hcju/isabelleps huggingface dataset:
arr_pr1_iff: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr (pr1 a b) = (ide a \<and> ide b)"
lipschitz: fixes t :: "real" assumes "t \<in> T" shows "B'-lipschitz_on (cball x0 r) (f t)"
abs_eq: fixes x :: "'a pmf" shows "st_vec_of_pmf x = Abs_st_vec (vec_lambda (pmf x))"
index_le_index_eq_nextl: shows "card (UNIV // eq_app_right language) \<le> card (UNIV // eq_nextl)"
C12: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y)\<^sup>\<star> = (1::'a) + x \<cdot> (y \<cdot> x)\<^sup>\<star> \<cdot> y"
span_post: fixes v :: "'a"   and r :: "'a"   and g :: "'a"   and t :: "'a" assumes "regular v"   and "v \<sqinter> top = v"   and "id v = id r \<sqinter> top"   and "v \<sqinter> - id v \<sqinter> g = bot"   and "t \<le> v \<sqinter> id v \<sqinter> g"   and "id r \<sqinter> top \<le> id r \<sqinter> top" shows "id v = id r \<sqinter> top"
assms_3: shows "list = fst (interp_ops (ops @ [(oid, ins)]))"
eval_sup: fixes P :: "'a pred"   and Q :: "'a pred" shows "eval (P \<squnion> Q) = eval P \<squnion> eval Q"
CRR_market_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure" shows "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G \<equiv> CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk \<and> CRR_market_axioms M geom_proc G"
cases: fixes x :: "('f, 'b) Term.term" obtains xa :: "'b" where "x = Var xa"   | f :: "'f" and ts :: "('f, 'b) Term.term list" where "x = Fun f ts"
D': shows "D' = derivation_shift D (length \<delta>) 0"
cod_simps_3: shows "\<nu>.cod.leg1 = \<nu>.leg1"
ID_distr_local_2: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and e :: "('a, 'b, 'c) expr" shows "RID\<^sub>L (\<sigma>, \<tau>, e) = RID\<^sub>S \<sigma> \<union> RID\<^sub>S \<tau> \<union> RID\<^sub>E e"
simps: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "top0 list"   and a5 :: "'b state \<times> (v sem_env, v) result" shows "evaluate_prog a1 a2 a3 a4 a5 = ((\<exists>ck env s. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = [] \<and> a5 = (s, Rval (make_sem_env nsEmpty nsEmpty))) \<or> (\<exists>ck s1 s2 s3 env top0 tops new_env r. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = top0 # tops \<and> a5 = (s3, combine_dec_result new_env r) \<and> evaluate_top ck env s1 top0 (s2, Rval new_env) \<and> evaluate_prog ck (extend_dec_env new_env env) s2 tops (s3, r)) \<or> (\<exists>ck s1 s2 env top0 tops err. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = top0 # tops \<and> a5 = (s2, Rerr err) \<and> evaluate_top ck env s1 top0 (s2, Rerr err)))"
assms_3: fixes xi :: "'a"   and x :: "'b"   and si :: "'d"   and s :: "'c" assumes "(xi, x) \<in> A"   and "(si, s) \<in> R" shows "fi xi si \<le> \<Down> R (f x s)"
safe_distance_axioms: shows "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"
R_lin_independent_ConsD2: fixes m :: "'v"   and ms :: "'v list"   and r :: "'f"   and rs :: "'f list" assumes "R_lin_independent (m # ms)"   and "set (r # rs) \<subseteq> UNIV"   and "(r # rs) \<bullet>\<cdot> (m # ms) = (0::'v)" shows "r = (0::'f)"
set_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list" assumes "f permutes {..<length xs}" shows "set (permute_list f xs) = set xs"
cpt_update: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and P :: "'c set"   and p :: "'c"   and env :: "'c \<Rightarrow> nat"   and \<pi>l :: "'d list"   and \<pi> :: "'d" assumes "Shallow S s0 \<delta> AP L AP'"   and "cpt (P - {p}) env \<pi>l" shows "cpt P (env(p := length \<pi>l)) (\<pi>l @ [\<pi>])"
