hcju/isabelleps huggingface dataset:
assert_all_state_indices: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and s :: "('i, 'a) state" assumes "\<triangle> t"   and "assert_all_state t as = s" shows "indices_state s \<subseteq> fst ` set as"
eq: fixes A :: "'b set"   and B :: "'c set" assumes "A \<in> sets S"   and "B \<in> sets T" shows "emeasure M {x \<in> space M. X x \<in> A \<and> Y x \<in> B} = \<integral>\<^sup>+x\<in>A. (\<integral>\<^sup>+y\<in>B. f (x, y)\<partial>T)\<partial>S"
intros_8: shows "CK_nf (C \<circ> K \<circ> C \<circ> K)"
infsetsum_cong_neutral: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'c)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'c)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "infsetsum f A = infsetsum g B"
gcd_zero_iff: fixes a :: "'a"   and b :: "'a" shows "(some_gcd a b = (0::'a)) = (a = (0::'a) \<and> b = (0::'a))"
monad_state_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return bind get put" shows "monad_state (return_env return) (bind_env bind) (get_env get) (put_env put)"
count_greater_eq_Suc_zero_iff: fixes M :: "'a multiset"   and x :: "'a" shows "(Suc 0 \<le> count M x) = (x \<in># M)"
mset_single_cases2: fixes s :: "'c"   and c :: "'c multiset"   and r' :: "'c"   and c' :: "'c multiset" assumes "add_mset s c = add_mset r' c'" obtains "s = r'" and "c = c'"   | "c' = c' - {#s#} + {#s#}" and "c = c - {#r'#} + {#r'#}" and "c - {#r'#} = c' - {#s#}"
unordered_pair_element_equality: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "{p, q} = {r, s}"   and "p = r" shows "q = s"
complete\<^sub>N_def: fixes N :: "'c quasinorm" shows "complete\<^sub>N N = (\<forall>u. cauchy_ine\<^sub>N N u \<longrightarrow> (\<exists>x. tendsto_ine\<^sub>N N u x))"
A: shows "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
VDM_validn_valid: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<forall>n.  \<Turnstile>\<^sub>n c : A " shows " \<Turnstile> c : A "
wellformed_transaction_unlabel_cases_5: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step" assumes "wellformed_transaction T"   and "x \<in> set (unlabel (transaction_send T))" shows "\<exists>t. x = send\<langle>t\<rangle>"
zero_less_iff_neq_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
rel_inject_1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and y1 :: "'c" shows "rel_behaviour R (Terminates x1) (Terminates y1) = R x1 y1"
def_ts: shows "e = Eq t s \<or> e = Eq s t"
mult_pos_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times a b) zero"
SafeArrowCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and call :: "'a call_expr"   and b :: "'a expr" assumes "\<Gamma> \<turnstile> Call src SafeArrowCall call \<Rrightarrow> b" obtains src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = Call (TypeOperationCall src\<^sub>2 ArrowCall SelectByKindOp (to_required_type \<sigma>)) ArrowCall call\<^sub>2" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2 : \<tau>" and "element_type \<tau> \<sigma>" and "OclVoid[?] \<le> \<sigma>" and "\<Gamma> \<turnstile>\<^sub>E TypeOperationCall src\<^sub>2 ArrowCall SelectByKindOp (to_required_type \<sigma>) : \<rho>" and "(\<Gamma>, \<rho>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2"
prod_prime_eq: fixes A :: "nat set"   and B :: "nat set" assumes "finite A"   and "finite B"   and "\<And>x. x \<in> A \<Longrightarrow> prime x"   and "\<And>x. x \<in> B \<Longrightarrow> prime x"   and "\<Prod>A = \<Prod>B" shows "A = B"
midpoint_plus_self: fixes a :: "'c"   and b :: "'c" shows "midpoint a b + midpoint a b = a + b"
content_cbox_if_cart: fixes a :: "(real, 'c) vec"   and b :: "(real, 'c) vec" shows "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>UNIV. b $ i - a $ i)"
