hcju/isabelleps huggingface dataset:
inline1_Inl_results_gpv: fixes x :: "'b"   and s' :: "'s"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "Inl (x, s') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "x \<in> results_gpv \<I> gpv \<and> I s'"
assms_1: shows "\<forall>r\<in>set rs. P (get_match r)"
divide_prod_assoc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x / (y * z) = x / y / z"
f_1_def: fixes x :: "int" shows "local.f_1 x = (THE y. local.P x y)"
inj_on_hom_induced_section_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and p :: "int" assumes "section_map X Y f" shows "inj_on (hom_induced p X {} Y {} f) (carrier (homology_group p X))"
mod: shows "LinearCombinations.submodule K (subspace_sum W1 W2) V"
less_prod_def: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "less_prod leq_a less_a less_b = (\<lambda>(x1, x2) (y1, y2). less_a x1 y1 \<or> leq_a x1 y1 \<and> less_b x2 y2)"
LENGTH: shows "length [] = id 0 \<and> (\<forall>h t. length (h # t) = Suc (length t))"
circ_decompose_6: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<star> * (y * x\<^sup>\<star>)\<^sup>\<star> = y\<^sup>\<star> * (x * y\<^sup>\<star>)\<^sup>\<star>"
i_last_message_hold_i_append_mult: fixes xs :: "'b message_af list"   and m :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "length xs = m * k" shows "(xs \<frown> f) \<longmapsto> k = xs \<longmapsto> k \<frown> f \<longmapsto> k"
cl10_d_var1: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d (x \<inter> NC) = 1\<^sub>\<sigma> \<inter> (x \<cdot> NC)"
Cn_imp_bis: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longrightarrow>. q) = (Cn (A \<union> {q}) \<subseteq> Cn (A \<union> {p}))"
antisym_conv: fixes y :: "'e list"   and x :: "'e list" assumes "prefix y x" shows "prefix x y = (x = y)"
is_expansion_uminus: 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 basis :: "(real \<Rightarrow> 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"   and "basis_wf basis"   and "is_expansion F basis" shows "is_expansion (uminus F) basis"
coreflexive_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c"   and j :: "'c" assumes "matrix_idempotent_semiring.coreflexive f"   and "i \<noteq> j" shows "f (i, j) = bot"
mbuf2t_take_add'_2: fixes f :: "'b option list set \<Rightarrow> 'b option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "'b option list set list"   and ys :: "'b option list set list"   and buf :: "'b option list set list \<times> 'b option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "'b option list set list \<times> 'b option list set list"   and nts' :: "nat list"   and \<sigma> :: "(char list \<times> 'b list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'b list set"   and \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula"   and j' :: "nat" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) [Monitor.progress \<sigma> \<phi> j..<Monitor.progress \<sigma> \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) [Monitor.progress \<sigma> \<psi> j..<Monitor.progress \<sigma> \<psi> j'] ys"   and "j \<le> j'" shows "wf_ts \<sigma> j' \<phi> \<psi> nts'"
bisim1BlockThrowSome: fixes P :: "'b prog"   and h :: "'heap"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and V :: "nat"   and T :: "ty"   and v :: "'addr val" assumes "P,e,h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "P,{V:T=\<lfloor>v\<rfloor>; e},h \<turnstile> (Throw a, xs) \<leftrightarrow> (stk, loc, Suc (Suc pc), \<lfloor>a\<rfloor>)"
generat_lub_def: fixes lub1 :: "'j set \<Rightarrow> 'g"   and lub2 :: "'k set \<Rightarrow> 'h"   and lub3 :: "'l set \<Rightarrow> 'i"   and A :: "('j, 'k, 'l) generat set" shows "generat_lub lub1 lub2 lub3 A = (if \<exists>x\<in>A. is_Pure x then Pure (lub1 (generat.result ` (A \<inter> {f. is_Pure f}))) else IO (lub2 (generat.output ` (A \<inter> {f. \<not> is_Pure f}))) (lub3 (continuation ` (A \<inter> {f. \<not> is_Pure f}))))"
eq_id_iff: fixes f :: "'q \<Rightarrow> 'q" shows "(\<forall>x. f x = x) = (f = id)"
hyp1: shows "fst (Gauss_Jordan_upt_k_det_P A k) = det (fst (Gauss_Jordan_upt_k_PA A k))"
dvd_div_ge_1: fixes a :: "nat"   and b :: "nat" assumes "1 \<le> a"   and "b dvd a" shows "1 \<le> a div b"
Inf_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" assumes "e \<in> funcset UNIV {- 1..1}" shows "Inf_aform X \<le> aform_val e X"
