hcju/isabelleps huggingface dataset:
client2_def: shows "client2 \<equiv> ''x'' ::= N 0;; WHILE Less (V ''x'') (N 4000000) DO (''x'' ::= Plus (V ''x'') (N 1);; IF BEq (V ''x'') (N 2000000) THEN ''low'' ::= V ''high'' ELSE SKIP)"
monotone_spmf: fixes x :: "'h" shows "monotone (ord_spmf (=)) (\<le>) (\<lambda>p. spmf p x)"
zero: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "module_hom s1 s2 f" shows "f (0::'b) = (0::'c)"
vec_to_iarray_matrix_matrix_mult: fixes A :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "vec_to_iarray (A *v x) = matrix_to_iarray A *iv vec_to_iarray x"
gt_subst: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "wary t"   and "wary s"   and "t >\<^sub>t s" shows "subst \<rho> t >\<^sub>t subst \<rho> s"
mult_column_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and n :: "'c"   and q :: "'b" shows "mult_column A n q = (\<chi>i j. if j = n then A $ i $ j * q else A $ i $ j)"
Plus_infinite_bij_betw: fixes A :: "'a set"   and g :: "'b \<Rightarrow> 'a"   and B :: "'b set" assumes "infinite A"   and "inj_on g B \<and> g ` B \<subseteq> A" shows "(\<exists>f. bij_betw f (A <+> B) A) \<and> (\<exists>h. bij_betw h (B <+> A) A)"
coupling_def: fixes Q :: "'c \<Rightarrow> nat option"   and V :: "'c \<Rightarrow> nat option"   and D :: "'c \<Rightarrow> enat"   and S :: "'c set" shows "coupling Q V D S \<equiv> D = Dijkstra_Abstract.enat_of_option \<circ> V ++ Q \<and> S = dom V \<and> dom V \<inter> dom Q = {}"
dg_Rel_Cod_vrange: fixes \<alpha> :: "V" shows "\<R>\<^sub>\<circ> (dg_Rel \<alpha>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> dg_Rel \<alpha>\<lparr>Obj\<rparr>"
Mg': shows "set_borel_measurable borel {a..b} g'"
nb\<^sub>1\<^sub>6: shows "dom ((\<phi>\<^sub>S \<Psi> s) ++ map_of (effect_to_assignments op')) = (\<Union>v\<in>{v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None}. {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v}) \<union> set (effect_of op) \<union> (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
comp_converter_assoc: fixes conv1 :: "('q, 'r, 'w, 'x) converter"   and conv2 :: "('w, 'x, 'u, 'v) converter"   and conv3 :: "('u, 'v, 's, 't) converter" shows "(conv1 \<odot> conv2) \<odot> conv3 = conv1 \<odot> conv2 \<odot> conv3"
islimpt_def: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<forall>T. x \<in> T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x))"
list_induct2': fixes P :: "'g list \<Rightarrow> 'h list \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" assumes "P [] []"   and "\<And>x xs. P (x # xs) []"   and "\<And>y ys. P [] (y # ys)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
bottom_eq: fixes L :: "('a, 'b) gorder_scheme"   and b :: "'a" assumes "bounded_lattice L"   and "b \<in> carrier L"   and "\<And>x. x \<in> carrier L \<Longrightarrow> b \<sqsubseteq>\<^bsub>L\<^esub> x" shows "b = \<bottom>\<^bsub>L\<^esub>"
on_curvep_imp_in_carrierp: fixes a :: "'a"   and b :: "'a"   and p :: "'a \<times> 'a \<times> 'a" assumes "on_curvep a b p" shows "in_carrierp p"
ranI: fixes m :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "m a = Some b" shows "b \<in> ran m"
inf_subs: shows "ground.Inf_from_q q (lifted_q_calc.\<G>_Fset N) \<subseteq> {\<iota>. \<exists>\<iota>'\<in>Inf_from N. \<G>_I_q q \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I_q q \<iota>')} \<union> Red_I_q q (lifted_q_calc.\<G>_Fset N)"
e_lam_elim: fixes v :: "val"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E (ELam x e) \<rho>" obtains f :: "(val \<times> val) list" where "v = VFun f" and "\<forall>v1 v2. (v1, v2) \<in> set f \<longrightarrow> v2 \<in> E e ((x, v1) # \<rho>)"
Upper_elemD: fixes u :: "'a"   and A :: "'a set"   and x :: "'a" assumes "u .\<in> Upper L A"   and "u \<in> carrier L"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "x \<sqsubseteq> u"
generator_pow_order: shows "\<^bold>g [^] order G = \<one>"
is_lub_def: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a" assumes "class.po below" shows "po.is_lub below S x = (po.is_ub below S x \<and> (\<forall>u. po.is_ub below S u \<longrightarrow> below x u))"
