hcju/isabelleps huggingface dataset:
mod_diff_cong: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (minus a b) c = modulo (minus a' b') c"
Quotient_rat: shows "Quotient ratrel Abs_Rat Rep_Rat cr_rat"
\<alpha>: shows "\<alpha> \<in> {0<..<1}"
simps: fixes \<eta> :: "nat"   and "oracle" :: "'b \<Rightarrow> 'grp \<Rightarrow> (bool list \<times> 'b) spmf"   and \<sigma> :: "'b"   and \<A> :: "('grp \<Rightarrow> ((bool list \<times> bool list) \<times> 'c, 'grp, bool list) gpv) \<times> ('grp \<times> bool list \<Rightarrow> 'c \<Rightarrow> (bool, 'grp, bool list) gpv)" shows "advantage \<eta> (oracle, \<sigma>) \<A> = \<bar>spmf (run_gpv oracle (game \<eta> \<A>) \<sigma>) True - 1 / 2\<bar>"
countable_finite_power: fixes A :: "'c set" assumes "countable A" shows "countable {B. B \<subseteq> A \<and> finite B}"
sum_mat_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and A :: "'a mat" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mat plus zero A = comm_monoid_add.sum plus zero (($$) A) ({0..<dim_row A} \<times> {0..<dim_col A})"
simps_6: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "delete\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = delete\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle>"
step_Normal_elim_cases_12: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Cond b c1 c2 # cs, css, Normal s) \<rightarrow> u" obtains "u = (c1 # cs, css, Normal s)" and "s \<in> b"   | "u = (c2 # cs, css, Normal s)" and "s \<notin> b"
map_ident: fixes t :: "'d \<Rightarrow> 'e" shows "(\<lambda>x. x) \<circ> t = t"
atrm_def: shows "atrm \<equiv> Collect atrmp"
distinct_12: fixes x7 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Rfork x7 \<noteq> VE x1"
disj_mono: fixes P1 :: "bool"   and Q1 :: "bool"   and P2 :: "bool"   and Q2 :: "bool" assumes "P1 \<longrightarrow> Q1"   and "P2 \<longrightarrow> Q2" shows "P1 \<or> P2 \<longrightarrow> Q1 \<or> Q2"
of_nat_eq_id: shows "semiring_1_class.of_nat = id"
BA: shows "similar_mat (add_col_sub_row val i l A) A"
3: shows "w\<^sup>T * - - v \<le> - - v \<squnion> - (w\<^sup>\<star> * e\<^sup>T * top)"
assms_1: shows "HT_mods \<pi> mods P c Q"
set_1: fixes x11 :: "nat"   and x12 :: "'a" shows "set_tree (Leaf x11 x12) = {x12}"
split_half_def: fixes a :: "('d \<times> 'e) array \<times> nat" shows "BTree_ImpSet.split_half a \<equiv> pfa_length a \<bind> (\<lambda>l. return (l div 2))"
Int_expand: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "{(S, S'). P S S' \<and> Q S S'} = {(S, S'). P S S'} \<inter> {(S, S'). Q S S'}"
div_by_0: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a zero = zero"
healthy: fixes x :: "'b"   and s :: "'a" assumes "x \<in> S s" shows "nearly_healthy (wlp (prog x))"
Length_SNOC: fixes x :: "'x"   and \<AA> :: "'i" shows "Length (SNOC x \<AA>) = Suc (Length \<AA>)"
d1_2: fixes y :: "'a" assumes "y \<in> cball x d1" shows "\<bar>t y\<bar> < e"
poly_lift_hom_extends_hom: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and r :: "'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "r \<in> carrier R" shows "poly_lift_hom R S \<phi> (to_polynomial R r) = to_polynomial S (\<phi> r)"
if_SE_D2: fixes P :: "'b \<Rightarrow> bool"   and \<sigma> :: "'b"   and B\<^sub>1 :: "'b \<Rightarrow> (bool \<times> 'b) option"   and B\<^sub>2 :: "'b \<Rightarrow> (bool \<times> 'b) option" assumes "\<not> P \<sigma>" shows "(\<sigma> \<Turnstile> if\<^sub>S\<^sub>E P B\<^sub>1 B\<^sub>2) = (\<sigma> \<Turnstile> B\<^sub>2)"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (M.\<ll> a)"
