hcju/isabelleps huggingface dataset:
simps_1: fixes g :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "iter g a 0 = a"
ordinal_oLog_base_1: fixes x :: "ordinal" shows "oLog (oSuc 0) x = 0"
assms: shows "continuous_on (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)"
no_write_to_read_only_memory_read_only_reads_eq: fixes \<S> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "no_write_to_read_only_memory \<S> sb"   and "\<forall>a\<in>outstanding_refs is_Write\<^sub>s\<^sub>b sb. a \<in> read_only \<S>' \<longrightarrow> a \<in> read_only \<S>" shows "no_write_to_read_only_memory \<S>' sb"
transLexLessRestricted: fixes Vbl :: "nat set" shows "trans (lexLessRestricted Vbl)"
intro: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "semilat (A, r, f)" shows "Semilat A r f"
hyps_1: shows "L (substL (lang e1) \<sigma>) = lang (subst e1 \<sigma>)"
lcdh_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "lcdh \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) \<bind> (\<lambda>zs. return_spmf (\<^bold>g [^] (x * y) \<in> zs))))"
other_def: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and I :: "'f set"   and \<sigma> :: "'f \<Rightarrow> 'e"   and \<sigma>' :: "'f \<Rightarrow> 'e" shows "other P I \<sigma> \<sigma>' \<equiv> \<forall>i. if i \<in> I then \<sigma>' i = \<sigma> i else P (\<sigma> i) (\<sigma>' i)"
length_def: fixes as :: "'a iarray" shows "IArray.length as = length (IArray.list_of as)"
vrat_uminus_def: shows "vrat_uminus = (\<lambda>x\<in>\<^sub>\<circ>\<rat>\<^sub>\<circ>. (- rat_of_vrat x)\<^sub>\<rat>)"
finite_dimensional_vector_space_pair_on_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set"   and basis2 :: "'e set" shows "finite_dimensional_vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1 basis2 \<equiv> finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1 \<and> finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 basis2"
assu1: shows "(\<lambda>n. real_of_int (a n) powr (1 / real_of_int (2 ^ n))) \<longlonglongrightarrow> A"
filter_filterR: fixes x :: "nat discr Box"   and r :: "R" shows "lfilter\<cdot>(neg oo Abs_cfun ((=\<^sub>B) x))\<cdot>(a2c\<cdot>r) = a2c\<cdot>(filterR\<cdot>x\<cdot>r)"
rcos_disjoint: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" assumes "Group.group G"   and "subgroup H G" shows "pairwise disjnt (rcosets\<^bsub>G\<^esub> H)"
1: shows "dims (subtensor (tensor0 ds) i) = dims (tensor0 (tl ds))"
inverse_renaming_preserves_literal_order: fixes A :: "'at set"   and L1 :: "'at Literal"   and L2 :: "'at Literal" assumes "literal_ordering (rename_literal A L1) (rename_literal A L2)" shows "literal_ordering L1 L2"
poincare_line_cvec_cmat_def: fixes z :: "complex \<times> complex"   and w :: "complex \<times> complex" shows "poincare_line_cvec_cmat z w = (let (z1, z2) = z; (w1, w2) = w; nom = w1 * cnj w2 * (z1 * cnj z1 + z2 * cnj z2) - z1 * cnj z2 * (w1 * cnj w1 + w2 * cnj w2); den = z1 * cnj z2 * cnj w1 * w2 - w1 * cnj w2 * cnj z1 * z2 in if den \<noteq> 0 then mk_poincare_line_cmat (Re (\<i> * den)) (\<i> * nom) else if z1 * cnj z2 \<noteq> 0 then mk_poincare_line_cmat 0 (\<i> * z1 * cnj z2) else if w1 * cnj w2 \<noteq> 0 then mk_poincare_line_cmat 0 (\<i> * w1 * cnj w2) else mk_poincare_line_cmat 0 \<i>)"
seq'_not_inj_on_card_UNIV: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "\<not> inj_on (fx0.seq' f x) {0..card UNIV}"
forward_terminating_iff1: fixes x :: "'a" assumes "path x" shows "forward_terminating x = (has_end_points x \<or> x = 0)"
fvi_And_Not: fixes b :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.fvi b (MFOTL.And_Not \<phi> \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
simps_1: fixes s :: "('c, 'd) fmap"   and pfx_a :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "rem_condless_act s pfx_a [] = pfx_a"
coversallAddSubset: fixes a :: "'b set"   and ts :: "'b set set" shows "\<Union> (insert a ts) = \<Union> (addSubsetSet a ts)"
totalize_Join: fixes F :: "'c program"   and G :: "'c program" shows "totalize F \<squnion> totalize G = totalize (F \<squnion> G)"