same_eval'': fixes At :: "atomUni"   and a :: "real"   and b :: "real"   and c :: "real" assumes "getPoly At = (a, b, c)"   and "a \<noteq> 0 \<or> b \<noteq> 0 \<or> c \<noteq> 0" shows "\<exists>x. \<forall>y<x. aEvalUni At y = aEvalUni At x"
varFIXvar: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" shows "PhiWhileP A b \<Phi> (s, t) = ((b, A, \<Phi>, s, t) \<in> var)"
finite_measure_subadditive: fixes A :: "'c set"   and M :: "'c pmf"   and B :: "'c set" assumes "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M (A \<union> B) \<le> measure_pmf.prob M A + measure_pmf.prob M B"
left_commute: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b \<sqinter> (a \<sqinter> c) = a \<sqinter> (b \<sqinter> c)"
hyps: fixes n :: "nat" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
diff_strict_left_mono: fixes 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 b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less b a" shows "less (minus c a) (minus c b)"
indep_var_rv1: fixes S :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and T :: "'b measure"   and Y :: "'a \<Rightarrow> 'b" assumes "indep_var S X T Y" shows "random_variable S X"
alternate_antimono: fixes f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" assumes "\<And>x. f x \<le> x"   and "\<And>x. g x \<le> x" shows "decseq (alternate f g)"
lfp_loop_fp: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "healthy (wp body)"   and "sound P" shows "\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body (\<lambda>s. bound_of P) s + \<guillemotleft> \<N> G \<guillemotright> s * P s \<tturnstile> \<lambda>s. bound_of P"
Inst: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and e :: "expr"   and Q :: "'b assn"   and T :: "ref_ty" assumes "G,A\<turnstile>{Normal P} e-\<succ> {\<lambda>Val:v:. \<lambda>s.. Q\<leftarrow>In1 (Bool (v \<noteq> Null \<and> G,s\<turnstile>v fits RefT T))}" shows "G,A\<turnstile>{Normal P} e InstOf T-\<succ> {Q}"
uminus_add_in_Ker_eq_eq_im: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G" shows "(- g + h \<in> Ker) = (T g = T h)"
simps_23: fixes r :: "'a rexp"   and s :: "'a rexp" shows "local.le_rexp (Star r) (Star s) = local.le_rexp r s"
f_aggregate_take_nth: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat"   and m :: "nat"   and ag :: "'b list \<Rightarrow> 'b" assumes "n < length xs div k"   and "n < m div k" shows "f_aggregate (xs \<down> m) k ag ! n = f_aggregate xs k ag ! n"
assert_s_def: shows "assert_s = Incremental_State_Ops_Simplex.assert_s assert_bound_code"
interv_sum_list_conv_sum_set_int: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and f :: "int \<Rightarrow> 'd"   and k :: "int"   and l :: "int" assumes "class.monoid_add plus zero" shows "sum_list (map f [k..l]) = sum f (set [k..l])"
case_prodE: fixes c :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool"   and p :: "'ap \<times> 'aq" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'ap" and y :: "'aq" where "p = (x, y)" and "c x y"
simps_14: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and m1 :: "'b common_primitive match_expr"   and m2 :: "'b common_primitive match_expr" shows "oiface_rewrite ipassmt (MatchAnd m1 m2) = MatchAnd (oiface_rewrite ipassmt m1) (oiface_rewrite ipassmt m2)"
sigma_algebra_pre_sigma: fixes T :: "'a \<Rightarrow> 't" assumes "stopping_time F T" shows "sigma_algebra \<Omega> {A. \<forall>t. {\<omega> \<in> A. T \<omega> \<le> t} \<in> sets (F t)}"
res_ri1: fixes y :: "'a"   and x :: "'a" shows "y \<cdot> (y \<rightarrow> x) \<le> x"
N\<alpha>: shows "(N, \<alpha>) \<in> \<RR>"
wt_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "Ik.wtE \<xi>"   and "Ik.wt T" shows "intT (Ik.tpOf T) (Ik.int \<xi> T)"