total_quasi_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "total_quasi_ordered_set A less_eq \<equiv> connex A less_eq \<and> transitive A less_eq"
diff_single_eq_union: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<in># M" shows "(M - {#x#} = N) = (M = add_mset x N)"
indicator_measurable_on: fixes S :: "'a set" assumes "S \<in> sets lebesgue" shows "indicat_real S measurable_on UNIV"
assms_1: shows "[a\<frown>b]P \<longmapsto> Rs"
llength_ltl: fixes xs :: "'a llist" shows "llength (ltl xs) = epred (llength xs)"
\<Delta>\<Phi>_insert: fixes x :: "'b"   and h :: "'b heap" shows "\<Phi> (Pairing_Heap_List1.insert x h) - \<Phi> h \<le> log 2 (real (size_hp h + 1))"
under_underS_trans: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "trans r"   and "antisym r"   and "a \<in> under r b"   and "b \<in> underS r c" shows "a \<in> underS r c"
seqE: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.seq g f" obtains "A_B.seq (fst g) (fst f)" and "A.seq (snd g) (snd f)"
lookup_map_of: fixes t :: "(('a \<times> 'b) \<times> 'c) tree"   and x :: "'a" assumes "sorted1 (Tree2.inorder t)" shows "lookup t x = AList_Upd_Del.map_of (Tree2.inorder t) x"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>0 (local.dom \<sigma>) s\<^sub>0 s\<^sub>1 (src s\<^sub>0)"
shiftr1_is_div_2: fixes x :: "'b word" shows "x >> 1 = x div 2"
abs_sin_le_one: fixes x :: "real" shows "\<bar>sin x\<bar> \<le> 1"
vec_vec_Tensor_elements: fixes y :: "'a list"   and x :: "'a list" assumes "y \<noteq> []" shows "\<forall>i<length x * length y. vec_vec_Tensor x y ! i = x ! (i div length y) * y ! (i mod length y)"
simps_3: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uu :: "char list set"   and Qannot :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and uv :: "char list set"   and uw :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set"   and C :: "acom" shows "varacom ({(P, uu)/(Qannot, uv)/uw} CONSEQ C) = support P \<union> varacom C \<union> support Qannot"
equivp_reflp_symp_transp: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "equivp R = (reflp R \<and> symp R \<and> transp R)"
for_in_RI: fixes x :: "'a"   and R :: "('a \<times> 'b) set" assumes "x \<in> Domain R" shows "(x, fun_of_rel R x) \<in> R"
right_add_zero_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "A + 0\<^sub>m nr nc = A"
map_mono_suffix: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "suffix xs ys" shows "suffix (map f xs) (map f ys)"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<tau> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* C" shows "Prop (P \<parallel> Q')"
even_diff_nat: fixes m :: "nat"   and n :: "nat" shows "even (m - n) = (m < n \<or> even (m + n))"
nonneg_cexpr_sem_integrate_vars: fixes \<rho> :: "nat \<Rightarrow> val"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and vs :: "nat list"   and e :: "cexpr" assumes "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "distinct vs"   and "set vs \<inter> V' = {}"   and "nonneg_cexpr (set vs \<union> V') \<Gamma> e"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> set vs \<union> V'" shows "0 \<le> extract_real (cexpr_sem \<rho> (integrate_vars \<Gamma> vs e))"
compact_convex_hull: fixes S :: "'a set" assumes "compact S" shows "compact (convex hull S)"
