hcju/isabelleps huggingface dataset:
assms_3: fixes \<eta> :: "nat" shows "bound' \<eta> * max (bound_conv \<eta>) 1 \<le> bound \<eta>"
pred_def: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c measure" shows "pred M P = ({x \<in> space M. P x} \<in> sets M)"
tauFresh: fixes x :: "name"   and P :: "('a, 'b, 'c) psi" shows "x \<sharp> \<tau>.P = x \<sharp> P"
assms_1: shows "openin (topology\<^sub>N N) U"
augment_edge_cf': fixes u :: "nat"   and v :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'capacity"   and \<Delta> :: "'capacity" assumes "(u, v) \<in> cfE_of f" shows "cf_of (augment_edge f (u, v) \<Delta>) = (cf_of f)((u, v) := cf_of f (u, v) - \<Delta>, (v, u) := cf_of f (v, u) + \<Delta>)"
strict_trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a <s b"   and "b <s c" shows "a <s c"
word_arith_wis_8: shows "1 = word_of_int 1"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (\<lambda>f. C'.T (C.cod \<iota>\<^sub>C, f))"
ns: shows "ns = (case a of Leq v c \<Rightarrow> LEQ_ns q c | Geq v c \<Rightarrow> GEQ_ns q c)"
\<G>_elim: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> \<G> \<phi> w" shows "w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"
extr_monotone_param: fixes T :: "atom set"   and U :: "atom set"   and H :: "chmsg set" assumes "T \<subseteq> U" shows "extr T H \<subseteq> extr U H"
L1: fixes Q' :: "pi" assumes "Q \<longmapsto> \<tau> \<prec> Q'" shows "(P, Q') \<notin> Rel"
de_morgan1: fixes x :: "'a"   and y :: "'a" shows "!(!x \<cdot> !y) = \<tt>\<tt> x \<oplus> \<tt>\<tt> y"
assms_4: shows "\<forall>m. X m \<in> borel_measurable (F m)"
record_simps_4: fixes f :: "nat \<Rightarrow> nat"   and x :: "'a state" shows "defined_types (update_clock f x) = defined_types x"
log_induct: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "0 < n"   and "P 1"   and "\<And>n. \<lbrakk>2 \<le> n; P (n div 2)\<rbrakk> \<Longrightarrow> P n" shows "P n"
finfun_dom_def: fixes f :: "'c \<Rightarrow>f 'd" shows "finfun_dom f = Abs_finfun (\<lambda>a. f $ a \<noteq> finfun_default f)"
comp_sym_sym: fixes a0 :: "'a"   and a1 :: "'a" assumes "ide a0"   and "ide a1" shows "\
[a1, a0] \<cdot> \
[a0, a1] = a0 \<otimes> a1"
dist_vs1: shows "distinct (ram1 # vs1 @ [ram2])"
mnt: shows "maintainedA (constant_rules C \<union> identity_rules L \<union> {top_rule S_Top, nonempty_rule}) G'"
inv_P'PAQQ': fixes A :: "'b mat"   and n :: "nat"   and P :: "'b mat"   and P' :: "'b mat"   and Q :: "'b mat"   and Q' :: "'b mat" assumes "A \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "inverts_mat P' P"   and "inverts_mat Q Q'"   and "Q \<in> carrier_mat n n"   and "P' \<in> carrier_mat n n"   and "Q' \<in> carrier_mat n n" shows "P' * (P * A * Q) * Q' = A"
that: fixes a :: "complex"   and b :: "complex" assumes "Re a < Re b"   and "Im a < Im b"   and "s \<in> box a b"   and "f analytic_on box a b" shows "thesis"
equal_mod_timpls_inv_3: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "equal_mod_timpls TI (Fun f T) (Fun g S)"   and "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> ((the_Abs f, the_Abs g) \<in> set TI \<or> (the_Abs g, the_Abs f) \<in> set TI \<or> (\<exists>ti\<in>set TI. (the_Abs f, snd ti) \<in> set TI \<and> (the_Abs g, snd ti) \<in> set TI))"
nempty: shows "{i. i < j \<and> k \<le> i \<and> C \<in> lnth Ns i} \<noteq> {}"