weak_unit_iff_self_source: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "weak_composition V H" shows "weak_composition.weak_unit V H a = (a \<in> weak_composition.sources V H a)"
restrict_in: fixes x :: "'a"   and A :: "'a set"   and m :: "'a \<Rightarrow> 'b option" assumes "x \<in> A" shows "(m |` A) x = m x"
eqn_hyp: shows "matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) *\<^sub>v (M_mat signs subsets *\<^sub>v w_vec p qs signs) = matr_option (dim_row (M_mat signs subsets)) (mat_inverse (M_mat signs subsets)) *\<^sub>v v_vec p qs subsets"
f'_inv: fixes y :: "'c" assumes "y \<in> dest.carrier" shows "f (f' y) = y"
msetext_huet_compat_list: fixes y :: "'b"   and x :: "'b"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and xs' :: "'b list" assumes "y \<noteq> x"   and "gt y x" shows "msetext_huet gt (xs @ y # xs') (xs @ x # xs')"
Continuous_def: fixes H :: "'a upred \<Rightarrow> 'a upred" shows "Continuous H = (\<forall>A. \<not> A = {} \<longrightarrow> H (\<Or>A) = \<Or>(H ` A))"
gen_iapath_triv_arc: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "gen_iapath (pverts G) u p v" shows "(u, v) \<in> parcs G"
ie: shows "real_of_int \<lfloor>Inum (real_of_int x # bs) e\<rfloor> = Inum (real_of_int x # bs) e"
s2_2: shows "results_in t2 s2 \<le> t1"
assms_7: shows "z \<in> circline_set H1 \<inter> circline_set H2"
hyp_CondL: fixes b :: "expr"   and bv :: "val"   and e1 :: "expr"   and e2 :: "expr" assumes "constVal b = Some bv"   and "the_Bool bv"   and "P b"   and "P e1" shows "P (b ? e1 : e2)"
subst_inj_on_is_bij_betw: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "inj_on \<theta> (subst_domain \<theta>) = bij_betw \<theta> (subst_domain \<theta>) (subst_range \<theta>)"
ev_Stream: fixes P :: "'b stream \<Rightarrow> bool"   and x :: "'b"   and s :: "'b stream" shows "ev P (x ## s) = (P (x ## s) \<or> ev P s)"
min_height_size1: fixes t :: "'c tree" shows "2 ^ min_height t \<le> size1 t"
hom_span: fixes gens :: "(bool \<times> 'd) list set" assumes "gens \<subseteq> carrier \<F>\<^bsub>I\<^esub>" shows "G.lift g ` \<langle>gens\<rangle>\<^bsub>\<F>\<^bsub>I\<^esub>\<^esub> = \<langle>G.lift g ` gens\<rangle>\<^bsub>G\<^esub>"
borel_measurable_LIMSEQ_metric: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>i. f i \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> (\<lambda>i. f i x) \<longlonglongrightarrow> g x" shows "g \<in> borel_measurable M"
erh_def: fixes R :: "('f, 'g) Ring_scheme"   and S :: "('f, 'h) Ring_scheme"   and X :: "'f"   and A :: "('e, 'i) Ring_scheme"   and B :: "('e, 'j) Ring_scheme"   and Y :: "'e"   and f :: "'f \<Rightarrow> 'e"   and n :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'f)" shows "erh R S X A B Y f n c = polyn_expr A Y n (cf_h f c)"
HOMCollection_Op: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" shows "HOMCollection (Op C) X Y = HOMCollection C Y X"
set_tag_name_impl: shows "set_tag_name = local.a_set_tag_name"
lt_monom_mult_le: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.lt (local.punit.monom_mult c t p) \<preceq> local.punit.splus t (punit.lt p)"
filter_in_nths: fixes xs :: "'b list"   and s :: "nat set" assumes "distinct xs" shows "filter (\<lambda>x. x \<in> set (nths xs s)) xs = nths xs s"
member_def: shows "member \<equiv> map_fun list_of_dlist id (equal_base.list_member ceq')"
saturated_saturate: fixes \<Gamma> :: "'b inference set"   and N :: "'b literal multiset set" shows "inference_system.saturated \<Gamma> (inference_system.saturate \<Gamma> N)"
