hcju/isabelleps huggingface dataset:
lspace_B: shows "(\<lambda>x. B) \<in> lspace (lebesgue_on S) (ennreal p)"
Sup_def: fixes A :: "'a set" shows "Sup A = Finite_Set.fold sup bot A"
comm_monoid_mset_axioms: shows "comm_monoid_mset (*) (1::'a)"
uniform_lim_inverse: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and b :: "real" assumes "uniform_limit S f l F"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> norm (l x)"   and "0 < b" shows "uniform_limit S (\<lambda>x y. inverse (f x y)) (inverse \<circ> l) F"
EquivE: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]" shows "[\<phi> in v] = [\<psi> in v]"
normalize_quot_aux_2: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "b = snd (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
neg: fixes A :: "fmla"   and i :: "name"   and x :: "trm" shows "(Robinson_Arithmetic.neg A)(i::=x) = Robinson_Arithmetic.neg (A(i::=x))"
Nil: shows "cs\<^bsup>\<pi>\<^esup> i = xs @ [x, x'] @ []"
l10_iff: fixes x :: "'a"   and y :: "'a" shows "(- x \<sqinter> y = bot) = (- x \<le> - y)"
FAccRedsNull: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<rangle> \<rightarrow>* \<langle>THROW NullPointer,s'\<rangle>"
inv_chart_inverse: fixes x :: "'b"   and c :: "('b, 'c) chart" assumes "x \<in> domain c" shows "inv_chart c (apply_chart c x) = x"
bounded_bilinear_def: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "bounded_bilinear prod \<equiv> ((\<forall>a a' b. prod (a + a') b = prod a b + prod a' b) \<and> (\<forall>a b b'. prod a (b + b') = prod a b + prod a b')) \<and> (\<forall>r a b. prod (r *\<^sub>R a) b = r *\<^sub>R prod a b) \<and> (\<forall>a r b. prod a (r *\<^sub>R b) = r *\<^sub>R prod a b) \<and> (\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K)"
deadlocked'I: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<And>t. not_final_thread s t \<Longrightarrow> t \<in> deadlocked s" shows "deadlocked' s"
if_false: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<not> eval\<^sub>B mem b" shows "(Stmt.If b t e, mem) \<leadsto>\<^sub>s (e, mem)"
polyfun_extremal_lemma: fixes e :: "real"   and c :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "0 < e" shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z ^ i) \<le> e * norm z ^ Suc n"
B: fixes as :: "('a poly \<times> nat) set"   and p :: "'a poly"   and n :: "nat" assumes "monic_factorization as p" shows "monic_factorization.B p n = (\<Prod>(a, i)\<in>as - UNIV \<times> {0..<n}. a)"
inorder_getmin': fixes t :: "'t"   and a :: "'a"   and b :: "'b" assumes "inv t"   and "sorted1 (inorder t)"   and "inorder t \<noteq> []"   and "getmin t = (a, b)" shows "is_min2 (a, b) (set (inorder t))"
not_out_bet: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "\<not> Tarski_neutral_dimensionless.Out Bet B A C" shows "Bet A B C"
add_diff_eq: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a +\<^sub>o\<^sub>w (b -\<^sub>o\<^sub>w c) = a +\<^sub>o\<^sub>w b -\<^sub>o\<^sub>w c"
digitToTree_list: fixes d :: "('a, 'b) Digit" shows "FingerTreeStruc.toList (FingerTreeStruc.digitToTree d) = FingerTreeStruc.digitToList d"
rel_prod_conv: fixes R1 :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_prod R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
comp_inv_arr': fixes f :: "'a" assumes "local.iso f" shows "local.inv f \<cdot> f = local.dom f"
DomInterAllowsMT_Ports: fixes p :: "int"   and po :: "int"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "p \<noteq> po" shows "dom (C (AllowPortFromTo a b p)) \<inter> dom (C (AllowPortFromTo c d po)) = {}"
p_nonpos: fixes t :: "real"   and s :: "'a"   and s' :: "'a" assumes "t \<le> 0" shows "p s s' t = of_bool (s = s')"
Confidentiality_Kas: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A \<lbrace>Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbV" shows "Key authK \<notin> analz (knows Spy evs)"
refl: fixes a :: "'i multiset" shows "a \<subseteq># a"