range_reducer_def: fixes p :: "nat"   and l :: "float" shows "range_reducer p l = (if l < 0 \<or> 2 * lb_pi p < l then approx p (Pi * Num (- 2) * Floor (Num (l * Float 1 (- 1)) / Pi)) [] else Some 0)"
lasso_v0_def: fixes L :: "('a, 'b) lasso_scheme" shows "lasso_v0 L \<equiv> case lasso_reach L of [] \<Rightarrow> lasso_va L | v0 # x \<Rightarrow> v0"
of_real_power_int: fixes x :: "real"   and n :: "int" shows "of_real (x powi n) = of_real x powi n"
frac_of_int: fixes z :: "int" shows "frac (of_int z) = (0::'d)"
length_not_greater_eq_2_iff: shows "(\<not> 2 \<le> LENGTH('a)) = (LENGTH('a) = 1)"
simps_2: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd"   and uu :: "tcolor \<times> 'c \<times> 'd"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "invc2 \<langle>l, (a, uu), r\<rangle> = (invc l \<and> invc r)"
map\<^sub>1_def: fixes f :: "'b" shows "map\<^sub>1 f = \<tau>\<^sub>1 (H f)"
left_hcomp_closed: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<omega> :: "'b"   and \<mu> :: "'b"   and x :: "'b"   and y :: "'b"   and \<nu> :: "'b"   and c :: "'b"   and d :: "'b" assumes "right_hom V H \<omega>"   and "partial_magma.in_hom (subcategory.comp V (weak_composition.right V H \<omega>)) \<mu> x y"   and "partial_magma.in_hom V \<nu> c d"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.in_hom (subcategory.comp V (weak_composition.right V H \<omega>)) (H \<nu> \<mu>) (H c x) (H d y)"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vint_uminus" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> vint_uminus). -\<^sub>\<int> a = b"
hcomp_in_vhomE: fixes \<nu> :: "'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" obtains "D.arr \<mu>" and "D.arr \<nu>" and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" and "src\<^sub>D \<mu> = src\<^sub>D f" and "src\<^sub>D \<mu> = src\<^sub>D g" and "trg\<^sub>D \<nu> = trg\<^sub>D f" and "trg\<^sub>D \<nu> = trg\<^sub>D g"
in_measure: fixes x :: "'c"   and y :: "'c"   and f :: "'c \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
add_comm: fixes x :: "'b"   and y :: "'b" shows "Sep_Eq.add2 x y = Sep_Eq.add2 y x"
null_part: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "\<exists>N. N \<in> null_sets M \<and> null_part M S \<subseteq> N"
valid_selection_exists: fixes G :: "nat \<times> ('d \<times> nat \<times> nat) set"   and Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set" assumes "worklist G Rs \<noteq> {}"   and "set_of_graph_rules Rs" shows "\<exists>L R f. valid_selection Rs G R f"
finite_Diff: fixes A :: "'g set"   and B :: "'g set" assumes "finite A" shows "finite (A - B)"
less: fixes a :: "'a operation"   and b :: "'a operation" shows "(a < b) = (if pos a \<noteq> pos b then pos a < pos b else if is_INS a \<noteq> is_INS b then is_INS a \<and> \<not> is_INS b else if op_idx a \<noteq> op_idx b then op_idx a < op_idx b else op_int a < op_int b)"
shconfD: fixes P :: "'b prog"   and h :: "heap"   and sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state" assumes "P,h \<turnstile>\<^sub>s sh \<surd>"   and "sh C = \<lfloor>(sfs, i)\<rfloor>" shows "P,h,C \<turnstile>\<^sub>s sfs \<surd>"
is_natural_2: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "\<phi> (C.CC.cod f) \<cdot>\<^sub>D T\<^sub>DoFF.map f = \<phi> f"
parallelD1: fixes x :: "'c list"   and y :: "'c list" assumes "x \<parallel> y" shows "\<not> prefix x y"
w': shows "1 \<le> Re (w - complex_of_real (\<sigma> - 1))"
strict_implies_order: fixes b :: "'a list"   and a :: "'a list" assumes "b <lex a" shows "b \<le>lex a"
eval_Assoc'_Ide: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Ide v"   and "Src t = Trg u"   and "Src u = Trg v" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = \<a>\<^sup>-\<^sup>1[\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>]"
Inf_ereal_def: fixes S :: "ereal set" shows "Inf S = (SOME x. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x))"
ramanujan_sum_def: fixes k :: "nat"   and n :: "nat" shows "ramanujan_sum k n = (\<Sum>m | m \<in> {1..k} \<and> coprime m k. unity_root k (int (m * n)))"
