hcju/isabelleps huggingface dataset:
plus_dim2: fixes A :: "'b tensor"   and B :: "'b tensor" assumes "dims A = dims B" shows "dims (A + B) = dims B"
assms_2: shows "f \<in> SEQ A \<or> g \<in> SEQ A"
app_css_steps: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and s :: "('d, 'f) xstate"   and cs' :: "('d, 'e, 'f) com list"   and css' :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list"   and t :: "('d, 'f) xstate"   and xs :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>+ (cs', css', t)" shows "\<Gamma>\<turnstile> (cs, css @ xs, s) \<rightarrow>\<^sup>+ (cs', css' @ xs, t)"
local: fixes p :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and \<sigma> :: "nat \<Rightarrow> 'b"   and a :: "'c seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'b"   and q :: "('b, 'c, 'a, 'd) seqp"   and l' :: "'d"   and pp :: "('b, 'c, 'a, 'd) seqp" assumes "p \<in> cterms \<Gamma>"   and "l \<in> labels \<Gamma> p"   and "P (\<sigma>, l)"   and "((\<sigma>, p), a, \<sigma>', q) \<in> oseqp_sos \<Gamma> i"   and "((\<sigma>, p), a, \<sigma>', q) \<in> automaton.trans A"   and "l' \<in> labels \<Gamma> q"   and "(\<sigma>, pp) \<in> oreachable A S U"   and "p \<in> sterms \<Gamma> pp"   and "(\<sigma>', q) \<in> oreachable A S U"   and "S \<sigma> \<sigma>' a" shows "P (\<sigma>', l')"
swap_psubst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "good X"   and "goodEnv rho" shows "X #[rho] #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[rho &[z1 \<and> z2]_zs]"
chop_add_length: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" assumes "R_Chop(r,s,t)" shows "\<parallel>r\<parallel> = \<parallel>s\<parallel> + \<parallel>t\<parallel>"
owner_seskey_secret: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_shakey \<inter> bad_prikey"   and "(Owner n, Crypt (SesK SK) (Pwd n)) \<in> s" shows "SesKey SK \<notin> spied s"
Cauchy_Schwarz_ineq: fixes x :: "'c"   and y :: "'c" shows "\<langle>x, y\<rangle> * \<langle>y, x\<rangle> \<le> \<langle>x, x\<rangle> * \<langle>y, y\<rangle>"
decidable_imp_semidecidable: fixes X :: "nat set" assumes "decidable X" shows "semidecidable X"
surj_on_mult: fixes x :: "nat"   and q :: "nat" assumes "coprime x q"   and "inj_on (\<lambda>b. x * b mod q) {..<q}" shows "(\<lambda>b. x * b mod q) ` {..<q} = {..<q}"
covector_comp_inf: 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 bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "times top z = z" shows "times x (inf y z) = inf (times x y) z"
check_extension_alt_def: fixes i :: "nat"   and A :: "(nat, int mpoly) fmap"   and v :: "nat"   and \<V> :: "nat set"   and p :: "int mpoly" shows "SPEC (\<lambda>b. b \<longrightarrow> i \<notin># dom_m A \<and> v \<notin> \<V>) \<bind> (\<lambda>b. if \<not> b then RETURN False else RETURN (p + Var v) \<bind> (\<lambda>p'. SPEC (\<lambda>b. b \<longrightarrow> vars p' \<subseteq> \<V>) \<bind> (\<lambda>b. if \<not> b then RETURN False else mult_poly_spec p' p' \<bind> (\<lambda>pq. let p' = - p' in add_poly_spec pq p' \<bind> (\<lambda>p. weak_equality p 0 \<bind> (\<lambda>eq. if eq then RETURN True else RETURN False)))))) \<le> check_extension A \<V> i v p"
mru_opt_trans_def: shows "mru_opt_trans = (\<Union>r Q S v. \<Union> (range (opt_mru_round r Q S v)))\<^sup>="
a1: shows "i < dim_row (M\<^sup>\<star> * N\<^sup>\<star>)"
