hcju/isabelleps huggingface dataset:
prod_list_complementary_nthss: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and A :: "nat set" shows "prod_list (map f xs) = prod_list (map f (nths xs A)) * prod_list (map f (nths xs (- A)))"
sorted_nth_mono: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sorted xs"   and "i \<le> j"   and "j < length xs" shows "xs ! i \<le> xs ! j"
F1set1_IF1set: fixes x :: "('d, 'd IF1, 'd IF2) F1" shows "F1set1 x \<subseteq> IF1set (ctor1 x)"
coinduct_strong: fixes R :: "('f, 'g, 'h) Resumption.resumption \<Rightarrow> ('f, 'g, 'h) Resumption.resumption \<Rightarrow> bool"   and resumption :: "('f, 'g, 'h) Resumption.resumption"   and resumption' :: "('f, 'g, 'h) Resumption.resumption" assumes "R resumption resumption'"   and "\<And>resumption resumption'. R resumption resumption' \<Longrightarrow> Resumption.resumption.is_Done resumption = Resumption.resumption.is_Done resumption' \<and> (Resumption.resumption.is_Done resumption \<longrightarrow> Resumption.resumption.is_Done resumption' \<longrightarrow> Resumption.resumption.result resumption = Resumption.resumption.result resumption') \<and> (\<not> Resumption.resumption.is_Done resumption \<longrightarrow> \<not> Resumption.resumption.is_Done resumption' \<longrightarrow> Resumption.resumption.output resumption = Resumption.resumption.output resumption' \<and> rel_fun (=) (\<lambda>resumption resumption'. R resumption resumption' \<or> resumption = resumption') (Resumption.resumption.resume resumption) (Resumption.resumption.resume resumption'))" shows "resumption = resumption'"
simps_4: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'f list \<Rightarrow> 'e" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
Inl_def: shows "Inl = Abs_sum \<circ> Inl_Rep"
antipar_1: shows "trg t\<^sub>0\<^sup>* = src t\<^sub>0"
biQuot_rQuot_lQuot: fixes x :: "'b"   and y :: "'b"   and A :: "'b list set" shows "biQuot x y A = rQuot y (lQuot x A)"
filterlim_split_at: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and x :: "'a" assumes "filterlim f F (at_left x)"   and "filterlim f F (at_right x)" shows "filterlim f F (at x)"
bot_uexpr_rep_eq: fixes b :: "'a" shows "\<lbrakk>\<bottom>\<rbrakk>\<^sub>e b = True"
cmp_extend_def: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" shows "cmp_extend D cmp \<equiv> cmp_combine D cmp UNIV univ_cmp"
separating_insert2: fixes a :: "'a"   and V :: "'a set"   and F :: "'a \<Rightarrow> 'b set"   and v :: "'a" assumes "separating (insert a V) F"   and "a \<notin> V"   and "v \<in> V" shows "F a \<inter> F v = {}"
lsorted_ltlI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist" assumes "ord.lsorted less_eq xs" shows "ord.lsorted less_eq (ltl xs)"
Par_ZObis_discrL2: fixes c2 :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and c1 :: "('test, 'atom) com" assumes "c2 \<approx>01 d"   and "discr c1" shows "c1 | c2 \<approx>01 d"
assms_1: shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
dvd_div_eq_cancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "divide a c = divide b c"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "a = b"
less_le_not_le: fixes x :: "nat list"   and y :: "nat list" shows "(x <\<^sub>v y) = (x \<le>\<^sub>v y \<and> \<not> y \<le>\<^sub>v x)"
map_transfer: fixes Rb :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and Sd :: "'d \<Rightarrow> 'f \<Rightarrow> bool" shows "((Rb ===> Sd) ===> llist_all2 Rb ===> llist_all2 Sd) lmap lmap"