WHILEI_lfp_body_def: fixes I :: "'b \<Rightarrow> bool"   and m :: "'c nres"   and c :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c nres" shows "WHILEI_lfp_body I m c f \<equiv> \<lambda>x. sup m (x \<bind> (\<lambda>s. ASSERT (I s) \<bind> (\<lambda>_. ASSUME (c s) \<bind> (\<lambda>_. f s))))"
insert: fixes A :: "'b fset"   and x :: "'b" assumes "A \<noteq> {||}" shows "fMax (finsert x A) = max x (fMax A)"
le_\<tau>_less: fixes \<sigma> :: "'a trace"   and i :: "nat"   and j :: "nat" assumes "\<tau> \<sigma> i \<le> \<tau> \<sigma> j"   and "j < i" shows "\<tau> \<sigma> i = \<tau> \<sigma> j"
times_lepoll_mono: fixes A :: "'b set"   and C :: "'c set"   and B :: "'d set"   and D :: "'e set" assumes "A \<lesssim> C"   and "B \<lesssim> D" shows "A \<times> B \<lesssim> C \<times> D"
member_le_L2_set: fixes A :: "'b set"   and i :: "'b"   and f :: "'b \<Rightarrow> real" assumes "finite A"   and "i \<in> A" shows "f i \<le> L2_set f A"
non_tickFree_implies_nonMt: fixes s :: "'c event list" assumes "\<not> tickFree s" shows "s \<noteq> []"
bigthetaD2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "f \<in> \<Omega>[F](g)"
inf_apply: fixes f :: "'h \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "inf f g x = inf (f x) (g x)"
strict_coboundedI1: fixes c :: "'g"   and a :: "'g"   and b :: "'g" assumes "c < a" shows "c < max a b"
unit_simps_1: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "src \<i>[a] = a"
simps: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat" shows "sequentially_consistent P (E, ws) = (\<forall>r\<in>read_actions E. P,E \<turnstile> r \<leadsto>mrw ws r)"
iso_iff_group_isomorphisms: fixes f :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set \<Rightarrow> 'a"   and H :: "('a, 'b) monoid_scheme" shows "(f \<in> Group.iso (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) H) = (\<exists>g. group_isomorphisms (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) H f g)"
intros_6: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"   and "\<Gamma> \<Rightarrow> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta> \<down> Suc n"
triangle_left_implies_right: assumes "(\<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g) \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = \<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]" shows "(g \<star> \<xi> \<cdot> (f \<star> local.inv \<eta> \<star> g) \<cdot> (local.inv \<xi> \<star> f \<star> g)) \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) = \<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]"
eq4: shows "\<phi> (\<omega> [^]\<^bsub>R\<^esub> 2 ^ (p - 2) \<oplus>\<^bsub>R\<^esub> \<omega>' [^]\<^bsub>R\<^esub> 2 ^ (p - 2)) = \<zero>\<^bsub>S\<^esub>"
sin_series_stream_def: shows "sin_series_stream = sin_series_stream_aux False 1 1"
l_get_host_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" shows "l_get_host_wf heap_is_wellformed known_ptr known_ptrs type_wf get_host \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_get_host_wf_axioms heap_is_wellformed known_ptrs type_wf get_host"
le_shiftr1: fixes u :: "'b word"   and v :: "'b word" assumes "u \<le> v" shows "shiftr1 u \<le> shiftr1 v"
simps_2: fixes cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and pc :: "nat"   and s :: "nat \<times> 'c"   and ss :: "(nat \<times> 'c) list"   and x :: "'c" shows "merge cert f r T pc (s # ss) x = merge cert f r T pc ss (let (pc', s') = s in if pc' = pc + 1 then s' \<squnion>\<^bsub>f\<^esub> x else if s' \<sqsubseteq>\<^bsub>r\<^esub> cert ! pc' then x else T)"
p_fun_comp: fixes x :: "'a"   and y :: "'a" assumes "is_p_fun x"   and "is_p_fun y" shows "is_p_fun (x ; y)"
between_g: fixes i :: "nat" shows "(f (j_of ((j_of ^^ i) 0)), f (Suc ((j_of ^^ i) 0))) \<in> S\<^sup>*"
scaleR_right_diff_distrib: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
freshEnv_psubstAbs_ident: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodAbs A"   and "goodEnv rho"   and "\<And>zs z. freshEnv zs z rho \<or> freshAbs zs z A" shows "A $[rho] = A"
dist: shows "dist_over_inf (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
