hcju/isabelleps huggingface dataset:
assoc_coherence: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<a>[b, c, a] \<cdot> \
[a, b \<otimes> c] \<cdot> \<a>[a, b, c] = (b \<otimes> \
[a, c]) \<cdot> \<a>[b, a, c] \<cdot> (\
[a, b] \<otimes> c)"
solovay_strassen_def: shows "solovay_strassen = primality_test solovay_strassen_witness"
invertible_inverse_invertible: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "invertible (inverse u)"
catch_env_def: fixes catch :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and m1 :: "('d, 'e) envT"   and m2 :: "('d, 'e) envT" shows "catch_env catch m1 m2 = EnvT (\<lambda>r. catch (run_env m1 r) (run_env m2 r))"
finite_distinct_list: fixes A :: "'d set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> distinct xs"
notin_sets_outer_measure_of_cover: fixes E :: "'a set" assumes "E \<subseteq> space M"   and "E \<notin> sets M" shows "\<exists>B\<in>sets M. 0 < emeasure M B \<and> emeasure M B < \<infinity> \<and> outer_measure_of M (B \<inter> E) = emeasure M B \<and> outer_measure_of M (B - E) = emeasure M B"
isomorphic_cards_2: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2" shows "card (uedges G\<^sub>1) = card (uedges G\<^sub>2)"
hequiv_trans_rel: fixes B :: "'c set"   and A :: "(('d, 'c) fm list \<times> 'c) set" assumes "Hintikka B A" shows "trans (hequiv_rel B A)"
Union_quotient: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "equiv A r" shows "\<Union> (A // r) = A"
assms_3: fixes x :: "'a" shows "f x \<le> ennreal C"
contra: shows "\<not> (\<exists>n'\<le>n. P n')"
funionM_assoc: fixes x :: "'m"   and y :: "'m"   and z :: "'m" shows "funionM (funionM x y) z = funionM x (funionM y z)"
repr: fixes v :: "'n" assumes "v \<in> span B" shows "(\<Sum>i<n. representation B v (b i) *\<^sub>R b i) = v"
neq_y: shows "(f ^^ (funpow_dist1 f x y - n)) x \<noteq> y"
internal_inv_Branching_Left: fixes b :: "bool list"   and l :: "tree"   and r :: "tree" shows "internal (Left # b) (Branching l r) = internal b l"
\<tau>exec_moveI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_move ci P t e h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'"   and "\<tau>move2 P h stk e pc xcp" shows "\<tau>exec_move ci P t e h (stk, loc, pc, xcp) s'"
countable_atoms: shows "countable {x. 0 < M1.prob {x}}"
bound: fixes k :: "nat" assumes "k < n" shows "norm (v1 k * v2 k) \<le> b1 * b2"
g_1: shows "g' = of_int_poly_i ff_ops g"
cases: fixes a :: "('sequent \<times> 'rule) stream" assumes "epath a" obtains steps :: "('sequent \<times> 'rule) stream" and sl :: "'sequent fset" where "a = steps" and "snd (shd steps) \<in> R" and "fst (shd (stl steps)) |\<in>| sl" and "effStep (shd steps) sl" and "epath (stl steps)"
pCons_induct: fixes P :: "'b poly \<Rightarrow> bool"   and p :: "'b poly" assumes "P 0"   and "\<And>a p. \<lbrakk>a \<noteq> (0::'b) \<or> p \<noteq> 0; P p\<rbrakk> \<Longrightarrow> P (pCons a p)" shows "P p"
hom_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula" shows "hom f x = (THE a. \<exists>S. hom_graph f S x a)"
divisors_pos_fun_int: shows "divisors_pos_fun divisors_int_pos"
cbox_sing: fixes a :: "'a" shows "cbox a a = {a}"
simps_12: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
assms_3: shows "\<rho> \<in> space (state_measure (Suc ` V') (case_nat t \<Gamma>))"
