hcju/isabelleps huggingface dataset:
select_convs_7: fixes igWls :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'f \<Rightarrow> 'c \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'b \<Rightarrow> 'h"   and igAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'c"   and igOp :: "'g \<Rightarrow> ('d \<Rightarrow> 'h option) \<Rightarrow> ('e \<Rightarrow> 'c option) \<Rightarrow> 'h"   and igFresh :: "'a \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSwapAbs :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubst :: "'a \<Rightarrow> 'h \<Rightarrow> 'b \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubstAbs :: "'a \<Rightarrow> 'h \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'i" shows "igFreshAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igFreshAbs"
mu_G_def: fixes A :: "('i \<Rightarrow> 'a) set" shows "\<mu>G A = (THE x. \<forall>J\<subseteq>I. finite J \<longrightarrow> (\<forall>X\<in>sets (Pi\<^sub>M J M). A = emb I J X \<longrightarrow> x = emeasure (P J) X))"
simps_1: fixes x :: "nat"   and v :: "exp"   and y :: "nat" shows "subst x v (EVar y) = (if x = y then v else EVar y)"
pairs_subset: fixes aps :: "bool \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and flag :: "bool"   and xs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "\<And>gs bs h ps. set (aps flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))" shows "set (pairs aps flag xs) \<subseteq> Pair flag ` (set xs \<times> set xs)"
abs_eq: fixes xa :: "'b"   and x :: "'c" shows "Poly_Mapping.single xa x = Abs_poly_mapping (\<lambda>k'. x when xa = k')"
le_zero_iff_pprt_id: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero a = (lattice_ab_group_add.pprt zero sup a = a)"
join_cont2: fixes Y :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "chain Y"   and "\<And>i. compatible x (Y i)" shows "x \<squnion> (\<Squnion>i. Y i) = (\<Squnion>i. x \<squnion> Y i)"
end_halt: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" assumes "0 < x"   and "inv_end x (Suc 0, l, r)" shows "\<exists>stp. is_final (steps0 (Suc 0, l, r) tcopy_end stp)"
se_preserves_finiteness2: fixes c :: "('d, 'e) conf"   and l :: "('d, 'e) label"   and c' :: "('d, 'e) conf" assumes "se c l c'"   and "finite (pred c)" shows "finite (pred c')"
sign_vec_def: fixes qs :: "rat poly list"   and x :: "real" shows "sign_vec qs x \<equiv> map (squash \<circ> (\<lambda>p. rpoly p x)) qs"
r_cancel_one: fixes x :: "nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "a \<in> carrier Zp" shows "(a \<oplus> x = x) = (a = \<zero>)"
runit_def: fixes a :: "'a" shows "MC.runit a \<equiv> THE f. \<guillemotleft>f : MC.tensor a MC.unity \<rightarrow> a\<guillemotright> \<and> MC.tensor f MC.unity = MC.tensor a \<iota> \<cdot> MC.assoc a MC.unity MC.unity"
lower_lift: fixes f :: "nat \<Rightarrow> real"   and p :: "real mpoly"   and i :: "nat"   and j :: "nat" shows "insertion f p = insertion f (lowerPoly i j (liftPoly i j p))"
plus_half_bot: fixes x :: "'a"   and y :: "'a" assumes "x * y * x = bot" shows "(x * y)\<^sup>+ = x * y"
cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f X Y \<tau> = f (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) \<tau>"
s_3: fixes z :: "complex" assumes "z \<in> s" shows "eval_fps F z = f z"
roots_1: shows "\<rho> u \<^sup>@ k = u"
set_bd_2: fixes x :: "('e, 'd, 'f) F1" shows "|F1set2 x| \<le>o bd_F1"
I'_def: shows "I' \<equiv> \<lambda>x. if x \<notin> Z then I x else if intv_const (I x) = k x then Greater (k x) else Intv (intv_const (I x))"
abs_eq: fixes x :: "int" shows "nat_of_integer (integer_of_int x) = nat x"
simps_2: fixes A :: "real mat"   and m :: "real mat convnet" shows "remove_weights (Conv A m) = Conv (dim_row A, dim_col A) (remove_weights m)"
f_def: fixes x :: "V" shows "f x = \<langle>x, r\<lparr>x\<rparr>\<rangle>"
