hcju/isabelleps huggingface dataset:
nospoof: fixes rs :: "'a common_primitive rule list"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option" assumes "simple_ruleset rs" shows "(\<forall>iface\<in>dom ipassmt. nospoof TYPE('b) iface ipassmt rs) = no_spoofing TYPE('b) ipassmt rs"
AxiomC: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term set" assumes "t \<in> M" shows "M \<turnstile>\<^sub>c t"
vconn_implies_hpar_2: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D \<mu> = trg\<^sub>D f"
normal_inv_iff: fixes N :: "'a set" shows "N \<lhd> G = (subgroup N G \<and> (\<forall>x\<in>carrier G. \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N))"
fds_truncate_mult: fixes m :: "nat"   and f :: "'b fds"   and g :: "'b fds" shows "fds_truncate m (fds_truncate m f * fds_truncate m g) = fds_truncate m (f * g)"
simps: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and \<sigma> :: "('sf, 'sc, 'sz) subst" shows "Rsubst (SG, C) \<sigma> = (map (\<lambda>\<phi>. Ssubst \<phi> \<sigma>) SG, Ssubst C \<sigma>)"
cases: fixes x :: "'form \<times> nat list" obtains c :: "'form" where "x = (c__, [])"   | c :: "'form" and uu_ :: "nat" and "is" :: "nat list" where "x = (c, uu_ # is)"
yD_1: shows "root_cond (p2, l2, r2) (real_of_1 (p2, l2, r2))"
image_mset_filter_swap2: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'b"   and D :: "'c multiset" shows "filter_mset Q (image_mset P D) = image_mset P {#C. C \<in># {#C \<in># D. Q (P C)#}#}"
GEM1_def: shows "GEM1 \<equiv> MM \<and> GEM1_axioms"
SpanI: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set ms \<subseteq> V"   and "V \<subseteq> R_scalar_mult.RSpan UNIV smult ms" shows "V = R_scalar_mult.RSpan UNIV smult ms"
row_mat: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "i < nr" shows "Matrix.row (Matrix.mat nr nc f) i = Matrix.vec nc (\<lambda>j. f (i, j))"
fmfilter_alt_defs_2: fixes A :: "'c set" shows "fmdrop_set A = fmfilter (\<lambda>a. a \<notin> A)"
Domain_Id_on: fixes A :: "'b set" shows "r2s (s2r A) = A"
select_convs_1: fixes carrier :: "'c set"   and eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and more :: "'d" shows "(\<sqsubseteq>\<^bsub>\<lparr>carrier = carrier, eq = eq, le = le, \<dots> = more\<rparr>\<^esub>) = le"
maxsimpchain_CConsI: fixes w :: "'a set"   and x :: "'a set"   and xs :: "'a set list" assumes "maxsimp w"   and "maxsimpchain (x # xs)"   and "w \<sim> x" shows "maxsimpchain (w # x # xs)"
prems_2: shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> S # Ss"
evaln_elim_cases_22: fixes G :: "prog"   and s :: "st"   and c1 :: "stmt"   and c2 :: "stmt"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (c1 Finally c2)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains x1 :: "abopt" and s1 :: "st" and s2 :: "state" where "x = \<diamondsuit>" and "s' = (if \<exists>err. x1 = Some (Error err) then (x1, s1) else abupd (abrupt_if (x1 \<noteq> None) x1) s2)" and "G\<turnstile>Norm s \<midarrow>c1\<midarrow>n\<rightarrow> (x1, s1)" and "G\<turnstile>Norm s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2"
x_in_mr_coset: fixes H :: "'a set"   and m :: "'a"   and x :: "'a" assumes "submodule R M H"   and "m \<in> carrier M"   and "x \<in> m \<uplus>\<^bsub>M\<^esub> H" shows "\<exists>h\<in>H. m \<plusminus> h = x"
Fun_comp: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" shows "Fun (local.comp g f) = restrict (Fun g \<circ> Fun f) (Dom f)"