has_derivative_ident: fixes F :: "'a filter" shows "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
document_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_document_ptr_kind ptr" shows "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr"
P1_cong: fixes tms :: "tm list"   and P :: "tm \<Rightarrow> fm"   and H :: "fm set"   and x :: "tm"   and x' :: "tm" assumes "\<And>i t x. atom i \<sharp> tms \<Longrightarrow> (P t)(i::=x) = P (subst i x t)"   and "H \<turnstile> x EQ x'" shows "H \<turnstile> P x IFF P x'"
sum_rationals_common_divisor: fixes A :: "'a set"   and g :: "'a \<Rightarrow> int"   and f :: "'a \<Rightarrow> int" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> g x \<noteq> 0" shows "\<exists>c. (\<Sum>x\<in>A. real_of_int (f x) / real_of_int (g x)) = real_of_int c / real_of_int (Lcm (g ` A))"
ignore_def: fixes \<Delta> :: "'state \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'b"   and vl1 :: "'c"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'a" shows "ignore \<Delta> s s1 vl1 a ou s' vl' \<equiv> \<not> \<gamma> (Trans s a ou s') \<and> \<Delta> s' vl' s1 vl1"
strict_implies_not_eq: fixes b :: "'c"   and a :: "'c" assumes "b < a" shows "a \<noteq> b"
base_case_sgas_alt: fixes qs :: "real poly list"   and p :: "real poly" assumes "length qs = 1"   and "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> {[1], [- 1]}"
ras_top_def: fixes s :: "'b array \<times> nat" shows "ras_top s \<equiv> ras_get s 0"
subset_insertI: fixes B :: "'c set"   and a :: "'c" shows "B \<subseteq> insert a B"
AR_imp_contractible: fixes S :: "'a set" assumes "AR S" shows "contractible S"
point_in_vector_or_pseudo_complement: fixes p :: "'a"   and v :: "'a" assumes "point p"   and "vector v" shows "p \<le> - - v \<or> p \<le> - v"
llist_all2_cons_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and h :: "'e \<Rightarrow> ('f, 'e) step"   and P :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and sg :: "'c"   and sh :: "'e" shows "llist_all2_cons g h P sg sh = llist_all2 P (lunstream g sg) (lunstream h sh)"
lub_option_def: fixes lub :: "'d set \<Rightarrow> 'c"   and Y :: "'d option set" shows "lub_option lub Y = (if Y \<subseteq> {None} then None else Some (lub {x. Some x \<in> Y}))"
ring_1_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.ring_1 one times plus zero minus uminus \<equiv> class.monoid_mult one times \<and> class.ring plus zero minus uminus times \<and> class.zero_neq_one one zero"
distinct_right: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and y :: "'b" assumes "all_distinct (Node l x False r)"   and "y \<in> set_of r" shows "x \<noteq> y"
set_iterator_genord_union_correct: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_a :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and it_b :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_b :: "'e set" assumes "set_iterator_genord it_a S_a R"   and "set_iterator_genord it_b S_b R"   and "S_a \<inter> S_b = {}"   and "\<And>a b. \<lbrakk>a \<in> S_a; b \<in> S_b\<rbrakk> \<Longrightarrow> R a b" shows "set_iterator_genord (set_iterator_union it_a it_b) (S_a \<union> S_b) R"
le_max_iff_disj: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z \<le> max x y) = (z \<le> x \<or> z \<le> y)"
bernoulli_denom_dvd: fixes n :: "nat" shows "bernoulli_denom n dvd 2 * (2 ^ n - 1)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>x r. P \<langle>\<langle>\<rangle>, x, r\<rangle>"   and "\<And>v va vb x r. (\<And>xa xb y. \<lbrakk>xa = del_left \<langle>v, va, vb\<rangle>; (xb, y) = xa\<rbrakk> \<Longrightarrow> P \<langle>v, va, vb\<rangle>) \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, x, r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
map_sel_3: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c tree" shows "right (map_tree f a) = map_tree f (right a)"
component_of_term_def: fixes v :: "'t" shows "tp.component_of_term v = snd (pair_of_term v)"