l1_inv4D: fixes x :: "l1_state"   and A :: "agent"   and B :: "agent"   and gnx :: "msg"   and Rb :: "fid_t" assumes "0 < signalsResp x (Commit A B (Exp gnx (NonceF (Rb $ ny))))"   and "x \<in> l1_inv4" shows "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xny, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Rb xgnx = Some gnx"
has_field_sees_aux: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and F :: "char list"   and b :: "staticb"   and T :: "ty" assumes "P \<turnstile> C has_fields FDTs"   and "map_of FDTs (F, C) = \<lfloor>(b, T)\<rfloor>" shows "map_of (map (\<lambda>((F, D), b, T). (F, D, b, T)) FDTs) F = \<lfloor>(C, b, T)\<rfloor>"
gHom: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e"   and y :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> carrier F"   and "y \<in> carrier F" shows "f (x \<cdot>\<^bsub>F\<^esub> y) = f x \<cdot>\<^bsub>G\<^esub> f y"
simple_extension_carrier: fixes x :: "'a" assumes "x \<in> carrier R" shows "simple_extension (carrier R) x = carrier R"
InvMsgInGlob_def: fixes c :: "('a, 'loc \<times> 't, 'b) Exchange.configuration_scheme" shows "cri.InvMsgInGlob c = (\<forall>p q t. c_msg c p q \<noteq> [] \<longrightarrow> t \<in>#\<^sub>z hd (c_msg c p q) \<longrightarrow> (\<exists>t'. t'\<le>\<^sub>pt \<and> 0 < zcount (c_glob c q) t'))"
swap_set_both_in: fixes a :: "atom"   and S :: "atom set"   and b :: "atom" assumes "a \<in> S"   and "b \<in> S" shows "(a \<rightleftharpoons> b) \<bullet> S = S"
irrefl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "ordering less_eq less" shows "\<not> less a a"
gtree_def: fixes g :: "'a growth" shows "gtree g = (case g of Stay t \<Rightarrow> t | Sprout l p r \<Rightarrow> Branch2 l p r)"
join: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and r' :: "'r"   and \<sigma>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau>' :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and v :: "('r, 'l, 'v) val" assumes "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = Some (\<sigma>', \<tau>', VE v)" shows "revision_step r s ((s(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r' := None))"
int_pow_neg: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> - i = inv\<^bsub>G\<^esub> (x [^]\<^bsub>G\<^esub> i)"
assms: shows "exe_sig_conds (ExeSignature cto tao sa)"
inner_eq_abs_times_sgn: fixes b :: "'a"   and u :: "'a" assumes "b \<in> Basis" shows "u \<bullet> b = \<bar>u\<bar> \<bullet> b * sgn (u \<bullet> b)"
filter_mset_cong2: fixes M :: "'c multiset"   and f :: "'c \<Rightarrow> bool"   and g :: "'c \<Rightarrow> bool"   and N :: "'c multiset" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x"   and "M = N" shows "filter_mset f M = filter_mset g N"
rep_eq: fixes x :: "integer" shows "Rep_uint16 (Uint16 x) = word_of_int (int_of_integer x)"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = Matrix.vec n (\<lambda>j. if j = i then 1::'b else (0::'b))"
mult_sgn_dvd_iff: fixes l :: "int"   and r :: "int"   and k :: "int" shows "(l * sgn r dvd k) = (l dvd k \<and> (r = 0 \<longrightarrow> k = 0))"
simps_2: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and s1 :: "'e stream"   and s2 :: "'f stream" shows "stl (smap2 f s1 s2) = smap2 f (stl s1) (stl s2)"
simps_1: fixes b :: "bit" shows "rem_initial b [] = []"
comm_monoid_set_axioms: shows "comm_monoid_set (lift_option (+)) (Some (0::'b))"
