hcju/isabelleps huggingface dataset:
coinduct: fixes X :: "'a treeFI \<Rightarrow> bool"   and x :: "'a treeFI" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>tr' tr. x = tr \<and> tr' \<in> set (sub tr) \<and> (X tr' \<or> infiniteTr tr')" shows "infiniteTr x"
secureI_secureFillI: fixes c :: "IMP"   and X :: "Var set"   and C :: "CtxtProg" assumes "secure c"   and "LOW X C"   and "LOW X Ctxt_Body"   and "body = Fill Ctxt_Body c" shows "secure (Fill C c)"
nonrestrict_def: fixes r :: "hf"   and A :: "hf" shows "nonrestrict r A = \<lbrace>z \<^bold>\<in> r. \<forall>x\<^bold>\<in>A. \<forall>y. z \<noteq> \<langle>x, y\<rangle>\<rbrace>"
rqs_cost_exp_eq: fixes n :: "nat" shows "rqs_cost_exp n = 2 * real (n + 1) * harm n - 4 * real n"
new_pre_not_Nil: shows "new_pre \<noteq> []"
smult_content_normalize_primitive_part: fixes p :: "'a poly" shows "smult (content p) (normalize (primitive_part p)) = normalize p"
monad_stateT: fixes return :: "'b \<times> 'c \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "monad return bind" shows "monad (return_state return) (bind_state bind)"
integrable_on_cmult_left_iff: fixes c :: "real"   and f :: "'a \<Rightarrow> 'b"   and s :: "'a set" assumes "c \<noteq> 0" shows "((\<lambda>x. of_real c * f x) integrable_on s) = (f integrable_on s)"
rb_spp_aux_alt: fixes args :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list \<times> 't list \<times> (('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_inv (fst (app_args vec_of args))" shows "rb_spp_aux args = app_args spp_of (rb_aux (app_args vec_of args))"
pderivs_simps_2: fixes s :: "'a list" shows "pderivs s One = (if s = [] then {One} else {})"
mem_box_cart_1: fixes x :: "(real, 'a) vec"   and a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "(x \<in> box a b) = (\<forall>i. a $ i < x $ i \<and> x $ i < b $ i)"
filterlim_at_top_strict_mono: fixes s :: "nat \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> nat"   and F :: "nat filter" assumes "strict_mono s"   and "strict_mono r"   and "filterlim s at_top F" shows "filterlim (s \<circ> r) at_top F"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "L_action.Act x1 \<noteq> Eff x2"
pell_def: fixes D :: "nat" shows "pell D \<equiv> \<not> is_square D"
ide_map\<^sub>0_obj: fixes a :: "'c" assumes "C.obj a" shows "C.ide (\<epsilon>\<^sub>0 a)"
prems: shows "ackloop_dom (ack (Suc m) n # l)"
stable_rank_at_ge: fixes n :: "nat"   and m :: "nat"   and x :: "nat" assumes "n \<le> m"   and "stable_rank_at x n" shows "stable_rank_at x m"
iso_integer_zeroth_homology_group_aux: fixes X :: "'b topology"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b"   and f' :: "(nat \<Rightarrow> real) \<Rightarrow> 'b" assumes "path_connected_space X"   and "singular_simplex 0 X f"   and "singular_simplex 0 X f'" shows "homologous_rel 0 X {} (frag_of f) (frag_of f')"
distinct_151: shows "GreaterOrEqual \<noteq> BinAnd"
hnr_ASSERT: fixes \<Phi> :: "bool"   and \<Gamma> :: "assn"   and c :: "'a Heap"   and \<Gamma>' :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and c' :: "'b nres" assumes "\<Phi> \<Longrightarrow> hn_refine \<Gamma> c \<Gamma>' R c'" shows "hn_refine \<Gamma> c \<Gamma>' R (ASSERT \<Phi> \<bind> (\<lambda>_. c'))"
Nml_Dom: fixes t :: "'a local.term" assumes "Nml t" shows "Nml (Dom t)"
fst_set_add_indices: fixes ns :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h"   and data' :: "nat \<times> 'h" shows "fst ` set (fst (add_indices ns data')) = fst ` set (fst ns)"
Pow_raw: fixes X :: "V set" assumes "X \<in> range elts" shows "inv elts ` Pow X \<in> range elts"
f_def: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and x :: "real" shows "f a b c d x = (d - c) / (b - a) * (x - a) + c"
edge'_simps_1: fixes x :: "'v node" shows "edge' SOURCE x = (x = Inner (source \<Delta>))"
pmult_singleton: fixes h1 :: "'a"   and p1 :: "'a list" shows "[h1] *** p1 = h1 %* p1"