distinct_13: shows "AbsOp \<noteq> numeric_unop.ToIntegerOp"
poly_eval_sum: fixes a :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and I :: "'e set" shows "poly_eval a (sum f I) = (\<Sum>i\<in>I. poly_eval a (f i))"
assms_1: fixes s :: "'a" assumes "s \<in> S" shows "is_orthogonal a s"
min_trms_def: fixes E :: "('a trm \<times> 'a trm) set" shows "min_trms E = {x. \<exists>pair. (pair \<in> E \<and> (\<forall>pair'\<in>E. (snd pair', snd pair) \<notin> trm_ord)) \<and> x = fst pair}"
measurable_inequality_set_1: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. f x \<le> g x} \<in> sets M"
q0: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set todo \<union> set old" shows "q \<noteq> 0"
simps_2: fixes a :: "'a"   and l :: "'a rose_tree list"   and r :: "'a rose_tree list"   and z :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list"   and t :: "'a rose_tree" shows "tree_of_zipper ((a, l, r) # z, t) = tree_of_zipper (z, Tree (a, l @ t # r))"
tllist_ord_bot: fixes xs :: "('c, 'b) tllist" shows "local.tllist_ord (TNil b) xs"
language_ranking_iff: fixes A :: "('d, 'c) nba"   and w :: "'d stream" assumes "finite (nodes A)" shows "(w \<notin> language A) = (\<exists>f. ranking A w f)"
locally_essentially_surjective: fixes a :: "'b"   and b :: "'b"   and g :: "'a" assumes "D.obj a"   and "D.obj b"   and "\<guillemotleft>g : E.G.map\<^sub>0 a \<rightarrow>\<^sub>C E.G.map\<^sub>0 b\<guillemotright>"   and "C.ide g" shows "\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright> \<and> D.ide f \<and> F.right_map f \<cong>\<^sub>C g"
\<tau>mRedT_thread_preserved: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "mthr.silent_moves s s'" shows "(thr s t = None) = (thr s' t = None)"
check_no_overlapI: fixes \<gamma> :: "'d set \<Rightarrow> 'e \<Rightarrow> bool"   and t :: "('d, 'f) flow_entry_match list" assumes "no_overlaps \<gamma> t" shows "check_no_overlap \<gamma> t"
map_sel_1: fixes a :: "'a llist"   and f :: "'a \<Rightarrow> 'b" assumes "\<not> lnull a" shows "lhd (lmap f a) = f (lhd a)"
project_constrains: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'f set"   and F :: "'f program"   and A :: "'d set"   and B :: "'d set" assumes "Extend h" shows "(project h C F \<in> A co B) = (F \<in> C \<inter> extend_set h A co extend_set h B \<and> A \<subseteq> B)"
assms_2: shows "constructive_security real middle sim_outer \<I>_real \<I>_middle \<I>_common bound_outer lossless_outer w2"
adapt_aexp_is_subst: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s" shows "adapt_aexp e s \<sigma>\<^sub>s\<^sub>y\<^sub>m = e \<sigma>"
offset_hmset_of_nat: fixes n :: "nat" shows "offset_hmset (of_nat n) = n"
compatible_up: fixes x :: "'b"   and y :: "'b" shows "compatible (up\<cdot>x) (up\<cdot>y) = compatible x y"
induct: fixes P :: "int poly \<Rightarrow> int poly list \<Rightarrow> bool"   and a0 :: "int poly"   and a1 :: "int poly list" assumes "\<And>f us. \<lbrakk>\<And>x xa xb x2 xc xd xe y. \<lbrakk>x = degree f; xa = x div 2; xb = find_map_filter (\<lambda>u. gcd f (LLL_short_polynomial pl (Suc xa) u)) (\<lambda>f2. let deg = degree f2 in 0 < deg \<and> deg < x) (filter (\<lambda>u. degree u \<le> xa) us); xb = Some x2; xc = f div x2; xd = List.partition (\<lambda>gi. p.dvdm gi xc) us; (xe, y) = xd\<rbrakk> \<Longrightarrow> P xc xe; \<And>x xa xb x2 xc xd xe y. \<lbrakk>x = degree f; xa = x div 2; xb = find_map_filter (\<lambda>u. gcd f (LLL_short_polynomial pl (Suc xa) u)) (\<lambda>f2. let deg = degree f2 in 0 < deg \<and> deg < x) (filter (\<lambda>u. degree u \<le> xa) us); xb = Some x2; xc = f div x2; xd = List.partition (\<lambda>gi. p.dvdm gi xc) us; (xe, y) = xd\<rbrakk> \<Longrightarrow> P x2 y\<rbrakk> \<Longrightarrow> P f us" shows "P a0 a1"