ext_inject: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b"   and I' :: "nat"   and M' :: "nat"   and N' :: "nat"   and S' :: "nat"   and more' :: "'b" shows "(\<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = \<lparr>vars.I = I', M = M', N = N', S = S', \<dots> = more'\<rparr>) = (I = I' \<and> M = M' \<and> N = N' \<and> S = S' \<and> more = more')"
geq0': shows "sterm_sem I (f1 fid2 vid1) (sol 0) < sterm_sem I (f1 fid1 vid1) (sol 0)"
simps: fixes a :: "('a, 'b, 'c) exp" shows "is_addr a = (\<exists>aa. a = addr aa)"
vconn_implies_hpar_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.in_hom V \<mu> f g" shows "trg g = trg f"
nocd: fixes l :: "nat" assumes "n \<le> l" shows "\<not> k cd\<^bsup>\<pi>\<^esup>\<rightarrow> l"
closed_others_def: fixes \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set" shows "closed_others \<R> \<equiv> \<forall>c\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C mem\<^sub>C'. (\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>x. mem\<^sub>C x \<noteq> mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<forall>x. dma\<^sub>C mem\<^sub>C x \<noteq> dma\<^sub>C mem\<^sub>C' x \<longrightarrow> \<not> var_asm_not_written mds\<^sub>C x) \<longrightarrow> (\<langle>c\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>"
simps: fixes a :: "A" shows "wrap\<cdot>(B\<cdot>a) = a"
semIntAll_prFreshAll: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prFreshAll (semInt SEM) (semIntAbs SEM) SEM"
rev_nth: fixes n :: "nat"   and xs :: "'f list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
digraph_isomorphism_subgraphI: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('a, 'b) pre_digraph" assumes "digraph_isomorphism hom"   and "subgraph H G" shows "pre_digraph.digraph_isomorphism H hom"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (\<tau>\<^sub>0 a)"
scaleR_sum_list: fixes a :: "real"   and xs :: "'b list" shows "a *\<^sub>R sum_list xs = sum_list (map ((*\<^sub>R) a) xs)"
bet__obtuse: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "A \<noteq> B"   and "B \<noteq> C" shows "Obtuse A B C"
law: fixes X :: "'a set" assumes "P X" shows "f (g X) = h (f X) \<and> P (g X)"
wellformed_token_def: fixes t :: "symbol \<times> character list" shows "wellformed_token t = is_terminal (terminal_of_token t)"
map_of_def: fixes lookup :: "'b \<Rightarrow> ('c, 'a option) state"   and heap :: "'c"   and k :: "'b" shows "state_mem_defs.map_of lookup heap k = fst (run_state (lookup k) heap)"
eq_implies_le: fixes \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" assumes "\<rho> \<approx> \<rho>'" shows "\<rho> \<sqsubseteq> \<rho>'"
simps_3: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and s :: "'s"   and s' :: "'s"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Trans p s s' \<mapsto> c' = (can_occur (Trans p s s') c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> (\<forall>i. msgs c' i = msgs c i) \<and> no_snapshot_change c c')"
cases: fixes x :: "'c list" obtains "x = []"   | v :: "'c" and va :: "'c list" where "x = v # va"
inject_2: fixes x2 :: "Secrets"   and y2 :: "Secrets" shows "(sE x2 = sE y2) = (x2 = y2)"
norm: shows "\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p}"
MP: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile>\<^sub>H F"   and "\<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G" shows "\<Gamma> \<turnstile>\<^sub>H G"
