hcju/isabelleps huggingface dataset:
regular_implies_setmonotone: fixes f :: "'a set \<Rightarrow> 'a set" assumes "regular f" shows "setmonotone f"
simps_2: fixes l :: "tree"   and r :: "tree" shows "treesize (Branching l r) = 1 + treesize l + treesize r"
ide_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "subcategory C Arr" shows "partial_magma.ide (subcategory.comp C Arr) a = (partial_magma.arr (subcategory.comp C Arr) a \<and> partial_magma.ide C a)"
to_int_mod_ring_def: shows "to_int_mod_ring \<equiv> (Rep_mod_ring ---> id) (\<lambda>x. x)"
irreducible_mult_imp_irreducible: fixes a :: "'a" assumes "a \<in> carrier R - {\<zero>}"   and "irreducible (Ring_Divisibility.mult_of R) a" shows "irreducible R a"
less_imp_not_eq2: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less"   and "less x y" shows "(y = x) = False"
converse_r_into_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "r y x" shows "equivclp r x y"
openin_subopen: fixes U :: "'d topology"   and S :: "'d set" shows "openin U S = (\<forall>x\<in>S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
Healthy_def: fixes P :: "'d upred"   and H :: "'d upred \<Rightarrow> 'd upred" shows "P is H \<equiv> H P = P"
remove1_mset_add_mset_If: fixes L' :: "'a"   and C :: "'a multiset"   and L :: "'a" shows "remove1_mset L (add_mset L' C) = (if L = L' then C else remove1_mset L C + {#L'#})"
min_product_finite: fixes A :: "'b fset set"   and B :: "'b fset set" assumes "finite A"   and "finite B" shows "finite (A \<otimes>\<^sub>m B)"
upright_obtain_support: fixes a :: "'c zmultiset"   and t :: "'c" assumes "upright a"   and "0 < zcount a t" obtains s :: "'c" where "s < t" and "zcount a s < 0" and "nonpos_upto a s"
gen_nchotomy: fixes J :: "'a set"   and P :: "'a set set"   and I :: "'a set" assumes "J \<in> P" shows "J \<subseteq> gen P I \<or> gen P I \<inter> J = {}"
adh_lemma: fixes x :: "'a llist"   and P :: "'a llist set" assumes "x \<in> A\<^sup>\<spadesuit>"   and "P \<subseteq> A\<^sup>\<spadesuit>" shows "(x adh P) = (\<forall>r\<in>pfinpref A x. \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P)"
ev_1: shows "\<forall>\<^sub>F x in at_top. 0 < \<pi> x"
res'_6: shows "E' = (\<Sum>\<^sub># (mset Cs') + DA') \<cdot> \<sigma>'''"
sound_reduction_generalized: fixes k :: "nat"   and \<S>\<^sub>1 :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list"   and \<S>\<^sub>2 :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list" assumes "\<S>\<^sub>1 \<hookrightarrow>\<^bsup>k\<^esup> \<S>\<^sub>2" shows "\<lbrakk> \<S>\<^sub>2 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<subseteq> \<lbrakk> \<S>\<^sub>1 \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
mem_plus_V_E: fixes l :: "V"   and x :: "V"   and y :: "V" assumes "l \<in> elts (x + y)" obtains "l \<in> elts x"   | z :: "V" where "z \<in> elts y" and "l = x + z"
not_prefix_cases: fixes ps :: "'c list"   and ls :: "'c list" assumes "\<not> ps \<le> ls" obtains "ps \<noteq> []" and "ls = []"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = a__ # as__" and "ls = x__ # xs__" and "x__ = a__" and "\<not> as__ \<le> xs__"   | a :: "'c" and as :: "'c list" and x :: "'c" and xs :: "'c list" where "ps = a # as" and "ls = x # xs" and "x \<noteq> a"
R1_xor_def: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "R1_xor A B = return_spmf A"
vsv_axioms: fixes a :: "V"   and b :: "V" shows "vsv (ZFC_in_HOL.set {\<langle>a, b\<rangle>})"
intros_6: fixes d :: "'d"   and u :: "'e \<Rightarrow> 'd"   and c :: "'e" assumes "d < u c" shows "u \<turnstile> GT c d"
