hcju/isabelleps huggingface dataset:
prj_yields_pullback: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "category_with_pullbacks C"   and "partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g" shows "limit_cone cospan_shape.comp C (cospan_diagram.map C f g) (partial_magma.dom C (category_with_pullbacks.prj1 C f g)) (cospan_diagram.mkCone C f g (category_with_pullbacks.prj1 C f g) (category_with_pullbacks.prj0 C f g))"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T"   and "mpt_axioms M T" shows "mpt M T"
simps_1: fixes x :: "brick list"   and ys :: "wall" shows "basic x \<circ> ys = x * ys"
NextRecvCap'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and t :: "'loc \<times> 't" shows "NextRecvCap' c0 c1 p t = (cri.next_recvcap' (exchange_config c0) (exchange_config c1) p t \<and> prop_config c1 = prop_config c0 \<and> init c1 = init c0)"
map_rec: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f xs = rec_list \<epsilon> (\<lambda>x _. (#) (f x)) xs"
Suc_le_mono: fixes n :: "nat"   and m :: "nat" shows "order.greater_eq (Suc m) (Suc n) = order.greater_eq m n"
prems: shows " t \<longrightarrow>\<^sub>1\<^sup>* t1"
linear_injective_isomorphism: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "\<exists>f'. Vector_Spaces.linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
finite_subset_induct: fixes F :: "'a set"   and A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
mod_and_dist: fixes P :: "assn"   and Q :: "assn"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> P \<and>\<^sub>A Q) = (h \<Turnstile> P \<and> h \<Turnstile> Q)"
absorbent: fixes x :: "'a"   and y :: "'a" assumes "x \<in> R"   and "y \<in> R"   and "x \<cdot> y \<in> I" shows "x \<in> I \<or> y \<in> I"
dense: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.dense_order less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
subgraphs_def: fixes G :: "nat set \<times> nat set set" shows "subgraphs G = {G'. subgraph G' G}"
polynomial_incl: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set"   and p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" assumes "polynomial\<^bsub>L\<lparr>carrier := {x \<in> carrier L. (Id.S.algebraic over indexed_const ` carrier R) x}\<rparr>\<^esub> K p" shows "set p \<subseteq> K"
inject: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool"   and y1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and y3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and y4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and y5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and y6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and y7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and y8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and y9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and y10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and y11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "(Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 = Direction y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4 \<and> x5 = y5 \<and> x6 = y6 \<and> x7 = y7 \<and> x8 = y8 \<and> x9 = y9 \<and> x10 = y10 \<and> x11 = y11)"
ltakeWhile_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "((A ===> (=)) ===> llist_all2 A ===> llist_all2 A) ltakeWhile ltakeWhile"
nvalid_augment_Faults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and n :: "nat"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>\<Turnstile>n:\<^bsub>/F'\<^esub> P c Q,A"
dur_mono: fixes i :: "nat"   and j :: "nat"   and xs :: "('c \<Rightarrow> real) stream" assumes "i \<le> j" shows "dur xs i \<le> dur xs j"
le_approx_lemma_F: fixes P :: "'c process"   and Q :: "'c process" assumes "P \<sqsubseteq> Q" shows "F Q \<subseteq> F P"