size_height: fixes kdt :: "'b kdt" shows "KD_Tree.size_kdt kdt \<le> 2 ^ height kdt"
N_def: shows "fs.gs.N = Max (sq_norm ` set (map of_int_hom.vec_hom fs_init))"
satisfies0_cong: fixes \<BB> :: "interp"   and a :: "presb"   and \<AA> :: "interp" assumes "wf0 (#\<^sub>V \<BB>) a"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>"   and "lformula0 a"   and "\<And>m k. m < #\<^sub>V \<BB> \<Longrightarrow> m\<^bsup>\<AA>\<^esup>k = m\<^bsup>\<BB>\<^esup>k" shows "\<AA> \<Turnstile>0 a = \<BB> \<Turnstile>0 a"
general_qe': fixes F :: "real \<Rightarrow> bool"   and eq :: "(real \<times> real \<times> real) list"   and les :: "(real \<times> real \<times> real) list"   and leq :: "(real \<times> real \<times> real) list"   and neq :: "(real \<times> real \<times> real) list"   and F\<epsilon> :: "real \<Rightarrow> bool"   and F\<^sub>i\<^sub>n\<^sub>f :: "bool"   and roots :: "real \<times> real \<times> real \<Rightarrow> real set"   and A :: "real set"   and B :: "real set"   and C :: "real set"   and D :: "real set" assumes "F = (\<lambda>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set leq. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set neq. a * x\<^sup>2 + b * x + c \<noteq> 0))"   and "F\<epsilon> = (\<lambda>r. (\<forall>(a, b, c)\<in>set eq. \<exists>y>r. \<forall>x\<in>{r<..y}. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. \<exists>y>r. \<forall>x\<in>{r<..y}. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set leq. \<exists>y>r. \<forall>x\<in>{r<..y}. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set neq. \<exists>y>r. \<forall>x\<in>{r<..y}. a * x\<^sup>2 + b * x + c \<noteq> 0))"   and "F\<^sub>i\<^sub>n\<^sub>f = ((\<forall>(a, b, c)\<in>set eq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> (\<forall>(a, b, c)\<in>set leq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set neq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"   and "roots = (\<lambda>(a, b, c). if a = 0 \<and> b \<noteq> 0 then {- c / b} else if a \<noteq> 0 \<and> 0 \<le> b\<^sup>2 - 4 * a * c then {(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} \<union> {(- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} else {})"   and "A = \<Union> (roots ` set eq)"   and "B = \<Union> (roots ` set les)"   and "C = \<Union> (roots ` set leq)"   and "D = \<Union> (roots ` set neq)" shows "(\<exists>x. F x) = (F\<^sub>i\<^sub>n\<^sub>f \<or> (\<exists>r\<in>A. F r) \<or> (\<exists>r\<in>B. F\<epsilon> r) \<or> (\<exists>r\<in>C. F r) \<or> (\<exists>r\<in>D. F\<epsilon> r))"
eq1: shows "(\<Sum>v\<in>keys p \<union> keys q. monomial (lookup p v) (decr_grading_term d n v)) = (\<Sum>v\<in>keys p. monomial (lookup p v) (decr_grading_term d n v))"
is_map_def: fixes x :: "'a" shows "is_map x \<equiv> is_p_fun x \<and> is_total x"
T'_I_T: fixes s :: "'s"   and as :: "'s list"   and os :: "'t list"   and o :: "'t" shows "emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o \<omega>'. \<omega> = (s, o) ## zip as os @- \<omega>'} = emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}"
lookup_def: fixes xs :: "('a, 'b) oalist_ntm" shows "OAlist_lookup_ntm xs = lookup_ko_ntm (list_of_oalist_ntm xs)"
small_set_vsubset: fixes A :: "V set"   and B :: "V" assumes "small A"   and "A \<subseteq> elts B" shows "set A \<subseteq>\<^sub>\<circ> B"
iso_unit_2: shows "Right_a.in_hom (some_unit a) (a \<star> a) a"
l2_inv4'I: fixes x :: "l2_state" assumes "\<And>Rb A B gnx. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progress (progress x Rb) xsk; guessed_frame Rb xgnx = Some gnx\<rbrakk> \<Longrightarrow> guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))" shows "x \<in> l2_inv4'"
that: fixes x' :: "'a" assumes "(x, x') \<in> R"   and "RETURN x' \<le> m" shows "thesis"
simp_elD: fixes b :: "'b list"   and G :: "'b list set"   and us :: "'b list list" assumes " b \<in>B G "   and "us \<in> lists G\<^sub>+"   and "concat us = b" shows "\<^bold>|us\<^bold>| = 1"