gt_compat_fun: fixes t :: "('s, 'v) tm"   and t' :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary t"   and "t' >\<^sub>t t" shows "App s t' >\<^sub>t App s t"
UN_UN_flatten: fixes C :: "'d \<Rightarrow> 'c set"   and B :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "\<Union> (C ` \<Union> (B ` A)) = (\<Union>y\<in>A. \<Union> (C ` B y))"
strict_prefix_map_inj: fixes f :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ys :: "'b list" assumes "inj_on f (ran\<^sub>l xs \<union> ran\<^sub>l ys)"   and "strict_prefix (map f xs) (map f ys)" shows "strict_prefix xs ys"
dist_in_closed_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> {a--b}" shows "dist x a \<le> dist a b \<and> dist x b \<le> dist a b"
cond_bind_pmf_def: shows "local.cond_bind_pmf = embed_pmf F"
plus_mono: fixes s1 :: "'lbl"   and s2 :: "'lbl"   and s3 :: "'lbl"   and s4 :: "'lbl" assumes "s1 \<le> s2"   and "s3 \<le> s4" shows "s1 + s3 \<le> s2 + s4"
set_update_by_pair: fixes xs :: "('a \<times> 'c) list"   and v :: "'c"   and k :: "'a" assumes "cmp2.oalist_inv_raw xs"   and "v \<noteq> (0::'c)" shows "set (cmp2.update_by_pair (k, v) xs) = insert (k, v) (set xs - range (Pair k))"
nfv_def: fixes \<phi> :: "'b MFOTL.formula" shows "MFOTL.nfv \<phi> = Max (insert 0 (Suc ` fv \<phi>))"
simps_5: fixes f1 :: "'b"   and f2 :: "'c set \<Rightarrow> 'b"   and x2 :: "'c set" shows "(case RES x2 of FAILi \<Rightarrow> f1 | RES x \<Rightarrow> f2 x) = f2 x2"
One_pdevs_def: shows "One_pdevs \<equiv> Abs_pdev One_pdevs_raw"
perm_2: fixes pi :: "(name \<times> name) list" shows "pi \<bullet> \<tau> = \<tau>"
assms_3: shows "nat \<lfloor>b\<rfloor> \<le> b'"
lambdabinder0_def: shows "\<^bold>\<lambda>\<^sup>0 \<equiv> map_fun eval\<o> make\<o> id"
ereal_diff_positive: fixes a :: "ereal"   and b :: "ereal" assumes "a \<le> b" shows "0 \<le> b - a"
hom_compose: fixes h :: "'a \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and i :: "'c \<Rightarrow> 'e"   and I :: "('e, 'f) monoid_scheme" assumes "h \<in> hom G H"   and "i \<in> hom H I" shows "compose (carrier G) i h \<in> hom G I"
rel_fun_mono: fixes X :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and Y :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "(X ===> A) f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "(Y ===> B) f g"
strict_implies_order: fixes b :: "'b"   and a :: "'b" assumes "b < a" shows "b \<le> a"
assms_2: shows "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> A"
append_take_drop_id: fixes n :: "nat"   and xs :: "'f list" shows "take n xs @ drop n xs = xs"
distinct_implies_inj_on: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and xs :: "'d list" assumes "bij_betw enum {0..<card A} A"   and "length xs = card A"   and "distinct xs" shows "inj_on (function_of A enum xs) A"
forward_terminating_path_end_points_2: fixes x :: "'a" assumes "forward_terminating_path x" shows "x\<^sup>T \<le> x\<^sup>\<star> ; end_points x"
segment_bound_2: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> closed_segment a b" shows "norm (x - b) \<le> norm (b - a)"
polyn_n_m1: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat"   and m :: "nat" assumes "pol_coeff S c"   and "n < m"   and "m \<le> fst c" shows "polyn_expr R X m c = polyn_expr R X n c \<plusminus> \<Sigma>\<^sub>f R (\<lambda>j. snd c j \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) Suc n m"
ltl_lSup: fixes A :: "'c llist set" shows "ltl (lSup A) = lSup (ltl ` (A \<inter> {xs. \<not> lnull xs}))"
weave_append2: fixes xs :: "'a list"   and ys :: "'a list"   and A :: "nat set"   and y :: "'a" assumes "length xs + length ys \<notin> A"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "weave A xs (ys @ [y]) = weave A xs ys @ [y]"
Seq: shows "Seq r c\<^sub>2 \<in> redexes c"
Not_Domain_rtrancl: fixes x :: "'a"   and R :: "('a \<times> 'a) set"   and y :: "'a" assumes "x \<notin> Domain R" shows "((x, y) \<in> R\<^sup>*) = (x = y)"