Ssegment_free: fixes a :: "'c"   and D :: "'c Order" assumes "a \<notin> carrier D" shows "Ssegment D a = carrier D"
wf_r_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and P :: "nat \<times> k \<times> f \<times> nat \<times> nat \<Rightarrow> bool" assumes "update_arg_wf r"   and "\<And>nhip. P (0, unk, val, Suc 0, nhip)"   and "\<And>dsn hops nhip. 0 < dsn \<Longrightarrow> P (dsn, kno, val, hops, nhip)" shows "P r"
com_rel_elim_cases_8: fixes c :: "('d, 'e, 'f) com"   and c1 :: "'d \<Rightarrow> ('d, 'e, 'f) com" assumes "(c, DynCom c1) \<in> com_rel" obtains x :: "'d" where "c = c1 x"
span_is_monotone: fixes S :: "'c set"   and T :: "'c set" assumes "S \<subseteq> T" shows "span S \<subseteq> span T"
ln_add: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "ln (x + y) = ln x + ln (1 + y / x)"
eff_Ch: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "eff (Ch ch c1 c2) s i = s"
hd_n: shows "head_\<omega> n1 < head_\<omega> n2"
simps_5: fixes e1 :: "expr"   and e2 :: "expr" shows "free_vars <e1 ,  e2> = free_vars e1 \<union> free_vars e2"
I: shows "card {i. i < CARD('nr) \<and> i \<in> I} = CARD('nr2)"
seq'_cons: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and s :: "state"   and t :: "state"   and rs :: "'c rule list"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], s\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, t\<rangle> \<Rightarrow> t'"   and "no_matching_Goto \<gamma> p [r]" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, s\<rangle> \<Rightarrow> t'"
add_diff_cancel_left': fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and fi :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "cancel_comm_semigroup f fi" shows "fi (f a b) a = b"
tail_events_def: fixes A :: "'b \<Rightarrow> 'a set set" shows "tail_events A = (\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))"
reachable_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v" shows "v \<in> verts (G \<restriction> {v. u \<rightarrow>\<^sup>* v})"
inf_def: fixes s1 :: "'a"   and s2 :: "'a" shows "s1 \<sqinter> s2 \<equiv> THE s. is_glb s s1 s2"
unat_def: fixes x :: "nat" shows "unat x \<equiv> Abs_unat (Some x)"
intpow_uminus_nat: fixes n :: "int"   and x :: "'b" assumes "0 \<le> n" shows "x ^\<^sub>Z - n = inverse (x ^ nat n)"
preserves_reflects_arr: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" shows "B.arr (E.map f) = A2_BxA2.arr f"
integral_sum: fixes I :: "'c set"   and M :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "LINT x|M. (\<Sum>i\<in>I. f i x) = (\<Sum>i\<in>I. integral\<^sup>L M (f i))"
at_within_def: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c"   and s :: "'c set" assumes "class.topological_space open" shows "topological_space.at_within open a s = inf (topological_space.nhds open a) (principal (s - {a}))"
holomorphic_on_power: fixes f :: "complex \<Rightarrow> complex"   and s :: "complex set"   and n :: "nat" assumes "f holomorphic_on s" shows "(\<lambda>z. f z ^ n) holomorphic_on s"
assms_3: shows "zs \<noteq> ys \<or> z \<notin> {y, y1}"
"case": fixes f :: "('b \<Rightarrow> 'c \<times> 'b) \<Rightarrow> 'a"   and x :: "'b \<Rightarrow> 'c \<times> 'b" shows "(case State x of State x \<Rightarrow> f x) = f x"
conjugate_dual_additive: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugate f g" shows "dual_additive (uminus \<circ> f)"
Cfun_app_eqvt: fixes \<pi> :: "perm"   and f :: "'d \<rightarrow> 'c"   and x :: "'d" shows "\<pi> \<bullet> f\<cdot>x = (\<pi> \<bullet> f)\<cdot>(\<pi> \<bullet> x)"
surjE: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'b" assumes "surj f" obtains x :: "'c" where "y = f x"
directed_filtered_dual: shows "directed \<circ> (`) \<partial> = filtered"
inverse_le_iff_le: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a"   and "less zero b" shows "less_eq (inverse a) (inverse b) = less_eq b a"