simplicial_chain_boundary: fixes p :: "nat"   and S :: "('c \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "simplicial_chain (p - 1) S (chain_boundary p c)"
pred_gpv_Done: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool"   and x :: "'d" shows "pred_gpv P Q (Generative_Probabilistic_Value.Done x) = P x"
mahler_approximation_main: fixes k :: "nat"   and d :: "nat"   and f :: "int poly"   and mm :: "int"   and c :: "int"   and g :: "int poly"   and dd :: "nat"   and kk :: "nat" assumes "k \<noteq> 0 \<Longrightarrow> \<lfloor>real d * mahler_measure f\<rfloor> \<le> mm"   and "c = (- 1) ^ degree f"   and "g = graeffe_poly_impl_main c f k"   and "dd = d ^ 2 ^ Suc k"   and "kk = 2 ^ Suc k" shows "\<lfloor>real d * mahler_measure f\<rfloor> \<le> local.mahler_approximation_main dd c g mm k kk"
a_neigh_def: shows "a_neigh \<equiv> SOME b. G.face_cycle_succ b = a"
extend_bound: fixes A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c"   and m :: "nat" shows "\<Union> (extend A S f ` {..m}) = extend A S f m"
openin_product_topology_alt: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "('c \<Rightarrow> 'd) set" shows "openin (product_topology X I) S = (\<forall>x\<in>S. \<exists>U. finite {i \<in> I. U i \<noteq> topspace (X i)} \<and> (\<forall>i\<in>I. openin (X i) (U i)) \<and> x \<in> Pi\<^sub>E I U \<and> Pi\<^sub>E I U \<subseteq> S)"
satisfies_nFAnd: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "(\<AA> \<Turnstile> nFAnd \<phi> \<psi>) = (\<AA> \<Turnstile> FAnd \<phi> \<psi>)"
invariance_of_domain_homeomorphism: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open S"   and "continuous_on S f"   and "DIM('b) \<le> DIM('a)"   and "inj_on f S" obtains g :: "'b \<Rightarrow> 'a" where "homeomorphism S (f ` S) f g"
ball_subseteq: fixes s1 :: "'a set"   and s2 :: "'a set" shows "(\<forall>x\<in>s1. x \<in> s2) = (s1 \<subseteq> s2)"
encode_Suc: fixes base :: "nat"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base" shows "encode base e (Suc n) = of_nat (Suc n mod base) * \<omega>^ (encode base 0 e) + encode base (e + 1) (Suc n div base)"
eval_def: fixes R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and phi :: "'a \<Rightarrow> 'b"   and s :: "'b" shows "eval R S phi s = (\<lambda>p\<in>carrier (UP R). \<Oplus>\<^bsub>S\<^esub>i\<in>{..deg R p}. phi (coeff (UP R) p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
homeomorphism_translation: fixes a :: "'a"   and S :: "'a set" shows "homeomorphism ((+) a ` S) S ((+) (- a)) ((+) a)"
is_sig_red_def: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "is_sig_red sing_reg top_tail F p = (\<exists>q. sig_red sing_reg top_tail F p q)"
sum_list_zero_nreduced: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and as :: "'b list"   and A :: "'b set" assumes "class.monoid_add plus zero"   and "as \<noteq> []"   and "monoid_add.sum_list plus zero as = zero" shows "\<not> monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero as) as"
two_not_one: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 cnj :: "'a \<Rightarrow> 'a" assumes "class.cnj divide inverse times one plus zero minus uminus cnj" shows "numeral.numeral one plus (num.Bit0 num.One) \<noteq> one"
at_phl_cond: fixes u :: "'a"   and v :: "'a"   and x :: "'a"   and z :: "'a"   and w :: "'a"   and y :: "'a" assumes "\<lbrace>\<alpha> u \<cdot> \<tau> v\<rbrace>x\<lbrace>z\<rbrace>"   and "\<lbrace>\<alpha> u \<cdot> \<tau> w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>\<alpha> u\<rbrace>\<tau> v \<cdot> x + \<tau> w \<cdot> y\<lbrace>z\<rbrace>"
limit_o: fixes a :: "'b"   and w :: "nat \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "a \<in> limit w" shows "f a \<in> limit (f \<circ> w)"
