hcju/isabelleps huggingface dataset:
simps_1: fixes x21 :: "'c tree"   and x22 :: "'c"   and x23 :: "'c tree"   and y21 :: "'c tree"   and y22 :: "'c"   and y23 :: "'c tree" shows "(\<langle>x21, x22, x23\<rangle> = \<langle>y21, y22, y23\<rangle>) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
PRIO_TAGI: fixes ma :: "int"   and mi :: "int" shows "PRIO_TAG ma mi"
2: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. transitive x \<and> linear_orderable_6 x)"
fct_gen_subalgebra_fct_measurable: fixes X :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "X \<in> space M \<rightarrow> space N" shows "X \<in> fct_gen_subalgebra M N X \<rightarrow>\<^sub>M N"
complete_development_prm: fixes A :: "'a trm"   and A' :: "'a trm"   and \<pi> :: "'a prm" assumes "A >>> A'" shows "\<pi> \<cdot> A >>> \<pi> \<cdot> A'"
LeadsTo_UN_UN: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> F \<in> A i \<longmapsto>w A' i" shows "F \<in> \<Union> (A ` I) \<longmapsto>w \<Union> (A' ` I)"
collapse: fixes BinomialTree :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.Node (BinomialHeapStruc.val BinomialTree) (BinomialHeapStruc.prio BinomialTree) (BinomialHeapStruc.rank BinomialTree) (BinomialHeapStruc.children BinomialTree) = BinomialTree"
poly_div_gcd_squarefree_2: fixes p :: "'a poly"   and x :: "'a" assumes "p \<noteq> 0" shows "(poly (p div gcd p (pderiv p)) x = (0::'a)) = (poly p x = (0::'a))"
tauCases: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Prop :: "freeRes \<Rightarrow> pi \<Rightarrow> bool" assumes "\<tau>.P \<longmapsto> \<alpha> \<prec> P'"   and "\<lbrakk>\<alpha> = \<tau>; P = P'\<rbrakk> \<Longrightarrow> Prop (\<tau>) P" shows "Prop \<alpha> P'"
nonzero_pow_nonzero: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "a [^] n \<noteq> \<zero>"
Not_Not: fixes r :: "'a language" shows "Coinductive_Language.Not (Coinductive_Language.Not r) = r"
measure_eqI: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" shows "M = N"
that: fixes l' :: "('a \<times> 'd) list" assumes "(tsl m, l') \<in> \<langle>Rk \<times>\<^sub>r Rv\<rangle>list_rel"   and "RETURN l' \<le> it_to_sorted_list (key_rel ordR) (map_to_set m')" shows "thesis"
prems_2: shows "InvariantWatchListsUniq (getWatchList state')"
committed_reads_see_committed_writes_def: fixes ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('a, 'b) justifying_execution" shows "committed_reads_see_committed_writes ws J = (\<forall>n. \<forall>r'\<in>read_actions (justifying_exec (J (Suc n))) \<inter> committed (J (Suc n)). let r = action_translation (J (Suc n)) r'; committed_n = action_translation (J n) ` committed (J n) in r \<in> committed_n \<or> action_translation (J (Suc n)) (justifying_ws (J (Suc n)) r') \<in> committed_n \<and> ws r \<in> committed_n)"
assms_2: shows "y \<in> zero_space\<^sub>N N"
simps: fixes i :: "nat"   and j :: "nat" shows "rec_ci_id i j = addition j i (i + 1)"
independent_distinct: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list" assumes "ring R"   and "subfield K R"   and "ring.independent R K Us" shows "distinct Us"
fls_base_factor_one: shows "fls_base_factor 1 = 1"
mcont_times_ennreal1: fixes x :: "ennreal" shows "mcont Sup (\<le>) Sup (\<le>) ((*) x)"
fresh_def: fixes a :: "atom"   and x :: "'a" shows "a \<sharp> x \<equiv> a \<notin> supp x"
ide_self_inverse: fixes a :: "'a" assumes "ide a" shows "inverse_arrows a a"
Blacks_def: fixes M :: "node list" shows "Blacks M \<equiv> \<lbrace>\<acute>(<) (length M) \<and> \<acute>((!) M) = Black\<rbrace>"
poly_mod_dvd_drop_smult: fixes u :: "int poly"   and p :: "int"   and c :: "int"   and l :: "nat"   and f :: "int poly" assumes "monic u"   and "prime p"   and "c \<noteq> 0"   and "\<bar>c\<bar> < p ^ l"   and "poly_mod.dvdm (p ^ l) u (smult c f)" shows "poly_mod.dvdm p u f"
variety_ofI_alt: fixes a :: "'d \<Rightarrow> 'c"   and F :: "(('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "poly_eval a ` F \<subseteq> {0::'c}" shows "a \<in> \<V> F"