coprime_of_int_polyI: fixes p :: "int poly"   and q :: "int poly" assumes "algebraic_semidom_class.coprime p q" shows "algebraic_semidom_class.coprime (of_int_poly p) (of_int_poly q)"
distinct_22: fixes x71 :: "key"   and x72 :: "msg"   and x2 :: "nat" shows "Crypt x71 x72 \<noteq> Number x2"
as_bs_2: shows "start_remove xs k [] = as @ remove_cycles bs k [k]"
iso_is_section: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "Maps.section f"
cr_fmap_def: shows "cr_fmap \<equiv> \<lambda>x y. x = ($$) y"
divide_less_cancel: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide a c) (divide b c) = ((less zero c \<longrightarrow> less a b) \<and> (less c zero \<longrightarrow> less b a) \<and> c \<noteq> zero)"
bin_to_bl_0: fixes bs :: "int" shows "bin_to_bl 0 bs = []"
cases: fixes a :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set" assumes "a \<in> Regions.region X I r" obtains u :: "'a \<Rightarrow> real" and X\<^sub>0 :: "'a set" where "a = u" and "\<forall>x\<in>X. 0 \<le> u x" and "\<forall>x\<in>X. Regions.intv_elem x u (I x)" and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d}" and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))"
EF_step: fixes x :: "'a"   and y :: "'a"   and f :: "'a set" assumes "x \<rightarrow>\<^sub>i y"   and "y \<in> f" shows "x \<in> EF f"
inject_2: fixes x21 :: "'a Trace"   and x22 :: "'a"   and y21 :: "'a Trace"   and y22 :: "'a" shows "(x21 \<leadsto> x22 = y21 \<leadsto> y22) = (x21 = y21 \<and> x22 = y22)"
kD_sqnf_is_proj3: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "ip \<in> kD rt" shows "sqnf rt ip = \<pi>\<^sub>3 (the (rt ip))"
assms_3: shows "\<forall>x\<in>s. f x = g x"
min_Suc_gt_2: fixes a :: "nat"   and b :: "nat" assumes "a < b" shows "min b (Suc a) = Suc a"
fv: shows "fv body \<subseteq> set (this # pns)"
2: shows "indoc a (u\<^sub>1 @ [a] @ u\<^sub>2)"
while_bot_true: fixes P :: "'b hrel" shows "while\<^sub>\<bottom> true do P od = \<mu> ((;;) P)"
simps: fixes a :: "'b list"   and C :: "'b list set"   and b :: "'b list" shows "(a \<in> no_head C b) = (a = \<epsilon> \<or> (\<exists>u. a = u \<and> u \<in> C \<and> u \<noteq> b) \<or> (\<exists>u v. a = u \<cdot> v \<and> u \<noteq> \<epsilon> \<and> u \<in> no_head C b \<and> v \<in> \<langle>C\<rangle>))"
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd" assumes "\<And>U I V s. \<lbrakk>open_cover_of_open_subset S is_open U I V; \<And>i. i \<in> I \<Longrightarrow> V i \<subseteq> U; s \<in> \<FF> U; \<And>i. i \<in> I \<Longrightarrow> \<rho> U (V i) s = zero_str (V i)\<rbrakk> \<Longrightarrow> s = zero_str U"   and "\<And>U I V s. \<lbrakk>open_cover_of_open_subset S is_open U I V; \<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> \<FF> (V i); \<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<rho> (V i) (V i \<inter> V j) (s i) = \<rho> (V j) (V i \<inter> V j) (s j)\<rbrakk> \<Longrightarrow> \<exists>t. t \<in> \<FF> U \<and> (\<forall>i. i \<in> I \<longrightarrow> \<rho> U (V i) t = s i)" shows "sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str"
limit_def: fixes f :: "'a set \<Rightarrow> 'a set"   and x :: "'a set" shows "limit f x = natUnion (\<lambda>n. funpower f n x)"
val_surj_n_val: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "\<exists>x\<in>carrier K. v x = 1" shows "n_val K v = v"
ltln_strong_weak_stable_words_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n (\<phi> and\<^sub>n F\<^sub>n \<psi>[\<F>\<G> \<psi> w]\<^sub>\<Sigma>\<^sub>1) W\<^sub>n \<psi>"
IH1: shows "interp_ins (before @ op_list) = xs @ zs"
seqE: fixes g :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.seq g f" obtains "S.arr f" and "S.arr g" and "S.dom g = S.cod f"
compact_eq_Heine_Borel: fixes S :: "'a set" shows "compact S = (\<forall>C. Ball C open \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