map_pmf_of_set_inj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)"
assms_3: shows "F = (\<lambda>x. \<forall>rel\<in>R. \<forall>(a, b, c)\<in>Atoms rel. rel (a * x\<^sup>2 + b * x + c) 0)"
p: shows "Ifm bbs (x # bs) (NDvd j (CN 0 c e))"
instInp_fmla: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and \<phi> :: "'c"   and t :: "'b" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "\<phi> \<in> fmla"   and "t \<in> trm" shows "Generic_Syntax.instInp var subst \<phi> t \<in> fmla"
in_components_self: fixes s :: "'a set" shows "(s \<in> components s) = (connected s \<and> s \<noteq> {})"
lm01: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "argmax f A = A \<inter> f -` {Max (f ` A)}"
new_document_no_child_nodes: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes (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>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_document_ptr) \<rightarrow>\<^sub>r []"
map_of_zip_map: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" shows "map_of (zip xs (map f xs)) = (\<lambda>x. if x \<in> set xs then Some (f x) else None)"
ccpo_Sup_least: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and z :: "'c" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
valid_sub_consts: fixes p :: "('b, 'c) form"   and e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and cs :: "'b list" assumes "\<forall>e f g. eval e f g p" shows "eval e f g (sub_consts cs p)"
simps_2: fixes x :: "'a"   and xs :: "[:'a:]"   and p :: "'a \<rightarrow> tr" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "sfilter\<cdot>p\<cdot>(x :# xs) = If p\<cdot>x then x :# sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs"
x\<^sub>0'_min_def: shows "x\<^sub>0'_min = max A (max X (max 1 (max a0 (max (real gx3) (real x\<^sub>1 + 1)))))"
simps_26: fixes vb :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(SKIP {vb} \<le> v;; va) = False"
coboundedI1: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<le> c" shows "a \<sqinter> b \<le> c"
Mp_unique_factorization_m: fixes m :: "int"   and f :: "int poly"   and cfs :: "int \<times> int poly multiset" assumes "poly_mod_2 m" shows "poly_mod.unique_factorization_m m (poly_mod.Mp m f) cfs = poly_mod.unique_factorization_m m f cfs"
intro: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ord_term_strict :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"
SV_inv4_sv_round: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and D :: "process \<Rightarrow> val option" shows "{SV_inv4 \<inter> (Vinv1 \<inter> Vinv2)} sv_round r S v D {> SV_inv4}"
low_indistinguishable_def: fixes mds :: "Mode \<Rightarrow> var\<^sub>C set"   and c\<^sub>1 :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>2 :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt" shows "conc.low_indistinguishable mds c\<^sub>1 c\<^sub>2 = (\<forall>mem\<^sub>1 mem\<^sub>2. conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<longrightarrow> conc.mm_equiv_abv \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>C \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>C)"
norm_prod_def: fixes x :: "'b \<times> 'c" shows "norm x = sqrt ((norm (fst x))\<^sup>2 + (norm (snd x))\<^sup>2)"
Rep_ref_induct: fixes y :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "y \<in> ref"   and "\<And>x. P (Rep_ref x)" shows "P y"
lookup_listsum: fixes "is" :: "nat list"   and ds :: "nat list"   and As :: "'a tensor list" assumes "is \<lhd> ds"   and "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "lookup (listsum ds As) is = (\<Sum>A\<leftarrow>As. lookup A is)"
tm_min_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t1 :: "taylor_model"   and y :: "real"   and t2 :: "taylor_model"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat" assumes "x \<in>\<^sub>i range_tm e t1"   and "y \<in>\<^sub>i range_tm e t2"   and "num_params (tm_poly t1) \<le> length I"   and "num_params (tm_poly t2) \<le> length I"   and "develops_at_within e a I" shows "min x y \<in>\<^sub>i range_tm e (tm_min prec I a t1 t2)"
deriv_bound_image: fixes r' :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> 'b) set" assumes "deriv_bound r' (f a) n"   and "\<And>a b. (a, b) \<in> r \<Longrightarrow> (f a, f b) \<in> r'\<^sup>+" shows "deriv_bound r a n"