upper_bound: fixes c :: "'c" assumes "c \<in> \<X>"   and "Regions.isIntv (I c)" shows "(u \<oplus> (1 - Max ({frac (u c) |c. c \<in> \<X> \<and> Regions.isIntv (I c)} \<union> {0})) / 2) c < real (intv_const (I c) + 1)"
is_valid_problem_sas_plus_then_1: fixes \<Psi> :: "('a, 'b) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). \<R>\<^sub>+ \<Psi> v \<noteq> {}"
assms_2: shows "v \<in> local.reads (\<pi> k)"
uses_att: fixes n :: "'n"   and f :: "('var \<Rightarrow> 'val) \<Rightarrow> 'out"   and \<sigma> :: "'var \<Rightarrow> 'val"   and \<sigma>' :: "'var \<Rightarrow> 'val" assumes "local.att n = Some f"   and "\<forall>v\<in>local.reads n. \<sigma> v = \<sigma>' v" shows "f \<sigma> = f \<sigma>'"
con_comp_failures_divergences: fixes xs :: "'e list"   and y :: "'e"   and ys :: "'e list"   and Y :: "'e set"   and P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e"   and zs :: "'e list" assumes "(xs @ y # ys, Y) \<in> con_comp_failures P Q p q"   and "y \<notin> range p"   and "y \<notin> range q" shows "\<exists>xs'. (\<exists>ys'. xs @ zs = xs' @ ys') \<and> set xs' \<subseteq> range p \<union> range q \<and> map (inv p) (filter (\<lambda>x. x \<in> range p) xs') \<in> divergences P \<and> map (inv q) (filter (\<lambda>x. x \<in> range q) xs') \<in> divergences Q"
wpt_mono: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "less_eq x y" shows "less_eq (mbt_algebra.wpt one times inf top x) (mbt_algebra.wpt one times inf top y)"
R3_def: shows "R3 = (\<lambda>x. sum_upto (\<lambda>x. real x powr - \<alpha>) x - x powr (1 - \<alpha>) / (1 - \<alpha>))"
c_k_def: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int"   and k :: "int" assumes "QI c_k c w"   and "i < j"   and "k \<in> {i + 1..j}" shows "c_k i j k = w i j + c i (k - 1) + c k j"
src_hcomp: fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_diff a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "head t >\<^sub>h\<^sub>d head s" and "chkvar t s" and "chksubs (>\<^sub>t) t s"
WTStaticCast_sub_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and D :: "char list" shows "WTStaticCast_sub P C D = (P \<turnstile> Path D to C unique \<or> (subcls1p P)\<^sup>*\<^sup>* C D \<and> (\<forall>Cs. P \<turnstile> Path C to D via Cs  \<longrightarrow> Subobjs\<^sub>R P C Cs))"
step: shows "\<Gamma>\<turnstile> (c # cs, css, s) \<rightarrow> (cs', css', t)"
pr0_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "cod (pr0 a b) = b"
fds_deriv_uminus: fixes f :: "'b fds" shows "fds_deriv (- f) = - fds_deriv f"
y1: shows "local.bad y1 \<and> p (local.worst_chain i) y1 \<and> (\<forall>z. local.bad z \<and> p (local.worst_chain i) z \<longrightarrow> \<not> gt y1 z)"
eq: shows "rep_list (spair p q) = local.punit.monom_mult ((1::'b) / punit.lc (rep_list p)) (l - punit.lt (rep_list p)) (rep_list p) + local.punit.monom_mult (- (1::'b) / punit.lc (rep_list q)) (l - punit.lt (rep_list q)) (rep_list q)"
Least_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
par_comp_prefix: fixes A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and M :: "('fun, 'var) Term.term set" assumes "par_comp (A @ B) M" shows "par_comp A M"
assms_2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2]))"
vimageD: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e"   and A :: "'e set" assumes "a \<in> f -` A" shows "f a \<in> A"
a_elemrcos_carrier: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and a :: "'e"   and a' :: "'e" assumes "abelian_subgroup H G"   and "a \<in> carrier G"   and "a' \<in> H +>\<^bsub>G\<^esub> a" shows "a' \<in> carrier G"
prems: fixes x :: "nat" assumes "x \<in> free_vars (IF b THEN e1 ELSE e2)" shows "\<sigma>1 x = \<sigma>2 x"
rep_eq: shows "status full_status = (\<lambda>(f, n). [0..<n])"
fempty_fminus: fixes A :: "'c fset" shows "{||} |-| A = {||}"
convolution_assoc: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> aezfun_set"   and "h \<in> aezfun_set" shows "convolution (convolution f g) h = convolution f (convolution g h)"
rtrancl3p_refl: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "rtrancl3p r a [] a"