partn_lst_greater_resource: fixes M :: "nat"   and \<alpha> :: "nat list"   and \<gamma> :: "nat"   and N :: "nat" assumes "partn_lst {..<M} \<alpha> \<gamma>"   and "M \<le> N" shows "partn_lst {..<N} \<alpha> \<gamma>"
hs: shows "hs = fst (add_indices aux (snd data))"
get_state_def: fixes f :: "'e \<Rightarrow> ('e, 'f) stateT" shows "get_state f = StateT (\<lambda>s. run_state (f s) s)"
is_ladder_not_empty: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list" assumes "is_ladder D L" shows "L \<noteq> []"
Od_compare: fixes S :: "'a set"   and S' :: "'a set"   and R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set" shows "(S = S' \<and> R = R') = (D\<lparr>carrier := S, rel := R\<rparr> = D\<lparr>carrier := S', rel := R'\<rparr>)"
eval_poly_add: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "a \<in> carrier R" shows "local.eval (poly_add p q) a = local.eval p a \<oplus> local.eval q a"
complement_overlap: fixes x :: "i"   and y :: "i" assumes "CMC"   and "\<not> O x y" shows "O x (\<midarrow> y)"
sum_incoming_extend: fixes u :: "nat"   and g :: "nat \<times> nat \<Rightarrow> 'b" assumes "\<And>v. \<lbrakk>v \<in> V; (v, u) \<notin> E\<rbrakk> \<Longrightarrow> g (v, u) = (0::'b)" shows "sum g (incoming u) = (\<Sum>v\<in>V. g (v, u))"
simps_2: fixes v :: "'b"   and va :: "'b list"   and \<alpha> :: "'b \<Rightarrow> 'c set \<times> 'c set" shows "v # va \<otimes>\<^bsub>\<alpha>\<^esub> [] = {v # va}"
vlrestriction_def: fixes D :: "V" shows "vlrestriction D = (\<lambda>\<langle>r, A\<rangle>\<in>\<^sub>\<circ>D. set {\<langle>a, b\<rangle> |a b. a \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
And_safe: fixes \<phi> :: "formula"   and \<psi> :: "formula" assumes "safe_formula \<phi>"   and "\<not> safe_assignment (fv \<phi>) \<psi>"   and "safe_formula \<psi>"   and "P \<phi>"   and "P \<psi>" shows "P (And \<phi> \<psi>)"
rp_def: fixes t :: "real" shows "rp t = t\<^sup>2 * p' t - 2 * real k * t * p t + p t"
ile: shows "i < length (list_of (ltake (enat (Suc (max i j))) lst))"
oracle_def: fixes \<sigma> :: "'c \<Rightarrow> bool list option"   and x :: "'c" shows "hash.oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set (nlists UNIV len_plain) \<bind> (\<lambda>bs. return_spmf (bs, \<sigma>(x \<mapsto> bs))) | Some bs \<Rightarrow> return_spmf (bs, \<sigma>))"
intro: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist a1 c1 = dist a2 c2"   and "dist b1 c1 = dist b2 c2"   and "angle b1 a1 c1 = angle b2 a2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2"   and "angle a1 c1 b1 = angle a2 c2 b2" shows "congruent_triangle a1 b1 c1 a2 b2 c2"
mem_carrier: fixes x :: "'a" assumes "x \<in> K" shows "x \<in> carrier B"
tiTable_ack_call: fixes req :: "nat \<Rightarrow> reqType list"   and x :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and t :: "nat" assumes "tiTable_SampleT req x stop lose st_in b ack i1 vc st_out"   and "ack t = [call]"   and "msg (Suc 0) x"   and "ts lose"   and "msg (Suc 0) stop" shows "st_in t = init_state \<and> req t = [init]"
nsum_minus: fixes A :: "('d, 'e) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> \<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A (\<lambda>x\<in>{j. j \<le> n}. -\<^sub>a\<^bsub>A\<^esub> f x) n"
Ide_in_Hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t" shows "t \<in> {t. monoidal_language.Arr C t \<and> monoidal_language.Dom C t = t \<and> monoidal_language.Cod C t = t}"
double_compl: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and x :: "'g" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
SFAssInitThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and v :: "val"   and h\<^sub>1 :: "heap"   and l\<^sub>1 :: "sfields"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C has F,Static:t in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, l\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
Ext: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and \<Psi>'' :: "'b" assumes "(\<Psi>', R, S) \<in> Rel" shows "(\<Psi>' \<otimes> \<Psi>'', R, S) \<in> Rel"
