hcju/isabelleps huggingface dataset:
weak_Nullstellensatz_aux_3: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and x :: "'c"   and X :: "'c set" assumes "F \<subseteq> P[insert x X]"   and "x \<notin> X"   and "1 \<notin> ideal F"   and "\<not> ideal F \<inter> P[{x}] \<subseteq> {0}" obtains a :: "'d" where "1 \<notin> ideal (poly_eval (\<lambda>_. monomial a 0) ` focus {x} ` F)"
no_ulen_sub: fixes x :: "'b word"   and y :: "'b word" shows "(x - y \<le> x) = (uint y \<le> uint x)"
analytic_on_open: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "open S" shows "(f analytic_on S) = (f holomorphic_on S)"
ex_cong1: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool" assumes "\<And>x. P x = P' x" shows "(\<exists>x. P x) = (\<exists>x. P' x)"
int_ops_6: fixes a :: "nat"   and b :: "nat" shows "int (a - b) = (if int a < int b then 0 else int a - int b)"
insert_e1_e2_V: shows "\<forall>(e1, e2)\<in>F'. insert e1 (insert e2 V) = V"
ih1: shows "opnet onp p\<^sub>1 \<Turnstile> (otherwith S (net_tree_ips p\<^sub>1) (oarrivemsg I), other U (net_tree_ips p\<^sub>1) \<rightarrow>) global (\<lambda>\<sigma>. \<forall>i\<in>net_tree_ips p\<^sub>1. P i \<sigma>)"
inte_Dirichlet_kernel_mul_expand: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and n :: "nat" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "S \<in> sets lebesgue" shows "LINT x|lebesgue_on S. Dirichlet_kernel n x * f x = LINT x|lebesgue_on S. sin ((real n + 1 / 2) * x) * f x / (2 * sin (x / 2)) \<and> integrable (lebesgue_on S) (\<lambda>x. Dirichlet_kernel n x * f x) = integrable (lebesgue_on S) (\<lambda>x. sin ((real n + 1 / 2) * x) * f x / (2 * sin (x / 2)))"
normalize_snd_quot_of_fract: fixes x :: "'b fract" shows "normalize (denominator x) = denominator x"
cBallE: fixes A :: "'f cset"   and P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "cBall A P" obtains "P x"   | "\<not> cin x A"
IH_1: fixes y :: "'a ifex" assumes "ro_ifex xb1"   and "ro_ifex y"   and "\<And>ass. val_ifex xb1 ass = val_ifex y ass" shows "xb1 = y"
inpt_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a" shows "inpt r x = (\<exists>y. r x y)"
stateful_policy_ext_def: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "\<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons stateful_policy_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple hosts (Record.iso_tuple_cons Record.tuple_iso_tuple flows_fix flows_state)) more"
Least_fin_cut_same: fixes xs :: "'b list"   and y :: "'b" shows "(LEAST n. drop n xs = replicate (length xs - n) y) = length xs - length (takeWhile (\<lambda>x. x = y) (rev xs))"
intrinsic_borderI: fixes ys :: "'a list" assumes "strict_border w ys" shows "strict_border (intrinsic_border ys) ys"
select_convs_2: fixes init :: "'a"   and prod :: "'a \<Rightarrow> ('b + 'a) list fset"   and more :: "'c" shows "cfg.prod \<lparr>init = init, prod = prod, \<dots> = more\<rparr> = prod"