Gramian_determinant_2: fixes k :: "nat" assumes "k \<le> length fs" shows "(0::'a) < d k"
inf_le1: fixes x :: "'b"   and y :: "'b" shows "inf x y \<le> x"
bisim1WhileThrow2: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and c :: "'addr expr1" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,while (c) e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, Suc (length (compE2 c) + pc), \<lfloor>a\<rfloor>)"
lincomb_distrib: fixes U :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a"   and c :: "'a" assumes "U \<subseteq> carrier_vec n"   and "a \<in> U \<rightarrow> UNIV"   and "c \<in> UNIV" shows "c \<cdot>\<^sub>v lincomb a U = lincomb (\<lambda>u. c * a u) U"
one_zeroI: assumes "UNIV = {0::'b}" shows "(1::'b) = (0::'b)"
mat2matofpoly_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "mat2matofpoly A = (\<chi>i j. [:A $h i $h j:])"
simps_3: fixes a :: "atom" shows "push_forall (fm.Atom a) = simp_atom a"
success_effectE: fixes c :: "'c Heap"   and h :: "heap" assumes "success c h" obtains r :: "'c" and h' :: "heap" where "effect c h h' r"
IH: fixes n :: "nat"   and l :: "nat list" assumes "ackloop_dom (ack m n # l)" shows "ackloop_dom (n # m # l)"
ConsCons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'a"   and ys :: "'a list" assumes "as = x # xs"   and "bs = y # ys" shows "P"
recurrent_class: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a" assumes "MC_syntax.recurrent K x" shows "(SIGMA x:UNIV. set_pmf (K x))\<^sup>* `` {x} = MC_syntax.communicating K `` {x}"
node_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_node_ptr_kind ptr" shows "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 (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr"
inf2I: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "A x y"   and "B x y" shows "(A \<sqinter> B) x y"
stutter_extend_sa: shows "sa (stutter_extend pid_system_automaton')"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root_safe :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_shadow_root_safe_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "type_wf = ShadowRootClass.type_wf"   and "get_shadow_root_safe = l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe get_shadow_root get_mode"   and "get_shadow_root_safe_locs = l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_shadow_root_safe_locs get_shadow_root_locs get_mode_locs" shows "l_get_shadow_root_safe\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf get_shadow_root_safe get_shadow_root_safe_locs get_shadow_root get_shadow_root_locs get_mode get_mode_locs"
arctan_less_iff: fixes x :: "real"   and y :: "real" shows "(arctan x < arctan y) = (x < y)"
nset_image_obtains: fixes X :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and k :: "nat" assumes "X \<in> [f ` A]\<^bsup>k\<^esup>"   and "inj_on f A" obtains Y :: "'b set" where "Y \<in> [A]\<^bsup>k\<^esup>" and "X = f ` Y"
fInf_distr: fixes f :: "'d \<Rightarrow> 'b" shows "Inf_pres (\<lambda>x. x \<circ> f)"
cauchy_ine\<^sub>N_I: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>n\<ge>M. \<forall>m\<ge>M. eNorm N (u n - u m) < e" shows "cauchy_ine\<^sub>N N u"
exhaust: fixes y :: "fmla" obtains x11 :: "trm" and x12 :: "trm" where "y = x11 EQ x12"   | x21 :: "fmla" and x22 :: "fmla" where "y = x21 OR x22"   | x3 :: "fmla" where "y = neg x3"   | x41 :: "name" and x42 :: "fmla" where "y = exi x41 x42"
append_eqvt: fixes pi :: "('e \<times> 'e) list"   and l1 :: "'d list"   and l2 :: "'d list" shows "pi \<bullet> (l1 @ l2) = pi \<bullet> l1 @ pi \<bullet> l2"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_inverse" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> vreal_inverse). a\<inverse>\<^sub>\<real> = b"
rGreaterT: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r2 < r1" shows "[Greater \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> True"
Comp: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and Q :: "'b assn"   and d :: "com"   and R :: "'b assn" assumes "G|-{P}. c .{Q}"   and "G|-{Q}. d .{R}" shows "G|-{P}. (c;; d) .{R}"