interchange: fixes h :: "'a"   and f :: "'a"   and k :: "'a"   and g :: "'a" assumes "seq h f"   and "seq k g" shows "local.prod h k \<cdot> local.prod f g = local.prod (h \<cdot> f) (k \<cdot> g)"
bernoulli_code: fixes n :: "nat" shows "bernoulli n = (if n = 0 then 1 else if n = 1 then - 1 / 2 else if odd n then 0 else akiyama_tanigawa n 0)"
homotopic_with_imp_continuous: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "homotopic_with_canon P X Y f g" shows "continuous_on X f \<and> continuous_on X g"
abupd_store_invariant: fixes f :: "abopt \<Rightarrow> abopt"   and s :: "state" shows "snd (abupd f s) = snd s"
sorted_list_of_set_UN_eq_concat: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "strict_mono_sets I f"   and "finite I"   and "\<And>i. finite (f i)" shows "list_of (\<Union> (f ` I)) = concat (map (list_of \<circ> f) (list_of I))"
assms_7: fixes x :: "real" assumes "x \<in> {a<..<b} - s" shows "(g has_vector_derivative g' x) (at x)"
simps_1: fixes K :: "clock"   and n :: "nat" shows "\<lbrakk> K \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. hamlet (Rep_run \<rho> n K)}"
subst_overrides: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and \<sigma> :: "'d \<Rightarrow> 'e option"   and \<tau> :: "'d \<Rightarrow> 'e option"   and mem :: "'d \<Rightarrow> 'e" assumes "sifum_security_init dma \<C>_vars eval"   and "dom \<sigma> = dom \<tau>" shows "subst \<sigma> (subst \<tau> mem) = subst \<sigma> mem"
simps_5: fixes r :: "real" shows "(- \<infinity> < ereal r) = True"
build_ip_partition_disjoint: fixes V1 :: "'a wordinterval"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and V2 :: "'a wordinterval" assumes "V1 \<in> set (build_ip_partition c rs)"   and "V2 \<in> set (build_ip_partition c rs)"   and "V1 \<noteq> V2" shows "wordinterval_to_set V1 \<inter> wordinterval_to_set V2 = {}"
refl: fixes b :: "'b"   and x :: "'b" shows "flat_ord b x x"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "blocking_on ds X h X''" shows "\<not> stable_on ds X"
ide_cod: fixes f :: "'a" assumes "Fop.Bop.arr f" shows "Fop.Bop.ide (Fop.Bop.cod f)"
assms: fixes z :: "real" assumes "x \<le> ereal z" shows "y \<le> ereal z"
Nom: fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "p at b in' set ((ps, a) # branch)"   and "Nom a at b in' set ((ps, a) # branch)"   and "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"   and "new p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
fps_times_conv_fls_times: fixes f :: "'a fps"   and g :: "'a fps" shows "f * g = fls_regpart (fps_to_fls f * fps_to_fls g)"
tree_edges_ssE: shows "tree_edges s \<subseteq> E"
resbar_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and s :: "state"   and p :: "PrIds" shows "resbar rmhist s p = (if S1 rmhist p s \<or> S2 rmhist p s then ires s p else if S3 rmhist p s then if rmhist s p = histA then ires s p else MemFailure else if S4 rmhist p s then if rmhist s p = histB \<and> ires s p = NotAResult then MemFailure else ires s p else if S5 rmhist p s then res (rmCh s p) else if S6 rmhist p s then if res (crCh s p) = RPCFailure then MemFailure else res (crCh s p) else NotAResult)"
e_cons_elementary: fixes e :: "nat"   and es :: "nat" shows "e_cons e es = (if es = 0 then Suc (prod_encode (0, e)) else Suc (triple_encode (e_length es) e (pdec2 (es - 1))))"
equivalent_objects_reflexive: fixes a :: "'b" assumes "B.obj a" shows "B.equivalent_objects a a"