l_get_attribute_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_attribute type_wf get_attribute get_attribute_locs \<equiv> ((\<forall>element_ptr k h h'. reads (get_attribute_locs element_ptr) (get_attribute element_ptr k) h h') \<and> (\<forall>h element_ptr k. type_wf h \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_attribute element_ptr k)) \<and> (\<forall>h element_ptr k. h \<turnstile> ok get_attribute element_ptr k \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h) \<and> (\<forall>element_ptr k h. pure (get_attribute element_ptr k) h)"
ring_simprules_9: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
elims: fixes x :: "'b pair_pre_digraph"   and xa :: "'b \<times> 'b"   and xb :: "'b"   and y :: "'b pair_pre_digraph" assumes "subdivide x xa xb = y" obtains G :: "'b pair_pre_digraph" and u :: "'b" and v :: "'b" and w :: "'b" where "x = G" and "xa = (u, v)" and "xb = w" and "y = \<lparr>pverts = pverts G \<union> {w}, parcs = parcs G - {(u, v), (v, u)} \<union> {(u, w), (w, u), (w, v), (v, w)}\<rparr>"
simps_2: fixes e :: "'b"   and h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "prefixToElement e (h # t) = (if element h = e then [h] else h # prefixToElement e t)"
assms_2: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p S'"
conj_imp_eq_imp_imp: fixes P :: "bool"   and Q :: "bool"   and R :: "prop" shows "(P \<and> Q \<Longrightarrow> PROP R) \<equiv> (\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> PROP R)"
hseq_char': fixes \<nu> :: "'c"   and \<mu> :: "'c" shows "C.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>C \<mu> \<noteq> C.null)"
distinct_2: fixes freeRes' :: "freeRes"   and pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "freeRes' \<prec> pi' \<noteq> name1<\<nu>name2> \<prec> pi"
path_single_Arr: fixes Obj :: "'a set"   and Arr :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "graph Obj Arr Dom Cod"   and "x \<in> Arr" shows "graph.path Obj Arr Dom Cod (Dom x) (Cod x) [x]"
irreducible_multD: fixes a :: "'a"   and b :: "'a" assumes "irreducible (a * b)" shows "a dvd (1::'a) \<and> irreducible b \<or> b dvd (1::'a) \<and> irreducible a"
intro: fixes sizes :: "int set"   and grouping :: "int" assumes "\<And>k. k \<in> sizes \<Longrightarrow> grouping \<le> k" shows "tBD_axioms grouping sizes"
simps_6: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and y :: "Vars.var"   and z :: "Vars.var" shows "(scrut ? e1 : e2)[y::=z] = (scrut[y::=z] ? e1[y::=z] : e2[y::=z])"
bal_del0: fixes t :: "'a tree23"   and k :: "int" assumes "bal t" shows "bal (del0 k t)"
basis_UNIV: fixes X :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)" shows "UNIV = {x. \<exists>g. (\<Sum>i\<in>UNIV. g i *s X $ i) = x}"
prime_nonprime_wlog: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a" assumes "\<And>p q. \<lbrakk>prime p; prime q\<rbrakk> \<Longrightarrow> P p q"   and "\<And>p q. \<not> prime p \<Longrightarrow> P p q"   and "\<And>p q. P p q \<Longrightarrow> P q p" shows "P p q"
vinsert_def: fixes y :: "V"   and B :: "V" shows "vinsert y B = set {x. x = y} \<union>\<^sub>\<circ> B"
le_any_fun_inv: fixes v :: "val"   and t :: "(val \<times> val) list" assumes "v \<sqsubseteq> VFun t" obtains t1 :: "(val \<times> val) list" where "v = VFun t1" and "t1 \<lesssim> t"
pseudominimal_in_def: fixes P :: "'a set"   and x :: "'a" shows "pseudominimal_in P x \<equiv> minimal_in (P - {bottom P}) x"