inject_1: fixes x2 :: "'f"   and y2 :: "'f" shows "(Skip x2 = Skip y2) = (x2 = y2)"
Red_I_\<G>_q_def: fixes q :: "'q"   and N :: "'f set" shows "no_labels.Red_I_\<G>_q q N = {\<iota> \<in> Inf_F. \<G>_I_q q \<iota> \<noteq> None \<and> the (\<G>_I_q q \<iota>) \<subseteq> Red_I_q q (no_labels.\<G>_Fset_q q N) \<or> \<G>_I_q q \<iota> = None \<and> \<G>_F_q q (concl_of \<iota>) \<subseteq> no_labels.\<G>_Fset_q q N \<union> Red_F_q q (no_labels.\<G>_Fset_q q N)}"
subcocycle_lim_birkhoff_1: fixes u :: "'a \<Rightarrow> real" assumes "integrable M u" shows "- \<infinity> < subcocycle_avg_ereal (birkhoff_sum u)"
lub_in_Field: fixes M :: "'a set" assumes "M \<in> Chains leq"   and "M \<noteq> {}" shows "lub M \<in> Field leq"
assms_5: shows "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])"
p1_3: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> p1" shows "K \<subseteq> s1"
nprv_clear3_2: fixes \<phi>1 :: "'fmla"   and \<phi>3 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>2 :: "'fmla" assumes "nprv {\<phi>1, \<phi>3} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3} \<psi>"
simps_2: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and x :: "'c"   and xs :: "'c list" shows "ipurge_tr_rev I D u (x # xs) = (if D x \<in> sources I D u (x # xs) then x # ipurge_tr_rev I D u xs else ipurge_tr_rev I D u xs)"
mthd_simp: fixes C :: "'c"   and m :: "'b" shows "mthd (C, m) = m"
order_trans: fixes y :: "'a list"   and x :: "'a list"   and z :: "'a list" assumes "y \<le>lex x"   and "z \<le>lex y" shows "z \<le>lex x"
hyps: assumes "b < degree f" shows "(\<Sum>i\<le>b. monom (g i) i) mod f = (\<Sum>i\<le>b. monom (g i) i)"
order_subst1: fixes a :: "'d"   and f :: "'e \<Rightarrow> 'd"   and b :: "'e"   and c :: "'e" assumes "a \<le> f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
h: assumes "freeIn 0 \<phi>'" shows "eval (lowerFm 0 1 \<phi>') xs = eval (AllQ \<phi>') xs"
card_le_Suc_iff: fixes n :: "nat"   and A :: "'a set" shows "(Suc n \<le> card A) = (\<exists>a B. A = a \<triangleright> B \<and> a \<notin> B \<and> n \<le> card B \<and> finite B)"
sums_iff: fixes f :: "nat \<Rightarrow> 'd"   and x :: "'d" shows "f sums x = (summable f \<and> suminf f = x)"
trnr\<^sub>\<eta>_def: fixes v :: "'a"   and \<mu> :: "'a" shows "trnr\<^sub>\<eta> v \<mu> \<equiv> (\<mu> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[v, f\<^sup>*, f] \<cdot> (v \<star> \<rho>.\<eta>) \<cdot> \<r>\<^sup>-\<^sup>1[v]"
cmult_def: fixes \<kappa> :: "V"   and \<mu> :: "V" shows "\<kappa> \<otimes> \<mu> \<equiv> vcard (vtimes \<kappa> \<mu>)"
permutes_natset_le: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S"   and "\<forall>i\<in>S. p i \<le> i" shows "p = id"
Int_space_eq1: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> null_sets M" shows "space M \<inter> x = x"
a_cs: fixes x :: "complex" assumes "x \<in> S" shows "closed_segment a x \<subseteq> S"
K: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a * c \<le> b * c"
eq: shows "OAlist_map_val_ntm (\<lambda>_. f) xs = OAlist_map_val_ntm (\<lambda>_ c. f c when c \<noteq> (0::'c)) xs"
symmetry1: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "symp R" shows "\<forall>x y. (x, y) \<in> {(x, y). R x y}\<^sup>* \<longrightarrow> (y, x) \<in> {(x, y). R x y}\<^sup>*"
bij_betw_comp_iff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and A' :: "'e set"   and f' :: "'e \<Rightarrow> 'f"   and A'' :: "'f set" assumes "bij_betw f A A'" shows "bij_betw f' A' A'' = bij_betw (f' \<circ> f) A A''"
adjoint_equivalence_axioms: shows "adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>'.map"
2: shows "\<guillemotleft>\<lparr>Chn = fgh.chine_assoc', Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr> : f \<star> g \<star> h \<Rightarrow> (f \<star> g) \<star> h\<guillemotright>"
Un_Int_eq_1: fixes S :: "'c set"   and T :: "'c set" shows "(S \<union> T) \<inter> S = S"
simps_1: fixes v2 :: "'b" shows "list_result (Rval v2) = Rval [v2]"
path_via: shows "P \<turnstile> Path C to C' via Cs "
ltlc_aps_list_def: fixes \<phi> :: "'b ltlc" shows "ltlc_aps_list \<phi> = ltlc_aps_list' \<phi> []"