prems_3: shows "\<forall>a\<in>set (a # l). case a of (v, S) \<Rightarrow> typ_ok \<Theta> (Tv v S)"
False_False_in_steps_atom: fixes a :: "'b"   and w :: "'b list" shows "(([False], [False]) \<in> NAe.steps (atom a) w) = (w = [])"
eintT_ntsem: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "eintT \<sigma> a" shows "eintT \<sigma> (ntsem \<sigma> a)"
cos_arccos: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "cos (arccos y) = y"
classes_above_sees_method: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and C' :: "char list" assumes "classes_above P C \<inter> classes_changed P P' = {}"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C'" shows "P' \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C'"
Berlekamp_gcd_step: fixes h :: "'b mod_ring poly"   and f :: "'b mod_ring poly" assumes "[h ^ CARD('b) = h] (mod f)"   and "monic f"   and "square_free f" shows "f = (\<Prod>c\<in>UNIV. gcd f (h - [:c:]))"
Torder_adjunction: fixes D :: "('c, 'd) Order_scheme"   and X :: "'c set"   and a :: "'c" assumes "Order D"   and "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq>\<^bsub>D\<^esub> a"   and "Torder (Iod D X)" shows "Torder (Iod D (X \<union> {a}))"
inj_map: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "inj ((\<circ>) f)"
lexordp_eq_conv_lexord: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp_eq xs ys = (xs = ys \<or> lexordp xs ys)"
vreal_le_def: shows "vreal_le = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> real_of_vreal a \<le> real_of_vreal b}"
right_neutral: fixes a :: "'a" shows "a \<sqinter> \<top> = a"
surjective: fixes r :: "('a, 'b) stateful_policy_scheme" shows "r = \<lparr>hosts = hosts r, flows_fix = flows_fix r, flows_state = flows_state r, \<dots> = stateful_policy.more r\<rparr>"
simps_1: fixes x :: "'b" shows "replicate 0 x = \<epsilon>"
eq_zero: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat" assumes "dcharacter n \<chi>"   and "\<not> coprime a n" shows "\<chi> a = 0"
integral_cos: fixes t :: "real"   and a :: "real"   and b :: "real" assumes "t \<noteq> 0" shows "LBINT x=ereal a..ereal b. cos (t * x) = sin (t * b) / t - sin (t * a) / t"
map_upt: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map (\<lambda>i. f (xs ! i)) [0..<length xs] = map f xs"
assms_3: shows "{\<lambda>tp. tp = ([], <args>)} p {\<lambda>tp. tp = (Bk \<up> m, <n> @ Bk \<up> k)}"
msetext_huet_snoc: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "msetext_huet gt (xs @ [x]) xs"
local_ring_morphism_axioms: shows "local_ring_morphism g B addB multB zeroB oneB C addC multC zeroC oneC"
tab_def_alt_1: shows "tab = (rep r \<star> tab\<^sub>0 r) \<cdot> tab\<^sub>0.trnr\<^sub>\<eta> (tab\<^sub>1 r) (tab\<^sub>1 r \<star> (tab\<^sub>0 r)\<^sup>*)"
alg_dvd_mult2: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a alg_dvd b"   and "algebraic_int c" shows "a alg_dvd b * c"
p_S2: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and s :: "'a" assumes "Reachability_Problem K S S1 S2"   and "s \<in> S2" shows "Reachability_Problem.p K S S1 S2 s = 1"
thm_2_3_7: fixes ord :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and ord_strict :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow> bool"   and X :: "'b set"   and fs :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "pm_powerprod ord ord_strict"   and "finite X"   and "set fs \<subseteq> P[X]"   and "pm_powerprod.is_GB_cofactor_bound ord (set fs) b" shows "(1 \<in> ideal (set fs)) = (1 \<in> set (ordered_term.Macaulay_list ord ord_strict (pm_powerprod.deg_shifts ord X b fs)))"
path_from_toE: fixes v :: "nat"   and xs :: "nat list"   and w :: "nat" assumes "v \<leadsto>xs\<leadsto> w" shows "path xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
\<psi>_altdef: fixes x :: "real" assumes "0 < x" shows "\<psi> x = sum_upto (\<lambda>m. prime_sum_upto (\<lambda>x. ln (real x)) (root m x)) (log 2 x)"