REFPHI: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma>' :: "'\<sigma>a"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "\<forall>x\<in>S - it. \<forall>y\<in>it. RR x y"   and "\<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R" shows "\<Phi> it \<sigma>"
P'RelQ': shows "(\<Psi>, P', rev p \<bullet> Q') \<in> Rel"
test_inf_semicommutative: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "x \<sqinter> y \<le> y \<sqinter> x"
assms: fixes i :: "'a" shows "continuous_on UNIV (f i)"
T2_axioms_def: fixes T :: "'b set set" shows "T2_axioms T \<equiv> \<forall>x\<in>carrier\<^bsub>T\<^esub>. \<forall>y\<in>carrier\<^bsub>T\<^esub>. x \<noteq> y \<longrightarrow> (\<exists>u\<in>nhds\<^bsub>T\<^esub> x. \<exists>v\<in>nhds\<^bsub>T\<^esub> y. u \<inter> v = {})"
cases: fixes x :: "'a interval \<times> 'a interval" obtains lx :: "'a" and ux :: "'a" and ly :: "'a" and uy :: "'a" where "x = (Interval lx ux, Interval ly uy)"
lookup_pair_filter: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and P :: "'a \<times> 'c \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (filter P xs) k = (let v = lookup_pair x xs k in if P (k, v) then v else (0::'c))"
find_non0_2: fixes l :: "nat"   and A :: "'a mat"   and m :: "nat" assumes "det_selection_fun sel_fun"   and "local.find_non0 l A = Some m" shows "l < m"
cnj_of_real: fixes x :: "real" shows "Octonions.cnj (octo_of_real x) = octo_of_real x"
dist_triangle4: fixes x :: "'b"   and t :: "'b"   and y :: "'b"   and z :: "'b" shows "dist x t \<le> dist x y + dist y z + dist z t"
substAbs_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodAbs A" shows "goodAbs (A $[X / x]_xs)"
path_connected_space_iff_components_eq: fixes X :: "'b topology" shows "path_connected_space X = (\<forall>C\<in>path_components_of X. \<forall>C'\<in>path_components_of X. C = C')"
continuous_on: fixes s :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. T (g x))"
landau_pair_axioms: shows "landau_pair bigo smallo bigo smallo bigo smallo (\<le>)"
\<psi>_\<phi>: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "Adj.D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<psi> x (\<phi> y f) = f"
msubsteq: fixes a :: "poly"   and r :: "tm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list"   and c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm" assumes "islin (Eq (CNP 0 a r))" shows "Ifm vs (x # bs) (msubsteq c t d s a r) = Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (2::'b) # bs) (Eq (CNP 0 a r))"
assms_1: fixes x :: "nat" assumes "lev \<le> x" shows "\<not> loose_bvar1 t x"
incoming_def: fixes G :: "('k, 'l) graph_scheme"   and y :: "'k" shows "\<^bold>I\<^bold>N\<^bsub>G\<^esub> y = {x. (x, y) \<in> \<^bold>E\<^bsub>G\<^esub>}"
inject: fixes x1 :: "'d option"   and x2 :: "('c \<times> ('c, 'd) trie) list"   and y1 :: "'d option"   and y2 :: "('c \<times> ('c, 'd) trie) list" shows "(Trie x1 x2 = Trie y1 y2) = (x1 = y1 \<and> x2 = y2)"
Ch_unilateral_substitutes: shows "\<forall>h. unilateral_substitutes_on UNIV (Ch h)"
action_proj_def: fixes a :: "('e, 'f) fmap \<times> ('e, 'g) fmap"   and vs :: "'e set" shows "action_proj a vs \<equiv> (fmrestrict_set vs (fst a), fmrestrict_set vs (snd a))"
locks_ticket_simps_2: fixes k :: "nat"   and k' :: "nat" shows "\<not> locks_ticket (WAIT k) k'"
Dag_unique: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t1 :: "dag"   and t2 :: "dag" assumes "Dag p l r t1"   and "Dag p l r t2" shows "t1 = t2"
pgen_lossless_gpv_Done: fixes fail :: "ennreal"   and \<I> :: "('m, 'n) \<I>"   and x :: "'o" shows "pgen_lossless_gpv fail \<I> (Done x)"
