hcju/isabelleps huggingface dataset:
While_2: shows "ir_valid P (WHILE x1 DO c) c' Q"
commute: fixes a :: "'t"   and b :: "'t" shows "ord_term_lin.max a b = ord_term_lin.max b a"
arccos_ubound_abs: fixes y :: "real" assumes "\<bar>y\<bar> \<le> 1" shows "arccos y \<le> pi"
type_less_x_Optional: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> < \<sigma>[?]" obtains \<rho> :: "'a basic_type" where "\<tau> = \<rho>__[1]" and "\<rho>__ \<le> \<sigma>"   | \<rho> :: "'a basic_type" where "\<tau> = \<rho>[?]" and "\<rho> < \<sigma>"
simps: fixes a :: "'b set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = insert aa A \<and> finite A))"
fls_lr_inverse_X_power_2: fixes n :: "nat"   and y :: "'a" shows "fls_right_inverse (fls_X ^ n) y = fls_shift (int n) (fls_const y)"
simps_2: fixes t1 :: "typ"   and t2 :: "typ" shows "mk_scheme (t1 -> t2) = mk_scheme t1 =-> mk_scheme t2"
stateinterpol_insert: fixes v :: "variable \<Rightarrow> real"   and w :: "variable \<Rightarrow> real"   and S :: "variable set"   and z :: "variable" shows "Vagree (stateinterpol v w S) (stateinterpol v w (insert z S)) (- {z})"
Sigma_Suc2: fixes m :: "nat"   and n :: "nat"   and A :: "'a set" assumes "m = n + 2" shows "A \<times> below m = A \<times> {n} \<union> A \<times> {n + 1} \<union> A \<times> below n"
A: shows "g_size_abort s m \<equiv> iterate_size_abort (iteratei m) s"
measurable_alw: fixes P :: "'a stream \<Rightarrow> bool"   and M :: "'a measure" assumes "Measurable.pred (stream_space M) P" shows "Measurable.pred (stream_space M) (alw P)"
projection_split_last: fixes n :: "nat"   and \<tau> :: "'b list"   and X :: "'b set" assumes "Suc n = length (\<tau> \<upharpoonleft> X)" shows "\<exists>\<beta> x \<alpha>. x \<in> X \<and> \<tau> = \<beta> @ [x] @ \<alpha> \<and> \<alpha> \<upharpoonleft> X = [] \<and> n = length ((\<beta> @ \<alpha>) \<upharpoonleft> X)"
joinpaths_def: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" shows "g1 +++ g2 = (\<lambda>x. if x \<le> 1 / 2 then g1 (2 * x) else g2 (2 * x - 1))"
join_le: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "dlat le"   and "le x z"   and "le y z" shows "le (dlat.join le x y) z"
sorted_wrt1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "sorted_wrt P [x] = True"
assms: fixes x :: "nat" assumes "x \<le> n" shows "f x = g x"
assms_2: shows "t \<subseteq> Collect (Domainp A)"
cod_comp: fixes g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "seq g f" shows "cod (g \<cdot> f) = cod g"
empty_not_insert_fset_1: fixes x :: "'b"   and S :: "'b fset" shows "{||} \<noteq> insert_fset x S"
L_2: shows "G, \<Gamma> \<Rightarrow> \<Delta>"
typ_of_mk_all: fixes A :: "Core.term"   and ty :: "typ"   and x :: "variable" assumes "typ_of A = Some propT" shows "typ_of (mk_all x ty A) = Some propT"
Delay_L4: fixes t :: "nat"   and y :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and x :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list" assumes "\<forall>t1\<le>t. y t1 = []"   and "Delay y i1 d x i2" shows "\<forall>t2\<le>t + d. x t2 = []"
elims: fixes x :: "val"   and y :: "nat" assumes "val_size x = y" obtains n :: "nat" where "x = VNat n" and "y = 0"   | t :: "(val \<times> val) list" where "x = VFun t" and "y = 1 + fun_size t"