CFG_SSA_wf_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and phis :: "'d \<Rightarrow> 'e \<times> 'g \<Rightarrow> 'g list option" shows "CFG_SSA_wf_axioms \<alpha>n invar inEdges' Entry defs uses phis \<equiv> (\<forall>v g n. v \<in> CFG_SSA_base.allUses \<alpha>n inEdges' uses phis g n \<longrightarrow> n \<in> set (\<alpha>n g) \<longrightarrow> CFG_SSA_base.defAss \<alpha>n invar inEdges' Entry defs phis g n v) \<and> (\<forall>g v. phis g (Entry g, v) = None)"
dim_gen_eigenspace: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and ev :: "'b"   and k :: "nat" assumes "jordan_nf A n_as" shows "dim_gen_eigenspace A ev k = sum_list (map (min k) (map fst (filter (\<lambda>(n, e). e = ev) n_as)))"
leftderives1_implies_derives1: fixes u :: "symbol list"   and v :: "symbol list" assumes "leftderives1 u v" shows "derives1 u v"
S_def: shows "S = {x. \<exists>N. \<forall>n\<ge>N. x < X n}"
float_plus_up: fixes x :: "float"   and y :: "float"   and prec :: "nat" shows "real_of_float x + real_of_float y \<le> real_of_float (float_plus_up prec x y)"
map_id: fixes p :: "'g spmf" shows "map_spmf id p = p"
compose_conv': fixes xs :: "('d \<times> 'f) list"   and ys :: "('f \<times> 'e) list" shows "map_of (compose xs ys) = map_of ys \<circ>\<^sub>m map_of xs"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (logical_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (logical_instr_sub1 instr_name result s2))"
f: shows "f = (\<lambda>stp. (abc_steps_l (length fap, xs @ x # rsx # anything) (fap @ B) stp, length fap, arity))"
n_def: shows "n = find_exponent p (2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound gs))"
merge_length: fixes a :: "'b option list"   and n :: "nat"   and b :: "'b option list" assumes "length a = n"   and "length b = n" shows "length (New_max.merge a b) = n"
xs: shows "(xs, X) \<in> clw_rel (\<langle>A\<rangle>ivl_rel)"
elims: fixes x :: "'b run"   and xa :: "clock"   and xb :: "nat"   and y :: "nat" assumes "#\<^sub>\<le> x xa xb = y" obtains \<rho> :: "'b run" and K :: "clock" where "x = \<rho>__" and "xa = K__" and "xb = 0" and "y = (if hamlet (Rep_run \<rho>__ 0 K__) then 1 else 0)"   | \<rho> :: "'b run" and K :: "clock" and n :: "nat" where "x = \<rho>" and "xa = K" and "xb = Suc n" and "y = (if hamlet (Rep_run \<rho> (Suc n) K) then 1 + #\<^sub>\<le> \<rho> K n else #\<^sub>\<le> \<rho> K n)"
00: shows "bprv (PPf (encPf prf) \<langle>\<phi>R\<rangle>)"
zip_eq_ConsE: fixes xs :: "'c list"   and ys :: "'d list"   and xy :: "'c \<times> 'd"   and xys :: "('c \<times> 'd) list" assumes "xs || ys = xy # xys" obtains x :: "'c" and xs' :: "'c list" and y :: "'d" and ys' :: "'d list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = xs' || ys'"
lookup_keys: fixes t :: "('a, 'b) RBT.rbt" shows "dom (lookup t) = set (keys t)"
TLA2: fixes P :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> $P \<and> P$ \<longrightarrow> A" shows "\<turnstile> \<box>P \<longrightarrow> \<box>A"
ideal_prod_eq_genideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "I \<cdot> J = Idl (I <#> J)"
pos_le_neg_iff: fixes A :: "'a"   and B :: "'a" shows "(Pos A \<le> Neg B) = (A \<le> B)"
tl_list_of: fixes xs :: "'c llist" assumes "lfinite xs" shows "tl (list_of xs) = list_of (ltl xs)"
cases: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" assumes "Lfilled_exact a1 a2 a3 a4" obtains lholed :: "Lholed" and es :: "e list" where "a1 = 0" and "a2 = lholed__" and "a3 = esa__" and "a4 = esa__" and "lholed__ = LBase [] []"   | vs :: "e list" and lholed :: "Lholed" and n :: "nat" and es' :: "e list" and l :: "Lholed" and es'' :: "e list" and k :: "nat" and es :: "e list" and lfilledk :: "e list" where "a1 = k + 1" and "a2 = lholed" and "a3 = es" and "a4 = vs @ [Label n es' lfilledk] @ es''" and "const_list vs" and "lholed = LRec vs n es' l es''" and "Lfilled_exact k l es lfilledk"