Unlock1SynchronizedNull: fixes xs :: "'addr val list"   and V :: "nat"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and v :: "'addr val"   and h :: "'heap" assumes "xs ! V = Null"   and "V < length xs" shows "uf,P,t \<turnstile>1 \<langle>insync\<^bsub>V\<^esub> (a) Val v,(h, xs)\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),(h, xs)\<rangle>"
reverse_Cons: fixes x :: "'b"   and xs :: "['b]" shows "reverse\<cdot>(x : xs) = reverse\<cdot>xs ++ [x]"
Sup_least: fixes A :: "('a \<times> 'b) set"   and z :: "'a \<times> 'b" assumes "\<And>x. s2p A x \<Longrightarrow> less_eq_prod x z" shows "less_eq_prod (Sup_prod A) z"
mono_neutral_right: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'c)" shows "prod g T = prod g S"
stat_ref_G: shows "\<forall>q\<in>Q. statically_complete_calculus Bot_G (Inf_G_q q) (entails_q q) (Red_I_q q) (Red_F_q q)"
Cons_2: shows "{a. case a of (t, X) \<Rightarrow> (t, X) \<in> set (x # xs) \<and> as \<in> Z X t} \<noteq> {}"
accs_reduce_f_acc: fixes \<delta> :: "('a, 'b) ta_rule set"   and t :: "'b tree"   and q :: "'a" assumes "accs \<delta> t q" shows "accs (reduce_rules \<delta> (f_accessible \<delta> {q})) t q"
count_cases: fixes y :: "'k \<Rightarrow> nat" assumes "y \<in> multiset" obtains x :: "'k multiset" where "y = count x"
r_def: shows "r \<equiv> \<lambda>x. if x \<in> W then r0 x else x"
At: shows "A\<^sup>T \<in> carrier_mat n n"
assms_2: shows "formulaEntailsLiteral (F1 @ F @ F2) l"
l3_inv6_derived: shows "l3_inv3 \<inter> l3_inv4 \<subseteq> l3_inv6"
not_ordLess_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r <o r'" shows "(r', r) \<notin> ordLeq"
NOTIN: shows "\<forall>M\<in>P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V - {u}}. u \<notin> M"
IH: shows "\<forall>x. (x, T) |\<notin>| X \<longrightarrow> open0_Var (x, T) t ::: U \<and> (\<forall>xa. open0_Var (x, T) t ::: xa \<longrightarrow> U = xa)"
com_sifum_secure_def: fixes cmd :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" shows "com_sifum_secure cmd \<equiv> case cmd of (c, mds\<^sub>s) \<Rightarrow> c \<sim>\<^bsub>mds\<^sub>s\<^esub> c"
m2: assumes "i \<in> set (take j xs)" shows "m2 \<in> {k. k < length xs \<and> k < j \<and> xs ! k = i}"
Im_divide_numeral: fixes z :: "complex"   and w :: "num" shows "Im (z / numeral w) = Im z / numeral w"
gt_mono: fixes x :: "real"   and bs :: "real list" shows "\<forall>a n s. Inum (x # bs) a = Inum (x # bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x # bs) (gt n s) = Ifm (x # bs) (Gt a)"
integer_of_uint32_def: shows "integer_of_uint32 = integer_of_int \<circ> int_of_uint32"
cball_def: fixes x :: "'a"   and e :: "real" shows "cball x e = {y. dist x y \<le> e}"
Bossy_CD_on_Cd: fixes X :: "'a set" shows "Bossy.CD_on ds X = Bossy.Cd (Xd d1h1) X \<union> Bossy.Cd (Xd d2h1) X \<union> Bossy.Cd (Xd d3h1) X"
A_FactGroup_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set" shows "G A_Mod H = add_monoid G Mod H"
compxEs2_stack_xlift_convs: fixes d :: "nat"   and es :: "'a expr1 list"   and pc :: "nat" assumes "0 < d" shows "compxEs2 es pc d = stack_xlift d (compxEs2 es pc 0)"
valid_consT: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "X \<sharp> ty_dom \<Gamma>"   and "T closed_in \<Gamma>" shows "\<turnstile> (TVarB X T # \<Gamma>) ok"
mk_minsky_copy1: fixes v :: "nat"   and w :: "nat" assumes "v \<noteq> w" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then 0 else if x = w then vs v else vs x))"