isNSCont_NSLIM: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" assumes "isNSCont f a" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a"
lset_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "lset (inf_llist f) = range f"
star_subdist_var_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.left_pre_kleene_algebra plus times one less_eq less star" shows "less_eq (times x y) (star (plus x y))"
has_derivative_inner_left: fixes g :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd"   and F :: "'c filter"   and b :: "'d" assumes "(g has_derivative g') F" shows "((\<lambda>x. g x \<bullet> b) has_derivative (\<lambda>x. g' x \<bullet> b)) F"
TAbs_type': fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and s :: "trm"   and S :: "ty"   and U' :: "ty"   and U :: "ty" assumes "\<Gamma> \<turnstile> (\<lambda>X<:S. s) : (\<forall>X<:U. U')"   and "X \<sharp> \<Gamma>"   and "X \<sharp> S"   and "X \<sharp> U" obtains S' :: "ty" where "\<Gamma>\<turnstile>U<:S" and "TVarB X U # \<Gamma> \<turnstile> s : S'" and "(TVarB X U # \<Gamma>)\<turnstile>S'<:U'"
group_left_inverse: fixes x :: "'a" shows "inverse x * x = (1::'a)"
f_Exec_Stream_expand_shrink_map_nth: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and f :: "'f \<Rightarrow> 'e message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "0 < k"   and "n < length xs" shows "map f (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) \<div> k ! n = last_message (map f (f_Exec_Comp_Stream trans_fun (xs ! n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (xs \<down> n \<odot> k) c)))"
\<tau>move2AAcc1: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and a :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option"   and i :: "'addr expr1" assumes "\<tau>move2 P h stk a pc xcp" shows "\<tau>move2 P h stk (a\<lfloor>i\<rceil>) pc xcp"
inside_bounded_complement_connected_empty: fixes S :: "'b set" assumes "connected (- S)"   and "bounded S" shows "inside S = {}"
x2_def: shows "x2 = (1 - v) *\<^sub>R ((1 - u) *\<^sub>R x0 + u *\<^sub>R x3) + v *\<^sub>R x3"
Gromov_product_at_diff3: fixes e :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" shows "\<bar>Gromov_product_at e x y - Gromov_product_at e x z\<bar> \<le> dist y z"
1: shows "to_polynomial Zp (\<ominus> (\<one> \<oplus> [p] \<cdot> \<one> [^] 3 \<otimes> a [^] 4)) \<bullet> \<alpha> = \<ominus> (\<one> \<oplus> [p] \<cdot> \<one> [^] 3 \<otimes> a [^] 4)"
assms_3: shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr) \<rightarrow>\<^sub>r owner_document"
endo_inj_surj: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" assumes "finite A"   and "f ` A \<subseteq> A"   and "inj_on f A" shows "f ` A = A"
bdd_belowI2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'b set"   and m :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m (f x)" shows "preorder.bdd_below less_eq (f ` A)"
small_arr_Rel: fixes \<alpha> :: "V" shows "small {T. arr_Rel \<alpha> T}"
sym_Int: fixes r :: "('e \<times> 'e) set"   and s :: "('e \<times> 'e) set" assumes "sym r"   and "sym s" shows "sym (r \<inter> s)"
F_inv: fixes p :: "pState" assumes "pState_inv prog p"   and "sidx = pState.idx p"   and "cl_inv (g, p)" shows "pState_inv prog (snd (foldl (\<lambda>(g, p) d. mkVarChannel d (apsnd \<circ> pState.vars_update) g p) (g, p) decls))"
gallery_def: fixes xs :: "'a set list" shows "folding.gallery xs \<equiv> Ball (set xs) folding.chamber \<and> adjacentchain xs"
r_neg: fixes x :: "'c" assumes "x \<in> carrier S" shows "x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> x = \<zero>\<^bsub>S\<^esub>"