det_def: fixes A :: "'b mat" shows "det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'b))"
A_eq: shows "A = (\<lambda>i. (UNIV - J i) \<times> X i \<union> Sigma (J i) (F i))"
ListSum_add: fixes V :: "'a list"   and f :: "'a \<Rightarrow> nat"   and g :: "'a \<Rightarrow> nat" shows "ListSum V f + ListSum V g = (\<Sum>\<^bsub>x\<in>V\<^esub> f x + g x)"
length_init_syzygy_list: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "length (init_syzygy_list bs) = length bs"
hisp_his_eq: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" shows "hisp t n nid = (\<lambda>x. x \<in> his t n nid)"
src_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (src\<^sub>B f)"
merge_commute: fixes I :: "'b set"   and J :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "I \<inter> J = {}" shows "merge I J (x, y) = merge J I (y, x)"
cSUP_least: fixes A :: "'k set"   and f :: "'k \<Rightarrow> 'l multiset"   and M :: "'l multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq># M" shows "Sup (f ` A) \<subseteq># M"
finite: shows "finite (dfa.states \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>)"
x_in: shows "x \<in> set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2))"
in_eqvt: fixes pi :: "(name \<times> name) list"   and x :: "'d"   and X :: "'d set" shows "pi \<bullet> (x \<in> X) = (pi \<bullet> x \<in> pi \<bullet> X)"
RModuleEnd_over_group_ring_is_FGModuleEnd: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "Group G"   and "RModuleEnd (aezfun_setspan G) smult V T" shows "FGModuleEnd G smult V T"
\<psi>_in_terms_of_\<epsilon>o: fixes x :: "'c"   and g :: "'d"   and y :: "'d" assumes "C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G x\<guillemotright>" shows "\<psi> x g = \<epsilon>o x \<cdot>\<^sub>C F g"
\<iota>\<^sub>0_in: shows "\<iota>\<^sub>0 \<in> G.Inf_from M (\<Union> (\<G>_F ` M))"
converse_tranclpE: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and z :: "'c" assumes "r\<^sup>+\<^sup>+ x z" obtains "r x z"   | y :: "'c" where "r x y" and "r\<^sup>+\<^sup>+ y z"
simps_1: fixes uu :: "predicate"   and uv :: "'a list" shows "is_predAtom (Atom (predAtm uu uv)) = True"
fps_divide_deriv: fixes b :: "'a fps"   and a :: "'a fps" assumes "b dvd a" shows "fps_deriv (a / b) = (fps_deriv a * b - a * fps_deriv b) / b\<^sup>2"
t_def: shows "t \<equiv> THE x. abss x = s' \<and> x \<in> set_pmf (action cfg)"
map_upds_upd_conv_last_index: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'b list"   and m :: "'a \<Rightarrow> 'b option"   and y :: "'b" assumes "x \<in> set xs"   and "length xs \<le> length ys" shows "m(xs [\<mapsto>] ys, x \<mapsto> y) = m(xs [\<mapsto>] ys[last_index xs x := y])"
max_reachable_set_axioms: shows "max_reachable_set G S"
assms_2: shows "\<forall>i<n. rs ! i \<in> set s0"
mult_cancel_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semiring_no_zero_divisors_cancel plus zero times" shows "(times c a = times c b) = (c = zero \<or> a = b)"
intro: fixes M1 :: "'c measure"   and M2 :: "'d measure" assumes "pair_sigma_finite M1 M2"   and "prob_space M1"   and "prob_space M2" shows "pair_prob_space M1 M2"
simps: fixes bs :: "atom set"   and x :: "'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'e" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
fvt: fixes t :: "'trm" assumes "set (xs @ us @ vs) \<subseteq> var"   and "set ts \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set ts" shows "set (getFrN (xs @ us @ vs) ts [\<phi>] (length xs)) \<inter> FvarsT t = {}"
composite_simps_5: shows "local.dom (cod \<mu> \<star> cod \<nu> \<star> cod \<pi>) = local.dom (cod \<mu>) \<star> local.dom (cod \<nu>) \<star> local.dom (cod \<pi>)"
fold_random_permutation_foldr: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (\<lambda>xs. foldr f xs x) (pmf_of_set (permutations_of_set A))"
insert_if: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and x :: "'d" assumes "finite A" shows "sum_0 g (insert x A) = (if x \<in> A then sum_0 g A else g x + sum_0 g A)"