B: fixes x :: "'a" assumes "x \<in> S" shows "norm (f (Suc n) x) \<le> B"
point_construction_different: fixes A :: "'p"   and B :: "'p" shows "\<exists>C. Bet A B C \<and> B \<noteq> C"
isomorphicE: fixes a :: "('a, 'a set) Maps.arr"   and a' :: "('a, 'a set) Maps.arr" assumes "Maps.isomorphic a a'" obtains f :: "('a, 'a set) Maps.arr" where "Maps.in_hom f a a' \<and> Maps.iso f"
uminus_zero: shows "~\<^sub>h 0\<^sub>h = 0\<^sub>h"
poly_deg_zero_imp_monomial: fixes p :: "('d \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'e" assumes "poly_deg p = (0::'c)" shows "monomial (lookup p 0) 0 = p"
disj_matrices_add: fixes A :: "'a matrix"   and B :: "'a matrix"   and C :: "'a matrix"   and D :: "'a matrix" assumes "disj_matrices A B"   and "disj_matrices C D"   and "disj_matrices A D"   and "disj_matrices B C" shows "(A + B \<le> C + D) = (A \<le> C \<and> B \<le> D)"
linear_comb_memTr: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "'e set"   and T :: "'e set"   and n :: "nat" assumes "R module M"   and "submodule R M N"   and "T \<subseteq> N" shows "\<forall>f s. f \<in> {j. j \<le> n} \<rightarrow> T \<and> s \<in> {j. j \<le> n} \<rightarrow> carrier R \<longrightarrow> l_comb R M n s f \<in> N"
order_pair_axioms: shows "order_pair (s_mul_ext NS S) (ns_mul_ext NS S)"
assms_2: shows "of_nat (card G) \<noteq> (0::'f)"
sigma_algebra_axioms: fixes i :: "nat" shows "sigma_algebra (space M) (A i)"
that: fixes q' :: "'b" assumes "q = \<delta> q' (w n)"   and "x \<in> configuration q' n" shows "thesis"
WtNFT_def: fixes cl :: "('d, 'e, 'f) cmd list" shows "WtNFT cl \<equiv> \<Sum>n\<in>theNFT cl. 1 / real (length cl)"
subcls1I: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list"   and rest :: "fdecl list \<times> 'a mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "subcls1p P C D"
inv: shows "\<forall>p\<in>P. \<exists>s\<in>V. p = {s} \<union> neighbors s"
ANR_retract_of_ANR: fixes T :: "'b set"   and S :: "'b set" assumes "ANR T"   and "S retract_of T" shows "ANR S"
lnth_lzip: fixes n :: "nat"   and xs :: "'c llist"   and ys :: "'d llist" assumes "enat n < llength xs"   and "enat n < llength ys" shows "lnth (lzip xs ys) n = (lnth xs n, lnth ys n)"
brn_index: fixes n :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list"   and i :: "nat" assumes "n < length cl"   and "i < brn (cl ! n)" shows "brnL cl n + i < brnL cl (length cl)"
cons_chainp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and S :: "nat \<Rightarrow> 'b" assumes "P x (S 0)"   and "\<forall>i. P (S i) (S (Suc i))" shows "\<forall>i. P (case i of 0 \<Rightarrow> x | Suc x \<Rightarrow> S x) (case Suc i of 0 \<Rightarrow> x | Suc x \<Rightarrow> S x)"
Rep_FingerTree_cases: fixes y :: "('a, 'b) FingerTreeStruc" assumes "y \<in> {t. FingerTreeStruc.ft_invar t}" obtains x :: "('a, 'b) FingerTree" where "y = Rep_FingerTree x"
add_cancel_right_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(a = plus b a) = (b = zero)"
of_bl_False: fixes xs :: "bool list" shows "of_bl (False # xs) = of_bl xs"
height_gt_0_notin_alphabet_imp_sibling_left: fixes t\<^sub>1 :: "'a tree"   and a :: "'a"   and w :: "nat"   and t\<^sub>2 :: "'a tree" assumes "0 < height t\<^sub>1"   and "a \<notin> alphabet t\<^sub>1" shows "sibling (Node w t\<^sub>1 t\<^sub>2) a = sibling t\<^sub>2 a"
END_BEGIN_before: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "END i \<lless> BEGIN j = ((i, j) \<in> b)"
replicate_0: fixes x :: "'g" shows "replicate 0 x = []"
rot_circle_top_edge_def: shows "rot_circle_top_edge = (- 1, \<lambda>y. (0::'b, d / 2))"
A': shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> Cop.\<i> Cop.trg Cop.src f' g \<eta>' \<epsilon>'"
less_eq_poly_mapping_def: shows "(\<le>) \<equiv> map_fun lookup (map_fun lookup id) (\<lambda>f g. less_fun f g \<or> f = g)"
inf1E: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(A \<sqinter> B) x" obtains "A x" and "B x"
X_def: shows "X \<equiv> abs_var (BPS (binsert (True, undefined) (binsert (False, undefined) bempty)))"