split_face_edges_f: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and vs1 :: "nat list"   and vs2 :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "vs1 = between (vertices f) ram1 ram2"   and "vs1 \<noteq> []"   and "vs2 = between (vertices f) ram2 ram1"   and "vs2 \<noteq> []" shows "\<E> f = {(last vs2, ram1), (ram1, hd vs1), (last vs1, ram2), (ram2, hd vs2)} \<union> Edges vs1 \<union> Edges vs2"
assms_1: fixes N :: "nat" shows "A N \<in> sets M"
leadsTo_wf_induct: fixes r :: "('c \<times> 'c) set"   and F :: "'d program"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set" assumes "wf r"   and "\<forall>m. F \<in> A \<inter> f -` {m} \<longmapsto> A \<inter> f -` r\<inverse> `` {m} \<union> B" shows "F \<in> A \<longmapsto> B"
B: shows "\<forall>x'\<in>set (xs @ [x]). (u, D x') \<in> I"
simps: fixes l :: "'b"   and u :: "'c"   and l' :: "'b"   and u' :: "'c" shows "Psi.subset (l, u) (l', u') = (l' \<le> l \<and> u \<le> u')"
cases: fixes a :: "'a treeFI" assumes "infiniteTr a" obtains tr' :: "'a treeFI" and tr :: "'a treeFI" where "a = tr" and "tr' \<in> set (sub tr)" and "infiniteTr tr'"
eq_trans: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and v1 :: "Value"   and v2 :: "Value"   and v3 :: "Value" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "eq a v1 v2"   and "eq a v2 v3" shows "eq a v1 v3"
loop_def: shows "dwa.loop \<equiv> while (dwa_cond (pa_det_algo H1 H2)) (dwa_step (pa_det_algo H1 H2)) (dwa_initial (pa_det_algo H1 H2))"
qTerm_induct: fixes phi :: "('f, 'g, 'd, 'e, 'h) qTerm \<Rightarrow> bool"   and phiAbs :: "('f, 'g, 'd, 'e, 'h) qAbs \<Rightarrow> bool"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and A :: "('f, 'g, 'd, 'e, 'h) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. \<lbrakk>\<And>Y. (X, Y) \<in> qSwapped \<Longrightarrow> phi Y; \<And>Y. qSkel Y = qSkel X \<Longrightarrow> phi Y\<rbrakk> \<Longrightarrow> phiAbs (qAbs xs x X)" shows "phi X \<and> phiAbs A"
Postdomination_axioms: shows "Postdomination sourcenode targetnode kind valid_edge (_Entry_) (_Exit_)"
leg1_composite: shows "fgh.\<nu>\<pi>.leg1 = fgh.\<nu>.leg1 \<cdot> fgh.\<nu>\<pi>.prj\<^sub>1"
c_d: shows "(c, d) \<in> thetaSCh ch c1 c2 d1 d2"
finsertCI: fixes a :: "'b"   and B :: "'b fset"   and b :: "'b" assumes "a |\<notin>| B \<Longrightarrow> a = b" shows "a |\<in>| finsert b B"
diagonal_to_Smith_i_preserves_previous: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'b"   and xs :: "nat list" assumes "is_bezout_ext bezout"   and "i < min (nrows A) (ncols A)"   and "to_nat a \<noteq> to_nat b"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)" shows "diagonal_to_Smith_i xs A i bezout $ a $ b = A $ a $ b"
eval_fds_at_top_tail_bound: fixes f :: "'a fds"   and c :: "real"   and s :: "'a" assumes "abs_conv_abscissa f < ereal c"   and "c \<le> s \<bullet> (1::'a)" shows "norm (eval_fds f s - fds_nth f 1) \<le> (\<Sum>n. norm (fds_nth f (n + 2)) / real (n + 2) powr c) * 2 powr c / 2 powr (s \<bullet> (1::'a))"
at_exists_fresh': fixes x :: "'c" assumes "at TYPE('b)"   and "finite (supp x)" shows "\<exists>c. c \<sharp> x"
open_dist: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.metric_space dist uniformity open" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
simps_42: shows "cospan_shape.comp cospan_shape.Null cospan_shape.BT = cospan_shape.Null"
B: shows "(zps, Z) \<in> futures (c_process step out s\<^sub>0) xps"