in_hhomI: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and a :: "'b"   and b :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>"   and "src \<mu> = a"   and "trg \<mu> = b" shows "horizontal_homs.in_hhom V src trg \<mu> a b"
inf_assoc: fixes x :: "'g multiset"   and y :: "'g multiset"   and z :: "'g multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"
induct: fixes P :: "int \<times> int \<Rightarrow> real \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "real"   and a2 :: "(int \<times> int) list" assumes "\<And>uu_ uv_. P uu_ uv_ []"   and "\<And>uw_ ux_ uy_. P uw_ ux_ [uy_]"   and "\<And>p \<delta> p\<^sub>0 v va. (\<not> \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p) \<Longrightarrow> P p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (v # va)) \<Longrightarrow> P p \<delta> (p\<^sub>0 # v # va)" shows "P a0 a1 a2"
equivalence_maps_compose: fixes f :: "'b"   and f' :: "'b" assumes "D.equivalence_map f"   and "D.equivalence_map f'"   and "src\<^sub>D f' = trg\<^sub>D f" shows "D.equivalence_map (f' \<star>\<^sub>D f)"
cblinfun_eqI: fixes x :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and y :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" assumes "\<And>i. x *\<^sub>V i = y *\<^sub>V i" shows "x = y"
bot_comp: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times bot x = bot"
simps_14: shows "op_sem pdf_operator.Ln = lift_RealVal safe_ln"
us: shows "us \<equiv> getFrN (map snd txs) (pls r1 r2 # map fst txs) [] (length txs)"
splitting_sublemma_2: fixes C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and H :: "'b set"   and K :: "'b set" assumes "short_exact_sequence C B A g f"   and "f ` carrier A = H"   and "subgroup K B"   and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}"   and "H <#>\<^bsub>B\<^esub> K = carrier B" shows "g \<in> Group.iso (subgroup_generated B K) C"
simps_2: fixes c :: "'b"   and f :: "'c hp \<Rightarrow> 'b"   and h :: "'c hp" shows "lift_hp c f (Some h) = f h"
assms: shows "\<lparr>!x \<cdot> !y\<rparr>z\<lparr>z \<cdot> !x\<rparr>"
correct_state_def: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list"   and t :: "'thread_id" shows "correct_state \<Phi> t \<equiv> \<lambda>(xp, h, frs). compP2 P,h \<turnstile> t \<surd>t \<and> hconf h \<and> preallocated h \<and> (case frs of [] \<Rightarrow> True | f # fs \<Rightarrow> let (stk, loc, C, M, pc) = f in \<exists>Ts T mxs mxl\<^sub>0 is xt \<tau>. compP2 P \<turnstile> C sees M: Ts\<rightarrow>T = \<lfloor>(mxs, mxl\<^sub>0, is, xt)\<rfloor> in C \<and> \<Phi> C M ! pc = \<lfloor>\<tau>\<rfloor> \<and> conf_f (compP2 P) h \<tau> is f \<and> conf_fs (compP2 P) h \<Phi> M (length Ts) T fs \<and> conf_xcp (compP2 P) h xp (is ! pc))"
\<psi>: shows "\<guillemotleft>\<psi> : F \<I>\<^sub>C \<rightarrow>\<^sub>D \<I>\<^sub>D\<guillemotright> \<and> D.iso \<psi>"
orbit_def: shows "orbit \<equiv> \<lambda>f x. {xa. orbitp f x xa}"
sup_sharp_homTr: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and L :: "('k, 'f, 'l) Module_scheme"   and u :: "'e \<Rightarrow> 'i"   and f :: "'i \<Rightarrow> 'k" assumes "R module M"   and "R module N"   and "R module L"   and "u \<in> mHom R M N"   and "f \<in> mHom R N L" shows "sup_sharp R M N L u f \<in> mHom R M L"
nullary_qtable_cases: fixes n :: "nat"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set" assumes "qtable n {} P Q X" shows "X = empty_table \<or> X = unit_table n"
