hcju/isabelleps huggingface dataset:
connected_arc_image: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "connected (path_image g)"
singular_relcycle_diff: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X S a"   and "singular_relcycle p X S b" shows "singular_relcycle p X S (a - b)"
sigma_finite_measure_axioms: shows "sigma_finite_measure (M \<Otimes>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M))"
wf: shows "wf (map_prod (\<lambda>(x, xs). x # xs) (\<lambda>(x, xs). x # xs) ` (r <*lex*> lexn r n))"
simTauChain: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q' :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel"   and "\<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> Rel \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>[Rel] Q" obtains P' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" and "(\<Psi>, P', Q') \<in> Rel"
add_col_sub_row_carrier_3: fixes A :: "'b mat"   and n :: "nat"   and a :: "'b"   and k :: "nat"   and l :: "nat" assumes "A \<in> carrier_mat n n" shows "add_col_sub_row a k l A \<in> carrier_mat n n"
Stirling_fps_def: fixes m :: "nat" shows "Stirling_fps m = fps_const (1 / fact m) * (fps_exp 1 - 1) ^ m"
V_def: shows "V = (\<lambda>x. {n. \<exists>l\<in>{k..n - s}. u n x - u (n - l) x \<le> - d * real l})"
intros_24: fixes v :: "v" shows "\<lparr>[$C v, $Drop]\<rparr> \<leadsto> \<lparr>[]\<rparr>"
powsum_def: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x\<^bsub>m\<^esub>\<^bsup>n\<^esup> = sum ((^) x) {m..n + m}"
one_neq_zero: fixes one :: "'b"   and zero :: "'b" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
countable_UN': fixes X :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "countable X"   and "A ` X \<subseteq> M" shows "\<Union> (A ` X) \<in> M"
is_singletonI': fixes A :: "'b set" assumes "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "is_singleton A"
simps_5: fixes yc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and more :: "'a" shows "yc_update yc' \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = \<lparr>xc = xc, yc = yc' yc, \<dots> = more\<rparr>"
grow_mono: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "bc n \<le> bc n'"
intro: fixes X :: "'b set set"   and C0 :: "'b set" assumes "ThinChamberComplex X"   and "ThinChamberComplexManyFoldings_axioms X C0" shows "ThinChamberComplexManyFoldings X C0"
between_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S" shows "B\<^sub>\<real> cart2_pt (endpoint_in_S a b) cart2_pt a cart2_pt (endpoint_in_S b a)"
\<D>_2: shows "\<Union> \<D> \<subseteq> cbox a b"
assms_1: shows "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne"
PRelQ: fixes A\<^sub>R :: "name list"   and \<Psi>\<^sub>R :: "'b" assumes "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q" shows "(\<Psi> \<otimes> \<Psi>\<^sub>R, P, Q) \<in> Rel"
pmf_map_spmf_None: fixes f :: "'i \<Rightarrow> 'h"   and p :: "'i spmf" shows "pmf (map_spmf f p) None = pmf p None"
std_dual_bij_betw: assumes "finite B" shows "bij_betw local.std_dual S (dual_space S)"
resolution_and_max_literal: fixes R :: "'at Literal set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and A :: "'at"   and M :: "'at Literal" assumes "R = resolvent_upon P1 P2 A"   and "strictly_maximal_literal P1 (Pos A)"   and "strictly_maximal_literal P2 (Neg A)"   and "strictly_maximal_literal R M" shows "(atom M, A) \<in> atom_ordering"
simps_3: fixes y :: "int"   and x :: "int" shows "(Num_arc y < Num_arc x) = (y < x)"
Y: assumes "distincts yss"   and "map set xss = map set yss"   and "map hd xss = map hd yss" shows "list_all2 (\<lambda>x ys. x \<in> set ys) yss (map cyc_permutations xss)"
compose_assoc: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and h :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "f \<in> A \<rightarrow> B" shows "compose A h (compose A g f) = compose A (compose B h g) f"
Mem_Neg_refl: fixes x :: "tm" shows "{} \<turnstile> Neg (x IN x)"
squarefree_part_nonzero: fixes n :: "'b" shows "squarefree_part n \<noteq> (0::'b)"