Elem_implode: fixes A :: "ZF set"   and x :: "ZF" assumes "A \<in> range explode" shows "Elem x (implode A) = (x \<in> A)"
flow_lipschitzE: fixes a :: "real"   and b :: "real"   and x :: "'a" assumes "{a..b} \<subseteq> existence_ivl0 x" obtains L :: "real" where "L-lipschitz_on {a..b} (flow0 x)"
Nonce_Exp: fixes X :: "nonce_t"   and X' :: "msg"   and Y' :: "msg" shows "Nonce X \<noteq> Exp X' Y'"
ex_finite_reduced_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" obtains G :: "('t \<Rightarrow>\<^sub>0 'c) set" where "G \<subseteq> dgrad_p_set d m" and "finite G" and "is_reduced_GB G" and "pmdl G = pmdl F"
better_T_LET: fixes x :: "W.var"   and \<Gamma> :: "(W.var \<times> tyS) list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "ty"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and " \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 "   and " (x, close \<Gamma> T\<^sub>1) # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 " shows " \<Gamma> \<turnstile> Let x be t\<^sub>1 in t\<^sub>2 : T\<^sub>2 "
u_def: shows "u = 142196984054132045946501548559032969 / 246305600854754639064867000000000000"
open_Collect_neg: fixes P :: "'a \<Rightarrow> bool" assumes "closed {x. P x}" shows "open {x. \<not> P x}"
induct: fixes P :: "'a bdd \<Rightarrow> bool list \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "'a bdd"   and a1 :: "bool list"   and a2 :: "'a" assumes "\<And>a x. P (Leaf a) [] x"   and "\<And>a w ws x. \<lbrakk>w \<Longrightarrow> P (Leaf a) ws x; \<not> w \<Longrightarrow> P (Leaf a) ws x\<rbrakk> \<Longrightarrow> P (Leaf a) (w # ws) x"   and "\<And>l r w ws x. \<lbrakk>w \<Longrightarrow> P r ws x; \<not> w \<Longrightarrow> P l ws x\<rbrakk> \<Longrightarrow> P (Branch l r) (w # ws) x"   and "\<And>v va c. P (Branch v va) [] c" shows "P a0 a1 a2"
prv_eql_subst_trm_rev: fixes t1 :: "'trm"   and t2 :: "'trm"   and \<phi> :: "'fmla"   and y :: "'var" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "\<phi> \<in> fmla"   and "y \<in> var" shows "prv (imp (eql t1 t2) (imp (subst \<phi> t2 y) (subst \<phi> t1 y)))"
c\<theta>_red: fixes C\<theta> :: "'a literal multiset"   and N' :: "('a literal multiset \<times> label) set" assumes "C\<theta> \<in> \<G>_F C" shows "C\<theta> \<in> G.Red_F (\<Union>D\<in>N'. \<G>_F (fst D))"
succeed\<^sub>R_def: fixes i :: "nat" shows "succeed\<^sub>R i = {uu_. \<exists>r \<nu> s q. uu_ = (r, \<nu>, s) \<and> r q = Some i \<and> q \<notin> F - {q\<^sub>0} \<and> \<delta> q \<nu> \<in> F}"
at_map_Sup_pres: shows "Sup_pres at_map"
diff_iff: fixes f :: "'b \<Rightarrow> 'b" shows "diff k charts2 charts2 f = (\<forall>x\<in>dest.carrier. \<exists>c1\<in>dest.atlas. \<exists>c2\<in>dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
rm_vars_empty_1: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "rm_vars {} s = s"
d: fixes x :: "'a" assumes "x \<in> s" shows "d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union> (d x)"
has_integral_periodic_offset_lemma: fixes f :: "real \<Rightarrow> 'a"   and b :: "real"   and a :: "real"   and c :: "real"   and i :: "'a" assumes "\<And>x. f (x + (b - a)) = f x"   and "has_bochner_integral (lebesgue_on {a..a + c}) f i" shows "has_bochner_integral (lebesgue_on {b..b + c}) f i"