simps_2: shows "PX3d D2 = linord_of_list [X3.Xd2]"
cases: fixes x :: "'c \<times> 'c \<times> nat" obtains s :: "'c" and t_end :: "'c" and n :: "nat" where "x = (s, t_end, n)"
interior_of_triangle: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "DIM('a) = 2" shows "interior (convex hull {a, b, c}) = convex hull {a, b, c} - (closed_segment a b \<union> closed_segment b c \<union> closed_segment c a)"
is_mapI: fixes M :: "('a \<times> 'b) set" assumes "\<And>a b\<^sub>1 b\<^sub>2. \<lbrakk>(a, b\<^sub>1) \<in> M; (a, b\<^sub>2) \<in> M\<rbrakk> \<Longrightarrow> b\<^sub>1 = b\<^sub>2" shows "is_map M"
simps_2: fixes e :: "'b"   and X :: "'b list" shows "all_partitions_list (e # X) = all_coarser_partitions_with_list e (all_partitions_list X)"
x0_in_I: shows "x0 \<in> {a<..<b}"
listrel1_iff_update: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" shows "((xs, ys) \<in> listrel1 r) = (\<exists>y n. (xs ! n, y) \<in> r \<and> n < length xs \<and> ys = xs[n := y])"
sel_2: fixes x21 :: "'b tree"   and x22 :: "'b"   and x23 :: "'b tree" shows "value \<langle>x21, x22, x23\<rangle> = x22"
subsumes_def: fixes C :: "'b Literal set"   and D :: "'b Literal set" shows "subsumes C D \<equiv> C \<subseteq> D"
qbf: shows "unit p = (fst (unit p), fst (snd (unit p)), snd (snd (unit p)))"
path_comp: fixes xs :: "'a list"   and ys :: "'a list" assumes "walk xs"   and "walk ys"   and "xs = [] \<or> ys = [] \<or> last xs \<rightarrow> hd ys"   and "distinct (xs @ ys)" shows "path (xs @ ys)"
expand: fixes option :: "'d option"   and option' :: "'d option" assumes "(option = None) = (option' = None)"   and "\<lbrakk>option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
FIG_def: shows "FIG.FIG = Var (FIG_alphabet.v, FIG_alphabet.v_update) (Schema FIG.Init `;` lfp ((`;`) (req \<rightarrow> Schema FIG.Out `;` out`!`list_select FIG_alphabet.v \<rightarrow> Skip \<box> read ret (FIG_alphabet.x, FIG_alphabet.x_update) (Schema FIG.Remove))))"
bot_eq_sup_iff: fixes x :: "'a"   and y :: "'a" shows "(bot = x \<squnion> y) = (x = bot \<and> y = bot)"
cone_image_times: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" shows "(*) a ` cone (h, U) = cone (a * h, U)"
RModuleHom_axioms: shows "RModuleHom UNIV (\<cdot>) V (\<star>) T"
swapidseq_ext_zero: fixes S :: "'a set" assumes "finite S" shows "swapidseq_ext S 0 id"
inverse_distrib_swap: fixes a :: "'b"   and b :: "'b" shows "- (a + b) = - b + - a"
axioms_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and \<mu> :: "'b" assumes "identity_arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>" shows "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>"
DSourcesA6_L1: shows "DSources level1 sA6 = {}"
ImageI: fixes a :: "'d"   and b :: "'e"   and r :: "('d \<times> 'e) set"   and A :: "'d set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
empty_eq_map_of_iff: fixes xys :: "('l \<times> 'm) list" shows "(Map.empty = map_of xys) = (xys = [])"
worklist_tree_state_inv: fixes succs :: "'d \<Rightarrow> 'd list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and ws :: "'d list"   and s :: "'c"   and s' :: "'c"   and I :: "'c \<Rightarrow> bool" assumes "worklist_tree succs f ws s = Some s'"   and "I s"   and "\<And>x s. I s \<Longrightarrow> I (f x s)" shows "I s'"
cases: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv"   and n :: "enat" assumes "interaction_bounded_by consider gpv n" obtains "interaction_bound consider gpv \<le> n"