assms_2: shows "ys \<noteq> [] \<longrightarrow> last xs = hd ys"
Euclidean_dist_upper: fixes i :: "'b"   and x :: "'b"   and y :: "'b" assumes "i \<in> Basis" shows "dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
Im_eq_0: fixes z :: "complex" assumes "\<bar>Re z\<bar> = cmod z" shows "Im z = 0"
convex_hull_linear_image: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "linear f" shows "f ` (convex hull s) = convex hull f ` s"
filter_exist: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and P :: "(nat \<Rightarrow> int) \<Rightarrow> bool"   and n :: "nat"   and m :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "\<forall>n. \<exists>m>n. P (s m)"   and "n \<le> m" shows "P (s (seq_filter s P n))"
project_set_ivl_def: fixes X :: "'b set"   and b :: "'b"   and y :: "real" shows "project_set_ivl X b y = CHECK (\<lambda>_. ()) (b \<in> set Basis_list \<or> - b \<in> set Basis_list) \<bind> (\<lambda>_. ivl_rep X \<bind> (\<lambda>(i, s). RETURN (op_atLeastAtMost_ivl (i + (y - i \<bullet> b) *\<^sub>R b) (s + (y - s \<bullet> b) *\<^sub>R b) ::: lvivl_rel)))"
dgrad_p_set_leI: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "\<And>f. f \<in> F \<Longrightarrow> dgrad_p_set_le d {f} G" shows "dgrad_p_set_le d F G"
upd_def: fixes G :: "Var \<Rightarrow> L"   and x :: "Var"   and p :: "L" shows "upd G x p = (\<lambda>y. if x = y then p else G y)"
mk_eqcl_snd_nth: fixes i :: "nat"   and zs :: "nat list"   and xs :: "nat option list"   and j :: "nat"   and T :: "bool list list" assumes "i < length zs" shows "snd (mk_eqcl xs zs j T) ! i = zs ! i"
minus: fixes x :: "complex"   and xa :: "'b" shows "- x *\<^sub>C xa = - (x *\<^sub>C xa)"
assms: shows "filter is_mw_Mark (sys_mem_store_buffers p s) = []"
assms_2: assumes "\<paragraph>''invar-initial''" shows "I s\<^sub>0"
cong_gcd_eq: fixes a :: "'b"   and b :: "'b"   and m :: "'b" assumes "[a = b] (mod m)" shows "gcd a m = gcd b m"
AndL_inv: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F \<^bold>\<and> G, \<Gamma> \<Rightarrow> \<Delta>" shows "F, G, \<Gamma> \<Rightarrow> \<Delta>"
rel_interior_UNIV: shows "rel_interior UNIV = UNIV"
maxsimpchain_append_reduce2: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "maxsimpchain (xs @ ys)" shows "maxsimpchain ys"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'c list" assumes "i < |xs|" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
integrable_const: fixes M :: "'a pmf"   and a :: "'b" shows "integrable (measure_pmf M) (\<lambda>x. a)"
eq_\<I>_gpv_mono': fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and \<I> :: "('j, 'k) \<I>"   and gpv :: "('h, 'j, 'k) gpv"   and gpv' :: "('i, 'j, 'k) gpv"   and A' :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and \<I>' :: "('j, 'k) \<I>" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "\<And>x y. A x y \<Longrightarrow> A' x y"   and "\<I> \<le> \<I>'" shows "eq_\<I>_gpv A' \<I>' gpv gpv'"
necessitation_averse_axiom_instance: fixes \<phi> :: "\<o>" assumes "[\<phi>]" shows "[\<phi> in dw]"