gt_tl_def: fixes ys :: "'a list"   and xs :: "'a list" shows "gt_tl ys xs = ext gt (tl ys) (tl xs)"
assms_2: fixes x :: "'a" assumes "x \<in> topspace X" shows "f x = g x"
mset_neg_minus: fixes Mp :: "'a multiset"   and Mn :: "'a multiset" shows "mset_neg (abs_zmultiset (Mp, Mn)) = Mn - Mp"
map_of_is_SomeI: fixes xys :: "('k \<times> 'l) list"   and x :: "'k"   and y :: "'l" assumes "distinct (map fst xys)"   and "(x, y) \<in> set xys" shows "map_of xys x = Some y"
compose_eq: fixes x :: "'b"   and A :: "'b set"   and g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd" assumes "x \<in> A" shows "(g \<circ> f \<down> A) x = g (f x)"
wlsAll_implies_goodAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and s :: "'e"   and X :: "('b, 'c, 'd, 'a, 'f) qTerm set"   and xs :: "'d"   and s' :: "'e"   and A :: "('b, 'c, 'd, 'a, 'f) qAbs set"   and delta :: "'f"   and inp :: "'b \<Rightarrow> ('b, 'c, 'd, 'a, 'f) qTerm set option"   and binp :: "'c \<Rightarrow> ('b, 'c, 'd, 'a, 'f) qAbs set option" assumes "FixSyn TYPE('a) Delta" shows "(FixSyn.wls Delta s X \<longrightarrow> FixVars.good X) \<and> (FixSyn.wlsAbs Delta (xs, s') A \<longrightarrow> FixVars.goodAbs A) \<and> (FixSyn.wlsInp Delta delta inp \<longrightarrow> FixVars.goodInp inp) \<and> (FixSyn.wlsBinp Delta delta binp \<longrightarrow> FixVars.goodBinp binp)"
exists_least_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m. less m n \<longrightarrow> \<not> P m))"
xor_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a XOR b) XOR c = a XOR b XOR c"
has_integral_setdiff: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and S :: "'a set"   and j :: "'b"   and T :: "'a set" assumes "(f has_integral i) S"   and "(f has_integral j) T"   and "negligible (T - S)" shows "(f has_integral i - j) (S - T)"
gEQAddcross: fixes A :: "('c, 'd) aGroup_scheme"   and l1 :: "'c"   and l2 :: "'c"   and r1 :: "'c"   and r2 :: "'c" assumes "aGroup A"   and "l1 \<in> carrier A"   and "l2 \<in> carrier A"   and "r1 \<in> carrier A"   and "r1 \<in> carrier A"   and "l1 = r2"   and "l2 = r1" shows "l1 \<plusminus>\<^bsub>A\<^esub> l2 = r1 \<plusminus>\<^bsub>A\<^esub> r2"
ltl_prop_equiv_quotient_equivp: shows "equivp (\<equiv>\<^sub>P)"
icard_image_le: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "icard (f ` A) \<le> icard A"
gbinomial_mult_fact': fixes a :: "'a"   and k :: "nat" shows "(a gchoose k) * fact k = (\<Prod>i = 0..<k. a - of_nat i)"
delete_def: fixes x :: "'b"   and t :: "'b tree23" shows "delete x t = treeD (del x t)"
ep_nb: shows "bound0 (evaldjf (\<upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))) (remdups (map simp_num_pair (map (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))))))))"
subfieldE_3: fixes K :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subfield K R" shows "K \<subseteq> carrier R"
empty_Q0_RP: shows "Q_of_state (lhd (lmap state_of_wstate Sts)) = {}"
Icc_subset_Ici_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and h :: "'b"   and l' :: "'b" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq l h \<subseteq> ord.atLeast less_eq l') = (\<not> less_eq l h \<or> less_eq l' l)"
AS: fixes p :: "('a, 'b) fm"   and q :: "('a, 'b) fm"   and r :: "('a, 'b) fm" shows "\<turnstile> (p \<^bold>\<longrightarrow> q \<^bold>\<longrightarrow> r) \<^bold>\<longrightarrow> (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> r"
Fvars_SS: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "yy \<notin> FvarsT t1" shows "Fvars (SS t1 t2) = FvarsT t1 \<union> FvarsT t2"