fps_compose_mult_distrib: fixes c :: "'b fps"   and a :: "'b fps"   and b :: "'b fps" assumes "c $ 0 = (0::'b)" shows "a * b oo c = (a oo c) * (b oo c)"
iso_swapar_def: shows "iso_swapar = (\<lambda>((sm, s1), s2). (s1, sm, s2), \<lambda>(s1, sm, s2). ((sm, s1), s2))"
Inter_subset: fixes A :: "'a set set"   and B :: "'a set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B"   and "A \<noteq> {}" shows "\<Inter> A \<subseteq> B"
PreFunctorFunctor: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "Functor F" shows "PreFunctor F"
inj_on_contraD: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e"   and y :: "'e" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
absorb_iff2: fixes b :: "'b"   and a :: "'b" shows "(b \<le> a) = (a \<sqinter> b = b)"
aforms_relp_def: fixes R :: "('a list \<times> 'b) set" shows "\<langle>R\<rangle>aforms_relp = aforms_rel O \<langle>R\<rangle>set_rel"
length_rows: fixes A :: "'b mat" shows "length (rows A) = dim_row A"
if_C_def: fixes c :: "'a control_state_scheme \<Rightarrow> bool"   and E :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and F :: "'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option" shows "_if_SECLEAN c E F = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else if c \<sigma> then E \<sigma> else F \<sigma>)"
M_def: shows "M = (SUP t\<in>{x - 1..x + 1}. \<bar>poly (pderiv p) t\<bar>)"
select_convs_2: fixes c_sinvar :: "'a graph \<Rightarrow> bool"   and c_offending_flows :: "'a graph \<Rightarrow> ('a \<times> 'a) set set"   and c_isIFS :: "bool"   and more :: "'b" shows "c_offending_flows \<lparr>c_sinvar = c_sinvar, c_offending_flows = c_offending_flows, c_isIFS = c_isIFS, \<dots> = more\<rparr> = c_offending_flows"
InitProcessing: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
total_orderI: assumes "\<And>x y. \<lbrakk>x \<in> UNIV; y \<in> UNIV\<rbrakk> \<Longrightarrow> x \<le> y \<or> y \<le> x" shows "total_order \<lparr>carrier = UNIV, eq = (=), le = (\<le>)\<rparr>"
geom_quot: fixes a :: "'a"   and b :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "p = monom P \<one> (Suc n) \<ominus>\<^bsub>P\<^esub> monom P (b [^] Suc n) 0"   and "g = UP_root_div p b" shows "a [^] Suc n \<ominus> b [^] Suc n = (a \<ominus> b) \<otimes> to_fun g a"
bind_left_unit: fixes c :: "'c"   and P :: "'c \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'a) option" shows "bind\<^sub>S\<^sub>E (unit\<^sub>S\<^sub>E c) P = P c"
closure_Int: fixes I :: "'c set set" shows "closure (\<Inter> I) \<subseteq> \<Inter> {closure S |S. S \<in> I}"
vote_setD: fixes rv :: "'c \<times> 'd"   and v_f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd option"   and a :: "'e" assumes "rv \<in> vote_set v_f {a}" shows "v_f (fst rv) a = Some (snd rv)"
cong3_transitivity: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A0 :: "'a"   and B0 :: "'a"   and C0 :: "'a"   and A1 :: "'a"   and B1 :: "'a"   and C1 :: "'a"   and A2 :: "'a"   and B2 :: "'a"   and C2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Cong3 Cong A0 B0 C0 A1 B1 C1"   and "Tarski_neutral_dimensionless.Cong3 Cong A1 B1 C1 A2 B2 C2" shows "Tarski_neutral_dimensionless.Cong3 Cong A0 B0 C0 A2 B2 C2"
Moebius_function_compose: fixes w1 :: "complex"   and w2 :: "complex"   and z :: "complex" assumes "- w1 = w2"   and "cmod w1 < 1"   and "cmod z < 1" shows "Moebius_function 0 w1 (Moebius_function 0 w2 z) = z"