underS_subset_under: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "underS r a \<subseteq> under r a"
append_Nil2: fixes xs :: "'k list" shows "xs @ [] = xs"
idom_is_ring: fixes R :: "('c, 'd) Ring_scheme" assumes "Idomain R" shows "Ring R"
reachablep_reachable_eq: fixes F :: "'b program" shows "reachablep F = (\<lambda>x. x \<in> reachable F)"
infinite: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "infinite A" shows "sum g A = (0::'b)"
compile_def: shows "Rewriting_Pterm.compile = (|`|) (map_prod id translate_rhs)"
distinct_sorted_wrt_imp_sorted_wrt_strict: fixes xs :: "'c list"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "distinct xs"   and "sorted_wrt rel xs" shows "sorted_wrt (\<lambda>x y. rel x y \<and> x \<noteq> y) xs"
gval_fold_gOr_rev: fixes l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gOr (rev l) g) s = gval (fold gOr l g) s"
G: shows "(xs @ zs, Z) \<in> seq_comp_failures P Q"
SEval_append: fixes m :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and xs :: "form list"   and ys :: "form list" shows "SEval m e (xs @ ys) = (SEval m e xs \<or> SEval m e ys)"
set_3: fixes x1 :: "'a" shows "vars_term (Var x1) = {x1}"
join_leftcomm: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
invertible: fixes u :: "'a" assumes "u \<in> R" shows "additive.invertible u"
hd_w_ss_in_A: shows "hd w # ss ! hd w \<in> A"
traces_failures: fixes xs :: "'e list"   and P :: "'e process" assumes "xs \<in> traces P" shows "(xs, {}) \<in> failures P"
assms: shows "set valuation1 \<subseteq> set valuation2"
sep_add_ac_4: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x y = sep_disj y x"
Gets_imp_knows: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> kerbIV_gets" shows "X \<in> knows B evs"
List_unique1: fixes h :: "'b \<Rightarrow> 'b ref"   and p :: "'b ref"   and as :: "'b list" assumes "List h p as" shows "\<exists>!as. List h p as"
strict_suffix_def: fixes xs :: "'c list"   and ys :: "'c list" shows "strict_suffix xs ys = (suffix xs ys \<and> xs \<noteq> ys)"
spr_tLast: fixes a :: "'a"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" assumes "spr_jview a t = spr_jview a t'" shows "envObs a (tLast t) = envObs a (tLast t')"
sep_conj_sep_impl2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a"   and R :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h"   and "\<And>h. P h \<Longrightarrow> sep_algebra.sep_impl plus sep_disj Q R h" shows "R h"
while_default_def: fixes d :: "'b"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and s :: "'b" shows "While_Default.while_default d b c s = (if \<forall>k. b ((c ^^ k) s) then d else the (while_option b c s))"
free_vars_cexpr_subst_aux: fixes x :: "nat"   and e :: "cexpr"   and e' :: "cexpr" shows "free_vars (cexpr_subst x e e') \<subseteq> (\<lambda>y. if x \<le> y then y + 1 else y) -` free_vars e' \<union> free_vars e"
map_reverse_iterateoi_I: fixes invar :: "'c \<Rightarrow> bool"   and ritoi :: "'c \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and \<alpha> :: "'c \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "\<And>m. invar m \<Longrightarrow> map_iterator_rev_linord (ritoi m) (\<alpha> m)" shows "map_reverse_iterateoi \<alpha> invar ritoi"
Collect_conj_eq: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