euler_criterion_int: fixes p :: "int"   and a :: "int" assumes "prime p"   and "2 < p" shows "[Legendre a p = a ^ ((nat p - 1) div 2)] (mod p)"
mult_closed: fixes x :: "'a"   and a :: "real" assumes "x \<in> V" shows "a \<cdot> x \<in> V"
rep_eq: fixes x :: "'b fls"   and xa :: "nat" shows "fls_nth (fls_compose_power x xa) = (\<lambda>n. if 0 < xa \<and> int xa dvd n then fls_nth x (n div int xa) else (0::'b))"
hyps: assumes "all_roots_real p" shows "all_roots_real (fcompose p [:a, b:] [:1, 1:])"
soundness: fixes r :: "'a rexp"   and s :: "'a rexp"   and v :: "'a \<Rightarrow> ('b \<times> 'b) set" assumes "lang r = lang s" shows "Relation_Interpretation.rel v r = Relation_Interpretation.rel v s"
right_neutral: fixes a :: "'c" shows "a \<squnion> bot = a"
dinvarC_add_def: shows "dinvarC_add \<equiv> \<lambda>(wl, res). qw.invar wl \<and> mr.invar res"
sset_smerge: fixes ss :: "'c stream stream" shows "sset (smerge ss) = \<Union> (sset ` sset ss)"
basic: fixes t :: "'b set"   and B :: "'b set set" assumes "t \<in> B" shows "t open\<^bsub>topo B\<^esub>"
pure_pure: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog" assumes "h \<turnstile> ok f"   and "pure f h" shows "h \<turnstile> f \<rightarrow>\<^sub>h h"
may_acquire_allD: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and ln :: "'c \<Rightarrow>f nat"   and l :: "'c" assumes "may_acquire_all ls t ln"   and "0 < ln $ l" shows "may_lock (ls $ l) t"
map_filter_def: fixes P :: "'c \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'd option" shows "map_filter P m = (\<lambda>x. if P x then m x else None)"
fmpred_iff: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap" shows "fmpred P m = (\<forall>x y. fmlookup m x = Some y \<longrightarrow> P x y)"
lists_fcs_def: shows "lists_fcs \<equiv> orbits_list (lists_fc_succ M)"
before_in_index1: fixes l :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 0 else 1) = index l x"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "\<epsilon> \<noteq> x21 # x22"
match: fixes L :: "('a, 'b) Term.term literal"   and K :: "('a, 'b) Term.term literal"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>"   and "is_pos L = is_pos K" shows "leq L K"
inline1'_fixp_induct: fixes P :: "(('b, 'call, 'ret) gpv \<Rightarrow> 's \<Rightarrow> ('b \<times> 's + 'call \<times> 'call' \<times> ('ret \<times> 's, 'call', 'ret') rpv \<times> ('b, 'call, 'ret) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. P inline1' \<Longrightarrow> P (\<lambda>gpv s. the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out' rpv'. return_spmf (Inr (out, out', rpv', rpv)))))" shows "P local.inline1'"
all_nth_imp_all_set: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>i<length xs. P (xs ! i)"   and "x \<in> ran\<^sub>l xs" shows "P x"
bounds_3: shows "\<B>\<^sub>u (l ! (j - 1)) = \<B>\<^sub>u (l ! i)"
once_none_always_none_aux: fixes j :: "state stream"   and e :: "((nat \<times> nat) \<times> transition) fset" assumes "\<exists>p r i. j = make_full_observation e None r p i" shows "alw (state_eq None) j"