powser_MSLCons: fixes c :: "real"   and cs :: "real msllist" assumes "convergent_powser (MSLCons c cs)" shows "\<forall>\<^sub>F x in nhds 0. powser (MSLCons c cs) x = x * powser cs x + c"
fmdrop_add_distrib: fixes a :: "'c"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "fmdrop a (m ++\<^sub>f n) = fmdrop a m ++\<^sub>f fmdrop a n"
bind_mono: fixes B :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'e option"   and C :: "'e \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'f option" assumes "monotone option.le_fun option_ord B"   and "\<And>y. monotone option.le_fun option_ord (C y)" shows "monotone option.le_fun option_ord (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
assms_2: fixes n :: "nat" shows "\<exists>N. \<forall>n0 n1. N < n0 \<and> N < n1 \<longrightarrow> s n0 n = s n1 n"
abs_summable_on_nat_iff: fixes f :: "nat \<Rightarrow> 'b"   and A :: "nat set" shows "(f abs_summable_on A) = summable (\<lambda>n. if n \<in> A then norm (f n) else 0)"
take_takeWhile: fixes n :: "nat"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "n \<le> length ys"   and "\<And>y. y \<in> set (take n ys) \<Longrightarrow> P y"   and "\<And>y. y \<in> set (drop n ys) \<Longrightarrow> \<not> P y" shows "take n ys = takeWhile P ys"
subst_upd_lens_plus: fixes \<sigma> :: "'e \<Rightarrow> 'f"   and x :: "'g \<Longrightarrow> 'f"   and y :: "'h \<Longrightarrow> 'f"   and u :: "'g"   and v :: "'h" shows "\<sigma>(x +\<^sub>L y \<mapsto>\<^sub>s \<guillemotleft>(u, v)\<guillemotright>) = \<sigma>(y \<mapsto>\<^sub>s \<guillemotleft>v\<guillemotright>, x \<mapsto>\<^sub>s \<guillemotleft>u\<guillemotright>)"
Blinfun_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. bounded_linear f}" shows "blinfun_apply (Blinfun y) = y"
Oops2_not_AKcryptSK: fixes evs :: "event list"   and A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and SK :: "nat" assumes "evs \<in> kerbV"   and "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs" shows "\<not> AKcryptSK servK SK evs"
eval_rest: shows "P \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and accu :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_Algorithm.filter_compliant_stateful_ACS_accu G M accu [] = accu"
fps_power_eqD: fixes f :: "'a fps"   and m :: "nat"   and g :: "'a fps" assumes "f ^ m = g ^ m"   and "fps_nth f 0 = fps_nth g 0"   and "fps_nth f 0 \<noteq> (0::'a)"   and "0 < m" shows "f = g"
arr_char: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data \<times> ('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.VVV.arr f = (Span.arr (fst f) \<and> Span.VV.arr (snd f) \<and> Span.src (fst f) = Span.trg (fst (snd f)))"
E\<^sub>1: shows "\<E> f\<^sub>1 = Edges (v # rev vs @ [u]) \<union> Edges (u # between (vertices f) u v @ [v])"
that: fixes s :: "'val \<Rightarrow> 'node option" assumes "g \<turnstile> ns\<Down>\<^sub>s s" shows "thesis"
poly_lim_neg_inf: fixes p :: "real poly" shows "\<forall>\<^sub>F x in at_bot. sgn (poly p x) = poly_neg_inf p"
power_minus_even: fixes n :: "nat"   and a :: "'a" assumes "even n" shows "(- a) ^ n = a ^ n"
beta_upright_alt: fixes va :: "'c zmultiset"   and vb :: "'c zmultiset" shows "beta_upright va vb = (\<forall>t. 0 < zcount va t \<longrightarrow> (\<exists>s<t. (zcount va s < 0 \<or> zcount vb s < 0) \<and> nonpos_upto va s))"
braking_movement_axioms: shows "braking_movement a\<^sub>o (other.s' \<delta>)"
pop_def: fixes xs :: "('d, 'c) bintree option list" shows "pop xs = (case find_min xs of None \<Rightarrow> (None, xs) | Some t \<Rightarrow> (Some (val t, priority t), delete_min xs))"
