrow> 'a" shows "Fix f = {x. f x = x}"
split: fixes P :: "'g \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and prod :: "'h \<times> 'i" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
greaterThanLessThan_iff: fixes i :: "'a word"   and l :: "'a word"   and u :: "'a word" shows "(i \<in> signed.greaterThanLessThan l u) = (l <s i \<and> i <s u)"
uint_bintrunc: fixes bin :: "num" shows "uint (numeral bin) = take_bit LENGTH('b) (numeral bin)"
simps: fixes a :: "'b list list" shows "map_head_side a = (\<exists>xs. a = xs \<and> (\<forall>x. x \<in> set xs \<longrightarrow> x \<noteq> []))"
nat_seg_image_imp_finite: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "A = f ` {i. i < n}" shows "finite A"
splitAtoms_negInfinity: fixes L :: "atomUni list"   and a :: "(real \<times> real \<times> real) list"   and b :: "(real \<times> real \<times> real) list"   and c :: "(real \<times> real \<times> real) list"   and d :: "(real \<times> real \<times> real) list"   and x :: "real" assumes "separateAtoms L = (a, b, c, d)" shows "(\<forall>l\<in>set L. evalUni (substNegInfinityUni l) x) = ((\<forall>(a, b, c)\<in>set a. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"
adopt_node_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> adopt_node document_ptr node_ptr \<rightarrow>\<^sub>h h'"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^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 document_ptr)|\<^sub>r"   and "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr)|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
comm_semiring_1_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "class.comm_semiring_1 times one plus zero \<equiv> class.comm_monoid_mult times one \<and> class.comm_semiring_0 plus zero times \<and> class.zero_neq_one one zero"
mk_regular2_def: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" shows "mk_regular2 P F I = I \<union> {F q x y |q x y. x \<in> I \<and> y \<in> I \<and> P q x y}"
simps_2: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "removeAll x (y # xs) = (if x = y then removeAll x xs else y # removeAll x xs)"
g_h: fixes x :: "'b" assumes "x \<in> B" shows "g x = h x"
set_def: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "local.set a = IMG ` hom unity a"
simps_20: fixes \<pi> :: "char list \<Rightarrow> com option"   and \<pi>' :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "small_step \<pi> (PScope \<pi>' SKIP, s) = Some (SKIP, s)"
Fix2: fixes \<phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "Monotone \<phi>"   and "\<phi> (FIX \<phi>) (s, t)" shows "FIX \<phi> (s, t)"
csk: shows "cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"
h1: shows "A \<leadsto> C \<and> C \<leadsto> B"
xt1_6: fixes y :: "'c"   and x :: "'c"   and z :: "'c" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
list_of_oalist_tc: fixes x :: "('a, 'b) oalist_tc" shows "list_of_oalist_tc x \<in> Collect tc.oalist_inv_raw"
vector_choose_size: fixes c :: "real" assumes "0 \<le> c" obtains x :: "'a" where "norm x = c"
add_indexed_Ints_vec: fixes x :: "'b vec"   and n :: "nat"   and y :: "'b vec"   and I :: "nat set" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "x \<in> indexed_Ints_vec I"   and "y \<in> indexed_Ints_vec I" shows "x + y \<in> indexed_Ints_vec I"
partGen_ex: fixes I :: "'a set"   and P :: "'a set set" assumes "I \<in> P" shows "\<exists>J\<in>partGen P. I \<subseteq> J"
invariant_cop_FD: fixes ds :: "'b set"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and fp :: "'a set" assumes "invariant ds P"   and "P ds fp" shows "P ds (cop_F ds fp)"
monad_fail_optionT: fixes return :: "'b option \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b option \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad return bind" shows "monad_fail (return_option return) (bind_option return bind) (fail_option return)"
simps_8: fixes \<phi> :: "atom fm" shows "push_forall (Neg \<phi>) = neg (push_forall \<phi>)"
order_neq_le_trans: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
one_step_implies_mult: fixes J :: "'b multiset"   and K :: "'b multiset"   and r :: "('b \<times> 'b) set"   and I :: "'b multiset" assumes "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(I + K, I + J) \<in> mult r"
simps_3: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "subs \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = subs"
p_a_relprime: shows "[a \<noteq> 0] (mod int p)"
lo1: shows "adjoint (x1 0) * x1 0 + adjoint (x1 1) * x1 1 = 1\<^sub>m d"
span_subspace: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'d set"   and A :: "'d set" assumes "module_on U\<^sub>M scale"   and "B \<subseteq> U\<^sub>M"   and "A \<subseteq> B"   and "B \<subseteq> module_on.span scale A"   and "module_on.subspace scale B" shows "module_on.span scale A = B"
set_borel_measurable_subset: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "set_borel_measurable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_borel_measurable M B f"
add_divide_eq_if_simps_4: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and z :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus a (divide b z) = (if z = zero then a else divide (minus (times a z) b) z)"
gcd_right_idem: fixes a :: "'a"   and b :: "'a" shows "gcd (gcd a b) b = gcd a b"