mult_all1_matrix: fixes i :: "nat"   and nr :: "nat"   and v :: "real vec" assumes "i < nr" shows "(all1_matrix nr (dim_vec v) *\<^sub>v v) $ i = sum_list (list_of_vec v)"
assms_3: fixes c :: "'a" assumes "\<And>b. b \<in> A \<Longrightarrow> b dvd c" shows "a dvd c"
assms_6: shows "kKS key \<notin> specKeysSecrets PQ"
unat_mult_power_lem: fixes k :: "nat"   and sz :: "nat" assumes "k < 2 ^ (LENGTH('b) - sz)" shows "unat (2 ^ sz * word_of_nat k) = 2 ^ sz * k"
prefix_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys" shows "length xs \<le> length ys"
run_asset_v_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_asset_v n s)"
ih_5: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
coeff_i_def: fixes ops :: "'b arith_ops_record" shows "coeff_i ops = nth_default (arith_ops_record.zero ops)"
ab0: shows "(a oo b) $ 0 \<noteq> (0::'a)"
dotprod_commute: fixes xs :: "nat list"   and ys :: "nat list" assumes "length xs = length ys" shows "xs \<bullet> ys = ys \<bullet> xs"
Q_Suc: fixes a :: "'a"   and b :: "'b" assumes "P a b"   and "Q a b" shows "Q a (g\<cdot>b)"
top_right_mult_increasing: fixes x :: "'a" shows "x \<le> x * top"
alphaInput: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"   and "y \<sharp> P" shows "P \<Longrightarrow>\<^sub>lu in [(x, y)] \<bullet> P''\<rightarrow>a<y> \<prec> P'"
deg_smult_decr: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier P" shows "deg R (a \<odot>\<^bsub>P\<^esub> f) \<le> deg R f"
iso_def: fixes f :: "'b" shows "local.iso f \<equiv> \<exists>g. inverse_arrows f g"
assms: shows "\<exists>n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n"
inj_vreal_of_real_impl: shows "inj vreal_of_real_impl"
some_third_chamberD_simplex: fixes X :: "'b set set"   and C :: "'b set"   and z :: "'b set"   and D :: "'b set" assumes "ThickChamberComplex X"   and "SimplicialComplex.maxsimp X C"   and "z \<lhd> C" shows "ThickChamberComplex.some_third_chamber X C D z \<in> X"
intro: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group H"   and "group_hom_axioms G H h" shows "group_hom G H h"
trans: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b"   and "b \<subseteq># c" shows "a \<subseteq># c"
simps_10: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and f3 :: "'e \<Rightarrow> 'd \<Rightarrow> 'c"   and x2 :: "'d" shows "(case Skip x2 of Done \<Rightarrow> f1 | Skip x \<Rightarrow> f2 x | Yield x xa \<Rightarrow> f3 x xa) = f2 x2"
cardsp_def: fixes s :: "event list"   and g :: "guest"   and k :: "key \<times> key" shows "cardsp s g k = (k \<in> cards s g)"
inf_Int_eq: fixes R :: "'b set"   and S :: "'b set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
\<tau>reds1r_cons_\<tau>reds1r: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and es :: "'addr expr1 list"   and xs :: "'addr val list"   and es' :: "'addr expr1 list"   and xs' :: "'addr val list"   and v :: "'addr val" assumes "\<tau>reds1gr uf P t h (es, xs) (es', xs')" shows "\<tau>reds1gr uf P t h (Val v # es, xs) (Val v # es', xs')"
int_cases2: fixes z :: "int" obtains n :: "nat" where "z = int n__"   | n :: "nat" where "z = - int n"
matrix_add_rdistrib: fixes B :: "(('a, 'd) vec, 'c) vec"   and C :: "(('a, 'd) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'd) vec" shows "(B + C) ** A = B ** A + C ** A"