infinite_Complement_set_less_eq: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "infinite UNIV" shows "\<not> - A \<sqsubseteq> B"
out\<alpha>_def: shows "out\<alpha> = snd\<^sub>L"
prime_factors_product: 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"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize (times x y)) = set_mset (factorial_semiring.prime_factorization zero times one normalize x) \<union> set_mset (factorial_semiring.prime_factorization zero times one normalize y)"
mapl_G_def: fixes l1 :: "'n \<Rightarrow> 'u"   and l2 :: "'o \<Rightarrow> 'v" shows "mapl_G l1 l2 = map_G l1 l2 id id id id"
seqsimilarI: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "first s = first t"   and "nextsuffix s \<approx> nextsuffix t" shows "s \<approx> t"
select_convs_2: fixes CatDom :: "('d, 'e, 'f) Category_scheme"   and CatCod :: "('a, 'b, 'c) Category_scheme"   and MapM :: "'e \<Rightarrow> 'b"   and more :: "'g" shows "CatCod \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatCod"
measurable_component_singleton: fixes i :: "'a"   and I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure" assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M M i"
valid_prefix_ipset_from_netmask_ipset_from_cidr: fixes pfx :: "'a prefix_match" shows "ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx) = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
Hermite_reduce_above_Least_eq_ge: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and ia :: "'c"   and ass :: "'a \<Rightarrow> 'a"   and res :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "echelon_form A"   and "\<not> is_zero_row i A"   and "\<not> is_zero_row ia A"   and "\<not> is_zero_row ia (Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res)"   and "ia < i" shows "(LEAST n. Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res $ ia $ n \<noteq> (0::'a)) = (LEAST n. A $ ia $ n \<noteq> (0::'a))"
set_swap: fixes n :: "nat"   and xs :: "'b list" shows "set (swap n xs) = set xs"
Int_atLeastAtMostR2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "ord.atLeast less_eq a \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq (ord.max less_eq a c) d"
zero_power: fixes n :: "nat" assumes "0 < n" shows "(0::'b) ^ n = (0::'b)"
head_\<omega>_subseteq: fixes M :: "hmultiset" shows "hmsetmset (head_\<omega> M) \<subseteq># hmsetmset M"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" shows "sorted_wrt P (x # ys) = (Ball (list.set ys) (P x) \<and> sorted_wrt P ys)"
T_on_on'': fixes s0 :: "'state"   and qs :: "'request list"   and A :: "('state \<Rightarrow> 'b) \<times> ('state \<times> 'b \<Rightarrow> 'request \<Rightarrow> 'answer \<times> 'b)" shows "T_on A s0 qs = T_on'' A s0 qs"
rat_uminus_code: fixes p :: "rat" shows "quotient_of (- p) = (let (a, b) = quotient_of p in (- a, b))"
comm_monoid_axioms: shows "comm_monoid (\<squnion>) \<bottom>"
bind_returns_result_E3: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and y :: "'c"   and x :: "'d" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "pure f h" shows "h \<turnstile> g x \<rightarrow>\<^sub>r y"
indRelR_def: fixes Enc :: "'a \<Rightarrow> 'b" shows "encoding.indRelR Enc \<equiv> _Collect (xa, x) (encoding.indRelRp Enc xa x)"
finiteVarsClause: fixes c :: "Literal list" shows "finite (vars c)"
VS_com_VDM: fixes t :: "TP"   and c :: "IMP"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "(t, c) \<in> VS_com" shows "(t = high \<longrightarrow> G \<rhd> c : twiddle) \<and> (t = low \<longrightarrow> (\<exists>A. G \<rhd> c : Sec A))"
times_int_code_1: fixes k :: "int" shows "k * 0 = 0"
bfilter_sound: fixes s :: "char list \<Rightarrow> int"   and S :: "'av Abs_State.st option"   and bv :: "bool"   and b :: "bexp" assumes "s \<in> \<gamma>\<^sub>o S"   and "bv = bval b s" shows "s \<in> \<gamma>\<^sub>o (bfilter b bv S)"