conj_distrib_axioms: shows "conj_distrib (\<iinter>) chaos"
adm_def: fixes P :: "'c \<Rightarrow> bool" shows "adm P = (\<forall>Y. chain Y \<longrightarrow> (\<forall>i. P (Y i)) \<longrightarrow> P (\<Squnion>i. Y i))"
SigmaE2: fixes a :: "'d"   and b :: "'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "(a, b) \<in> Sigma A B" obtains "a \<in> A" and "b \<in> B a"
proj2_pt_hyp2_rep: fixes p :: "hyp2" shows "proj2_pt (hyp2_rep p) = Rep_hyp2 p"
cong__le: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Cong A B C D" shows "Tarski_neutral_dimensionless.Le Bet Cong A B C D"
ereal_plus_eq_MInfty: fixes a :: "ereal"   and b :: "ereal" shows "(a + b = - \<infinity>) = ((a = - \<infinity> \<or> b = - \<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
WT_real: fixes \<eta> :: "nat" shows "\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res real_resource \<eta> \<surd>"
prod_emb_iff: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and K :: "'d set"   and X :: "('d \<Rightarrow> 'e) set" shows "(f \<in> prod_emb I M K X) = (f \<in> extensional I \<and> restrict f K \<in> X \<and> (\<forall>i\<in>I. f i \<in> space (M i)))"
f: shows "(f \<longlongrightarrow> (0::'b)) F"
le_natceiling_iff: fixes n :: "nat"   and r :: "real" assumes "real n \<le> r" shows "n \<le> nat \<lceil>r\<rceil>"
preserves_trg: fixes x :: "'a" shows "(if B.arr (trg\<^sub>B x) then trg\<^sub>B x else B.null) = trg\<^sub>B (if B.arr x then x else B.null)"
u: shows "Arr u \<and> Ide (Dom u) \<and> Ide (Cod u) \<and> Ide \<^bold>\<lfloor>Dom u\<^bold>\<rfloor> \<and> Ide \<^bold>\<lfloor>Cod u\<^bold>\<rfloor> \<and> arr \<lbrace>u\<rbrace> \<and> arr \<lbrace>Dom u\<rbrace> \<and> ide \<lbrace>Dom u\<rbrace> \<and> arr \<lbrace>Cod u\<rbrace> \<and> ide \<lbrace>Cod u\<rbrace>"
elims: fixes x :: "'upf_key"   and xa :: "bool"   and xb :: "unit"   and xc :: "bool list \<times> bool list + bool list \<times> bool list \<times> 'hash"   and y :: "(((bool list \<times> bool list \<times> 'hash) option + bool list option) \<times> unit, bool list, bool list) gpv" assumes "intercept_prf x xa xb xc = y" obtains uu_ :: "'upf_key" and uv_ :: "bool" and uw_ :: "unit" and ux_ :: "bool list \<times> bool list \<times> 'hash" where "x = uu_" and "xa = uv_" and "xb = uw_" and "xc = Inr ux_" and "y = Generative_Probabilistic_Value.Done (Inr None, ())"   | k :: "'upf_key" and b :: "bool" and uy_ :: "unit" and m1 :: "bool list" and m0 :: "bool list" where "x = k" and "xa = b" and "xb = uy_" and "xc = Inl (m1, m0)" and "y = (case length m1 = prf_clen \<and> length m0 = prf_clen of True \<Rightarrow> lift_spmf (spmf_of_set prf_domain) \<bind> (\<lambda>x. Generative_Probabilistic_Value.Pause x Generative_Probabilistic_Value.Done \<bind> (\<lambda>p. let c = p [\<oplus>] (if b then m1 else m0); t = upf_fun k (x @ c) in Generative_Probabilistic_Value.Done (Inl (Some (x, c, t)), ()))) | False \<Rightarrow> Generative_Probabilistic_Value.Done (Inl None, ()))"
integrable_norm: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b" assumes "integrable M f" shows "integrable M (\<lambda>x. norm (f x))"
Rel_AND: shows "((x0, x1), w0, w1) \<in> Rel_AND"
CapN: 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 enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "N (enc \<phi>) = enc (Syntax_with_Connectives_False.neg imp fls \<phi>)"
div_enat_def: fixes a :: "enat"   and b :: "enat" shows "a div b \<equiv> case a of enat x \<Rightarrow> case b of enat y \<Rightarrow> enat (x div y) | \<infinity> \<Rightarrow> 0 | \<infinity> \<Rightarrow> case b of enat 0 \<Rightarrow> 0 | enat (Suc n) \<Rightarrow> \<infinity> | \<infinity> \<Rightarrow> \<infinity>"
E_def: shows "cf.E \<equiv> {(u, v). cf (u, v) \<noteq> (0::'capacity)}"