vec_eq_iff: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(x = y) = (\<forall>i. x $h i = y $h i)"
sdiv_int_div_0: fixes x :: "int" shows "x sdiv 0 = 0"
assms_2: shows "(\<box>(\<lambda>s. P s \<longrightarrow> (\<circle>P) s)) \<sigma>"
bisimSubstOutputPushRes: fixes x :: "name"   and M :: "'a"   and N :: "'a"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> M"   and "x \<sharp> N" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P) \<sim>\<^sub>s M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P"
assms: shows "X \<subseteq> Bossy''.CD_on ds X'"
mono_Span: fixes Us :: "'a list"   and u :: "'a" assumes "set Us \<subseteq> carrier R"   and "u \<in> carrier R" shows "Span K Us \<subseteq> Span K (u # Us)"
inv_all_uppertD: fixes A :: "'a mat"   and j :: "nat"   and i :: "nat" assumes "local.inv_all local.uppert A"   and "j < i"   and "i < n" shows "A $$ (i, j) = (0::'a)"
closure_complement: fixes S :: "'c set" shows "closure (- S) = - interior S"
alw_ev_shift: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs1 :: "'c stream"   and xl :: "'c list" assumes "alw \<phi> xs1" shows "ev (alw \<phi>) (xl @- xs1)"
ex_nonempty_set: fixes P :: "'c \<Rightarrow> bool" assumes "\<exists>a. P a" shows "{x. P x} \<noteq> {}"
nat_pow_eone: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "x [^]\<^bsub>P\<^esub> 1 = x"
less_eq_action: fixes P :: "('a, 'b) action"   and Q :: "('a, 'b) action" shows "P \<le> Q \<equiv> P \<sqsubseteq> Q"
FL_bisimilar_eqvt: fixes F :: "'effect fs_set"   and P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "P \<sim>\<cdot>[F] Q" shows "p \<bullet> P \<sim>\<cdot>[p \<bullet> F] p \<bullet> Q"
set_2: fixes x2 :: "'c" shows "set_option (Some x2) = {x2}"
i: shows "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
gInduct_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gInduct MOD \<equiv> \<forall>phi phiAbs s X us s' A. (\<forall>xs x. phi (asSort xs) (gVar MOD xs x)) \<and> (\<forall>delta inp' inp binp' binp. wlsInp delta inp' \<and> wlsBinp delta binp' \<and> gWlsInp MOD delta inp \<and> gWlsBinp MOD delta binp \<and> liftAll2 phi (arOf delta) inp \<and> liftAll2 phiAbs (barOf delta) binp \<longrightarrow> phi (stOf delta) (gOp MOD delta inp' inp binp' binp)) \<and> (\<forall>xs s x X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<and> phi s X \<longrightarrow> phiAbs (xs, s) (gAbs MOD xs x X' X)) \<longrightarrow> (gWls MOD s X \<longrightarrow> phi s X) \<and> (gWlsAbs MOD (us, s') A \<longrightarrow> phiAbs (us, s') A)"
abst_inv: fixes y :: "'b \<times> 'c" shows "repr isom (abst isom y) = y"
mset_factorsE: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and p :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p" obtains "\<And>f. f \<in># F \<Longrightarrow> comm_semiring_1.irreducible times one zero f" and "F \<noteq> {#}" and "comm_monoid_mult.prod_mset times one F = p"
OUT_a: shows "d_OUT f a < weight (bipartite_web_of \<Gamma>) a"
reify_eqs_5: fixes \<phi> :: "atom fm"   and xs :: "'a list" shows "QEdlo_ex.interpret (Neg \<phi>) xs = (\<not> QEdlo_ex.interpret \<phi> xs)"
self_domain_iff_ide: fixes a :: "(bool, bool list) concrete_category.arr" shows "(a \<in> domains a) = ide a"
not_finite_existsD: fixes P :: "'e \<Rightarrow> bool" assumes "infinite {a. P a}" shows "\<exists>a. P a"