bind_pause: fixes out :: "'o"   and c :: "'i \<Rightarrow> 'm"   and f :: "'a \<Rightarrow> 'm" shows "bind (pause out c) f = pause out (\<lambda>i. bind (c i) f)"
ReprF: fixes f :: "'trm \<Rightarrow> 'trm"   and n :: "'trm" assumes "f \<in> F"   and "n \<in> num" shows "prveqlPT (instInp (encF f) n) (f n)"
length_concat_merge_adj: fixes xss :: "'b list list" shows "length (concat (merge_adj xss)) = length (concat xss)"
countable_product_event_const: fixes F :: "'c \<Rightarrow> 'd set"   and b :: "'d" assumes "\<And>i. countable (F i)" shows "countable {f. (\<forall>i. f i \<in> F i) \<and> finite {i. f i \<noteq> b}}"
spr_sim_rep_abs_syn: fixes ec :: "'s odlist"   and t :: "'s Trace" assumes "spr_simAbs ec = spr_jkbpCSt t" shows "set (toList ec) = spr_abs t"
e2_def: shows "e2 = lcm d (g x) div g x"
abs_eq: shows "1 = Abs_coreflexive (1::'b)"
assms_3: shows "\<forall>h. F h x = g (h x)"
eint_ord_simps_3: fixes q :: "eint" shows "q \<le> \<infinity>"
mapi_branch_head_add_oob: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and ps :: "('b, 'c) fm list"   and xs :: "(nat \<times> nat) set"   and f :: "('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and a :: "'c" shows "mapi_branch (mapper f ({(length branch, length ps)} \<union> xs)) ((ps, a) # branch) = mapi_branch (mapper f xs) ((ps, a) # branch)"
exhaust: fixes y :: "'a Box" obtains "y = \<bottom>"   | u :: "'a\<^sub>\<bottom>" where "y = Box\<cdot>u" and "u \<noteq> \<bottom>"
pathfinish_flow_to_path: fixes x :: "'a"   and t :: "real"   and t' :: "real" shows "pathfinish (flow_to_path x t t') = flow0 x t'"
Ord_ordermap: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and x :: "'c" assumes "wf r"   and "trans r" shows "Ord (ordermap A r x)"
assms_2: shows "\<pi> =\<^bsub>n\<^esub> \<pi>'"
lt_ord_p: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "lpp p \<prec> lpp q" shows "punit.ord_strict_p p q"
antisym_order_consistent_self: fixes r :: "('b \<times> 'b) set" assumes "antisym r" shows "order_consistent r r"
prime_elem_multiplicity_power_distrib: fixes p :: "'a"   and x :: "'a"   and n :: "nat" assumes "prime_elem p"   and "x \<noteq> (0::'a)" shows "multiplicity p (x ^ n) = n * multiplicity p x"
hnorm_triangle_ineq3: fixes x :: "'a star"   and y :: "'a star" shows "\<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
eq_height_0: fixes kdt :: "'b kdt" shows "(height kdt = 0) = (\<exists>p. kdt = Leaf p)"
sterms_choice_disj: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'g, 'e, 'h) seqp"   and p :: "('f, 'g, 'e, 'h) seqp"   and p1 :: "('f, 'g, 'e, 'h) seqp"   and p2 :: "('f, 'g, 'e, 'h) seqp" assumes "wellformed \<Gamma>" shows "(p \<in> sterms \<Gamma> (p1 \<oplus> p2)) = (p \<in> sterms \<Gamma> p1 \<or> p \<in> sterms \<Gamma> p2)"
B_def: shows "B = (\<lambda>x. set_pmf (p x))"
scale_zero_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a" assumes "Modules.module scale" shows "scale a (0::'b) = (0::'b)"
max_y_def: fixes x :: "nat list"   and j :: "nat" shows "max_y x j = (if j < n \<and> Ej j x \<noteq> {} then Min (Ej j x) else Max (set a))"
well_base_image_inj: fixes M :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "local.well_base M"   and "inj_on f (set_mset M)" shows "local.well_base (image_mset f M)"
endomorph: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "ModuleEnd smult M T" shows "T ` M \<subseteq> M"
sumE: fixes s :: "'d + 'e" obtains x :: "'d" where "s = Inl x"   | y :: "'e" where "s = Inr y"
lm030: fixes pair :: "'c \<times> 'd set" shows "card (omega pair) = card (snd pair)"
VPowI: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> B" shows "A \<in>\<^sub>\<circ> VPow B"
iMOD_modD: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]" shows "x mod m = r mod m"