prime_factors_primorial': fixes k :: "nat" shows "prime_factors (primorial' k) = nth_prime ` {..<k}"
elims_3: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list" assumes "\<not> abc_notfinal x xa" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s, lm)" and "xa = p" and "\<not> s < length p"
finsum_empty: fixes G :: "('a, 'b) ring_scheme"   and f :: "'c \<Rightarrow> 'a" assumes "abelian_monoid G" shows "finsum G f {} = \<zero>\<^bsub>G\<^esub>"
elims_2: fixes x :: "('b \<times> color) tree" assumes "invc x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> color) tree" and a :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "x = \<langle>l, (a, c), r\<rangle>" and "(c = Red \<longrightarrow> color l = Black \<and> color r = Black) \<and> invc l \<and> invc r"
field_divide_inverse: fixes a :: "'a"   and b :: "'a" shows "a / b = a \<cdot> inverse b"
simps_2: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool"   and i :: "nat" shows "\<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> i\<^esup> = {\<rho>. \<forall>n\<ge>i. R (time (Rep_run \<rho> n K\<^sub>1), time (Rep_run \<rho> n K\<^sub>2))}"
cases: fixes addr2thread_id :: "'e \<Rightarrow> 'f"   and thread_id2addr :: "'f \<Rightarrow> 'e"   and spurious_wakeups :: "bool"   and empty_heap :: "'g"   and allocate :: "'g \<Rightarrow> htype \<Rightarrow> ('g \<times> 'e) set"   and typeof_addr :: "'g \<Rightarrow> 'e \<Rightarrow> htype option"   and heap_read :: "'g \<Rightarrow> 'e \<Rightarrow> addr_loc \<Rightarrow> 'e val \<Rightarrow> bool"   and heap_write :: "'g \<Rightarrow> 'e \<Rightarrow> addr_loc \<Rightarrow> 'e val \<Rightarrow> 'g \<Rightarrow> bool"   and extTA :: "('e, 'f, 'g) external_thread_action \<Rightarrow> ('e, 'f, 'h, 'g) Jinja_thread_action"   and P :: "'e J_prog"   and t :: "'f"   and a1 :: "'e expr list"   and a2 :: "'g \<times> (String.literal \<Rightarrow> 'e val option)"   and a3 :: "('e, 'f, 'h, 'g) Jinja_thread_action"   and a4 :: "'e expr list"   and a5 :: "'g \<times> (String.literal \<Rightarrow> 'e val option)" assumes "J_heap_base.reds addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write extTA P t a1 a2 a3 a4 a5" obtains e :: "'e expr" and s :: "'g \<times> (String.literal \<Rightarrow> 'e val option)" and ta :: "('e, 'f, 'h, 'g) Jinja_thread_action" and e' :: "'e expr" and s' :: "'g \<times> (String.literal \<Rightarrow> 'e val option)" and es :: "'e expr list" where "a1 = e # es__" and "a2 = s__" and "a3 = ta__" and "a4 = e' # es__" and "a5 = s'__" and "J_heap_base.red addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write extTA P t e s__ ta__ e' s'__"   | es :: "'e expr list" and s :: "'g \<times> (String.literal \<Rightarrow> 'e val option)" and ta :: "('e, 'f, 'h, 'g) Jinja_thread_action" and es' :: "'e expr list" and s' :: "'g \<times> (String.literal \<Rightarrow> 'e val option)" and v :: "'e val" where "a1 = Val v # es" and "a2 = s" and "a3 = ta" and "a4 = Val v # es'" and "a5 = s'" and "J_heap_base.reds addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write extTA P t es s ta es' s'"
assms: shows "\<forall>asset\<in>support_set pf. integrable M (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"
widen_refl: fixes 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 T :: "ty" shows "subtype P T T"
comp_orA_def: fixes p2 :: "'f \<Rightarrow> 'e decision option"   and p1 :: "'d \<Rightarrow> 'f decision option" shows "p2 \<circ>\<^sub>\<or>\<^sub>A p1 \<equiv> map_option flat_orA \<circ> (lift p2 \<circ>\<^sub>m p1)"
denote_while_def: fixes dims :: "nat list"   and M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat" shows "state_sig.denote_while dims M0 M1 DS \<rho> = matrix_inf_sum (state_sig.d dims) (\<lambda>n. denote_while_n M0 M1 DS n \<rho>)"
intros_2: fixes x :: "real" assumes "x \<in> radical_sqrt" shows "- x \<in> radical_sqrt"
init_le_def: fixes G :: "prog"   and n :: "nat" shows "(G\<turnstile>init\<le>n) = (\<lambda>s. card (nyinitcls G s) \<le> n)"