IsConfSet_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and C :: "'d set" shows "IsConfSet F G C \<equiv> C \<subseteq> \<Union> (States ` F) \<and> RootExSem F G C \<and> UniqueSucStates F G C"
exr_iff: shows "(\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec_copr (qs1 @ qs2) r) = (\<exists>r\<in>set (characterize_root_list_p p). x = consistent_sign_vec_copr qs1 r @ consistent_sign_vec_copr qs2 r)"
lUnionM_def: fixes ms :: "'m list" shows "lUnionM ms \<equiv> foldr lunionM ms (return [])"
amin_commute: fixes x :: "ant"   and y :: "ant" shows "amin x y = amin y x"
generated_ringE: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set" assumes "semiring_of_sets \<Omega> M"   and "a \<in> semiring_of_sets.generated_ring M" obtains C :: "'c set set" where "finite C" and "disjoint C" and "C \<subseteq> M" and "a = \<Union> C"
shows_sep_cong: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and g :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and sep :: "char list \<Rightarrow> char list" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> f x = g x" shows "shows_sep f sep xs = shows_sep g sep ys"
lossless_S: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> lossless_spmf (local.etp.S (fst y))"
disj1: shows "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> Xout i \<inter> Xout j = {}"
simps_8: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<rbrace> = \<rho>' \<lbrace>t\<rbrace>"
trms\<^sub>s\<^sub>s\<^sub>t_subst: fixes S :: "('d, 'c) stateful_strand_step list"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "trms\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<theta>) = trms\<^sub>s\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
trunc_ell2_ket_cspan: fixes S :: "'c set"   and x :: "'c ell2" assumes "finite S" shows "trunc_ell2 S x \<in> cspan (range ket)"
dvd_mult_cancel_left: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times c a) (times c b) = (c = zero \<or> dvd.dvd times a b)"
u_simps_3: shows "trg (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) = src (f \<star> g)"
assms_2: shows "\<not> stateful_strand_step.is_Receive x"
arr_MkArr: fixes A :: "bool"   and B :: "bool"   and f :: "unit" assumes "A \<in> UNIV"   and "B \<in> UNIV"   and "f \<in> (if A = B then {()} else {})" shows "arr (MkArr A B f)"
basis_col_space_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "basis_col_space A = {row i (Gauss_Jordan (Finite_Cartesian_Product.transpose A)) |i. row i (Gauss_Jordan (Finite_Cartesian_Product.transpose A)) \<noteq> 0}"
rep_eq: fixes x :: "perm"   and xa :: "'e set['f]" shows "set_bset (x \<bullet> xa) = x \<bullet> set_bset xa"
indep: shows "indep_vars (\<lambda>_. borel) X UNIV"
uminus_Inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "class.complete_boolean_algebra Inf Sup inf less_eq less sup bot top minus uminus" shows "uminus (Inf A) = Sup (uminus ` A)"
not_call_simps_2: fixes l :: "'d"   and fa :: "'a \<Rightarrow> 'a"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}\<lbrakk>fa\<rbrakk> p)"
Rel_def: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Transfer.Rel r \<equiv> r"
assms_1: shows "type_update ts (to_ct_list t_in) (Type t_out) = ts'"
cases_scheme: fixes r :: "('a, 'b, 'c, 'd) configuration_scheme" obtains states :: "'a \<Rightarrow> 'b" and msgs :: "nat \<Rightarrow> 'c message list" and process_snapshot :: "'a \<Rightarrow> 'b option" and channel_snapshot :: "nat \<Rightarrow> 'c list \<times> recording_state" and more :: "'d" where "r = \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr>"
A: shows "(add_mset s c, e, c') \<in> gtr gtrs"
tsp__ncop1: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B CTSP P Q" shows "\<not> Coplanar A B C P"