PP_deff: fixes t :: "'trm" shows "PP t = subst P t xx"
that: shows "\<I>,\<I>' \<turnstile>\<^sub>C conv' \<sim> conv"
if_Assertion_assumption: fixes p :: "'b Assertion"   and x :: "'b"   and y :: "'b" shows "{\<cdot> p } * x \<squnion> {\<cdot> - p } * y = [\<cdot> p ] * x \<sqinter> ([\<cdot> - p ] * y)"
comb: fixes rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma>' :: "(name, value) fmap"   and u :: "sterm"   and u' :: "value"   and env :: "(name, value) fmap"   and uu :: "Term_Class.term"   and rhs :: "sterm"   and val :: "value" assumes "rs, \<Gamma> \<turnstile>\<^sub>v t \<down> Vabs cs \<Gamma>'"   and "rs, \<Gamma> \<turnstile>\<^sub>v u \<down> u'"   and "vfind_match cs u' = Some (env, uu, rhs)"   and "rs, \<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val" shows "rs, \<Gamma> \<turnstile>\<^sub>v t $\<^sub>s u \<down> val"
singular_chain_boundary: fixes p :: "nat"   and X :: "'c topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X c" shows "singular_chain (p - Suc 0) X (chain_boundary p c)"
float_val_ge_bottomfloat: fixes a :: "('a, 'b) IEEE.float" assumes "1 < LENGTH('a)"   and "is_finite a" shows "valof (- topfloat) \<le> valof a"
CSP1_idem: shows "CSP1 = CSP1 \<circ> CSP1"
uop_def: shows "uop \<equiv> map_fun id (map_fun Rep_uexpr mk\<^sub>e) (\<lambda>f e b. f (e b))"
disc_7: shows "\<not> is_SAT_PUSH' RELABEL'"
H: shows "ptrm_infer_type \<Gamma> (PFn x T A) = Some \<tau>"
smallstepP_generates_proofstream: fixes \<pi>\<^sub>1 :: "Syntax.term list"   and eP :: "Syntax.term"   and \<pi>\<^sub>2 :: "Syntax.term list"   and eP' :: "Syntax.term" assumes "\<lless>\<pi>\<^sub>1, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>2, eP'\<ggreater>" obtains \<pi> :: "Syntax.term list" where "\<pi>\<^sub>2 = \<pi>\<^sub>1 @ \<pi>"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "C'.arr f"   and "C'.dom f = a" shows "C' f a = f"
True_in_image_Bex: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a set" shows "(True \<in> P ` A) = Bex A P"
Lim_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b" shows "(f \<longlongrightarrow> l) at_infinity = (\<forall>e>0. \<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> dist (f x) l < e)"
minus_equality: fixes y :: "'c"   and x :: "'c" assumes "y \<oplus>\<^bsub>V\<^esub> x = \<zero>\<^bsub>V\<^esub>"   and "x \<in> carrier V"   and "y \<in> carrier V" shows "\<ominus>\<^bsub>V\<^esub> x = y"
grow_mono: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "hbc n \<le> hbc n'"
isinceE: fixes I :: "nat set"   and P' :: "nat \<Rightarrow> bool"   and P :: "nat \<Rightarrow> bool" assumes "P' t'. t' \<S> t I. P t" obtains t :: "nat" where "t \<in> I" and "P t"
that: assumes "0 \<le> max 0 (C * exp (3 / 2 * pi * \<bar>Im s\<bar>))"   and "\<And>r z. \<lbrakk>0 < r; r < pi; z \<in> sphere 0 r\<rbrakk> \<Longrightarrow> cmod (f s z) \<le> max 0 (C * exp (3 / 2 * pi * \<bar>Im s\<bar>)) * r powr (- Re s - 1)" shows "thesis"
cases: fixes x :: "chan" obtains A :: "agent" and B :: "agent" and M :: "msg" where "x = Insec Aa__ Ba__ Ma__"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Confid Ab__ Bb__ Mb__"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Auth Ac__ Bc__ Mc__"   | A :: "agent" and B :: "agent" and M :: "msg" where "x = Secure A B M"
sel_1: fixes x1 :: "'c set"   and x2 :: "'d set"   and x3 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd set"   and x4 :: "'d \<Rightarrow> bool" shows "alphabet (nba x1 x2 x3 x4) = x1"
netmap_subnets_samef: fixes s1 :: "'d net_state"   and f :: "nat \<Rightarrow> 'd option"   and s2 :: "'d net_state" assumes "netmap s1 = f"   and "netmap s2 = f" shows "netmap (SubnetS s1 s2) = f"
oldest_token_bounded: fixes q :: "'b"   and n :: "nat"   and x :: "nat" assumes "oldest_token q n = Some x" shows "x \<le> n"
eval_inverse: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (inverse F) = (\<lambda>x. inverse (eval F x))"
graph_eq: fixes g :: "'b ugraph" shows "graph (nodes g) (edges g) = g"