smult_one: fixes x :: "'a Matrix.vec" assumes "x \<in> carrier_vec n" shows "(1::'a) \<cdot>\<^sub>v x = x"
IH_2: fixes xa :: "'v \<times> 'v" assumes "e \<in> set (cycle_edges (enum_cycle j))" shows "h_minus_aux j xa \<le> h' xa"
inf_img_fin_dom': fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite (f ` A)"   and "infinite A" shows "\<exists>y\<in>f ` A. infinite (f -` {y} \<inter> A)"
DIAndProof_def: shows "DIAndProof = (DIAndConcl, [(0, Rrule ImplyR 0), (0, Lrule AndL 0), (0, Rrule ImplyR 0), (0, Cut DIAndCutP1), (1, Cut DIAndSG1), (0, Rrule CohideR 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc (Suc 0)), CloseId 1 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc 0), CloseId 0 0), (Suc (Suc 0), Cut DIAndCut34Elim1), (0, Lrule ImplyL 0), (Suc (Suc (Suc 0)), Lrule ImplyL 0), (0, Rrule CohideRR 0), (0, Rrule CohideRR 0), (Suc 0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc (Suc 0)))), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc (Suc (Suc 0)))), Lrule AndL 0), (Suc (Suc (Suc (Suc (Suc 0)))), CloseId 0 0), (Suc (Suc (Suc 0)), AxSubst AK DIAndSubst341), (Suc (Suc 0), CloseId 0 0), (Suc 0, CloseId 0 0), (0, Cut DIAndCut12Intro), (Suc 0, Rrule CohideRR 0), (Suc (Suc 0), AxSubst AK DIAndSubst12), (0, Lrule ImplyL 0), (1, Lrule ImplyL 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, Cut DIAndCutP12), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc 0))), AxSubst AK DIAndCurry12), (Suc (Suc (Suc 0)), Rrule CohideRR 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc 0), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc (Suc 0))), Rrule ImplyR 0), (Suc (Suc (Suc (Suc 0))), Rrule AndR 0), (Suc (Suc (Suc (Suc (Suc 0)))), CloseId 0 0), (Suc (Suc (Suc (Suc 0))), CloseId 1 0), (Suc (Suc 0), CloseId 0 0), (Suc 0, Cut DIAndCut34Elim2), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc (Suc 0))), AxSubst AK DIAndSubst342), (Suc (Suc (Suc 0)), Rrule CohideRR 0), (Suc (Suc (Suc 0)), G), (0, Rrule ImplyR 0), (Suc (Suc (Suc 0)), Lrule AndL 0), (Suc (Suc (Suc 0)), CloseId 1 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc 0, CloseId 0 0), (1, Cut DIAndSG2), (0, Lrule ImplyL 0), (0, Rrule CohideRR 0), (Suc (Suc (Suc 0)), CloseId 4 0), (Suc (Suc 0), Lrule ImplyL 0), (Suc (Suc (Suc 0)), CloseId 0 0), (Suc (Suc (Suc 0)), CloseId 0 0), (1, CloseId 1 0)])"
obtain_shortest_path: fixes u :: "nat"   and v :: "nat" assumes "cf.connected u v" obtains p :: "(nat \<times> nat) list" where "cf.isShortestPath u p v"
lexordp_eq_simps_1: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "ord.lexordp_eq less [] ys = True"
R_a0im1_ir_def: shows "R_a0im1_ir \<equiv> {(s, t). signals s = ir_runs2sigs (runs t) \<and> corrupted s = Domain (m1_state.leak t) \<times> UNIV}"
tauStepChainEqvt: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and p :: "(name \<times> name) list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<Longrightarrow>\<^sub>\<tau> p \<bullet> P'"
SCF_R1: fixes xs :: "'c option list"   and P :: "'c option process"   and X :: "'c option set"   and Q :: "'c option process" assumes "xs \<notin> sentences P"   and "(xs, X) \<in> failures P"   and "None \<notin> set xs" shows "(xs, X) \<in> seq_comp_failures P Q"
simps_2: fixes e :: "'a expr" shows "fin1 (throw e) = throw (fin1 e)"
Struct_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "CM.Struct wtFsym wtPsym arOf resOf intT intF intP \<equiv> M.Struct wtFsym wtPsym arOf resOf intT intF intP"