check_single_imp_weakening: fixes \<C> :: "t_context"   and e :: "b_e"   and t1s :: "t list"   and ctm :: "checker_type"   and ctn :: "checker_type"   and t2s :: "t list" assumes "check_single \<C> e (Type t1s) = ctm"   and "ctm \<noteq> Bot"   and "c_types_agree ctn t1s"   and "c_types_agree ctm t2s" shows "\<exists>ctm'. check_single \<C> e ctn = ctm' \<and> c_types_agree ctm' t2s"
in_fold_Deriv: fixes v :: "'a list"   and w :: "'a list"   and L :: "'a list set" shows "(v \<in> fold Deriv w L) = (w @ v \<in> L)"
u_5: fixes x :: "'a" shows "(SUP i. u i x) = f x"
Square_spec: shows "\<forall>n. \<Gamma>\<turnstile> \<lbrace>\<acute>N = n\<rbrace> \<acute>R :== PROC Square(\<acute>N) \<lbrace>\<acute>R = n * n\<rbrace>"
blinfun_compose_zero_1: shows "(o\<^sub>L) 0 = (\<lambda>_. 0)"
dvd_minus_iff: fixes 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 x :: "'b"   and y :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "dvd.dvd times x (uminus y) = dvd.dvd times x y"
obsE: fixes ns' :: "'node list"   and ns :: "'node list"   and S :: "'node set" assumes "ns' \<in> obs ns S"   and "Ball (set (tl ns)) return_node" obtains nsx :: "'node list" and n :: "'node" and nsx' :: "'node list" and n' :: "'node" where "ns = nsx @ n # nsx'" and "ns' = n' # nsx'" and "n' \<in> obs_intra n S" and "\<forall>nx\<in>set nsx'. \<exists>nx'. call_of_return_node nx nx' \<and> nx' \<in> S" and "\<forall>xs x xs'. nsx = xs @ x # xs' \<and> obs_intra x S \<noteq> {} \<longrightarrow> (\<exists>x''\<in>set (xs' @ [n]). \<exists>nx. call_of_return_node x'' nx \<and> nx \<notin> S)"
huets_us_def: fixes y :: "nat list"   and i :: "nat" shows "\<^bold>u y i = (inc y 0 ^^ Suc i) (zeroes (length y))"
LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
S: shows "M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<i]) \<in> carrier_vec n"
prod_list_replicate: fixes n :: "nat"   and a :: "'b" shows "prod_list (replicate n a) = a ^ n"
seqsim_empty_iff_empty: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t" shows "emptyseq s = emptyseq t"
perp_col0: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A B Perp C D"   and "X \<noteq> Y"   and "Col A B X"   and "Col A B Y" shows "C D Perp X Y"
aform_err_acc_err_leI: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and err :: "real"   and fx :: "real"   and p :: "nat" assumes "aform_val e (fst X) - (snd X + err) \<le> fx"   and "fx \<le> aform_val e (fst X) + (snd X + err)" shows "fx \<in> aform_err e (acc_err p X err)"
set_spmf_spmf_of_pmf: fixes p :: "'c pmf" shows "set_spmf (spmf_of_pmf p) = set_pmf p"
IH: fixes Ks :: "('a, 'c) Term.term literal list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "subsumes_list Ls (filter P Ks) \<sigma>" shows "subsumes_list Ls Ks \<sigma>"
cnvgnt_def: fixes T :: "'b set set"   and F :: "'b set set" shows "(F convergent\<^bsub>T\<^esub>) = (\<exists>x\<in>carrier\<^bsub>T\<^esub>. F \<longlongrightarrow>\<^bsub>T\<^esub> x)"
simps: fixes p :: "'b \<Rightarrow> bool"   and ws :: "'b list"   and uu :: "'b set" shows "While_Combinator.rtrancl_while_test p (ws, uu) = (ws \<noteq> [] \<and> p (hd ws))"
sset_range: fixes s :: "'c stream" shows "sset s = range ((!!) s)"
card_character_extensions: fixes H :: "'a set"   and \<chi> :: "'a \<Rightarrow> complex" assumes "subgroup H G"   and "character (G\<lparr>carrier := H\<rparr>) \<chi>" shows "card {\<chi>'' \<in> characters G. \<forall>x\<in>H. \<chi>'' x = \<chi> x} * card H = order G"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<Z> \<alpha>"