\<psi>: shows "bij_betw \<psi> (Collect arr) S"
no_real_roots: shows "\<forall>x\<in>proots p. Im x \<noteq> 0"
pickT: fixes \<sigma> :: "'tp" shows "eintT \<sigma> (pickT \<sigma>)"
closed_clw_rel_iplane_rel: fixes xi :: "((real list \<times> real list) \<times> real list sctn) list"   and x :: "'f set" assumes "(xi, x) \<in> clw_rel (\<langle>lvivl_rel, \<langle>lv_rel\<rangle>plane_rel\<rangle>Refine_Intersection.inter_rel)" shows "closed x"
abs_eq: fixes x :: "'b \<times> 'b" shows "set_uprod (abs_uprod x) = (case x of (a, b) \<Rightarrow> {a, b})"
simps_2: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and t :: "'a"   and ts :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f (t # ts) = term.list_comb app (app f t) ts"
eucl_rel_poly_unique_mod: fixes x :: "'a poly"   and y :: "'a poly"   and q1 :: "'a poly"   and r1 :: "'a poly"   and q2 :: "'a poly"   and r2 :: "'a poly" assumes "eucl_rel_poly x y (q1, r1)"   and "eucl_rel_poly x y (q2, r2)" shows "r1 = r2"
deg_ltrm: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "deg R (monom P (f (deg R f)) (deg R f)) = deg R f"
\<G>_I_def: fixes M :: "'a literal multiset set"   and \<iota> :: "'a literal multiset inference" shows "\<G>_I M \<iota> = {Infer (prems_of \<iota> \<cdot>\<cdot>cl \<rho>s) (concl_of \<iota> \<cdot> \<rho>) |\<rho> \<rho>s. is_ground_subst_list \<rho>s \<and> is_ground_subst \<rho> \<and> Infer (prems_of \<iota> \<cdot>\<cdot>cl \<rho>s) (concl_of \<iota> \<cdot> \<rho>) \<in> G_Inf M}"
assms: shows "(A, B) \<in> ns_mul_ext ns s"
kD_sqnf_is_proj3: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt" shows "sqnf rt ip = \<pi>\<^sub>3 (the (rt ip))"
hyps: fixes ys :: "'a list"   and thesis :: "bool" assumes "\<And>ys'. \<lbrakk>ys' \<in> listset xs; ys = map f ys'\<rbrakk> \<Longrightarrow> thesis"   and "ys \<in> listset (map ((`) f) xs)" shows "thesis"
scale_zero_left: fixes x :: "'b" shows "scale (0::'a) x = (0::'b)"
left_neutral: fixes a :: "'a ppath set" shows "pp_prod pp_one a = a"
path_connected_component: fixes S :: "'a set" shows "path_connected S = (\<forall>x\<in>S. \<forall>y\<in>S. path_component S x y)"
chainTransitionAppend_3: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'"   and "x \<sharp> P" shows "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"
P_component: fixes M :: "'a measure"   and i :: "nat" assumes "shift_operator_ergodic M" shows "distr (Pi\<^sub>M UNIV (\<lambda>_. M)) M (\<lambda>f. f i) = M"
Cons: shows "p_fusion ps (p_fusion qs rs_) = p_fusion (p_fusion ps qs) rs_"
negligible_imp_measurable: fixes S :: "'a set" assumes "negligible S" shows "S \<in> lmeasurable"
m_comm: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
singular_simplex_mono: fixes p :: "nat"   and X :: "'b topology"   and T :: "'b set"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'b"   and S :: "'b set" assumes "singular_simplex p (subtopology X T) f"   and "T \<subseteq> S" shows "singular_simplex p (subtopology X S) f"
build_split: fixes w :: "nat \<Rightarrow> 'a" shows "w = w 0 ## w [1..]"