f1: shows "(\<lambda>p. x (Suc (n + p)) - x n) \<longlonglongrightarrow> l - x n"
that: fixes X :: "(real \<times> real) \<times> real \<times> real"   and M' :: "real" assumes "X \<in> set Xs"   and "intersect_segment_xline prec X x = Some (m, M')"   and "\<And>X m' M'. \<lbrakk>X \<in> set Xs; intersect_segment_xline prec X x = Some (m', M')\<rbrakk> \<Longrightarrow> m \<le> m'" shows "thesis"
pp_def: shows "pp = (\<lambda>n. real (nth_prime (2 * n) - nth_prime n) / sqrt (real (nth_prime n)))"
basic_type_less_right_simps_5: fixes \<tau> :: "'a basic_type" shows "(\<tau> < Integer) = (\<tau> = UnlimitedNatural \<or> \<tau> = OclVoid)"
A4: shows "(b, c) \<in> R\<^sup>*"
wf_trm_subst_singleton: fixes t :: "('fun, 'b) Term.term"   and t' :: "('fun, 'b) Term.term"   and v :: "'b" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "wf\<^sub>t\<^sub>r\<^sub>m t'" shows "wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> Var(v := t'))"
undirected: fixes v :: "nat"   and w :: "nat" shows "v \<rightarrow> w = w \<rightarrow> v"
extremum_uniqueI: fixes a :: "'b word" assumes "a \<le> 0" shows "a = 0"
diagonal_sum: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> {..n + m} \<rightarrow> carrier R"   and "g \<in> {..n + m} \<rightarrow> carrier R" shows "(\<Oplus>k\<in>{..n + m}. \<Oplus>i\<in>{..k}. f i \<otimes> g (k - i)) = (\<Oplus>k\<in>{..n + m}. \<Oplus>i\<in>{..n + m - k}. f k \<otimes> g i)"
last_lift_not_None: fixes Q :: "'b com"   and x :: "'b com option \<times> 'b"   and xs :: "('b com option \<times> 'b) list" shows "fst (lift Q ((x # xs) ! length xs)) \<noteq> None"
Rep_Ainteg_inverse: fixes x :: "ant" shows "Abs_Ainteg (Rep_Ainteg x) = x"
elims_2: fixes x :: "('a, 'b) graph"   and xa :: "'a"   and xb :: "('a \<times> 'b \<times> 'a) list"   and xc :: "'a" assumes "is_path_undir x xa xb xc" obtains G :: "('a, 'b) graph" and v :: "'a" and v' :: "'a" where "x = Ga__" and "xa = v__" and "xb = []" and "xc = v'__" and "v__ = v'__ \<and> v'__ \<in> nodes Ga__"   | G :: "('a, 'b) graph" and v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and p :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = G" and "xa = v" and "xb = (v1, w, v2) # p" and "xc = v'" and "v = v1 \<and> ((v1, w, v2) \<in> edges G \<or> (v2, w, v1) \<in> edges G) \<and> is_path_undir G v2 p v'"
block_exists: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>A. a \<in> A \<and> A \<in> P"
assms_3: shows "(m, lam) \<in> set n_as"
fold_pls'_eq: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "g ` A \<subseteq> S" shows "Finite_Set.fold (pls' \<circ> g) z A = Finite_Set.fold ((\<oplus>\<^sub>o\<^sub>w) \<circ> g) z A"
assms: shows "low_equal s1 s2 \<and> load_word_mem s1 addra 8 = load_word_mem s2 addra 8"
finite_cyclic_subgroup_order: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "finite (carrier (subgroup_generated (relative_homology_group (int n) (nsphere n) (equator n)) {x})) = (GE.ord x \<noteq> 0)"
IH2': shows "frechet I t2 (fst \<nu>) (snd \<nu>) = frechet J t2 (fst \<nu>') (snd \<nu>')"
s_max: fixes x :: "real" shows "ego2.s x \<le> ego2.s_stop"
collapse_1: fixes pr_operation :: "pr_operation" assumes "is_PUSH pr_operation" shows "pr_operation = PUSH"
a: shows "d (d x \<cdot> ad y) = ad y \<cdot> d x"
f_closed': shows "\<forall>x\<in>U. \<forall>y\<in>U. local.min x y \<in> U"
1: shows "llength (ltake (enat (Suc k)) w) = eSuc (enat k)"
enum_ind_less_CARD: fixes x :: "'b" shows "enum_ind x < CARD('b)"
active_subset_def: fixes M :: "('a literal multiset \<times> label) set" shows "FL.active_subset M = {CL \<in> M. snd CL = Old}"
a0i_commit_def: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "a0i_commit A B M \<equiv> {(s, s'). signals s (Commit A B M) < signals s (Running A B M) \<and> s' = s\<lparr>signals := addSignal (signals s) (Commit A B M)\<rparr>}"