div_free_DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P: shows "D (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
PPs_closed_plus: fixes s :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t \<in> .[X]" shows "s + t \<in> .[X]"
refl_onD: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and a :: "'c" assumes "refl_on A r"   and "a \<in> A" shows "(a, a) \<in> r"
strong_tricolour_invD: fixes x :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and y :: "'e" assumes "black x s"   and "(x points_to y) s"   and "obj_at (\<lambda>s. True) y s"   and "strong_tricolour_inv s" shows "marked y s"
lfilter_hdtl: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" assumes "lfilter P lst \<noteq> LNil" shows "\<exists>n. LCons (lhd (lfilter P lst)) LNil = lfilter P (ltake (enat n) lst) \<and> ltl (lfilter P lst) = lfilter P (ldrop (enat n) lst)"
find_path1_def: fixes E :: "('b \<times> 'b) set"   and u0 :: "'b"   and P :: "'b \<Rightarrow> bool" shows "find_path1 E u0 P \<equiv> ASSERT (finite (E\<^sup>* `` {u0})) \<bind> (\<lambda>_. SPEC (\<lambda>p. case p of None \<Rightarrow> \<forall>v\<in>E\<^sup>+ `` {u0}. \<not> P v | Some (p, v) \<Rightarrow> path E u0 p v \<and> P v \<and> p \<noteq> []))"
pow_irr_mult_of: fixes x :: "'a mod_ring poly"   and n :: "nat" shows "x [^]\<^bsub>mult_of R\<^esub> n = x ^ n mod f"
pres_wf_exec: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w"   and "w \<noteq> []" shows "\<forall>(q, \<tau>)\<in>set (step (hd w) (ss ! hd w)). \<tau> \<in> A"
Inv_in_Hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a monoidal_language.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Inv C t \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = monoidal_language.Cod C t \<and> monoidal_language.Cod C t = monoidal_language.Dom C t}"
3: fixes r :: "nat" assumes "r \<in> set (tl rs)" shows "0 < r"
strip_step': fixes S :: "'av Abs_State.st option"   and c :: "'av Abs_State.st option acom" shows "strip (local.step' S c) = strip c"
poly_zero: fixes p :: "'a list" shows "(poly p = poly []) = (\<forall>c\<in>set p. c = (0::'a))"
coerce_prj: fixes x :: "udom" assumes "DEFL('a) \<sqsubseteq> DEFL('b)" shows "COERCE('b, 'a)\<cdot>(PRJ('b)\<cdot>x) = PRJ('a)\<cdot>x"
config_n_bv_2: fixes s0 :: "'b list"   and qs :: "'b list" shows "map_pmf (snd \<circ> snd) (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs) = return_pmf s0 \<and> map_pmf (fst \<circ> snd) (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs) = bv (length s0)"
enf: shows "\<forall>G. c_sinvar m G = (\<forall>e\<in>edges G. P e)"
map_12: fixes f :: "'c \<Rightarrow> 'b"   and x121 :: "'c ltlc"   and x122 :: "'c ltlc" shows "map_ltlc f (x121 R\<^sub>c x122) = map_ltlc f x121 R\<^sub>c map_ltlc f x122"
low_out_def: fixes \<tau> :: "trace" shows "low_out \<tau> \<equiv> the (final \<tau> ''lowOut'')"
frac_def: fixes r :: "'a"   and s :: "'a" shows "local.frac r s \<equiv> equivalence.Class (R \<times> (R\<setminus>I)) {(x, y). (x, y) \<in> (R \<times> (R\<setminus>I)) \<times> R \<times> (R\<setminus>I) \<and> local.rel x y} (r, s)"
code: fixes f :: "('c \<Rightarrow> 'b) tree"   and x :: "'c tree" shows "f \<diamondop> x = Node (root f (root x)) (left f \<diamondop> left x) (right f \<diamondop> right x)"