image_image: fixes f :: "'o \<Rightarrow> 'n"   and g :: "'p \<Rightarrow> 'o"   and A :: "'p set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
emb_step_chop: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b chop t"
subst_sound: fixes I :: "'a set"   and C :: "'a literal multiset"   and \<eta> :: "'s"   and \<rho> :: "'s" assumes "\<And>\<sigma>. is_ground_subst \<sigma> \<Longrightarrow> I \<Turnstile> C \<cdot> \<sigma>"   and "is_ground_subst \<eta>" shows "I \<Turnstile> C \<cdot> \<rho> \<cdot> \<eta>"
mult_le_cancel_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (times c a) (times c b) = ((less zero c \<longrightarrow> less_eq a b) \<and> (less c zero \<longrightarrow> less_eq b a))"
z_def: shows "z = pbasepoint pl \<leadsto> direction2 pl"
DirProds_def: fixes G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme"   and I :: "'d set" shows "DirProds G I = \<lparr>carrier = Pi\<^sub>E I (carrier \<circ> G), monoid.mult = \<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i, one = \<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>\<rparr>"
siso0_transT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "siso0 c"   and "mustT c s"   and "mustT c t"   and "(c, s) \<rightarrow>t s'"   and "s \<approx> t" shows "\<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'"
expand_inv_upd_incoming: fixes \<phi> :: "'b ltlr"   and n :: "('b, 'c) node_scheme"   and ns :: "('b, 'd) node_scheme set" assumes "expand_inv \<phi> (n, ns)" shows "expand_inv_result \<phi> (upd_incoming n ns)"
induct: fixes P :: "prim_ty \<Rightarrow> bool"   and prim_ty :: "prim_ty" assumes "P Void"   and "P Boolean"   and "P Integer" shows "P prim_ty"
bi_unique_alt_def2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_unique R = (R ===> R ===> (=)) (=) (=)"
of_bool_eq_2: fixes one :: "'a"   and zero :: "'a" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero True = one"
distinct_133: shows "GreaterThan \<noteq> BinXor"
abs_eq: fixes x :: "int \<times> int" assumes "ratrel x x" shows "- Abs_Rat x = Abs_Rat (- fst x, snd x)"
L2_set_right_distrib: fixes r :: "real"   and f :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "0 \<le> r" shows "r * L2_set f A = L2_set (\<lambda>x. r * f x) A"
normalize_ratfps_def: fixes x :: "'b ratfps" shows "normalize x = (if x = 0 then 0 else ratfps_of_poly (monom (1::'b) (ratfps_subdegree x)))"
ereal_1_times: fixes x :: "ereal" shows "ereal 1 * x = x"
hom_plus: fixes k :: "'a" assumes "k \<in> keys p \<union> keys q" shows "f k (lookup p k + lookup q k) = f k (lookup p k) + f k (lookup q k)"
rk_couple: fixes A :: "Points"   and B :: "Points" assumes "A \<noteq> B" shows "rk {A, B} = 2"
pcr_float_def: shows "pcr_float \<equiv> (=) OO cr_float"
lcliff_greek_less1: fixes r :: "('a \<times> 'a) set"   and as :: "(accent \<times> 'a) list"   and a :: "'a"   and b' :: "(accent \<times> 'a) list"   and b :: "'a"   and cs :: "(accent \<times> 'a) list"   and a' :: "(accent \<times> 'a) list"   and bs :: "(accent \<times> 'a) list" assumes "trans r"   and "snd ` set as \<subseteq> under r a"   and "b' = [(Macron, b)]"   and "snd ` set cs \<subseteq> under r a"   and "a' \<in> {[(Acute, a)], []}"   and "snd ` set bs \<subseteq> under r a \<inter> under r b" shows "(as @ b' @ cs @ a' @ bs, [(Acute, a), (Macron, b)]) \<in> greek_less r"
insert_is_Un: fixes a :: "'b"   and A :: "'b set" shows "insert a A = \<eta> a \<union> A"