finite_Keys: fixes A :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "finite A" shows "finite (Keys A)"
limit_cone_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" shows "limit_cone J C D a \<chi> \<equiv> (category C \<and> category J) \<and> diagram J C D \<and> Limit.cone J C D a \<chi> \<and> limit_cone_axioms J C D a \<chi>"
l\<^sub>2C1': shows "l\<^sub>2 \<in> C\<^sub>1'"
zero_local_ring_at_def: shows "zero_local_ring_at \<equiv> cq.zero_rel"
fmimage_drop: fixes a :: "'d"   and m :: "('d, 'c) fmap"   and A :: "'d fset" shows "fmimage (fmdrop a m) A = fmimage m (A |-| {|a|})"
C_eq_RS2_mr: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and p :: "((int \<times> int) set set, int) Combinators list" shows "applied_rule_rev C x (removeShadowRules2 p) = applied_rule_rev C x p"
take_stake: fixes n :: "nat"   and m :: "nat"   and s :: "'d stream" shows "take n (stake m s) = stake (min n m) s"
simps: fixes a :: "'c"   and r :: "('c \<times> 'd) set" shows "(a \<in> Domain r) = (\<exists>aa b. a = aa \<and> (aa, b) \<in> r)"
stable_pair_onE: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set" assumes "stable_pair_on ds XD_XH" obtains "- Rf CH (snd XD_XH) = fst XD_XH" and "- Rf (CD_on ds) (fst XD_XH) = snd XD_XH"
unique_dimension: fixes K :: "'a set"   and E :: "'a set" assumes "subfield K R"   and "finite_dimension K E" shows "\<exists>!n. dimension n K E"
adjust_preserves_span: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "(w \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us)) = (adjuster n w us + w \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set us))"
assms: shows "moebius_circline M H1 = moebius_circline M H2"
map_permI: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs <~~> ys" shows "map f xs <~~> map f ys"
bit_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a 0 = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
naturality: fixes f :: "'a \<times> 'a" assumes "C.VV.arr f" shows "\<Phi> (C.VV.cod f) \<cdot>\<^sub>D H\<^sub>DoFF.map f = FoH\<^sub>C.map f \<cdot>\<^sub>D \<Phi> (C.VV.dom f)"
SubstWellDefined: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a"   and d :: "('a, 'b) Expression"   and C :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> e : B"   and "Sig S \<triangleright> Vx : B \<turnstile> d : C" shows "Sig S \<triangleright> Vx : A \<turnstile> sub e in d : C"
mod_ring_eq: shows "(mod_ring_rel ===> mod_ring_rel ===> (=)) (=) (=)"
p: shows "p = r' + (\<Sum>g\<in>G. q' g \<odot> g)"
module_ow_axioms: shows "module_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (+\<^sub>V\<^sub>S\<^sub>_\<^sub>1) 0\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (-\<^sub>V\<^sub>S\<^sub>_\<^sub>1) uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 (*s\<^sub>o\<^sub>w\<^sub>_\<^sub>1)"
abs: fixes cs :: "(Term_Class.term \<times> sterm) list" assumes "t = Sabs cs" shows "thesis"
b: shows "\<forall>\<^sub>F K in sequentially. max c0 d2 \<le> real K"
set_def: shows "set \<equiv> ctor_fold_list (\<lambda>z. set1_pre_list z \<union> \<Union> (set2_pre_list z))"
simps_1: shows "Pairing_Heap_Tree.del_min \<langle>\<rangle> = \<langle>\<rangle>"
emeasure_PiM_emb: fixes M :: "'b \<Rightarrow> 'c measure"   and J :: "'b set"   and I :: "'b set"   and X :: "'b \<Rightarrow> 'c set" assumes "product_prob_space M"   and "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)" shows "emeasure (Pi\<^sub>M I M) (prod_emb I M J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. emeasure (M i) (X i))"
order_less_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x < y"   and "y < z" shows "x < z"
lr_appFLv: fixes R :: "('a \<Rightarrow> ValD) set"   and f :: "ValD"   and args :: "('a \<Rightarrow> ValD) list" assumes "logical_relation R"   and "(\<lambda>i. f) \<in> R"   and "set args \<subseteq> R" shows "appFLv f args \<in> R"
mod_mult_cong_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a"   and d :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo c (times b a)) d a = unique_euclidean_semiring.cong modulo c d a"
rb_aux_shape: fixes args :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_dom args" obtains bs :: "('t \<Rightarrow>\<^sub>0 'b) list" and ss :: "'t list" and z :: "nat" where "rb_aux args = ((bs, ss, []), z)"