induct: fixes P :: "'b btree \<Rightarrow> bool"   and a0 :: "'b btree" assumes "P Leaf"   and "\<And>ts t. \<lbrakk>\<And>x. x \<in> set (subtrees ts) \<Longrightarrow> P x; P t\<rbrakk> \<Longrightarrow> P (Node ts t)" shows "P a0"
CompleteLattice_def: shows "CompleteLattice \<equiv> {cl \<in> PartialOrder. (\<forall>S\<subseteq>pset cl. \<exists>L. isLub S cl L) \<and> (\<forall>S\<subseteq>pset cl. \<exists>G. isGlb S cl G)}"
that: fixes h' :: "'a" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "thesis"
assms: shows "take (max_Var_floatarith ra) ys = take (max_Var_floatarith ra) zs"
has_default_route_alt: fixes rt :: "'h routing_rule list" shows "has_default_route rt = (\<exists>r\<in>set rt. routing_prefix r = 0)"
groupI: fixes G :: "('f, 'g) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
length_trace_parallel_plan_gt_0: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "0 < length (trace_parallel_plan_strips I \<pi>)"
simps: fixes b :: "bool"   and r :: "(bool \<times> 'b) rexp" shows "final_a (b, r) = (final r \<or> b \<and> nullable r)"
lmeasurable_negligible_symdiff: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> lmeasurable"   and "negligible (sym_diff S T)" shows "T \<in> lmeasurable"
0: shows "a k \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> (inv a) k = 1"
analz_NI_I_K_synth_analz_NI_E: fixes implem :: "chan \<Rightarrow> msg"   and K :: "msg set"   and Bad :: "agent set"   and NI :: "msg set"   and I :: "msg set" assumes "semivalid_implem implem"   and "Keys_bad K Bad"   and "NI \<subseteq> payload"   and "K \<subseteq> range LtK"   and "I \<subseteq> basic_implem.valid implem" shows "analz (NI \<union> I \<union> K \<union> Tags) \<subseteq> synth (analz (extr Bad NI (semivalid_implem.abs implem I))) \<union> - payload"
disjSD_no_p_in_both: fixes x :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "disjSD_2 x y"   and "\<not> member DenyAll x"   and "\<not> member DenyAll y"   and "p \<in> dom (C x)"   and "p \<in> dom (C y)" shows "False"
\<psi>_\<phi>: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a"   and b :: "'a"   and a :: "'a" assumes "hom_functor C S \<phi>"   and "partial_magma.in_hom C f b a" shows "hom_functor.\<psi> C \<phi> (b, a) (\<phi> (b, a) f) = f"
l2_inv4'D: fixes Rb :: "fid_t"   and B :: "agent"   and A :: "agent"   and x :: "l2_state"   and gnx :: "msg" assumes "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "in_progress (progress x Rb) xsk"   and "guessed_frame Rb xgnx = Some gnx"   and "x \<in> l2_inv4'" shows "guessed_frame Rb xsk = Some (Exp gnx (NonceF (Rb $ ny)))"
ball_def: fixes x :: "'b"   and e :: "real" shows "ball x e = {y. dist x y < e}"
full_0_elim: fixes t :: "'a tree23" assumes "full 0 t" obtains "t = Empty"
equal_pmf_def: fixes p :: "'a pmf"   and q :: "'a pmf" shows "equal_class.equal p q = (mapping_of_pmf p = mapping_of_pmf q)"
Inf_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "\<Sqinter>A = (\<lambda>x. INF f\<in>A. f x)"
repr_independence: fixes G :: "('a, 'b) monoid_scheme"   and y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "Group.group G"   and "y \<in> H #>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H G" shows "H #>\<^bsub>G\<^esub> x = H #>\<^bsub>G\<^esub> y"
sinks_elem: fixes v :: "'e"   and I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xs :: "'f list" assumes "v \<in> sinks I D u xs" shows "\<exists>x\<in>set xs. v = D x"
either_sym: fixes t :: "'a ttree"   and t2 :: "'a ttree" shows "t \<oplus>\<oplus> t2 = t2 \<oplus>\<oplus> t"