Suc_n_is_len_\<tau>X: shows "Suc n = length (\<tau> \<upharpoonleft> X)"
accept\<^sub>R_simp: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set) set"   and w :: "nat \<Rightarrow> 'e" shows "accept\<^sub>R (\<delta>, q\<^sub>0, \<alpha>) w = (\<exists>(Fin, Inf)\<in>\<alpha>. limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> Fin = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> Inf \<noteq> {})"
assms: fixes i :: "'a" shows "space (M i) = UNIV"
properties_for_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "mset ys = mset xs"   and "linorder.sorted less_eq ys" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = ys"
assms_1: shows "\<turnstile> {\<lambda>s. P s \<and> bval b s} c {P}"
modelAppendTautologicalFormula: fixes valuation :: "Literal list"   and F :: "Literal list list"   and F' :: "Literal list list" assumes "model valuation F"   and "\<forall>c. c el F' \<longrightarrow> clauseTautology c"   and "vars F \<union> vars F' \<subseteq> vars valuation" shows "model valuation (F @ F')"
nv2L_varsL: fixes x :: "Preliminaries.var"   and l :: "('b, 'psym) lit" assumes "x \<in> nv2L l" shows "x \<in> varsL l"
monad_alt_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and alt :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "monad_alt return bind alt" shows "monad_alt (return_state return) (bind_state bind) (alt_state alt)"
vdomain_vconst_on: fixes A :: "V"   and c :: "V" shows "\<D>\<^sub>\<circ> (vconst_on A c) = A"
Scon_def: fixes \<phi>s :: "'aprop dfmla set" shows "Scon \<phi>s \<equiv> foldr Con (asList \<phi>s) Tr"
inverse_le_imp_le_neg: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less b zero" shows "less_eq b a"
one_int_def: shows "1 \<equiv> Abs_Integ (1, 0)"
distr_return: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and x :: "'d" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "x \<in> space M" shows "distr (return M x) N f = return N (f x)"
shunting_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- x \<sqinter> - y \<le> - z) = (- x \<le> - z \<squnion> - - y)"
subd_power_uminus: fixes p :: "nat"   and k :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" shows "subd p (frag_cmul ((- 1) ^ k) c) = frag_cmul ((- 1) ^ k) (subd p c)"
path_components_of_disjoint: fixes C :: "'a set"   and X :: "'a topology"   and C' :: "'a set" assumes "C \<in> path_components_of X"   and "C' \<in> path_components_of X" shows "disjnt C C' = (C \<noteq> C')"
sd_path_Nil_iff: fixes e :: "'b \<times> 'b"   and w :: "'b"   and p :: "('b \<times> 'b) list" shows "(sd_path e w p = []) = (p = [])"
ESS_2: shows "(edges g\<^sub>2)\<^sup>* \<subseteq> (edges (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2)))\<^sup>*"
order_pair_def: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" shows "order_pair S NS \<equiv> pre_order_pair S NS \<and> compat_pair S NS"
QAN_Normal: fixes t :: "trm" assumes "t \<leadsto>|" shows "t \<Down> t"
inf_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "\<Sqinter>\<^bsub>L\<^esub>A \<in> carrier L"
asymp_equiv_add_rightI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<sim>[F] g"   and "h \<in> o[F](g)" shows "(\<lambda>x. f x + h x) \<sim>[F] g"
\<epsilon>: shows "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> F \<epsilon> = unit (src\<^sub>C g) \<cdot>\<^sub>D \<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>) \<cdot>\<^sub>D D.inv (\<Phi> (f, g))"
assms: fixes x :: "nat" assumes "x \<in> set xs" shows "max_Var_floatarith f \<le> x"
segment_as_ball_2: fixes a :: "'a"   and b :: "'a" shows "open_segment a b = affine hull {a, b} \<inter> ball ((a + b) /\<^sub>R 2) (norm (b - a) / 2)"
reduced_word_for_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero" shows "monoid_add.reduced_word_for plus zero A a as \<equiv> is_arg_min length (\<lambda>as. as \<in> lists A \<and> monoid_add.sum_list plus zero as = a) as"
beta_lc: fixes t :: "sterm"   and t' :: "sterm" assumes "t -> t'" shows "lc t \<and> lc t'"