nsgTr3: fixes N :: "'a set"   and a :: "'a"   and b :: "'a"   and a1 :: "'a"   and b1 :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a1 \<in> carrier G"   and "b1 \<in> carrier G"   and "a \<cdot> \<rho> a1 \<in> N"   and "b \<cdot> \<rho> b1 \<in> N" shows "a \<cdot> b \<cdot> \<rho> (a1 \<cdot> b1) \<in> N"
inv: shows "inverse (\<alpha> i) = inverse (f (i - 1)) ^ (Suc (length (coeffs (F (i - 2)))) - length (coeffs (F (i - 1))))"
wpt_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.wpt one times inf top x = inf (times x top) one"
FOREACH_body_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c nres" shows "FOREACH_body f \<equiv> \<lambda>(xs, \<sigma>). let x = hd xs in f x \<sigma> \<bind> (\<lambda>\<sigma>'. RETURN (tl xs, \<sigma>'))"
maxBlk_in_nonInitBlks: fixes b :: "DiskBlock"   and s :: "state"   and p :: "Proc" assumes "b \<in> nonInitBlks s p"   and "Inv1 s" shows "maxBlk s p \<in> nonInitBlks s p \<and> (\<forall>c\<in>nonInitBlks s p. bal c \<le> bal (maxBlk s p))"
not_finite_existsD: fixes P :: "'k \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
measD: fixes d :: "'a set" assumes "d \<in> \<D>" shows "d \<in> lmeasurable"
supp_fun_def: fixes f :: "'c \<Rightarrow> 'd" shows "supp_fun f = {x. f x \<noteq> (0::'d)}"
IH: assumes "\<And>x. (0::'a) \<le> f x" shows "(0::'a) \<le> \<Sum>\<^sub># (image_mset f A)"
eval_notfree_unchanged: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and V :: "char list" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"   and "V \<notin> fv e" shows "l' V = l V"
assms_1: shows "(f has_vderiv_on f') (g ` T)"
exists_UnionP: fixes z :: "name"   and x :: "tm"   and y :: "tm"   and H :: "fm set" assumes "atom z \<sharp> (x, y)" shows "H \<turnstile> SyntaxN.Ex z (UnionP x y (Var z))"
inf_sup_aci_5: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
simps_2: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and c :: "'b"   and v1 :: "'b val" shows "injval (Plus r1 r2) c (val.Left v1) = val.Left (injval r1 c v1)"
measurable_If_set: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "g \<in> M \<rightarrow>\<^sub>M M'"   and "A \<inter> space M \<in> sets M" shows "(\<lambda>x. if x \<in> A then f x else g x) \<in> M \<rightarrow>\<^sub>M M'"
invar_insert: fixes h :: "'b hp option"   and x :: "'b" assumes "invar h" shows "invar (insert x h)"
more_temp_simps3_4: fixes F :: "'a \<Rightarrow> bool" shows "\<not> \<box>F \<equiv> \<diamond>\<not> F"
R_def: shows "R = {p \<in> set ys. fst m < fst p}"
perm_restrict_diff_cyclic: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and A :: "'b set" assumes "f permutes S"   and "cyclic_on f A" shows "perm_restrict f (S - A) permutes S - A"
Cauchy_altdef2: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'd" assumes "class.metric_space dist uniformity open" shows "uniform_space.Cauchy uniformity s = (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (s n) (s N) < e)"
assms: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "\<And>i. (f i, f (Suc i)) \<in> r" shows "\<exists>i. (f i, a) \<notin> r"
comp_ide_self: fixes a :: "'a" assumes "Right_a.ide a" shows "Right_a.comp a a = a"
Cons_3: shows "non_volatile_owned_or_read_only pending_write \<S> \<O> (x # sb)"
build_cons: fixes a :: "'a"   and u :: "'a list"   and v :: "nat \<Rightarrow> 'a" shows "(a # u) \<frown> v = a ## u \<frown> v"
filterlim_at: fixes f :: "'c \<Rightarrow> 'a"   and b :: "'a"   and s :: "'a set"   and F :: "'c filter" shows "filterlim f (at b within s) F = ((\<forall>\<^sub>F x in F. f x \<in> s \<and> f x \<noteq> b) \<and> (f \<longlongrightarrow> b) F)"
set_iterator_rev_linord_def: fixes it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S0 :: "'a set" shows "set_iterator_rev_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. y \<le> x)"
LPDi_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex"   and n :: "nat" assumes "ok m mr"   and "ms \<in> LPDi n mr" shows "ok m ms"
invariantI: fixes F :: "'d program"   and A :: "'d set" assumes "Init F \<subseteq> A"   and "F \<in> stable A" shows "F \<in> invariant A"