incl: shows "A \<subseteq> (T ^^ i) --` B"
mod_add_left_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) b) c = modulo (plus a b) c"
assms_3: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L1"   and "y \<in> carrier L1"   and "x \<sqsubseteq>\<^bsub>L1\<^esub> y" shows "f x \<sqsubseteq>\<^bsub>L2\<^esub> f y"
assms_3: shows "K \<inter> synth (analz H) = {}"
supp_fset: fixes S :: "'a fset" shows "supp (fset S) = supp S"
segment_image_interval_2: fixes a :: "'b"   and b :: "'b" shows "open_segment a b = (if a = b then {} else (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0<..<1})"
subst_ident: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X" shows "X #[Var zs z / z]_zs = X"
vec_mat_Tensor_length: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list"   and ys :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "length (mult.vec_mat_Tensor f xs ys) = length ys"
prod_relI: fixes a :: "'a"   and a' :: "'b"   and R1 :: "('a \<times> 'b) set"   and b :: "'c"   and b' :: "'d"   and R2 :: "('c \<times> 'd) set" assumes "(a, a') \<in> R1"   and "(b, b') \<in> R2" shows "((a, b), a', b') \<in> R1 \<times>\<^sub>r R2"
ring_id_pow_closed: fixes n :: "nat" shows "ring_id [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n \<in> carrier (function_ring (carrier Zp) Zp)"
prems_2: fixes S' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
hyps_2: fixes g :: "'a ugraph" assumes "edges g \<subset> edges ga__"   and "Undirected_Graph.connected g" shows "Ex (is_spanning_tree g)"
pick_swapAbs_qSwapAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "goodAbs A" shows "pick (A $[x1 \<and> x2]_xs) $= pick A $[[x1 \<and> x2]]_xs"
step_Normal_elim_cases_4: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b \<times> 'b) set"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Spec r, Normal s) \<rightarrow> u" obtains t :: "'b" where "u = (Skip, Normal t)" and "(s, t) \<in> r"   | "u = (Skip, Stuck)" and "\<forall>t. (s, t) \<notin> r"
cartesian_product: fixes C :: "('b \<times> 'c) set"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite C"   and "{a. \<exists>b. g a b \<noteq> (0::'d)} \<times> {b. \<exists>a. g a b \<noteq> (0::'d)} \<subseteq> C" shows "(\<Sum>a. Sum_any (g a)) = (\<Sum>(a, b). g a b)"
from_ns_def: fixes vl :: "(nat, QDelta) mapping"   and cs :: "QDelta ns_constraint list" shows "from_ns vl cs \<equiv> let \<delta> = \<delta>0_val_min cs \<langle>vl\<rangle> in Mapping.tabulate (vars_list_constraints cs) (\<lambda>var. val (\<langle>vl\<rangle> var) \<delta>)"
inj_prod_decode: fixes A :: "nat set" shows "inj_on prod_decode A"
fst_e_in_S: shows "fst3 e |\<subseteq>| S"
STL6: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>(F \<and> G) = (\<diamond>\<box>F \<and> \<diamond>\<box>G)"
finite_measure_eq_AE: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a pmf" assumes "AE x in measure_pmf M. (x \<in> A) = (x \<in> B)"   and "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A = measure_pmf.prob M B"
assms_1: shows "f \<in> borel_measurable (F n)"
psubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "psubst \<phi> txs \<in> fmla"
assms_2: shows "node_deliver_messages evts = msgs @ [last_msg]"
subgroups_Inter_pair: fixes I :: "'a set"   and J :: "'a set" assumes "subgroup I (add_monoid R)"   and "subgroup J (add_monoid R)" shows "subgroup (I \<inter> J) (add_monoid R)"