filter_tab_def: fixes c :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd option" shows "filter_tab c t = (\<lambda>k. case t k of None \<Rightarrow> None | Some x \<Rightarrow> if c k x then Some x else None)"
factor_bound_ge_0: fixes f :: "int poly"   and n :: "nat" assumes "f \<noteq> 0" shows "0 \<le> factor_bound f n"
axioms_2: 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" assumes "Limit.cone J C D a \<chi>" shows "category J"
neutr: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and x :: "'b" assumes "class.linordered_ab_monoid_add plus less_eq less neutral" shows "plus x neutral = x"
Array_widen: fixes P :: "'b prog"   and A :: "ty"   and T :: "ty" assumes "P \<turnstile> A\<lfloor>\<rceil> \<le> T" shows "(\<exists>B. T = B\<lfloor>\<rceil> \<and> P \<turnstile> A \<le> B) \<or> T = Class Object"
AE_symmetric: fixes P :: "'b \<Rightarrow> 'c"   and Q :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "AE x in M. P x = Q x" shows "AE x in M. Q x = P x"
\<phi>_\<psi>: fixes t' :: "'t" assumes "t' \<in> S'.Univ" shows "\<phi> (\<psi> t') = t'"
CFG_SSA_Transformed_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and phis :: "'a \<Rightarrow> 'b \<times> 'e \<Rightarrow> 'e list option"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd" shows "CFG_SSA_Transformed_axioms \<alpha>n invar inEdges' oldDefs oldUses defs uses phis var \<equiv> ((\<forall>g n. oldDefs g n = var g ` defs g n) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> oldUses g n = var g ` uses g n)) \<and> (\<forall>g n ns m v x v'. graph_path_base.path2 \<alpha>n invar inEdges' g n ns m \<longrightarrow> n \<notin> set (tl ns) \<longrightarrow> v \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v \<in> CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g m \<longrightarrow> x \<in> set (tl ns) \<longrightarrow> v' \<in> CFG_SSA_base.allDefs defs phis g x \<longrightarrow> var g v' \<noteq> var g v) \<and> (\<forall>g n v vs v'. phis g (n, v) = Some vs \<longrightarrow> v' \<in> set vs \<longrightarrow> var g v' = var g v) \<and> (\<forall>n g v v'. n \<in> set (\<alpha>n g) \<longrightarrow> v \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v' \<in> CFG_SSA_base.allDefs defs phis g n \<longrightarrow> v \<noteq> v' \<longrightarrow> var g v' \<noteq> var g v)"
eta_cfun: fixes f :: "'a \<rightarrow> 'b" shows "(\<Lambda> x. f\<cdot>x) = f"
cases: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'d" assumes "r\<^sup>*\<^sup>* a1 a2" obtains a :: "'d" where "a1 = a__" and "a2 = a__"   | a :: "'d" and b :: "'d" and c :: "'d" where "a1 = a" and "a2 = c" and "r\<^sup>*\<^sup>* a b" and "r b c"
cases: fixes xs :: "'d list"   and a :: "'d list" assumes "cyclic1 xs a" obtains "a = rotate1 xs"
da_elim_cases_25: fixes Env :: "env"   and B :: "lname set"   and T :: "ty"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>New T[e]\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
t_twice_len_plust_14: shows "t_twice_len + 14 = 14 + length (shift t_twice 12) div 2"
listassoc_wfactorsD: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "as [\<sim>]\<^bsub>Ring_Divisibility.mult_of R\<^esub> bs"   and "wfactors (Ring_Divisibility.mult_of R) as a"   and "wfactors (Ring_Divisibility.mult_of R) bs b"   and "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "b \<in> carrier (Ring_Divisibility.mult_of R)"   and "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)"   and "set bs \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "a \<sim>\<^bsub>Ring_Divisibility.mult_of R\<^esub> b"