HomFtorId: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,id\<^bsub>C\<^esub> Y] = id\<^bsub>SET\<^esub> (Hom\<^bsub>C\<^esub> X Y)"
simps_2: fixes p :: "'b"   and i :: "'b memref"   and "is" :: "'b memref list" shows "hd_prog p (i # is) = (case i of Prog\<^sub>s\<^sub>b p' x xa \<Rightarrow> p' | _ \<Rightarrow> hd_prog p is)"
subgroup_axioms: shows "subgroup Translations transformation.Sym (compose G) (identity G)"
higherPowers_def: shows "higherPowers \<equiv> map_fun id (map_fun id (map_fun lookup Abs_poly_mapping)) (\<lambda>b i p x. p (inc_above b i x))"
npow_suc: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c"   and n :: "nat" shows "x^\<^bsup>R Suc n\<^esup> = x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
LBI: shows "Mapping.lookup (LBI dir s') x\<^sub>i = Some (i, l\<^sub>i)"
eq_ffilter: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b fset"   and Q :: "'b \<Rightarrow> bool" shows "(ffilter P A = ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x = Q x)"
H'_3: shows "H = {f \<turnstile> \<C>, g \<turnstile> \<C>}"
elims_1: fixes x :: "('c, 'd) tm"   and xa :: "dir list"   and y :: "bool" assumes "position_of x xa = y" obtains uu_ :: "('c, 'd) tm" where "x = uu_" and "xa = []" and "y"   | uv_ :: "('c, 'd) hd" and uw_ :: "dir" and ux_ :: "dir list" where "x = Hd uv_" and "xa = uw_ # ux_" and "\<not> y"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t__ sa__" and "xa = dir.Left # ds__" and "y = position_of t__ ds__"   | t :: "('c, 'd) tm" and s :: "('c, 'd) tm" and ds :: "dir list" where "x = App t s" and "xa = dir.Right # ds" and "y = position_of s ds"
simps_1: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "Some f <*> Some x = Some (f x)"
row_matrix_matrix_mult: fixes P :: "(('a, 'c) vec, 'd) vec"   and i :: "'d"   and A :: "(('a, 'b) vec, 'c) vec" shows "P $ i v* A = (P ** A) $ i"
star2: fixes x :: "'a" shows "((1::'a) + x)\<^sup>\<star> = x\<^sup>\<star>"
dgr_demonic_def: fixes Q :: "'g \<Rightarrow> 'f \<Rightarrow> 'e"   and ij :: "'g" shows "dgr_demonic Q ij = [: Q ij :]"
refines_def: fixes e1 :: "db"   and e2 :: "db" shows "e1 \<unlhd> e2 \<equiv> \<forall>C. \<exists>V1. C<e1> \<Down> V1 \<longrightarrow> (\<exists>V2. C<e2> \<Down> V2 \<and> V1 \<sim> V2)"
app\<^sub>i_Invoke: fixes M :: "char list"   and n :: "nat"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invoke M n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length ST \<and> (ST ! n \<noteq> NT \<longrightarrow> (\<exists>C D Ts T m. ST ! n = Class C \<and> P \<turnstile> C sees M: Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts)))"
simps: fixes p :: "'a Point"   and q :: "'a Point" shows "from p to q = \<lparr>tdir = tval q - tval p, xdir = xval q - xval p, ydir = yval q - yval p, zdir = zval q - zval p\<rparr>"
sdrop_snil: fixes i :: "Integer" assumes "i \<noteq> \<bottom>" shows "sdrop\<cdot>i\<cdot>[::] = [::]"
spread_def: fixes k :: "'a"   and P :: "(real, 'a) vec set" shows "spread k P = (if P = {} then 0 else let V = (\<lambda>p. p $ k) ` P in Max V - Min V)"
sigma_sets_INTER: fixes A :: "'b set set"   and sp :: "'b set"   and S :: "nat set"   and a :: "nat \<Rightarrow> 'b set" assumes "A \<subseteq> Pow sp"   and "\<And>i. i \<in> S \<Longrightarrow> a i \<in> sigma_sets sp A"   and "S \<noteq> {}" shows "\<Inter> (a ` S) \<in> sigma_sets sp A"
incidence_system_axioms: shows "incidence_system (\<V> \<union> bl) (add_block bl)"
op_map_update_def: shows "op_map_update \<equiv> \<lambda>k v m. m(k \<mapsto> v)"