empty_stack_imp_empty_tj_stack: assumes "stack s = []" shows "tj_stack s = []"
middles: fixes x :: "'a"   and y :: "'a" shows "\<exists>z. dist x z = dist x y / 2 \<and> dist z y = dist x y / 2"
hd_word': fixes w :: "'b list" assumes "w \<noteq> \<epsilon>" shows "[hd w] \<cdot> tl w = w"
a: shows "a = (\<Sum>f0\<in>F0. q0 f0 * f0)"
staysae: fixes s :: "nat list"   and xs :: "nat list" assumes "s \<in> {[a, e], [e, a]}"   and "set xs \<subseteq> {a, e}" shows "fst (Partial_Cost_Model.config' (\<lambda>s. [], TS_step_d) (s, []) xs) \<in> {[a, e], [e, a]}"
tendsto_fst: fixes f :: "'a \<Rightarrow> 'b \<times> 'c"   and a :: "'b \<times> 'c"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. fst (f x)) \<longlongrightarrow> fst a) F"
not_numeral_less_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) one"
wp1_eq: fixes p :: "('a, 'b) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1 p"
cinner_extensionality: fixes \<psi> :: "'b"   and \<phi> :: "'b" assumes "\<And>\<gamma>. \<langle>\<gamma>, \<psi>\<rangle> = \<langle>\<gamma>, \<phi>\<rangle>" shows "\<psi> = \<phi>"
Prefixes_star: fixes A :: "'a list set" assumes "A \<noteq> {}" shows "Prefixes (star A) = star A @@ Prefixes A"
group_def: fixes G :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d" shows "Group_Theory.group G composition unit \<equiv> Group_Theory.monoid G composition unit \<and> Group_Theory.group_axioms G composition unit"
aform_rell_def: shows "aform_rell = aforms_rel"
continuous_sinh: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "continuous F f" shows "continuous F (\<lambda>x. sinh (f x))"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "ide (E a)"
tm_union_range_left: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat"   and t2 :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t1"   and "num_params (tm_poly t1) \<le> length I"   and "develops_at_within e a I" shows "x \<in>\<^sub>i range_tm e (tm_union prec I a t1 t2)"
c_snd_le_c_sum: fixes u :: "nat" shows "c_snd u \<le> c_sum u"
compow_similar_diag_mat_eq: fixes P :: "(('d, 'e) vec, 'f) vec"   and A :: "(('d, 'e) vec, 'e) vec"   and f :: "'f \<Rightarrow> 'd"   and n :: "nat" assumes "invertible P"   and "A = P\<^sup>-\<^sup>1 ** diag_mat f ** P" shows "((**) A ^^ n) (mat (1::'d)) = P\<^sup>-\<^sup>1 ** diag_mat (\<lambda>i. f i ^ n) ** P"
sup_comute1: fixes a :: "'a"   and b :: "'a" shows "a \<squnion> b = b \<squnion> a"
elims_3: fixes x :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<not> invc x" obtains l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and c :: "tcolor" and uu_ :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = \<langle>l, (a, c, uu_), r\<rangle>" and "\<not> (invc l \<and> invc r \<and> (c = Red \<longrightarrow> color l = Black \<and> color r = Black))"
tr_j_i: shows "trace (S t (j + 1)) (take (i - (j + 1)) (drop (j + 1) t)) (S t i)"
assms_1: shows "tms \<equiv> image_mset to_mini_formula"
hyps: fixes \<sigma>0 :: "'b" assumes "distinct S"   and "I (set S) \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> set S; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {x}) (f \<sigma> x)" shows "I {} (foldl f \<sigma>0 S)"
exhaust: fixes y :: "Vertex" obtains "y = A"   | "y = B"   | "y = C"   | "y = D"
fbox_dom: fixes x :: "'a" shows " |x] (zero_class.zero::'a) = ad x"