join_empty_right: fixes t :: "'b option list set"   and pos :: "bool" shows "join t pos {} = (if pos then {} else t)"
limsup_subseq_lim: fixes u :: "nat \<Rightarrow> 'b" shows "\<exists>r. strict_mono r \<and> (u \<circ> r) \<longlonglongrightarrow> limsup u"
map_fixed_idle_closed: fixes t :: "Syntax.term" assumes "closed t" shows "map_fixed undefined [] t = t"
notin: shows "\<nexists>k'. cs\<^bsup>\<pi> @\<^bsup>k\<^esup> \<pi>l\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'"
l_get_parent_get_host_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_get_parent_get_host_wf heap_is_wellformed get_parent get_shadow_root get_host get_child_nodes \<equiv> (\<forall>h P ptr. heap_is_wellformed h \<longrightarrow> (\<forall>parent. (\<forall>children child. h \<turnstile> get_child_nodes parent \<rightarrow>\<^sub>r children \<longrightarrow> child \<in> set children \<longrightarrow> P (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 child)) \<longrightarrow> (\<forall>shadow_root host. parent = cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host \<longrightarrow> h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root \<longrightarrow> P (cast\<^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\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root)) \<longrightarrow> P parent) \<longrightarrow> P ptr) \<and> (\<forall>h P ptr. heap_is_wellformed h \<longrightarrow> (\<forall>child. (\<forall>parent child_node. child = 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 child_node \<longrightarrow> h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<longrightarrow> P parent) \<longrightarrow> (\<forall>host shadow_root. child = cast\<^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\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root \<longrightarrow> h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host \<longrightarrow> P (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)) \<longrightarrow> P child) \<longrightarrow> P ptr)"
le_fininfE: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" obtains v\<^sub>2 :: "'item list" where "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2" and "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
rv: fixes i :: "'i" assumes "i \<in> I" shows "random_variable (M' i) (X i)"
simps_1: fixes cmp :: "'b comparator"   and y :: "'b" shows "Sorting_Algorithms.insort cmp y [] = [y]"
game_split: fixes g :: "game" shows "g = Game (left_options g) (right_options g)"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "semiring_hom hom" shows "Ring_Hom.ring_hom hom"
set_tag_name_locs_impl: shows "set_tag_name_locs = a_set_tag_name_locs"
assms_1: shows "matchs ps ts\<^sub>2 = Some env\<^sub>2"
the_gpv_lift_resumption: fixes r :: "('d, 'e, 'f) Resumption.resumption" shows "the_gpv (lift_resumption r) = (if Resumption.resumption.is_Done r then if Option.is_none (Resumption.resumption.result r) then return_pmf None else return_spmf (Pure (the (Resumption.resumption.result r))) else return_spmf (IO (Resumption.resumption.output r) (lift_resumption \<circ> Resumption.resumption.resume r)))"
tconf_def: fixes thread_id2addr :: "'a \<Rightarrow> 'b"   and typeof_addr :: "'c \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'c"   and t :: "'a" shows "heap_base.tconf thread_id2addr typeof_addr P h t \<equiv> \<exists>C. typeof_addr h (thread_id2addr t) = \<lfloor>Class_type C\<rfloor> \<and> P \<turnstile> C \<preceq>\<^sup>* Thread"
line_extension_smult_closed: fixes E :: "'a set"   and a :: "'a"   and k :: "'a"   and u :: "'a" assumes "\<And>k v. \<lbrakk>k \<in> K; v \<in> E\<rbrakk> \<Longrightarrow> k \<otimes> v \<in> E"   and "E \<subseteq> carrier R"   and "a \<in> carrier R"   and "k \<in> K"   and "u \<in> line_extension K a E" shows "k \<otimes> u \<in> line_extension K a E"
less_imp_not_eq2: fixes x :: "'a list"   and y :: "'a list" assumes "strict_prefix x y" shows "(y = x) = False"
generator: fixes x :: "('f, 'e) generator" shows "generator x \<in> {g. terminates g}"