dests_inv: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PAodv-:15, PPkt-:7, PRreq-:11, PRreq-:24, PRrep-:10, PRerr-:1} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> vD (rt \<xi>))) \<and> (l \<in> {PAodv-:16..PAodv-:19} \<union> {PPkt-:8..PPkt-:11} \<union> {PRreq-:12..PRreq-:15} \<union> {PRreq-:25..PRreq-:28} \<union> {PRrep-:11..PRrep-:14} \<union> {PRerr-:2..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> iD (rt \<xi>) \<and> the (dests \<xi> ip) = sqn (rt \<xi>) ip)) \<and> (l = PPkt-:14 \<longrightarrow> dip \<xi> \<in> iD (rt \<xi>)))"
log: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Log], Undecided\<rangle> \<Rightarrow> Undecided"
count_space_PiM_finite: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "finite A"   and "\<And>i. countable (B i)" shows "Pi\<^sub>M A (\<lambda>i. count_space (B i)) = count_space (Pi\<^sub>E A B)"
lset_ltakeWhileD: fixes x :: "'c"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" assumes "x \<in> lset (ltakeWhile P xs)" shows "x \<in> lset xs \<and> P x"
Vf: fixes w :: "'a" assumes "w \<in> A" shows "open (Vf w) \<and> w \<in> Vf w \<and> S \<inter> Vf w \<subseteq> - B \<and> (\<forall>e>0. \<exists>f\<in>R. f ` S \<subseteq> {0..1} \<and> (\<forall>x\<in>S \<inter> Vf w. f x < e) \<and> (\<forall>x\<in>S \<inter> B. 1 - e < f x))"
nat_pow_one: fixes n :: "nat" shows "\<one> [^] n = \<one>"
distinct_keys: fixes rbt :: "'c set_rbt" assumes "ID ccompare \<noteq> None" shows "distinct (RBT_Set2.keys rbt)"
subst_Var_NF: fixes t :: "dB"   and i :: "nat"   and j :: "nat" assumes "NF t" shows "NF (t[Var i/j])"
real_root_zero: fixes n :: "nat" shows "root n 0 = 0"
degree_m_eq_lead_coeff: fixes f :: "int poly" assumes "degree_m f = degree f" shows "lead_coeff (Mp f) = M (lead_coeff f)"
evaln_CompI: fixes G :: "prog"   and s :: "state"   and c1 :: "stmt"   and n :: "nat"   and s1 :: "state"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>s \<midarrow>c1\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>s \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
continuous_map_on_empty: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "topspace X = {}" shows "continuous_map X Y f"
ide_compE: fixes g :: "'a"   and f :: "'a" assumes "C.ide (C g f)" obtains "C.seq g f" and "C.seq f g" and "C g f = C.dom f" and "C g f = C.cod g"
square_part_unit: fixes x :: "'b" assumes "is_unit x" shows "square_part x = (1::'b)"
ccINF_mono: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "countable A"   and "countable B"   and "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<le> g m" shows "Inf (f ` A) \<le> Inf (g ` B)"
norm_suminf_le: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> real" assumes "\<And>n. norm (f n) \<le> g n"   and "summable g" shows "norm (suminf f) \<le> suminf g"
ESem_ignores_fresh_restr: fixes ESem :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<rightarrow> 'f"   and S :: "'e set"   and e :: "'d"   and \<rho> :: "'e \<Rightarrow> 'f" assumes "has_ignore_fresh_ESem ESem"   and "atom ` S \<sharp>* e" shows "ESem e\<cdot>\<rho> = ESem e\<cdot>(\<rho> f|` (- S))"
emb_equiv: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "(t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s) = (\<exists>ps. emb_at ps t = s)"
lset_lmember: fixes x :: "'a"   and xs :: "'a llist" shows "(x \<in> lset xs) = lmember x xs"
simps: fixes f :: "nat \<Rightarrow> ('b, 'c) fm \<Rightarrow> ('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and i :: "'c" shows "mapi_block f (ps, i) = (mapi f ps, i)"