kummers_test_divergence: fixes f :: "nat \<Rightarrow> real"   and p :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. 0 < f n"   and "\<forall>\<^sub>F n in sequentially. 0 < p n"   and "\<not> summable (\<lambda>n. inverse (p n))"   and "limsup (\<lambda>n. ereal (p n * f n / f (Suc n) - p (Suc n))) < 0" shows "\<not> summable f"
select_convs_3: fixes NTDom :: "('d, 'f, 'g, 'e, 'h, 'i) Functor"   and NTCod :: "('d, 'f, 'g, 'e, 'h, 'i) Functor"   and NatTransMap :: "'d \<Rightarrow> 'e"   and more :: "'j" shows "NatTransMap \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NatTransMap"
S2: shows "c_pair 2 0 = Suc (Suc 3)"
strict_iff_order: fixes b :: "'b"   and a :: "'b" shows "(b < a) = (b \<le> a \<and> a \<noteq> b)"
path_split2: fixes v :: "'d"   and v' :: "'d"   and pth1 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list"   and e :: "('d \<times> 'c) \<times> 'd \<times> 'b"   and pth2 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" shows "path v v' (pth1 @ e # pth2) = (path v (edge_begin e) pth1 \<and> path (edge_begin e) v' (e # pth2))"
less_imp_triv: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y" shows "(less y x \<longrightarrow> P) = True"
p_fun_triple: fixes x :: "'a" assumes "is_p_fun x" shows "x = x ; x\<^sup>T ; x"
assms_2: shows "open_cover_of_open_subset S ind_is_open U I C"
linear_construct: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B" shows "linear (*a) (*b) (construct B f)"
hom_induced_id: fixes c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and X :: "'a topology"   and S :: "'a set" assumes "c \<in> carrier (relative_homology_group p X S)" shows "hom_induced p X S X S id c = c"
comp_ide_arr: fixes b :: "'j"   and f :: "'j" assumes "J.ide b"   and "J.seq b f" shows "b \<cdot>\<^sub>J f = f"
Upair_comm: fixes a :: "ZF"   and b :: "ZF" shows "Upair a b = Upair b a"
set_supteq_into_supt: fixes t :: "('d, 'e) Term.term"   and ts :: "('d, 'e) Term.term list"   and s :: "('d, 'e) Term.term"   and f :: "'d" assumes "t \<in> set ts"   and "t \<unrhd> s" shows "Fun f ts \<rhd> s"
bij_betw_map_lists: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "bij_betw (map f) (lists A) (lists B)"
uni_set_samp: fixes y :: "'index \<times> 'trap" assumes "y \<in> set_spmf I" shows "map_spmf (F (fst y)) (local.etp.S (fst y)) = local.etp.S (fst y)"
simps_6: fixes w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "balance (Branch R Empty w x (Branch R b s t c)) y z (Branch B va vb vc vd) = Branch R (Branch B Empty w x b) s t (Branch B c y z (Branch B va vb vc vd))"
assms_4: fixes a :: "'e"   and b :: "'f" assumes "a \<in> set3_FGl x"   and "b \<in> set3_FGl y"   and "L3 a b" shows "L3' a b"
reduced_GB_is_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "is_Groebner_basis (reduced_GB F)"
split_le: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "((\<lambda>(x, y). f x y) \<le> (\<lambda>(x, y). g x y)) = (f \<le> g)"
finite_UNIV_card_ge_0: assumes "finite UNIV" shows "0 < card UNIV"
T\<^sub>p_split: fixes qs1 :: "'a list"   and as1 :: "(nat \<times> nat list) list"   and s :: "'a list"   and qs2 :: "'a list"   and as2 :: "(nat \<times> nat list) list" assumes "length qs1 = length as1" shows "T\<^sub>p s (qs1 @ qs2) (as1 @ as2) = T\<^sub>p s qs1 as1 + T\<^sub>p (Partial_Cost_Model.steps s qs1 as1) qs2 as2"
conv2: shows "\<I>' \<turnstile>g run_converter conv a \<surd>"
normalize_quot_aux_3: fixes b :: "'b"   and a :: "'b" assumes "b \<noteq> (0::'b)" shows "gcd a b * unit_factor b dvd a"
compact_norm_comb': shows "compact (norm ` comb' ` sphere 0 1)"
trivial_ker_imp_inj: assumes "a_kernel (K [X]) (Rupt A P) (rupture_surj A P) = {\<zero>\<^bsub>K [X]\<^esub>}" shows "inj_on (rupture_surj A P) (carrier (K [X]))"
restrict_top_ifex_ordered_invar: fixes b :: "'b ifex"   and var :: "'b"   and val :: "bool" assumes "ifex_ordered b" shows "ifex_ordered (restrict_top b var val)"
contract_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div> \<phi> = Cn (K \<div> \<phi>)"