that: fixes x :: "'a"   and y :: "'a" assumes "sphere a r = {x, y} \<and> dist x y = 2 * r" shows "thesis"
inv_into_into: fixes x :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "x \<in> f ` A" shows "inv_into A f x \<in> A"
conj_def: fixes p :: "fm"   and q :: "fm" shows "Cooper.conj p q = (if p = F \<or> q = F then F else if p = T then q else if q = T then p else And p q)"
assms_2: shows "insertion (nth_default 0 (xs' @ x # xs)) a = A"
Inl_Inr_iff: fixes a :: "hf"   and b :: "hf" shows "(HF.Inl a = HF.Inr b) = False"
char_poly_similar: fixes A :: "'a mat"   and B :: "'a mat" assumes "similar_mat A B" shows "char_poly A = char_poly B"
get_child_nodes_new_character_data: fixes ptr' :: "(_) object_ptr"   and new_character_data_ptr :: "(_) character_data_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^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 new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "r \<in> get_child_nodes_locs ptr'" shows "r h h'"
rep_eq: fixes x :: "'b \<times> 'c \<Rightarrow> bool"   and xa :: "('b, 'c) alist" shows "alist.impl_of (DAList.filter x xa) = filter x (alist.impl_of xa)"
ctr_simps_UNIV_I: fixes x :: "'b" shows "x \<in> UNIV"
greaterThanLessThan_empty_iff2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less" shows "({} = ord.greaterThanLessThan less a b) = less_eq b a"
rev_iterateoi_def: fixes list_rev_it :: "'f \<Rightarrow> (('d \<times> 'e) list \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> ('d \<times> 'e) list \<Rightarrow> ('d \<times> 'e) list) \<Rightarrow> ('d \<times> 'e) list \<Rightarrow> ('d \<times> 'e) list"   and S :: "'f" shows "poly_map_rev_iterateoi_defs.rev_iterateoi list_rev_it S \<equiv> it_to_it (list_rev_it S)"
nn_transfer_operator_intg_Tn: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (nn_transfer_operator ^^ n) g x * f x \<partial>M = \<integral>\<^sup>+ x. g x * f ((T ^^ n) x) \<partial>M"
that: fixes HM :: "'b pre_map" assumes "digraph_map H HM"   and "pre_digraph_map.euler_genus H HM \<le> pre_digraph_map.euler_genus G GM" shows "thesis"
lin_indpt_list_nonzero: fixes G :: "rat vec list" assumes "gs.lin_indpt_list G" shows "0\<^sub>v n \<notin> set G"
simps_1: fixes S :: "(char list \<Rightarrow> int) set"   and P :: "(char list \<Rightarrow> int) set" shows "step S (SKIP {P}) = SKIP {S}"
nth_Cons': fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "(x # xs) ! n = (if n = 0 then x else xs ! (n - 1))"
Rat_not_open: fixes T :: "real set" assumes "open T"   and "T \<noteq> {}" shows "\<not> T \<subseteq> \<rat>"
lm097: fixes xs :: "'c list"   and Y :: "'d list"   and x :: "'c" assumes "set (injectionsAlg xs Y) = injections (set xs) (set Y)" shows "set (injectionsAlg (x # xs) Y) = (\<Union>f\<in>injections (set xs) (set Y). {f \<union> {(x, y)} |y. y \<in> set Y - Range f})"
axioms_terms_ok: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term" assumes "wf_theory thy"   and "A \<in> axioms thy" shows "term_ok thy A"
ipurge_ref_aux_def: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "ipurge_ref_aux I D U xs X \<equiv> {x \<in> X. \<forall>v\<in>sinks_aux I D U xs. (v, D x) \<notin> I}"
nN_1: fixes p :: "ref"   and no :: "ref"   and not :: "dag" assumes "ordered lt var"   and "Dag p low high lt"   and "Dag no low high not"   and "no \<in> set_of lt" shows "ordered not var"