cmult: fixes c :: "'b"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "\<Omega>[F](\<lambda>x. c * f x) = \<Omega>[F](f)"
rev_iterateoi_def: fixes S :: "'s" shows "reverse_iterateoi S \<equiv> it_to_it (list_rev_it S)"
assms: shows "(x, x') \<in> \<langle>R\<rangle>infty_rel"
par_commute: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice" shows "(a::'a::refinement_lattice) \<parallel> (b::'a::refinement_lattice) = b \<parallel> a"
le_approx2: fixes P :: "'a process"   and Q :: "'a process"   and s :: "'a event list"   and X :: "'a event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F Q) = ((s, X) \<in> F P)"
binop_axioms: shows "binop \<rat>\<^sub>\<circ> vrat_mult"
primmultgreater2: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "prime q"   and "p \<noteq> q" shows "2 < p * q"
ideduct_hom_mono: fixes M :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and M' :: "('fun, 'var) Term.term set" assumes "\<langle>M;\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"   and "M \<subseteq> M'" shows "\<langle>M';\<A>;Sec\<rangle> \<turnstile>\<^sub>h\<^sub>o\<^sub>m t"
le_ordinal_of_nat: fixes x :: "ordinal"   and n :: "nat" shows "(x \<le> ordinal_of_nat n) = (\<exists>m. x = ordinal_of_nat m \<and> m \<le> n)"
pair_map_spmf1: fixes f :: "'j \<Rightarrow> 'h"   and p :: "'j spmf"   and q :: "'i spmf" shows "pair_spmf (map_spmf f p) q = map_spmf (apfst f) (pair_spmf p q)"
e_if: shows "if f z = 0 then 0 < zn else zn = 0"
sums_If_finite_set': fixes g :: "nat \<Rightarrow> 'a"   and S :: "'a"   and A :: "nat set"   and S' :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "g sums S"   and "finite A"   and "S' = S + (\<Sum>n\<in>A. f n - g n)" shows "(\<lambda>n. if n \<in> A then f n else g n) sums S'"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H (del_arc u)" shows "u \<rightarrow>\<^sup>*\<^bsub>del_arc u\<^esub> v"
inj_image_eq_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "inj f" shows "(f ` A = f ` B) = (A = B)"
exhaust_sel: fixes dining_event :: "dining_event" obtains "dining_event = picks (phil dining_event) (fork dining_event)"   | "dining_event = putsdown (phil dining_event) (fork dining_event)"
simps: fixes n :: "'b"   and l :: "'b tree"   and r :: "'b tree" shows "mkt_bal_r n l r = (if ht r = ht l + 2 then case r of MKT rn rl rr x \<Rightarrow> if ht rr < ht rl then case rl of MKT rln rll rlr x \<Rightarrow> mkt rln (mkt n l rll) (mkt rn rlr rr) else mkt rn (mkt n l rl) rr else mkt n l r)"
degree_convert_eq: fixes var :: "nat"   and p :: "real mpoly"   and xs :: "real list"   and a :: "real \<times> real \<times> real" assumes "convert_poly var p xs = Some a" shows "MPoly_Type.degree p var < 3"
mhomom3Tr0: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (\<iota>m\<^bsub>M H,K\<^esub>) \<in> mHom R (mdl M H) (mdl M (H \<minusplus> K) /\<^sub>m K)"
elims: fixes x :: "V list"   and y :: "V" assumes "\<omega>_sum x = y" obtains "x = []" and "y = 0"   | \<alpha> :: "V" and \<alpha>s :: "V list" where "x = \<alpha> # \<alpha>s" and "y = \<omega> \<up> \<alpha> + \<omega>_sum \<alpha>s"
MapsToCompI: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g" shows "f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> dom\<^bsub>C\<^esub> f to cod\<^bsub>C\<^esub> g"
one_subst_aux_automatic: fixes N :: "lam"   and N' :: "lam"   and x :: "name"   and M :: "lam" assumes " N \<longrightarrow>\<^sub>1 N'" shows " M[x::=N] \<longrightarrow>\<^sub>1 M[x::=N']"
G2: fixes l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "pre ({A} WHILE b DO C) Q l s" shows "Q l (postQ ({A} WHILE b DO C) s)"
bodies_def: shows "bodies \<equiv> [(Even, evn), (Odd, EvenOdd.odd)]"