h1: shows "dim_col (1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A))) = length (pivot_positions A)"
compute_zero_pp: shows "0 = Pm_fmap fmempty"
onlI: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'h, 'f, 'e) seqp"   and p :: "('g, 'h, 'f, 'e) seqp"   and P :: "'i \<times> 'e \<Rightarrow> bool"   and \<xi> :: "'i" assumes "\<And>l. l \<in> labels \<Gamma> p \<Longrightarrow> P (\<xi>, l)" shows "onl \<Gamma> P (\<xi>, p)"
Osum_Preorder: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Preorder r"   and "Preorder r'" shows "Preorder (r \<union>o r')"
nn_set_integral_cong: fixes f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and A :: "'b set" assumes "AE x in M. f x = g x" shows "set_nn_integral M A f = set_nn_integral M A g"
prefixes_at_Suc': fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat"   and b :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>= b" shows "eval (total_cons.r_prefixes s) [Suc t, j] = eval (total_cons.r_next s) [b]"
l_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(x \<oplus> a = x) = (a = \<zero>)"
step_Normal_elim_cases_16: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Throw # cs, css, Normal s) \<rightarrow> u" obtains "u = (cs, css, Abrupt s)"
b: shows "(f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> b = MkIde (Cod f)"
mult_adjugate_det: fixes A :: "(('a, 'b) vec, 'b) vec" shows "A ** adjugate A = mat (det A)"
dom_delete_subset: fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "fst ` set (delete k al) \<subseteq> fst ` set al"
nz_impl_b_least_le: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" assumes "f x y \<noteq> 0" shows "b_least f x \<le> y"
discI: fixes list :: "'c list"   and x21 :: "'c"   and x22 :: "'c list" assumes "list = x21 # x22" shows "list \<noteq> []"
distinct_2: fixes b' :: "'b"   and a :: "'a" shows "Inr b' \<noteq> Inl a"
interp_alt_factorization_def: shows "interp_alt_factorization = (\<lambda>(a, cs). smult a (\<Prod>(c, n)\<leftarrow>cs. [:1::'b, - c:] ^ Suc n))"
brI: fixes a :: "'a"   and \<alpha> :: "'b \<Rightarrow> 'a"   and c :: "'b"   and I :: "'b \<Rightarrow> bool" assumes "a = \<alpha> c"   and "I c" shows "(c, a) \<in> br \<alpha> I"
if_then_else_Error_2: fixes a :: "bool"   and c :: "('a, 'b) result"   and b :: "'b"   and d :: "'a" shows "((if a then c else Ok b) = Error d) = (a \<and> c = Error d)"
element_ptr_kinds_eq_h: shows "element_ptr_kinds h2 = element_ptr_kinds h"
map_respect_cyclic: shows "rel_fun (=) (rel_fun cyclic cyclic) map map"
uint_plus_simple: fixes x :: "'b word"   and y :: "'b word" assumes "x \<le> x + y" shows "uint (x + y) = uint x + uint y"
orbit_def: fixes G :: "('e, 'f) partial_object_scheme"   and \<phi> :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" shows "orbit G \<phi> x = {\<phi> g x |g. g \<in> carrier G}"
foldli_mono_dres_aux2: fixes f :: "'c \<Rightarrow> 'd dres \<Rightarrow> 'd dres"   and f' :: "'c \<Rightarrow> 'd dres \<Rightarrow> 'd dres"   and l :: "'c list"   and c :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d dres" assumes "\<And>x. f x dSUCCEED = dSUCCEED"   and "\<And>x. f' x dFAIL = dFAIL"   and "\<And>a x x'. x \<le> x' \<Longrightarrow> f a x \<le> f' a x'" shows "foldli l (case_dres False False c) f \<sigma> \<le> foldli l (case_dres False False c) f' \<sigma>"
leq3: fixes n :: "nat" shows "trace (denote_while_n_iter M0 M1 DS (Suc n) \<rho>) \<le> trace (denote_while_n_comp M0 M1 DS n \<rho>)"
le_mask_imp_and_mask: fixes x :: "'a word"   and n :: "nat" assumes "x \<le> mask n" shows "x && mask n = x"
ex_psi_in_U: shows "\<exists>j. \<psi> j \<in> U\<^bsub>LIM-CONS\<^esub>"
assms_2: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> X" shows "R \<leadsto>[(X \<union> bisim)] S"
assms_2: shows "y \<noteq> 0\<^sub>h"
g_deriv: fixes t :: "real" assumes "t \<in> {a..b}" shows "(g has_real_derivative g' t) (at t within {a..b})"
psubstAbs_idEnv: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodAbs A" shows "A $[idEnv] = A"