space_uniform_count_measure: fixes A :: "'a set" shows "space (uniform_count_measure A) = A"
Target_ChiRel_HAInit_StepConf: fixes S :: "'d"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and C :: "'d set" assumes "S \<in> Target TS"   and "(S, T) \<in> ChiRel A"   and "T \<in> HAInitStates A" shows "T \<in> StepConf A C TS"
find_graph_occurence_vertices: fixes S :: "nat \<Rightarrow> ('e, 'f) labeled_graph"   and V :: "'g set"   and f :: "('g \<times> 'f) set" assumes "chain S"   and "finite V"   and "univalent f"   and "f `` V \<subseteq> vertices (chain_sup S)" shows "\<exists>i. f `` V \<subseteq> vertices (S i)"
emeasure_eq_measure: fixes A :: "'a set" shows "emeasure N A = ennreal (Sigma_Algebra.measure N A)"
uminus_mult_right_mat: fixes A :: "'c mat"   and B :: "'c mat" assumes "dim_col A = dim_row B" shows "A * - B = - (A * B)"
unique_expression3: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and l :: "nat" assumes "H \<subseteq> carrier M"   and "f \<in> {k. k \<le> Suc n} \<rightarrow> H"   and "s \<in> {k. k \<le> Suc n} \<rightarrow> carrier R"   and "l \<le> Suc n"   and "f l \<notin> f ` ({k. k \<le> Suc n} - {l})" shows "\<exists>g m t. g \<in> {k. k \<le> m} \<rightarrow> H \<and> inj_on g {k. k \<le> m} \<and> t \<in> {k. k \<le> m} \<rightarrow> carrier R \<and> l_comb R M (Suc n) s f = l_comb R M m t g \<and> t m = s l \<and> g m = f l"
mat_plus_gt_left_mono: fixes sd :: "nat"   and n :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "sd \<le> n"   and "mat_gt (\<succ>) sd A B"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n" shows "mat_gt (\<succ>) sd (A + C) (B + C)"
strand_sem_split'_3: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d' \<theta>" shows "\<lbrakk>M; S\<rbrakk>\<^sub>d' \<theta>"
BC_2: shows "P' * Q' = 1\<^sub>m n"
B: shows "\<forall>b\<in>set B. \<exists>a\<in>set A. \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
eq: shows "{Lam r |r. Lam r \<in> A} = A"
assms_3: fixes v :: "'a"   and v' :: "'a" assumes "x = Some v"   and "(v, v') \<in> Id" shows "fs v \<le> \<Down> R (fs' v')"
starfun_hnorm: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b star" shows "hnorm ((*f* f) x) = (*f* (\<lambda>x. norm (f x))) x"
lambda: fixes s :: "trm"   and t :: "trm"   and T :: "Syntax.type" assumes "s \<longlonglongrightarrow> t" shows "\<lambda>T:s \<longlonglongrightarrow> \<lambda>T:t"
nonstop_t_uhalt_eq: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'a"   and stp :: "nat"   and a :: "nat"   and b :: "cell list"   and c :: "cell list" assumes "tm_wf0 tp"   and "steps0 (Suc 0, Bk \<up> l, <lm>) tp stp = (a, b, c)"   and "\<not> TSTD (a, b, c)" shows "rec_exec rec_nonstop [code tp, bl2wc (<lm>), stp] = Suc 0"
A: shows "g_to_list m = iterate_to_list (iteratei m)"
face_cycle_succ_a_neigh: shows "G.face_cycle_succ a_neigh = a"
TermEquivCl'_def: fixes T :: "('a, 'b) Axioms"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a" shows "TermEquivCl' T A e B \<equiv> \<lparr>TDomain = A, TExprSet = TermEquivClGen T A e B, TCodomain = B\<rparr>"
hunion_hempty_right: fixes A :: "hf" shows "A \<squnion> 0 = A"
smult_ket_vec: fixes x :: "complex"   and v :: "complex vec" shows "|x \<cdot>\<^sub>v v\<rangle> = x \<cdot>\<^sub>m |v\<rangle>"
oalist_inv_raw_sort_oalist: fixes xs :: "('a \<times> 'c) list" shows "oalist_inv_raw (sort_oalist xs)"
terminates_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step" shows "terminates g = (terminates_on g = UNIV)"
Cinfinite_csum1: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" assumes "Cinfinite r1" shows "Cinfinite (r1 +c r2)"
simps_1: fixes a :: "'a" shows "a ^ 0 = (1::'a)"
ib1_def: shows "ib1 \<equiv> local.interaction_bound"