rel_optionT_def: shows "rel_optionT = rel_optionT' \<top>"
surj_iff_all: fixes f :: "'d \<Rightarrow> 'c" shows "surj f = (\<forall>x. f (inv f x) = x)"
cut_le_subset: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t \<subseteq> I"
iter_tensor_of_H_fst_neg: fixes n :: "nat"   and i :: "nat"   and j :: "nat" assumes "2 ^ n \<le> i \<and> 2 ^ n \<le> j"   and "i < 2 ^ (n + 1) \<and> j < 2 ^ (n + 1)" shows "Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>Suc n\<^esub>  j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (- 1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)"
holf: fixes n :: "nat" shows "\<F> n holomorphic_on S"
inv_cancelR_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units R"   and "y \<in> carrier R"   and "z \<in> carrier R"   and "y = x \<otimes> z" shows "y \<otimes> inv x = z"
zero_le_arctan_iff: fixes x :: "real" shows "(0 \<le> arctan x) = (0 \<le> x)"
small1_big_continue: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and cs' :: "com \<times> (char list \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int" assumes "cs \<rightarrow> cs'"   and "cs' \<Rightarrow> t" shows "cs \<Rightarrow> t"
direct_arc_swapped: fixes u :: "'a"   and v :: "'a" shows "direct_arc (u, v) = direct_arc (v, u)"
simps_7: fixes v :: "'c rexp" shows "toplevel_summands (Star v) = {Star v}"
osum_minus_Id1: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r \<subseteq> Id" shows "r +o r' - Id \<subseteq> Inl ` Field r \<times> Inr ` Field r' \<union> map_prod Inr Inr ` (r' - Id)"
TESL_interp_homo_append: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<inter> \<lbrakk>\<lbrakk> \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
order_N_def: fixes a :: "'a N"   and b :: "'a N" shows "a \<le> b \<equiv> a l\<rightarrow> b = 1"
subsetI: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. s2p A x \<Longrightarrow> s2p B x" shows "A \<subseteq> B"
sum_inverse_squares_remainder_bound: fixes N :: "nat" assumes "0 < N" shows "\<exists>C. \<forall>n\<ge>1. norm (EM_remainder (2 * N + 1) (\<lambda>x. - fact (2 * N + 2) / x ^ (2 * N + 3)) (int n)) \<le> C / real n ^ (2 * N + 2)"
Tree\<^sub>\<alpha>_rep_abs: fixes t :: "('b, 'c, 'd) Tree" shows "rep_Tree\<^sub>\<alpha> (abs_Tree\<^sub>\<alpha> t) =\<^sub>\<alpha> t"
lang_INTERSECT: fixes n :: "nat"   and xs :: "'b rexp list" shows "lang n (INTERSECT xs) = (if xs = [] then lists (\<Sigma> n) else \<Inter> (lang n ` set xs))"
image_uminus_atLeast: fixes x :: "'b" shows "uminus ` {x..} = {..- x}"
Rep_CoCall: fixes x :: "CoCalls" shows "Rep_CoCall x \<in> {G. sym G}"
simps: fixes t :: "'c" shows "convert_term t = term_cases const free (\<lambda>t u. app (convert_term t) (convert_term u)) undefined t"
subst_term_combine': fixes fresh_idns :: "variable list"   and insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term" assumes "length fresh_idns = length insts"   and "distinct fresh_idns"   and "distinct (map fst insts)"   and "\<forall>idn\<in>set fresh_idns. idn \<notin> fst ` (fv t \<union> \<Union> (fv ` snd ` set insts) \<union> fst ` set insts)" shows "subst_term insts t = fold (\<lambda>single. subst_term [single]) (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (fold (\<lambda>single. subst_term [single]) (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t)"
is_interval_uminus: fixes x :: "'b set" shows "is_interval (uminus ` x) = is_interval x"
INT_extend_simps_10: fixes B :: "'e \<Rightarrow> 'c set"   and f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "(\<Inter>a\<in>A. B (f a)) = \<Inter> (B ` f ` A)"