that: fixes \<zeta> :: "'b"   and R :: "nat set"   and \<zeta>' :: "'b"   and R' :: "nat set" assumes "s = NodeS i \<zeta> R"   and "s' = NodeS i \<zeta>' R'" shows "thesis"
not_empty_eq_Ici_eq_empty: fixes l :: "'a" shows "{} \<noteq> {l..}"
0: shows "prv (imp (ldsj \<psi>s) (imp (scnj F) \<chi>))"
has_contour_integral_integral: fixes f :: "complex \<Rightarrow> complex"   and i :: "real \<Rightarrow> complex" assumes "f contour_integrable_on i" shows "(f has_contour_integral contour_integral i f) i"
rows_to_keep_def: fixes A :: "'b mat" shows "rows_to_keep A = map snd (pivot_positions (gauss_jordan_single A\<^sup>T))"
unit_in_vhom: shows "\<guillemotleft>\<sigma> : src h \<Rightarrow> k \<star> h\<guillemotright>"
At0: fixes x0 :: "'a"   and At :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and Follows :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "Chain x0 At Follows" shows "At x0 0"
proj_stoch_proc_geom_open_set: fixes n :: "nat" shows "\<forall>r\<in>range (proj_stoch_proc geom_proc n) \<inter> space (stream_space borel). \<exists>A\<in>sets (stream_space borel). range (proj_stoch_proc geom_proc n) \<inter> A = {r}"
distinct_filterD: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and n :: "nat"   and m :: "nat"   and x :: "'b" assumes "distinct (filter P xs)"   and "n < length xs"   and "m < length xs"   and "P x"   and "xs ! n = x"   and "xs ! m = x" shows "m = n"
rbt_eq_iff: fixes t1 :: "('c, 'd) mapping_rbt"   and t2 :: "('c, 'd) mapping_rbt" shows "(t1 = t2) = (impl_of t1 = impl_of t2)"
upper_ivl_of_aform_err: fixes p :: "nat"   and Rd :: "(real \<times> real pdevs) \<times> real" shows "upper (ivl_of_aform_err p Rd) = Sup_aform_err p Rd"
id_act: fixes x :: "'b" shows "\<one> \<odot> x = x"
free_thread_id_iff: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f" shows "free_thread_id ts t = (ts t = None)"
ideI: fixes a :: "'a" assumes "\<guillemotleft>a : a \<rightarrow>\<^sub>C a'\<guillemotright>"   and "C.ide a" shows "hhom\<^sub>C.ide a"
Abs_asTerm_asAbs_qAbs: fixes qX :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "qGood qX" shows "Abs xs x (asTerm qX) = asAbs (qAbs xs x qX)"
distF: shows "dist_over_sup (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
pp_surjective_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_p_algebra.dense (mtop \<odot> f) = (\<forall>j. \<exists>i. f (i, j) \<noteq> bot)"
vsubst_Var_simp: fixes ys :: "'varSort"   and y1 :: "'var"   and y :: "'var"   and xs :: "'varSort"   and x :: "'var" shows "Var xs x #[y1 // y]_ys = Var xs (x @xs[y1 / y]_ys)"
t_def: shows "t \<equiv> SOME t. degree (b * f - t * #v) < degree v"
star_left_unfold_equal: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_idempotent_semiring.Rf x (x\<^sup>\<odot>) = x\<^sup>\<odot>"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (*\<^sub>C) CBasis"
length_eq_card_rows: fixes A :: "(('d, 'e) vec, 'f) vec" shows "IArray.length (matrix_to_iarray A) = CARD('f)"
HVZK_def: fixes Rel :: "('a \<times> 'b) set"   and valid_pub :: "'a set"   and init :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and response :: "'c \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'f spmf"   and check :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and S_raw :: "'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and challenge_space :: "'e set" assumes "\<Sigma>_protocols_base Rel valid_pub" shows "\<Sigma>_protocols_base.HVZK init response check Rel S_raw challenge_space valid_pub \<equiv> \<forall>e\<in>challenge_space. (\<forall>(h, w)\<in>Rel. \<Sigma>_protocols_base.R init response h w e = \<Sigma>_protocols_base.S S_raw h e) \<and> (\<forall>h\<in>valid_pub. \<forall>(a, z)\<in>set_spmf (S_raw h e). check h a e z)"