cofinite_extremally_disconnected: fixes X :: "nat cofinite set" shows "extremally_disconnected X"
simps_7: fixes a :: "'c"   and b :: "'c" shows "dbm_add (Lt a) (Lt b) = Lt (a + b)"
poly_reflect_eq_0_iff: fixes p :: "'a poly"   and x :: "'a" shows "(poly (reflect_poly p) x = (0::'a)) = (p = 0 \<or> x \<noteq> (0::'a) \<and> poly p ((1::'a) / x) = (0::'a))"
less_eq_scene_def: fixes X :: "'a scene"   and Y :: "'a scene" shows "X \<subseteq>\<^sub>S Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 on Y \<oplus>\<^sub>S s\<^sub>3 on X = s\<^sub>1 \<oplus>\<^sub>S s\<^sub>2 \<oplus>\<^sub>S s\<^sub>3 on X on Y)"
llist_partial_function_definitions: shows "partial_function_definitions (\<sqsubseteq>) lSup"
insert_subset: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "order.greater_eq B (insert x A) = (x \<in> B \<and> order.greater_eq B A)"
simps_1: fixes t :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>t\<rangle>) = fv t"
subdegree_eq_0_left: fixes f :: "'b fps" assumes "\<exists>g. 1 = f * g" shows "subdegree f = 0"
irreducible_imp_prime_elem: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.irreducible times one zero x" shows "comm_semiring_1.prime_elem times one zero x"
sum_id: fixes f :: "nat \<Rightarrow> 'a" shows "sum f {0..<nr} = sum_list (map f [0..<nr])"
trss_def: shows "trss \<equiv> \<lambda>fg. {((xa, x), xd, xc, xb). trssp fg xa x xd xc xb}"
not_le: fixes y :: "'b"   and x :: "'b" shows "(\<not> y \<le> x) = (x < y)"
trace_rest_eq_def: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf" shows "trace_rest_eq rest1 rest2 IA IU p q = (\<forall>tr. set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV \<longrightarrow> rel_prod (rel_fun (eq_onp (\<lambda>ia. ia \<in> IA)) (=)) (rel_fun (eq_onp (\<lambda>iu. iu \<in> IU)) (=)) (trace_rest' rest1 p tr) (trace_rest' rest2 q tr))"
lin_redE: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "lin_red F p q" obtains f :: "'t \<Rightarrow>\<^sub>0 'd" where "f \<in> F" and "red_single p q f (0::'a)"
sets_T': fixes I :: "'s pmf" shows "sets (T' I) = sets (stream_space (count_space UNIV))"
stopType_lemma: fixes a :: "nat \<Rightarrow> stopType list"   and t :: "nat" assumes "msg (Suc 0) a" shows "a t = [] \<or> a t = [stop_vc]"
vsnd_conv: fixes a :: "V"   and b :: "V" shows "vsnd \<langle>a, b\<rangle> = b"
aext_uop: fixes f :: "'f \<Rightarrow> 'c"   and u :: "('f, 'e) uexpr"   and a :: "'e \<Longrightarrow> 'd" shows "uop f u \<oplus>\<^sub>p a = uop f (u \<oplus>\<^sub>p a)"
differentiable_on_subset: fixes f :: "'c \<Rightarrow> 'd"   and t :: "'c set"   and s :: "'c set" assumes "f differentiable_on t"   and "s \<subseteq> t" shows "f differentiable_on s"
disj_upred_def: shows "(\<or>) = (\<sqinter>)"
dterms_no_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p1 :: "('b, 'c, 'a, 'd) seqp"   and p2 :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "p1 \<oplus> p2 \<notin> dterms \<Gamma> p"
inf_sup_ord_4: fixes inf :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and y :: "'x"   and x :: "'x" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
uniform_discrete_finite_iff: fixes S :: "'a set" shows "(uniform_discrete S \<and> bounded S) = finite S"
set_less_eq_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq> B"   and "B \<sqsubseteq> A" shows "A = B"
concrete_category_axioms: shows "concrete_category {A. |A| <o \<AA>} (\<lambda>A B. extensional A \<inter> (A \<rightarrow> B)) (restrict (\<lambda>x. x)) (\<lambda>C B. compose)"