dy_fake_chan_mono2: fixes bad :: "agent set"   and IK :: "msg set"   and chan :: "chan set"   and y :: "msg set"   and z :: "chan set" assumes "extr bad IK chan \<subseteq> synth (analz y)"   and "chan \<subseteq> fake bad (synth (analz y)) z" shows "dy_fake_chan bad IK chan \<subseteq> fake bad (synth (analz y)) z"
Rep_formula_inject: fixes x :: "('a, 'b, 'c, 'd) formula"   and y :: "('a, 'b, 'c, 'd) formula" shows "(Rep_formula x = Rep_formula y) = (x = y)"
mu_def: shows "mu = IArray.of_fun (\<lambda>i. IArray.of_fun (d\<mu> fs i) i) m"
conj_disj_distr: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool" shows "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)"
hcomp_is_functor: shows "functor VV.comp (\<bullet>) (\<lambda>\<nu>\<mu>. fst \<nu>\<mu> \<star> snd \<nu>\<mu>)"
UNC_E: fixes A :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "UNC A" obtains "a = b"   | "a \<notin> NF A"   | "b \<notin> NF A"   | "(a, b) \<notin> A\<^sup>\<leftrightarrow>\<^sup>*"
gauss_jordan_single_1: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'a mat"   and x :: "'a vec" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C"   and "x \<in> carrier_vec nc" shows "(A *\<^sub>v x = 0\<^sub>v nr) = (C *\<^sub>v x = 0\<^sub>v nr)"
max2_equals2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(local.max2 a b = b) = ((a, b) \<in> r)"
idplus: fixes m :: "'n"   and n :: "'n" assumes "m \<noteq> n" shows "P (\<lambda>x. \<chi>i. if i = m then x $ m + x $ n else x $ i)"
rbt_min_opt_Branch: fixes t1 :: "('b, unit) RBT_Impl.rbt"   and c :: "color"   and k :: "'b"   and t2 :: "('b, unit) RBT_Impl.rbt" assumes "t1 \<noteq> rbt.Empty" shows "rbt_min_opt (Branch c t1 k () t2) = rbt_min_opt t1"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and o\<^sub>1 :: "'b option"   and o\<^sub>2 :: "'b option" shows "Opt.le r o\<^sub>1 o\<^sub>2 = (case o\<^sub>2 of None \<Rightarrow> o\<^sub>1 = None | Some y \<Rightarrow> case o\<^sub>1 of None \<Rightarrow> True | Some x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
matrix_mult_sum: fixes A :: "(('a, 'c) vec, 'b) vec"   and x :: "('a, 'c) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s column i A)"
dvdm_factor: fixes m :: "int"   and f :: "int poly"   and g :: "int poly"   and h :: "int poly" assumes "poly_mod.dvdm m f g" shows "poly_mod.dvdm m f (g * h)"
set_spmf_G_domain_rel: fixes h :: "'pub_input"   and w :: "'witness" assumes "(h, w) \<in> set_spmf G" shows "h \<in> Domain Rel"
simps_2: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "flows_fix \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_fix"
is_generalisation_of_same_arity: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" assumes "is_generalisation_of t' t i r" shows "Arity t = Arity t'"
simps_3: fixes t :: "ty"   and e :: "expr" shows "constVal (Cast t e) = None"
ext_inject: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c"   and hosts' :: "'b set"   and flows_fix' :: "('b \<times> 'b) set"   and flows_state' :: "('b \<times> 'b) set"   and more' :: "'c" shows "(\<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = \<lparr>hosts = hosts', flows_fix = flows_fix', flows_state = flows_state', \<dots> = more'\<rparr>) = (hosts = hosts' \<and> flows_fix = flows_fix' \<and> flows_state = flows_state' \<and> more = more')"
transp_inf: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "transp r"   and "transp s" shows "transp (inf r s)"
Rats_mult: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a * b \<in> \<rat>"
calc_ideal_points_different: fixes H :: "circline" assumes "is_poincare_line H" shows "\<exists>i1\<in>calc_ideal_points H. \<exists>i2\<in>calc_ideal_points H. i1 \<noteq> i2"
nsqn_quality_increases: fixes i :: "nat"   and \<xi> :: "state"   and \<xi>' :: "state" assumes "i \<in> kD (rt \<xi>)"   and "quality_increases \<xi> \<xi>'" shows "nsqn (rt \<xi>) i \<le> nsqn (rt \<xi>') i"
assms_4: shows "min_r \<equiv> Min (pre_ranks (\<lambda>q. state_rank q n) (w n) q)"
circ_right_unfold_1: fixes x :: "'b" shows "Rf (x\<^sup>\<star>) x = x\<^sup>\<star>"