Insert: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and t :: "('b, 'a) Term.term" assumes "\<theta> \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set"   and "v \<notin> subst_domain \<theta>"   and "v \<notin> range_vars \<theta>"   and "fv t \<inter> insert v (subst_domain \<theta>) = {}" shows "\<theta>(v := t) \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set"
Par_transT_simp: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" shows "\<not> (c1 | c2, s) \<rightarrow>t s'"
assms_2: shows "i \<subseteq>\<^sub>\<circ> j"
6: shows "f (l, l)\<^sup>\<star> * g (l, m) \<le> g (l, m)"
Group_Qg: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" assumes "Group G"   and "G \<triangleright> N" shows "Group (G / N)"
permute_set_eq_vimage: fixes p :: "perm"   and X :: "'b set" shows "p \<bullet> X = permute (- p) -` X"
one_closed: shows "\<one>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub> \<in> carrier \<F>\<^bsub>{()}\<^esub>"
s_Tr_2: assumes "\<pi>\<^sub>2 = []"   and "\<lless>[], e\<^sub>1\<ggreater> I\<rightarrow>i \<lless>[], e\<^sub>2\<ggreater>"   and "{$$} \<turnstile> e\<^sub>1, eP, eV : \<tau>"   and "{$$} \<turnstile> e\<^sub>1, eP, eV : \<tau>" shows "\<exists>\<pi> eP' eV'. \<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater> \<and> \<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>[], eV'\<ggreater> \<and> {$$} \<turnstile> e\<^sub>2, eP', eV' : \<tau>"
sw_trans: fixes us :: "'a"   and zs :: "'a"   and u :: "'b"   and y :: "'b"   and z :: "'b"   and x :: "'b" assumes "us \<noteq> zs \<or> u \<notin> {y, z}" shows "u @us[y \<and> x]_zs @us[z \<and> y]_zs = u @us[z \<and> x]_zs"
Smith_normal_form_mat_intro: fixes A :: "'b mat" assumes "\<forall>a. a + 1 < min (dim_row A) (dim_col A) \<longrightarrow> A $$ (a, a) dvd A $$ (a + 1, a + 1)"   and "isDiagonal_mat A" shows "Smith_normal_form_mat A"
list_rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
isomorphic_max_density: 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"   and "nonempty_graph G\<^sub>1"   and "nonempty_graph G\<^sub>2"   and "finite_graph G\<^sub>1"   and "finite_graph G\<^sub>2" shows "max_density G\<^sub>1 = max_density G\<^sub>2"
prv_imp_triv: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv \<psi>" shows "bprv (imp \<phi> \<psi>)"
singleton_rev_conv: fixes x :: "'b"   and xs :: "'b list" shows "([x] = rev xs) = (xs = [x])"
rep_eq: fixes x :: "uint8" shows "Rep_uint64 (uint64_of_uint8 x) = ucast (Rep_uint8 x)"
is_empty_def: fixes A :: "'f set" shows "Set.is_empty A = (A = {})"
of_nat_eq_0_iff: fixes m :: "nat" shows "(of_nat m = (0::'a)) = (m = 0)"
IH_2: fixes x :: "'a list list"   and xa :: "'a list"   and xb :: "nat"   and xc :: "'a"   and xd :: "'a list \<times> 'a list \<times> 'a list"   and xe :: "'a list"   and y :: "'a list \<times> 'a list"   and xda :: "'a list"   and ya :: "'a list"   and xdb :: "nat"   and xdc :: "nat" assumes "\<not> length xs \<le> 20"   and "x = chop 5 xs"   and "xa = map slow_median x"   and "xb = ((length xs + 4) div 5 - 1) div 2"   and "xc = mom_select xb xa"   and "xd = partition3 xc xs"   and "(xe, y) = xd"   and "(xda, ya) = y"   and "xdb = length xe"   and "xdc = length xda"   and "\<not> k < xdb"   and "\<not> k < xdb + xdc" shows "T_mom_select (k - xdb - xdc) ya \<le> T'_mom_select (length ya)"
charts_submanifold: shows "c_manifold (charts_submanifold S) k"
lemma_4_13_i: fixes b :: "'a"   and a :: "'a" assumes "b l\<rightarrow> a = a" shows "a = (1::'a) \<or> b = (1::'a)"
sheaf_spec_on_open_is_comm_ring: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring (comm_ring.sheaf_spec R addition multiplication zero unit U) (comm_ring.add_sheaf_spec R addition multiplication zero U) (comm_ring.mult_sheaf_spec R addition multiplication zero U) (comm_ring.zero_sheaf_spec R addition multiplication zero unit U) (comm_ring.one_sheaf_spec R addition multiplication zero unit U)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "local.a_known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some x \<Rightarrow> is_shadow_root_ptr x))"