line_extension_def: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and a :: "'a"   and E :: "'a set" assumes "ring R" shows "ring.line_extension R K a E = K #>\<^bsub>R\<^esub> a <+>\<^bsub>R\<^esub> E"
C_4: shows "\<forall>a\<in>C. emeasure M a \<noteq> \<infinity>"
bij_betw_ide_Obj_4: fixes a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "a \<in> Collect J_C.ide" shows "J_C.MkIde (J_C.Dom a) = a"
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>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr" shows "cast document_ptr = (case document_ptr of document_ptr.Ref x \<Rightarrow> Map.empty x | document_ptr.Ext (Inl shadow_root_ptr) \<Rightarrow> Some shadow_root_ptr | document_ptr.Ext (Inr x) \<Rightarrow> Map.empty x)"
path_image_subpath_commute: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "path_image (subpath u v g) = path_image (subpath v u g)"
piecewise_continuous_on_integrable: fixes a :: "real"   and b :: "real"   and I :: "real set"   and f :: "real \<Rightarrow> 'a" assumes "piecewise_continuous_on a b I f" shows "f integrable_on {a..b}"
DirProds_finite_group_iff: fixes I :: "'e set"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme" assumes "finite I" shows "finite_group (DirProds G I) = (\<forall>i\<in>I. finite_group (G i))"
lhd_sswSts: shows "lhd sswSts = lhd wSts"
NI_unrelated_def: shows "NI_unrelated \<equiv> \<forall>execs a n. run n (Some s0) execs \<rightharpoonup> (\<lambda>s_f. run n (Some s0) (purge execs (current s_f)) \<rightharpoonup> (\<lambda>s_f2. output_f s_f a = output_f s_f2 a \<and> current s_f = current s_f2))"
tail_in_verts: fixes e :: "'b" assumes "e \<in> arcs H" shows "tail H e \<in> verts H"
Q1'_def: shows "Q1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q1"
term_variants_pred_dense': fixes a :: "'a"   and b :: "'a"   and u :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and c :: "'a" assumes "term_variants_pred ((\<lambda>_. [])(a := [b])) u t" shows "\<exists>s. term_variants_pred ((\<lambda>_. [])(a := [c])) u s \<and> term_variants_pred ((\<lambda>_. [])(c := [b])) s t"
map_disj_comp: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 |` (UNIV - dom h\<^sub>0)"
load_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>load_tmps is. i < t"
cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a tree" assumes "class.linorder less_eq less" obtains "x = \<langle>\<rangle>"   | l :: "'a tree" and m :: "'a" and r :: "'a tree" where "x = \<langle>l, m, r\<rangle>"
assms_1: shows "f \<in> elts (VPi A B)"
evpm2: shows "\<forall>\<^sub>F x in at x0 within S. \<forall>s>0. s \<le> h \<longrightarrow> flow0 x s \<notin> p2"
x: fixes s :: "real" assumes "s \<in> {t..u}" shows "x s \<in> X"
Eq_cong1: fixes t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" shows "{} \<turnstile> t EQ t' AND u EQ u' IMP t EQ u IMP t' EQ u'"
ANR_imp_absolute_neighbourhood_extensor: fixes S :: "'a set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and U :: "'b set" assumes "ANR S"   and "continuous_on T f"   and "f ` T \<subseteq> S"   and "closedin (top_of_set U) T" obtains V :: "'b set" and g :: "'b \<Rightarrow> 'a" where "T \<subseteq> V" and "openin (top_of_set U) V" and "continuous_on V g" and "g ` V \<subseteq> S" and "\<And>x. x \<in> T \<Longrightarrow> g x = f x"
list_length_hint2: fixes x :: "'a list" assumes "length x = Suc 0" shows "[hd x] = x"
liouville_lambda_def: fixes n :: "nat" shows "liouville_lambda n = (if n = 0 then 0::'b else (- (1::'b)) ^ size (prime_factorization n))"
map_def: fixes f :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" shows "S.map f = (if S.arr f then f else S.null)"
CRR_market_fair_price: fixes pyf :: "bool stream \<Rightarrow> real"   and matur :: "nat" assumes "pyf \<in> borel_measurable (G matur)" shows "fair_price Mkt (\<Sum>w\<in>range (pseudo_proj_True matur). prod (prob_component ((1 + r - d) / (u - d)) w) {0..<matur} * discounted_value r (\<lambda>m. pyf) matur w) pyf matur"