inf_bot_right: fixes x :: "'h multiset" shows "x \<inter># {#} = {#}"
simps: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c :: "com"   and a1 :: "char list \<Rightarrow> int \<Rightarrow> int"   and a2 :: "nat" shows "count_it \<pi> b c a1 a2 = ((\<exists>s. a1 = s \<and> a2 = 0 \<and> \<not> bval b s) \<or> (\<exists>s s' n. a1 = s \<and> a2 = Suc n \<and> bval b s \<and> \<pi>: (c, s) \<Rightarrow> s' \<and> count_it \<pi> b c s' n))"
subsetbdd_set_of_bv: fixes N :: "bool list bdd list \<times> bool list"   and ws :: "bool list"   and q :: "bool list" assumes "wf_nfa N (length ws)"   and "nfa_is_node N q" shows "set_of_bv (bdd_lookup (subsetbdd (fst N) q (nfa_emptybdd (length q))) ws) = (\<Union>i\<in>set_of_bv q. set_of_bv (bdd_lookup (fst N ! i) ws))"
Par2F: fixes Q :: "pi"   and \<alpha> :: "Late_Semantics.freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>l \<alpha> \<prec> Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<alpha> \<prec> P \<parallel> Q'"
p: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "stutter_sampler f \<sigma>"   and "stutter_sampler g \<tau>"   and "\<sigma> \<circ> f = \<tau> \<circ> g" shows "P"
butlast_rest_bin: shows "butlast (bin_to_bl (length bl) (bl_to_bin bl)) = bin_to_bl (length bl - 1) (bl_to_bin bl div 2)"
1: shows "card ((#) B ` {l. length l = n - Suc 0 \<and> valid l}) = lcount (n - 1)"
dvd_abs_iff: 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 m :: "'a"   and k :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "dvd.dvd times m (abs k) = dvd.dvd times m k"
order_iff: fixes a :: "'b upred"   and b :: "'b upred" shows "(b \<sqsubseteq> a) = (a = (a \<and> b))"
subst_Var_IT: fixes r :: "dB"   and i :: "nat"   and j :: "nat" assumes "IT r" shows "IT (r[Var i/j])"
rsa_inv_set_spmf_I: fixes N :: "nat"   and e :: "nat"   and d :: "nat"   and x :: "nat" assumes "((N, e), d) \<in> set_spmf I" shows "(x ^ e mod N) ^ d mod N = x mod N"
fin: fixes i :: "nat" shows "f -` {enat i} \<inter> space M \<in> sets M"
unl_con: fixes c :: "'b"   and w :: "'c" shows "(#c) w \<equiv> c"
cong: fixes x :: "'a"   and x' :: "'a"   and y :: "'b"   and y' :: "'b" assumes "\<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor>"   and "\<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>"   and "P \<lfloor>x\<rfloor> \<lfloor>y\<rfloor>"   and "P \<lfloor>x'\<rfloor> \<lfloor>y'\<rfloor>" shows "g x y = g x' y'"
methods_f_def: fixes P :: "cld list"   and ty :: "ty" shows "methods_f P ty = (case find_path_ty_f P ty of None \<Rightarrow> None | Some ctxclds \<Rightarrow> Some (methods_in_path_f (map (\<lambda>(ctx, cld). cld) ctxclds)))"
norm_triangle_ineq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (plus x y) \<le> norm x + norm y"
vw: shows "adjuster n w us + w \<in> carrier_vec n"
similar_mat_jordan_matrix_smult: fixes k :: "'b"   and n_as :: "(nat \<times> 'b) list" assumes "k \<noteq> (0::'b)" shows "similar_mat (k \<cdot>\<^sub>m jordan_matrix n_as) (jordan_matrix (map (\<lambda>(n, a). (n, k * a)) n_as))"
wf_not_refl: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "wf r" shows "(a, a) \<notin> r"
simps_2: fixes a :: "'c"   and as :: "'c list"   and f :: "nat \<Rightarrow> nat"   and bs :: "'c list" shows "permut_aux (a # as) f bs = bs ! f 0 # permut_aux as (\<lambda>n. f (Suc n)) bs"