simps: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b" shows "wf_disc_sel (disc, sel) C = ((\<forall>a. disc a \<longrightarrow> C (sel a) = a) \<and> (\<forall>a. sel (C a) = a))"
mC_C_wt_def: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set" shows "mC_C_wt c d s t P F \<equiv> \<forall>I\<in>P. sum (wt c s) I = sum (wt d t) (F I)"
surjective: fixes r :: "'a state_scheme" shows "r = \<lparr>state.id = state.id r, no = no r, nhid = nhid r, msg = msg r, num = num r, sid = sid r, \<dots> = state.more r\<rparr>"
finite_list_upto: fixes m :: "int"   and n :: "int" shows "finite_list (upto\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n))"
equivalence_map_extends_to_adjoint_equivalence: fixes f :: "'a" assumes "equivalence_map f" shows "\<exists>g \<eta> \<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
tickFree_implies_front_tickFree: fixes s :: "'c event list" assumes "tickFree s" shows "front_tickFree s"
vdomain_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "\<D>\<^sub>\<circ> (vinsert \<langle>a, b\<rangle> r) = vinsert a (\<D>\<^sub>\<circ> r)"
A2: shows "\<lfloor>(\<lambda>w. universal (\<lambda>x xa. (\<P> x \<^bold>\<and> \<^bold>\<box>(\<lambda>v. \<forall>xb. (existsAt xb \<^bold>\<rightarrow> x xb \<^bold>\<rightarrow> xa xb) v) \<^bold>\<rightarrow> \<P> xa) w))\<rfloor>"
divide_le_cancel: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide a c) (divide b c) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
star_decompose_1: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y)\<^sup>\<odot> = (x\<^sup>\<odot> \<odot> y\<^sup>\<odot>)\<^sup>\<odot>"
abs_square_eq_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(power.power one times x 2 = one) = (abs x = one)"
dim_ty_sem_Intensity_def: fixes uu :: "J itself" shows "dim_ty_sem uu = \<^bold>J"
bisimilar_transp: shows "transp (\<sim>\<cdot>)"
simps_2: fixes n :: "nat"   and p :: "'b pmf" shows "replicate_pmf (Suc n) p = p \<bind> (\<lambda>x. replicate_pmf n p \<bind> (\<lambda>xs. return_pmf (x # xs)))"
b: shows "b \<in> smallest_ccdi_sets \<Omega> M"
sred_lift: fixes s :: "dB"   and t :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>s t" shows "lift s i \<rightarrow>\<^sub>s lift t i"
typ_ok_Ty: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and n :: "String.literal"   and Ts :: "typ list" assumes "typ_ok \<Theta> (Ty n Ts)" shows "list_all (typ_ok \<Theta>) Ts"
matrix_vector_mult_0_right: fixes A :: "(('c, 'e) vec, 'd) vec" shows "A *v 0 = 0"
inv_def: fixes x :: "'a" shows "local.inv x = (THE y. x ** y = one \<and> y ** x = one)"
snd_synthetic_divmod: fixes p :: "'a poly"   and c :: "'a" shows "snd (synthetic_divmod p c) = poly p c"
length_0_conv: fixes xs :: "'c list" shows "(length xs = 0) = (xs = [])"
compose_Nil: fixes x :: "brick list"   and ys :: "wall" shows "basic x \<circ> ys = x * ys"
lemma_W2: fixes a :: "'a G"   and b :: "'a G" shows "(a r\<rightarrow> b) l\<rightarrow> b = (b r\<rightarrow> a) l\<rightarrow> a"
image_Un: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
ArrDom_def: shows "ArrDom = 1\<^sub>\<nat>"
AE_T_enabled: fixes s :: "'a cfg" shows "almost_everywhere (MC.T s) (MC.enabled s)"
simps_2: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and S :: "('b, 'a) strand_step list" shows "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S) = (fv t \<subseteq> V \<and> wf\<^sub>s\<^sub>t V S)"
IH: fixes w :: "complex" assumes "w \<in> ball z r" shows "((\<lambda>u. f u / (u - w) ^ Suc k) has_contour_integral complex_of_real (2 * pi) * \<i> / fact k * (deriv ^^ k) f w) (circlepath z r)"
derives_implies_leftderives: fixes a :: "symbol list"   and b :: "symbol list" assumes "derives a b"   and "is_word b" shows "leftderives a b"