path_decomp_2: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and ys :: "'c list" assumes "Graph G"   and "Graph.walk G (xs @ ys) \<and> distinct (xs @ ys)" shows "Graph.walk G ys \<and> distinct ys"
other_otherD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'b"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a" assumes "other P {i} \<sigma> \<sigma>'" shows "\<forall>j. j \<noteq> i \<longrightarrow> P (\<sigma> j) (\<sigma>' j)"
lnth_ltake: fixes m :: "nat"   and n :: "enat"   and xs :: "'b llist" assumes "enat m < n" shows "ltake n xs ?! m = xs ?! m"
ket_plus_def: shows "ket_plus = Matrix.vec 2 (\<lambda>k. 1 / csqrt 2)"
add_mono_thms_linordered_semiring_2: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i = j \<and> k \<le> l" shows "i + k \<le> j + l"
substEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Z :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "goodEnv rho"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "rho &[Y / y]_ys &[Z / z]_zs = rho &[Z / z]_zs &[Y #[Z / z]_zs / y]_ys"
ground_fterms_of_hterms: fixes ts :: "hterm list" shows "ground\<^sub>t\<^sub>s (fterms_of_hterms ts)"
K_funs: shows "\<forall>i>K. num_funs (g i) = num_funs (g (Suc i))"
gets_the_def: fixes f :: "'b \<Rightarrow> 'c option" shows "gets_the f \<equiv> gets f >>= assert_opt"
starset_mem: fixes x :: "'c"   and A :: "'c set" shows "(star_of x \<in> *s* A) = (x \<in> A)"
count_mset_set': fixes A :: "'b set"   and x :: "'b" shows "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
cases: fixes a1 :: "Value"   and a2 :: "CValue" assumes "similar'_base a1 a2" obtains "a1 = \<bottom>" and "a2 = \<bottom>"
power_zero_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and k :: "num" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (numeral k) = zero"
read_actions_iff: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "(a \<in> read_actions E) = (a \<in> actions E \<and> (\<exists>ad al v. action_obs E a = NormalAction (ReadMem ad al v)))"
two_set_categories_bij_betw_Univ_axioms: shows "two_set_categories_bij_betw_Univ (\<cdot>\<acute>) (\<cdot>) \<AA> \<psi>"
frontier_Int_core: fixes s :: "'a set" shows "frontier s \<inter> core s = {}"
perm_eq_iff: fixes f :: "'a perm"   and g :: "'a perm" shows "(f = g) = (\<forall>a. f \<langle>$\<rangle> a = g \<langle>$\<rangle> a)"
f4: fixes b :: "'b"   and c :: "'c" shows "put\<^bsub>X /\<^sub>L Y\<^esub> (get\<^bsub>Y\<^esub> b) c = get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> b c)"
\<II>\<nn>\<ll>_inj_on: shows "inj_on \<II>\<nn>\<ll> \<II>\<nn>\<ll>_dom"
sourcenodes_is_n_Cons_butlast_targetnodes: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" shows "CFG.sourcenodes sourcenode as = n # butlast (CFG.targetnodes targetnode as)"
WTDynCast_ex_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and D :: "char list"   and C :: "char list" shows "WTDynCast_ex P D C = (\<exists>Cs. P \<turnstile> Path D to C via Cs )"
redT_updT_upd: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h"   and xw :: "'f \<times> 'g \<Rightarrow>f nat"   and ta :: "('h, 'f, 'i) new_thread_action"   and xw' :: "'f \<times> 'g \<Rightarrow>f nat" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_ok ts ta" shows "redT_updT ts ta(t \<mapsto> xw') = redT_updT (ts(t \<mapsto> xw')) ta"
gen_set_def: fixes emp :: "'g"   and ins :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and l :: "'h list" shows "gen_set emp ins l = fold ins l emp"
match_Null_def: shows "match_Null \<equiv> \<Lambda> x k. case x of Null \<Rightarrow> k | Node\<cdot>a\<cdot>tree1\<cdot>tree2 \<Rightarrow> Fixrec.fail"
Rep_seqauto_tuple: fixes SA :: "('d, 'e, 'f) seqauto" shows "Rep_seqauto SA = (States SA, InitState SA, Labels SA, Delta SA)"
excised_chain_exists: fixes X :: "'b topology"   and U :: "'b set"   and T :: "'b set"   and S :: "'b set"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "X closure_of U \<subseteq> X interior_of T"   and "T \<subseteq> S"   and "singular_chain p (subtopology X S) c" obtains n :: "nat" and d :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" and e :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" where "singular_chain p (subtopology X (S - U)) d" and "singular_chain p (subtopology X T) e" and "(singular_subdivision p ^^ n) c = d + e"