elims_2: fixes x :: "'a simple_match" assumes "empty_match x" obtains uu_ :: "iface" and uv_ :: "iface" and uw_ :: "'a word \<times> nat" and ux_ :: "'a word \<times> nat" and uy_ :: "protocol" and sps1 :: "16 word" and sps2 :: "16 word" and dps1 :: "16 word" and dps2 :: "16 word" where "x = \<lparr>iiface = uu_, oiface = uv_, src = uw_, dst = ux_, proto = uy_, sports = (sps1, sps2), dports = (dps1, dps2)\<rparr>" and "sps2 < sps1 \<or> dps2 < dps1"
monad_alt_writerT: fixes return :: "'d \<times> 'e list \<Rightarrow> 'f"   and bind :: "'f \<Rightarrow> ('d \<times> 'e list \<Rightarrow> 'f) \<Rightarrow> 'f"   and alt :: "'f \<Rightarrow> 'f \<Rightarrow> 'f" assumes "monad_alt return bind alt" shows "monad_alt (return_writer return) (bind_writer return bind) (alt_writer alt)"
A0: shows "A0 \<in> carrier_mat (n - 1) 1"
Union_Pow_eq: fixes A :: "'g set" shows "\<Union> (Pow A) = A"
coboundedI2: fixes c :: "'b"   and b :: "'b"   and a :: "'b" assumes "c \<le> b" shows "c \<le> sup a b"
vproduct_vsingletonD_3: fixes f :: "V"   and c :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {c}) A" shows "f = set {\<langle>c, f\<lparr>c\<rparr>\<rangle>}"
CondF: fixes s :: "'b"   and b :: "'b set"   and P1 :: "'b com"   and P2 :: "'b com" assumes "s \<notin> b" shows "(Some (Cond b P1 P2), s) -c\<rightarrow> (Some P2, s)"
to_nat_mono': fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less_eq a b" shows "mod_type.to_nat Rep a \<le> mod_type.to_nat Rep b"
dim_2: shows "dim_col (four_block_mat A B (0\<^sub>m m n) D) = n + m"
Lxy_mono: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "{x, y} \<subseteq> set xs"   and "distinct xs" shows "x < y in xs = x < y in Lxy xs {x, y}"
finmap_topological_basis: shows "topological_basis local.basis_finmap"
aff_dim_insert: fixes a :: "'b"   and S :: "'b set" shows "aff_dim (insert a S) = (if a \<in> affine hull S then aff_dim S else aff_dim S + 1)"
dgrad_p_set_closed_fst_crit_pair: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_grading d"   and "p \<in> dgrad_p_set d m"   and "q \<in> dgrad_p_set d m" shows "fst (crit_pair p q) \<in> dgrad_p_set d m"
\<b>_card_X: fixes m :: "nat"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and i :: "nat" assumes "exact_decomp m ps"   and "Suc (card X) \<le> i" shows "\<b> ps i = \<a> ps"
refine: shows "(impl_red_dfs, red_dfs) \<in> \<langle>Id\<rangle>slg_rel \<rightarrow> \<langle>Id\<rangle>dflt_rs_rel \<rightarrow> \<langle>Id\<rangle>dflt_rs_rel \<rightarrow> Id \<rightarrow> \<langle>\<langle>Id\<rangle>dflt_rs_rel \<times>\<^sub>r red_wit_rel Id\<rangle>nres_rel"
underS_def: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "underS r a \<equiv> {b. b \<noteq> a \<and> (b, a) \<in> r}"
simps_1: fixes left :: "'c \<Rightarrow> 'f \<Rightarrow> ('a \<times> 'c, 'd, 'e) gpv"   and right :: "'c \<Rightarrow> 'g \<Rightarrow> ('b \<times> 'c, 'd, 'e) gpv"   and s :: "'c"   and x :: "'f" shows "plus_intercept left right s (Inl x) = map_gpv (apfst Inl) id (left s x)"
simps_3: fixes t1 :: "('c, 'd) SkewBinomialTree"   and bq1 :: "('c, 'd) SkewBinomialTree list"   and t2 :: "('c, 'd) SkewBinomialTree"   and bq2 :: "('c, 'd) SkewBinomialTree list" shows "meldUniq (t1 # bq1) (t2 # bq2) = (if rank t1 < rank t2 then t1 # meldUniq bq1 (t2 # bq2) else if rank t2 < rank t1 then t2 # meldUniq (t1 # bq1) bq2 else ins (link t1 t2) (meldUniq bq1 bq2))"
fs_def: shows "fs = (\<lambda>k x. (- 1) ^ k * fact k / x ^ Suc k)"
errorT_cases: fixes y :: "'e\<cdot>('g, 'f) errorT" obtains k :: "'e\<cdot>'f error\<cdot>'g" where "y = ErrorT\<cdot>k"