ltl_to_generalized_rabin\<^sub>C_af_correct: fixes w :: "nat \<Rightarrow> 'a set"   and \<Sigma> :: "'a set list"   and \<phi> :: "'a ltl" assumes "range w \<subseteq> set \<Sigma>" shows "w \<Turnstile> \<phi> = accept\<^sub>G\<^sub>R_LTS (ltl_to_generalized_rabin\<^sub>C_af \<Sigma> \<phi>) w"
\<phi>_eq: fixes x :: "real" assumes "x \<in> {(real_of_int i' - 1) / 2 ^ (l' + 1)..(real_of_int i' + 1) / 2 ^ (l' + 1)}" shows "\<phi> (l, i) x = real_of_int i + 1 - 2 ^ (l + 1) * x"
mk_v_def: fixes I :: "('b, 'c, 'a) interp"   and ODE :: "('b, 'a) ODE"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and sol :: "(real, 'a) vec" shows "mk_v I ODE \<nu> sol = (THE \<omega>. Vagree \<omega> \<nu> (- semBV I ODE) \<and> Vagree \<omega> (mk_xode I ODE sol) (semBV I ODE))"
intro: fixes x :: "'d \<Longrightarrow> 'c" assumes "\<And>\<sigma> v. get\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) = v" shows "weak_lens x"
adm_wo_def: fixes H :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" shows "local.adm_wo H \<equiv> \<forall>f g x. (\<forall>y\<in>local.underS x. f y = g y) \<longrightarrow> H f x = H g x"
prod: fixes g :: "'a \<Rightarrow> nat"   and A :: "'a set" shows "\<chi> (prod g A) = (\<Prod>x\<in>A. \<chi> (g x))"
Jacobi_eq_0_imp_not_coprime: fixes p :: "int"   and n :: "int" assumes "p \<noteq> 0"   and "p \<noteq> 1"   and "Jacobi n p = 0" shows "\<not> coprime n p"
\<K>'_def: fixes s :: "'a" shows "\<K>' s \<equiv> case map_of (map (\<lambda>(s, \<mu>). (s, pmf_of_alist \<mu>)) \<K>) s of None \<Rightarrow> return_pmf (hd \<K>\<^sub>s) | Some s \<Rightarrow> s"
prod_tuple: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "span f g"   and "seq h f"   and "seq k g" shows "ECC.prod h k \<cdot> ECC.tuple f g = ECC.tuple (h \<cdot> f) (k \<cdot> g)"
a_consistent_app\<^sub>1: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, App e x, S)" shows "ArityAnalysisLetSafe.a_consistent Aexp (ae, inc\<cdot>a, as) (\<Gamma>, e, Arg x # S)"
L_transform_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes f :: "'d"   and \<alpha> :: "'c"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" shows "L_transform_Tree\<^sub>\<alpha> (FL_Formula.Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha>) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Act\<^sub>\<alpha> (L_action.Act \<alpha>) (L_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>))"
Max_wrt_among_of_weak_ranking_Cons2: fixes x :: "'b set"   and xs :: "'b set list"   and A :: "'b set" assumes "is_weak_ranking (x # xs)"   and "x \<inter> A \<noteq> {}" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = x \<inter> A"
fold_cong: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and s :: "'c"   and t :: "'c"   and B :: "'b set" assumes "comp_fun_commute f"   and "comp_fun_commute g"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "s = t"   and "A = B" shows "Finite_Set.fold f s A = Finite_Set.fold g t B"
C: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s C"
higher_differentiable_on: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and n :: "nat" assumes "bounded_linear f" shows "higher_differentiable_on S f n"
induct: fixes P :: "nat list \<Rightarrow> bool"   and a0 :: "nat list" assumes "\<And>n l. P (Suc n # l) \<Longrightarrow> P (n # 0 # l)"   and "\<And>m l. P (1 # m # l) \<Longrightarrow> P (0 # Suc m # l)"   and "\<And>n m l. P (n # Suc m # m # l) \<Longrightarrow> P (Suc n # Suc m # l)"   and "\<And>m. P [m]"   and "P []" shows "P a0"