slice2_ret_slice2: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice2 n"   and "n s-p\<rightarrow>\<^bsub>ret\<^esub> n'" shows "nx \<in> sum_SDG_slice2 n'"
Int_space_eq1: fixes x :: "'c set"   and M :: "'c measure" assumes "x \<in> sets M" shows "space M \<inter> x = x"
k_def: fixes i :: "nat"   and x :: "real" shows "k i x = insertion (nth_default 0 (L[var := x])) (isolate_variable_sparse p var i)"
assms_1: shows "CT;\<Gamma> \<turnstile>+ es : Cs"
restrict_singleton_list: fixes k :: "'d"   and f :: "'d \<times> 'd \<Rightarrow> 'c"   and ls :: "'d list"   and i :: "'d"   and j :: "'d" shows "([k] \<langle>f\<rangle> ls) (i, j) = (if i = k \<and> List.member ls j then f (i, j) else bot)"
nn_integral_mono_AE: fixes u :: "'h \<Rightarrow> ennreal"   and v :: "'h \<Rightarrow> ennreal"   and M :: "'h measure" assumes "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
unitary_mult: fixes A :: "'b \<Rightarrow> real"   and B :: "'b \<Rightarrow> real" assumes "unitary A"   and "unitary B" shows "unitary (\<lambda>s. A s * B s)"
personal_quorums_def: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool" shows "personal_quorums quorum_of \<equiv> \<forall>Q p p'. quorum_of p Q \<longrightarrow> p' \<in> Q \<longrightarrow> quorum_of p' Q"
cong: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" shows "comm_monoid_mult.Prod_any times one \<equiv> comm_monoid_mult.Prod_any times one"
assms_4: shows "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>t S = {}"
LN_4: fixes \<C> :: "t_context"   and ts :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> lfilledk : [] _> ts"   and "return \<C> = Some tvs" shows "\<exists>lholed' vs \<C>'. Lfilled j lholed' (vs @ [$Return]) lfilledk \<and> \<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs \<and> const_list vs"
oracle2: fixes s1 :: "'s1"   and s2 :: "'s2"   and x :: "'a" assumes "X s1 s2"   and "x \<in> outs_\<I> \<I>" shows "oracle2 s2 x = map_spmf snd (joint_oracle s1 s2 x)"
agree_times1: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<nu>' :: "(real, 'd) vec \<times> (real, 'd) vec"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" assumes "Vagree \<nu> \<nu>' (FVDiff (Times t1 t2))" shows "Vagree \<nu> \<nu>' (FVDiff t1)"
SourcesA4_L0: shows "Sources level0 sA4 = {sA1, sA2, sA3, sA4}"
step_op_ubx: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and opubx :: "'opubx"   and op :: "'op"   and ar :: "nat"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and x :: "('dyn, 'ubx1, 'ubx2) unboxed"   and H :: "'henv"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = IOpUbx opubx"   and "\<DD>\<ee>\<II>\<nn>\<ll> (\<BB>\<oo>\<xx> opubx) = op"   and "\<AA>\<rr>\<ii>\<tt>\<yy> op = ar"   and "ar \<le> length \<Sigma>"   and "\<UU>\<bb>\<xx>\<OO>\<pp> opubx (take ar \<Sigma>) = Some x" shows "Subx.step (State F H (Frame f pc \<Sigma> # st)) (State F H (Frame f (Suc pc) (x # drop ar \<Sigma>) # st))"
assms_4: shows "index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q"
right_diff_distrib: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec"   and z :: "(real, 7) vec" shows "x \<times>\<^sub>7 (y - z) = x \<times>\<^sub>7 y - x \<times>\<^sub>7 z"
is_red_image_monic: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red (monic ` B) p = is_red B p"
group_closure_abs_iff: fixes s :: "int"   and S :: "int set" shows "(\<bar>s\<bar> \<in> group_closure S) = (s \<in> group_closure S)"