simps_5: fixes va :: "'b match_expr" shows "normalized_nnf_match (MatchNot (MatchNot va)) = False"
axioms_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "conservative_mpt M T" shows "mpt M T"
simps_2: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr" shows "(P,E,h \<turnstile> e :\<^bsub>NT\<^esub> Boolean) = (P,E,h \<turnstile> e : Boolean)"
g_nondec: fixes x :: "real"   and y :: "real" assumes "a < ereal x"   and "x \<le> y"   and "ereal y < b" shows "g x \<le> g y"
intro: fixes invar :: "'c \<Rightarrow> bool"   and \<alpha> :: "'c \<Rightarrow> 'd set" assumes "\<And>s. invar s \<Longrightarrow> finite (\<alpha> s)" shows "finite_set \<alpha> invar"
TESL_interpretation_homo: fixes \<phi> :: "'b TESL_atomic"   and \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk> \<phi> \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L \<inter> \<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<phi> # \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
matrix_to_lp_vec_to_lpoly_row: fixes i :: "nat"   and A :: "rat mat" assumes "i < dim_row A" shows "matrix_to_lpolies A ! i = vec_to_lpoly (row A i)"
strict_trans: fixes b :: "'t"   and a :: "'t"   and c :: "'t" assumes "lt_of_nat_term_order cmp_term b a"   and "lt_of_nat_term_order cmp_term c b" shows "lt_of_nat_term_order cmp_term c a"
power_strict_increasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and N :: "nat"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n < N"   and "less one a" shows "less (power.power one times a n) (power.power one times a N)"
rel_spmf_try_spmf: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h spmf"   and p' :: "'i spmf"   and q :: "'h spmf"   and q' :: "'i spmf" assumes "rel_spmf R p p'"   and "\<not> lossless_spmf p' \<Longrightarrow> rel_spmf R q q'" shows "rel_spmf R (TRY p ELSE q) (TRY p' ELSE q')"
eOp_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'c option"   and ebinp :: "'bindex \<Rightarrow> 'd withERR option" assumes "\<not> igWlsInp MOD delta inp" shows "eOp MOD delta (OKI inp) ebinp = ERR"
subdivFace'_incr: fixes Q :: "graph \<Rightarrow> graph \<Rightarrow> bool"   and P :: "graph \<Rightarrow> graph \<Rightarrow> bool"   and g :: "graph"   and f' :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and ovl :: "nat option list" assumes "\<And>x y z. \<lbrakk>Q x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>f g. \<lbrakk>f \<in> \<F> g; \<not> final f\<rbrakk> \<Longrightarrow> P g (makeFaceFinal f g)"   and "\<And>g u v f vs. pre_splitFace g u v f vs \<Longrightarrow> Q g (snd (snd (splitFace g u v f vs)))"   and "pre_subdivFace' g f' v' v n ovl"   and "minGraphProps g"   and "f' \<in> \<F> g" shows "P g (subdivFace' g f' v n ovl)"
val2FormRemoveAll: fixes literal :: "Literal"   and valuation :: "Literal list" shows "removeAll [literal] (val2form valuation) = val2form (removeAll literal valuation)"
finfun_apply_inject: fixes x :: "'a \<Rightarrow>f 'b"   and y :: "'a \<Rightarrow>f 'b" shows "(($) x = ($) y) = (x = y)"
default_node_properties_def: shows "SINVAR_BLPtrusted.default_node_properties \<equiv> \<lparr>security_level = 0, trusted = False\<rparr>"
Id: shows "ring_hom_ring (K [X]) (A [X]) id"
ignore: fixes x :: "'call"   and s :: "'s" assumes "\<not> consider' x" shows "interaction_bound consider (callee s x) = 0"
hyps: assumes "\<And>x. x \<in> poly_vars p \<Longrightarrow> \<alpha> x = \<beta> x" shows "eval_poly \<alpha> p = eval_poly \<beta> p"