inject: fixes x1 :: "'b"   and x2 :: "'b tree list"   and y1 :: "'b"   and y2 :: "'b tree list" shows "(Node x1 x2 = Node y1 y2) = (x1 = y1 \<and> x2 = y2)"
all_two_cubes_have_four_distict_edges: shows "\<forall>twoCube\<in>two_chain. card (boundary twoCube) = 4"
merge_conv_merge0: fixes I' :: "('a \<times> 'b) list"   and I :: "('a \<times> 'b) list" shows "merge I' I = merge0 I' I"
fBex_def: shows "fBex \<equiv> map_fun fset id Bex"
ord_isom_inj: fixes E :: "('c, 'd) Order_scheme"   and f :: "'a \<Rightarrow> 'c"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "(a = b) = (f a = f b)"
Stop_is_CSP: shows "is_CSP_process (R (true \<turnstile> \<lambda>(A, A'). tr A' = tr A \<and> wait A'))"
hd_Cons_tl: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "lead_coeff xs # tl xs = xs"
has_codomain_iff_arr: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" shows "(A_BxA.codomains f \<noteq> {}) = A_BxA.arr f"
nodes_on_st_path: shows "\<forall>v\<in>V. connected s v \<and> connected v t"
invertible_right_cancel: fixes x :: "'a set"   and y :: "'a set"   and z :: "'a set" assumes "quotient.invertible x"   and "x \<in> Partition"   and "y \<in> Partition"   and "z \<in> Partition" shows "(y [\<cdot>] x = z [\<cdot>] x) = (y = z)"
euler_genus_iso: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "digraph_map G M"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph_map.euler_genus (app_iso hom G) (pre_digraph_map.map_iso G M hom) = pre_digraph_map.euler_genus G M"
agent_abs_def: fixes a :: "'a"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "agent_abs a t \<equiv> {(tFirst t', tLast t') |t'. t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t}"
modset_def: shows "modset \<equiv> ANALYZE (lhsv \<pi> c)"
matchT_T_def: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" shows "matchT_T c d \<equiv> \<forall>s t s'. s \<approx> t \<and> (c, s) \<rightarrow>t s' \<longrightarrow> (\<exists>t'. (d, t) \<rightarrow>t t' \<and> s' \<approx> t')"
CRYPT: fixes X :: "freemsg"   and X' :: "freemsg"   and K :: "nat" assumes "X \<sim> X'" shows "CRYPT K X \<sim> CRYPT K X'"
rel_fun_relcompp: fixes R1 :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and S1 :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and a :: "'g \<Rightarrow> 'h"   and b :: "'k \<Rightarrow> 'l" assumes "((R1 ===> S1) OO (R2 ===> S2)) a b" shows "(R1 OO R2 ===> S1 OO S2) a b"
reduced_run_ind: fixes q :: "'state"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action stream"   and l :: "'action list"   and w :: "'action list"   and w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and u :: "'action stream" assumes "reduced_run q v\<^sub>1 v\<^sub>2 l w w\<^sub>1 w\<^sub>2 u" shows "Ind (set w\<^sub>2) (sset u)"
h2: shows "h \<turnstile> (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent node) \<rightarrow>\<^sub>h h2"
mon_ww_cil: fixes w :: "('a, 'b) label list list"   and w1 :: "('a, 'b) label list list"   and \<alpha> :: "('a, 'b) label list \<Rightarrow> 'c set \<times> 'c set"   and w2 :: "('a, 'b) label list list"   and fg :: "('e, 'a, 'f, 'd, 'g) flowgraph_rec_scheme" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "mon_ww fg w = mon_ww fg w1 \<union> mon_ww fg w2"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
prod_result_cases: fixes r :: "'b \<times> ('c, 'd) result" obtains s :: "'b" and v :: "'c" where "r = (sa__, Rval v)"   | s :: "'b" and err :: "'d error_result" where "r = (s, Rerr err)"