CONSTRAINT_D: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "CONSTRAINT P x" shows "P x"
abs_summable_in_conv_radius: fixes z :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "ereal (norm z) < conv_radius f" shows "summable (\<lambda>n. norm (f n * z ^ n))"
to_Gromov_completion_def: fixes x :: "'b" shows "to_Gromov_completion x = abs_Gromov_completion (\<lambda>n. x)"
nrm_reverse: fixes a :: "'a"   and x :: "'a"   and y :: "'a" shows "a \<bullet> rot (x - y) = - a \<bullet> rot (y - x)"
alphaAll_sym: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and A :: "('a, 'b, 'varSort, 'var, 'c) qAbs" shows "(\<forall>X'. X #= X' \<longrightarrow> X' #= X) \<and> (\<forall>A'. A $= A' \<longrightarrow> A' $= A)"
simps_8: fixes x :: "'sz"   and y :: "'sz"   and a :: "('sf, 'sc, 'sz) hp" shows "PUrename x y a** = PUrename x y a**"
mulex_on_union: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" assumes "mulex_on P A M N"   and "K \<in> multisets A" shows "mulex_on P A (K + M) (K + N)"
induct: fixes x :: "Vertex \<Rightarrow> Vertex"   and P :: "(Vertex \<Rightarrow> Vertex) \<Rightarrow> bool" assumes "x \<in> complex_rotations"   and "\<And>r. r \<in> simple_rotations \<Longrightarrow> P r"   and "\<And>r s. \<lbrakk>r \<in> simple_rotations; s \<in> complex_rotations; P s\<rbrakk> \<Longrightarrow> P (r \<circ> s)" shows "P x"
subgame_ParityGame: fixes V' :: "'a set" shows "ParityGame (G'.subgame V')"
assms_2: shows "(A, z) \<in> foldSetD D f e"
ax_escape: fixes P :: "'b assn"   and G :: "prog"   and A :: "'b triple set"   and t :: "Term.term"   and Q :: "'b assn" assumes "\<forall>Y s Z. P Y s Z \<longrightarrow> G,A\<turnstile>{\<lambda>Y' s' Z'. (Y', s') = (Y, s)} t\<succ> {\<lambda>Y s Z'. Q Y s Z}" shows "G,A\<turnstile>{P} t\<succ> {Q}"
enat_add_sub_same: fixes a :: "enat"   and b :: "enat" assumes "a \<noteq> \<infinity>" shows "a + b - a = b"
cong: fixes a :: "nat"   and b :: "nat" assumes "[a = b] (mod n)" shows "\<chi>\<^sub>0 a = \<chi>\<^sub>0 b"
invim_of_ideal: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and J :: "'c set set" assumes "Ring R"   and "ideal R I"   and "ideal (R /\<^sub>r I) J" shows "ideal R (rInvim R (R /\<^sub>r I) (pj R I) J)"
inf_zhmultiset_def: fixes A :: "zhmultiset"   and B :: "zhmultiset" shows "inf A B = (if A < B then A else B)"
is_NegChecks_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "is_NegChecks stateful_strand_step = (\<exists>x71 x72 x73. stateful_strand_step = \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>)"
tensor_product_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and MN :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "M \<^bsub>MN\<^esub>\<Otimes>\<^bsub>R\<^esub> N = MN /\<^sub>m (TR\<^bsub>R\<^esub> M N MN)"
unitary_adjoint: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "unitary (adjoint A)"
C_eq_RS3: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "not_MT Cp p" shows "Cp (list2FWpolicy (rm_MT_rules Cp p)) = Cp (list2FWpolicy p)"
discrL_intro: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "\<And>c. c \<in> set cl \<Longrightarrow> discr c" shows "discrL cl"
rep_eq: fixes x :: "real_alg_3" shows "rep_real_alg_3 (tighten_bounds_3 x) = tighten_bounds_2 (rep_real_alg_3 x)"
cUN_csingleton: fixes A :: "'d cset" shows "cUnion (cimage csingle A) = A"
add_pred_entailment: fixes P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and p :: "'BExp" shows "P +\<^sub>\<S> p \<turnstile> P"
delta: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'b" assumes "comm_monoid_set f z"   and "finite S" shows "comm_monoid_set.F f z (\<lambda>k. if k = a then b k else z) S = (if a \<in> S then b a else z)"
simps_2: fixes \<mu> :: "'c" shows "Can \<^bold>\<langle>\<mu>\<^bold>\<rangle> = C.ide \<mu>"
norm_le_l1_cart: fixes x :: "(real, 'a) Finite_Cartesian_Product.vec" shows "norm x \<le> (\<Sum>i\<in>UNIV. \<bar>x $h i\<bar>)"