sets_lebesgue_inner_closed: fixes S :: "'c set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'c set" where "closed T" and "T \<subseteq> S" and "S - T \<in> lmeasurable" and "emeasure lebesgue (S - T) < ennreal e"
idom_divide_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "idom_divide_hom hom \<equiv> idom_hom hom \<and> idom_divide_hom_axioms hom"
Sup_uexpr_def: shows "Sup \<equiv> map_fun ((`) Rep_uexpr) mk\<^sub>e (\<lambda>PS A. \<Sqinter>P\<in>PS. P A)"
NN_neg: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and N :: "'c"   and \<phi> :: "'c" assumes "Repr_Neg var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv enc N"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (Repr_Neg.NN var Var FvarsT Fvars subst N (enc \<phi>) (enc (Syntax_with_Connectives_False.neg imp fls \<phi>)))"
emeasure_embed_measure_image: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and M :: "'c measure" assumes "inj f"   and "X \<in> sets M" shows "emeasure (embed_measure M f) (f ` X) = emeasure M X"
simps: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "flow \<Delta> f = ((\<forall>x. f x \<le> capacity \<Delta> x) \<and> (\<forall>x. x \<noteq> source \<Delta> \<longrightarrow> x \<noteq> sink \<Delta> \<longrightarrow> KIR f x))"
mexecT_eq_mexecdT: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and s' :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "correct_state_ts \<Phi> (thr s) (shr s)" shows "P \<turnstile> s -t\<triangleright>ta\<rightarrow>\<^bsub>jvm\<^esub> s' = P \<turnstile> s -t\<triangleright>ta\<rightarrow>\<^bsub>jvmd\<^esub> s'"
le_cases: fixes x :: "'b"   and y :: "'b" obtains "x \<le> y"   | "y \<le> x"
vimage_eq_UN: fixes f :: "'e \<Rightarrow> 'f"   and B :: "'f set" shows "f -` B = (\<Union>y\<in>B. f -` {y})"
hd_filter_in_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "filter P xs \<noteq> []" shows "hd (filter P xs) \<in> set (filter P xs)"
fw_upd_leI: fixes m' :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and j :: "nat" assumes "m' i k \<le> m i k"   and "m' k j \<le> m k j"   and "m' i j \<le> m i j" shows "fw_upd m' k i j i j \<le> fw_upd m k i j i j"
mon_pl_set: fixes w :: "('b set \<times> 'b set) list" shows "mon_pl w = \<Union> {fst e \<union> snd e |e. e \<in> set w}"
sortedConcStart: fixes a :: "('b, 'c) Combinators"   and aa :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "FWNormalisationCore.sorted (a # aa # p) l"   and "in_list a l"   and "in_list aa l"   and "all_in_list p l"   and "singleCombinators [a]"   and "singleCombinators [aa]"   and "singleCombinators p" shows "FWNormalisationCore.sorted (a # p) l"
hom_one: shows "map_poly complex_of_real 1 = 1"
le_imp_inv: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.add_linorder plus minus zero less_eq less"   and "less_eq a b" shows "b = plus a (minus b a)"
pot_comp_pp_def: shows "pot_comp = (\<lambda>cmp. cmp)"
case_prod_eta: fixes f :: "'e \<times> 'f \<Rightarrow> 'g" shows "(\<lambda>(x, y). f (x, y)) = f"
generate_topology_on_eq: fixes S :: "'c set set" shows "generate_topology_on S = arbitrary union_of finite' intersection_of (\<lambda>x. x \<in> S)"
rel_refl_strong: fixes x :: "'a \<times> 'b"   and R1a :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2a :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> Basic_BNFs.fsts x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> Basic_BNFs.snds x \<Longrightarrow> R2a z2 z2" shows "rel_prod R1a R2a x x"
symKey_compromise: fixes evs :: "event list" assumes "evs \<in> set_pur" shows "\<forall>SK KK. SK \<in> symKeys \<longrightarrow> (\<forall>K\<in>KK. K \<notin> range priEK) \<longrightarrow> (Key SK \<in> analz (Key ` KK \<union> knows Spy evs)) = (SK \<in> KK \<or> Key SK \<in> analz (knows Spy evs))"
horner_sum_simps_1: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'a" shows "horner_sum f a [] = (0::'a)"
right_residual: fixes a :: "'a"   and x :: "'a"   and b :: "'a" shows "(a * x \<le> b) = (x \<le> a r\<rightarrow> b)"
lift_types: fixes e :: "nat \<Rightarrow> LambdaType.type"   and ts :: "dB list"   and Ts :: "LambdaType.type list"   and i :: "nat"   and U :: "LambdaType.type" assumes "e \<tturnstile> ts : Ts" shows "e\<langle>i:U\<rangle> \<tturnstile> map (\<lambda>t. lift t i) ts : Ts"
Diagonal_to_Smith_row_i_PQ_def: fixes i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q) = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"