path_connectedin_subtopology: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set" shows "path_connectedin (subtopology X S) T = (path_connectedin X T \<and> T \<subseteq> S)"
domains_def: fixes f :: "Arr" shows "domains f \<equiv> {a. ide a \<and> local.comp f a \<noteq> null}"
is_Inf_binary: fixes x :: "'b"   and y :: "'b"   and inf :: "'b" shows "is_Inf {x, y} inf = is_inf x y inf"
fix_ind: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a \<rightarrow> 'a" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (F\<cdot>x)" shows "P (fix\<cdot>F)"
Resp_def: fixes p :: "'b"   and ou :: "'d"   and s :: "('a, 'b, 'c) SLin_state"   and s' :: "('a, 'b, 'c) SLin_state" shows "Resp p ou s s' \<equiv> status s p = Pending \<and> initialized s \<and> contains (dstate s) (pending s p) \<and> ou = \<gamma> (dstate s) (pending s p) \<and> s' = s\<lparr>status := (status s)(p := Ready)\<rparr>"
circle_y_x_coord_bounds: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "0 \<le> circle_y (x_coord x) \<and> circle_y (x_coord x) \<le> 1 / 2"
inject_2: fixes x2 :: "name"   and y2 :: "name" shows "(BoundOutputS x2 = BoundOutputS y2) = (x2 = y2)"
NatPredicateTippingPoint: fixes n2 :: "nat"   and Pr :: "nat \<Rightarrow> bool" assumes "0 < n2"   and "Pr 0"   and "\<not> Pr n2" shows "\<exists>n<n2. Pr n \<and> \<not> Pr (Suc n)"
simps_5: fixes f1 :: "CTypeId \<Rightarrow> 'a"   and f2 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and x1 :: "CTypeId" shows "(case new_instance x1 of new_instance x \<Rightarrow> f1 x | new_array x xa \<Rightarrow> f2 x xa) = f1 x1"
sel_spec_def: fixes sel :: "(('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) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> ((('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" shows "sel_spec sel = (\<forall>gs bs ps data. ps \<noteq> [] \<longrightarrow> sel gs bs ps data \<noteq> [] \<and> set (sel gs bs ps data) \<subseteq> set ps)"
up_filter: fixes x :: "'a" shows "ord.filter (\<lambda>x y. y \<le> x) (\<down>x)"
connected_Un_clopen_in_complement: fixes S :: "'c set"   and U :: "'c set"   and T :: "'c set" assumes "connected S"   and "connected U"   and "S \<subseteq> U"   and "openin (top_of_set (U - S)) T"   and "closedin (top_of_set (U - S)) T" shows "connected (S \<union> T)"
union_assoc: fixes M :: "'a multiset"   and N :: "'a multiset"   and K :: "'a multiset" shows "M + N + K = M + (N + K)"
max_channels_def: shows "max_channels \<equiv> 65535::'a"
minus_0_r: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<ominus> \<zero> = x"
assms_1: shows "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"
check_ipresIGWlsAbs: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model" shows "ipresIGWlsAbs check (errMOD MOD) MOD"
zero_fls_def: shows "0 \<equiv> Abs_fls (\<lambda>_. 0::'a)"
has_measure_differentiable_image: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and m :: "real" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "inj_on f S" shows "(f ` S \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue (f ` S) = m) = ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral m) S"
doc_tokens_def: fixes p :: "(symbol \<times> character list) list" shows "doc_tokens p = (wellformed_tokens p \<and> is_prefix (chars p) Doc)"
assms: shows "cs\<^bsup>\<pi>\<^esup> i = xs @ [x, x'] @ ys"
induct: fixes P :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a option" assumes "\<And>x. P None x"   and "\<And>v. P (Some v) None"   and "\<And>x y. P (Some x) (Some y)" shows "P a0 a1"
to_name_def: fixes of_name :: "name \<Rightarrow> 'c" assumes "class.is_name of_name" shows "is_name.to_name of_name = inv of_name"
norm_20: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(P \<W> (\<lambda>s. False)) = \<box>P"