h12: shows "(\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
var_renaming_of_def: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set" shows "var_renaming_of C\<^sub>1 C\<^sub>2 = (instance_of\<^sub>l\<^sub>s C\<^sub>1 C\<^sub>2 \<and> instance_of\<^sub>l\<^sub>s C\<^sub>2 C\<^sub>1)"
poly_prod_list_zero_iff: fixes ps :: "'a poly list"   and x :: "'a" shows "(poly (prod_list ps) x = (0::'a)) = (\<exists>p\<in>set ps. poly p x = (0::'a))"
antisym_conv3: fixes y :: "'a"   and x :: "'a" assumes "\<not> y < x" shows "(\<not> x < y) = (x = y)"
assms_3: shows "T' \<subseteq> existence_ivl t0 x0"
Aexp_Let: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "ABinds \<Gamma>\<cdot>(Aheap \<Gamma> e\<cdot>a) \<squnion> \<A>\<^bsub>a\<^esub> e \<sqsubseteq> Aheap \<Gamma> e\<cdot>a \<squnion> \<A>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e)"
splitAtRec_notRam: fixes ram :: "'c"   and vs :: "'c list"   and us :: "'c list" assumes "ram \<notin> set vs" shows "splitAtRec ram us vs = (us @ vs, [])"
fun_lub_def: fixes L :: "'d set \<Rightarrow> 'c"   and A :: "('b \<Rightarrow> 'd) set" shows "fun_lub L A = (\<lambda>x. L {y. \<exists>f\<in>A. y = f x})"
get_child_nodes_impl: shows "get_child_nodes = local.a_get_child_nodes"
rep_in_ARR: fixes f :: "'c local.term set" assumes "ARR f" shows "rep f \<in> f"
relto_into_relto_fun: fixes a :: "'a"   and b :: "'a"   and B :: "('a \<times> 'a) set"   and A :: "('a \<times> 'a) set" assumes "(a, b) \<in> B\<^sup>* O A O B\<^sup>*" shows "\<exists>as sel m. relto_fun A B (Suc 0) as sel m (a, b)"
mod_ring_unit_factor: shows "(mod_ring_rel ===> mod_ring_rel) (\<lambda>x. x) unit_factor"
op': shows "higher_differentiable_on (S \<inter> f -` {0<..}) (\<lambda>x. 1::'b) n"
Rel_abs: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" assumes "\<And>x y. Transfer.Rel A x y \<Longrightarrow> Transfer.Rel B (f x) (g y)" shows "Transfer.Rel (A ===> B) f g"
interior_Union_subset_cbox: fixes f :: "'a set set"   and t :: "'a set" assumes "finite f"   and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a b"   and "\<And>s. s \<in> f \<Longrightarrow> interior s \<subseteq> t"   and "closed t" shows "interior (\<Union> f) \<subseteq> t"
valid_plan_valid_head: fixes h :: "'c"   and as :: "'c list"   and PROB :: "'c set" assumes "h # as \<in> valid_plans PROB" shows "h \<in> PROB"
solve_step: fixes \<sigma> :: "nat \<Rightarrow> 'a state_scheme"   and \<sigma>' :: "nat \<Rightarrow> 'b state_scheme" assumes "\<forall>j. no (\<sigma> j) \<le> no (\<sigma>' j)"   and "num (\<sigma> i) \<le> no (\<sigma> (sid (\<sigma> i)))" shows "num (\<sigma> i) \<le> no (\<sigma>' (sid (\<sigma> i)))"
det_code_def: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat" shows "det_code sel_fun mf A = (if dim_row A = dim_col A then case triangulize sel_fun mf A of (m, A') \<Rightarrow> prod_list (diag_mat A') div m else (0::'a))"
nth_solution_complete: fixes z :: "nat \<times> nat" assumes "solution z" shows "z \<in> range nth_solution"
F_sets: fixes i :: "nat" shows "F i \<in> sets M"
simps_3: fixes x21 :: "color"   and x22 :: "('b, 'c) rbt"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "('b, 'c) rbt" shows "Branch x21 x22 x23 x24 x25 \<noteq> rbt.Empty"
extremum_uniqueI: fixes a :: "'k multiset" assumes "a \<subseteq># {#}" shows "a = {#}"
vec_of_basis_enum_zero: shows "vec_of_basis_enum (0::'a) = 0\<^sub>v (length canonical_basis)"
RPDi_Plus: fixes i :: "nat" shows "RPDi i (MPlus r s) \<subseteq> {MPlus r s} \<union> RPDi i r \<union> RPDi i s"
comm_add_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A + B = B + A"
simps_3: fixes x2 :: "'d" shows "Msg x2 \<noteq> Marker"
cases: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('e, 'i, 'j, 'k) SLin_state_scheme" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('e, 'i, 'j, 'k) SLin_state_scheme" where "x = (s1, s2)"