snd_fst_flip: fixes xy :: "'h \<times> 'g" shows "snd xy = (fst \<circ> (\<lambda>(x, y). (y, x))) xy"
pmc_success: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> Cn {}" shows "\<phi> \<notin> K \<div>\<^bsub>\<gamma>\<^esub> \<phi>"
DistinctNetsDenyAllow: fixes b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int" assumes "DenyAllFromTo b c \<in> set p"   and "AllowPortFromTo a d po \<in> set p"   and "allNetsDistinct p"   and "dom (Cp (DenyAllFromTo b c)) \<inter> dom (Cp (AllowPortFromTo a d po)) \<noteq> {}" shows "b = a \<and> c = d"
cee: assumes "e \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "e O e = e"
subst_wellformed: fixes t :: "'b"   and env :: "(name, 'b) fmap" assumes "pre_strong_term_class.wellformed t"   and "wellformed_env env" shows "pre_strong_term_class.wellformed (subst t env)"
filter_mono: fixes F1 :: "'b filter"   and F2 :: "'b filter"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "F1 \<le> F2"   and "f \<in> O[F2](g)" shows "f \<in> O[F1](g)"
alw: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<phi> xs"   and "alw \<phi> (stl xs)" shows "alw \<phi> xs"
prime_elem_not_zeroI: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p" shows "p \<noteq> zero"
compass_reflect_compass_9: shows "apply_cltn2 south equator_reflect = north"
pell_cnj_def: shows "pell_cnj = (\<lambda>(a, b). (a, - b))"
foo2: fixes i :: "int"   and s :: "'a \<Rightarrow> 'b" shows "s\<langle>a := i\<rangle>\<cdot>a = i"
get_dom_component_impl: shows "get_dom_component = local.a_get_dom_component"
map_scale_distrib_left: fixes k :: "'b"   and s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "k \<cdot> (s + t) = k \<cdot> s + k \<cdot> t"
simps_2: fixes f :: "nat list \<Rightarrow> nat"   and g :: "nat list \<Rightarrow> nat"   and x :: "nat"   and l :: "nat list" shows "PREC f g (x # l) = rec_nat (f l) (\<lambda>y r. g (r # y # l)) x"
LIMSEQ_incseq_SUP: fixes X :: "nat \<Rightarrow> 'b" assumes "bdd_above (range X)"   and "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"
sub_eq_zero: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "z -\<^sub>h z = 0\<^sub>h"
sv01_4: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1"   and "n \<noteq> 0"   and "the (inconsist (amalg01 f) (f \<triangleright> n)) < Suc n"   and "pdec1 (the (parallel (the (f 0)) (the (f 1)) (the (inconsist (amalg01 f) (f \<triangleright> n))))) \<noteq> 0" shows "s\<^bsub>01\<^esub> (f \<triangleright> n) = f 0"
infsetsum_all_0: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = (0::'d)" shows "infsetsum f A = (0::'d)"
emp_no_periodN: fixes n :: "nat" shows "\<not> periodN \<epsilon> n"
snd_crit_pair_below_lcs: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.ord_strict_p (snd (punit.crit_pair p q)) (monomial (1::'c) (fst (lcs (punit.lp p) (punit.lp q), local.punit.component_of_term (punit.lt p))))"
ev_3: shows "\<forall>\<^sub>F x in at_right 0. (f0 has_real_derivative f' x) (at x)"
order_iff: fixes b :: "'k"   and a :: "'k" shows "(b \<le> a) = (a = max a b)"
pdevs_val_msum_pdevs: fixes f :: "'a pdevs"   and n :: "nat"   and e :: "nat \<Rightarrow> real"   and g :: "'a pdevs" assumes "degree f \<le> n" shows "pdevs_val e (msum_pdevs n f g) = pdevs_val e f + pdevs_val (\<lambda>i. e (i + n)) g"
thetaIfW_Wretr: shows "thetaIfW \<subseteq> Wretr (thetaIfW \<union> Wbis)"