rep_eq: fixes x :: "'b pdevs"   and xa :: "nat"   and xb :: "'b" shows "pdevs_apply (pdev_upd x xa xb) = (pdevs_apply x)(xa := xb)"
NSLIMSEQ_LIMSEQ: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L" shows "X \<longlonglongrightarrow> L"
val_principalTr2: fixes v :: "'a \<Rightarrow> ant"   and c :: "ant"   and d :: "ant" assumes "valuation K v"   and "c \<in> v ` (carrier K - {\<zero>}) \<and> (\<forall>w\<in>v ` carrier K. \<exists>a. w = a * c) \<and> 0 < c"   and "d \<in> v ` (carrier K - {\<zero>}) \<and> (\<forall>w\<in>v ` carrier K. \<exists>a. w = a * d) \<and> 0 < d" shows "c = d"
same_class_from_restrict: fixes U :: "'a set set"   and V :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "U \<subseteq> V"   and "s \<in> \<O> V"   and "\<pp> \<in> U" shows "st.class_of V s = st.class_of U (sheaf_spec_morphisms V U s)"
valid_plans_def: fixes prob :: "'c set" shows "valid_plans prob \<equiv> {as. set as \<subseteq> prob}"
poly_mult_one_1: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "polynomial K p" shows "poly_mult [\<one>] p = p"
bind_lunit: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "\<lfloor>x\<rfloor> \<bind> f = f x"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and aa :: "name" and P :: "pi" where "a = aa \<and> [b].Pb__ = [x].P" and "Rs = aa<x> \<prec> P" and "x \<noteq> aa"
Sigma_0: fixes n :: "nat"   and f :: "nat list \<Rightarrow> nat"   and xs :: "nat list" assumes "\<forall>i\<le>n. f (xs @ [i]) = 0" shows "UF.Sigma f (xs @ [n]) = 0"
strategy_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "G'.strategy p \<sigma> \<equiv> \<forall>v\<in>G'.VV p. \<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w) \<longrightarrow> v \<rightarrow>\<^bsub>subgame V'\<^esub> \<sigma> v"
del_list_sorted1: fixes xs :: "('c \<times> 'd) list"   and a :: "'c"   and b :: "'d"   and ys :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 (xs @ (a, b) # ys)"   and "a \<le> x" shows "del_list x (xs @ (a, b) # ys) = xs @ del_list x ((a, b) # ys)"
cod_char: shows "Span.cod = (\<lambda>\<mu>. if Span.arr \<mu> then \<lparr>Chn = span_in_category.apex (\<odot>) (Cod \<mu>), Dom = Cod \<mu>, Cod = Cod \<mu>\<rparr> else Span.null)"
carrier_relative_homology_group: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "carrier (relative_homology_group (int p) X S) = homologous_rel_set p X S ` singular_relcycle_set p X S"
FIXvar: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "PhiWhileP A b \<Phi> (s, t)" shows "(b, A, \<Phi>, s, t) \<in> var"
verit_eq_simplify_25: fixes a :: "'a"   and b :: "'a" shows "(- a = - b) = (a = b)"
AE_embed_measure: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and M :: "'a measure" assumes "inj f" shows "almost_everywhere (embed_measure M f) P = (AE x in M. P (f x))"
\<B>: fixes T :: "'a set" assumes "openin (top_of_set S) T" shows "\<exists>\<U>\<subseteq>\<B>. T = \<Union> \<U>"
simps_9: fixes bs :: "real list"   and i :: "int"   and b :: "MIR.num" shows "Ifm bs (Dvd i b) = (real_of_int i rdvd Inum bs b)"
rep_eq: fixes x :: "'a st_mat"   and xa :: "'a st_vec" shows "st_vec (x *st xa) = st_mat x *v st_vec xa"
u: fixes C :: "'a set" assumes "C \<in> Chains r"   and "C \<noteq> {}" shows "\<exists>u\<in>Field r. \<forall>a\<in>C. (a, u) \<in> r"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c llist \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c llist" shows "(case LCons x21 x22 of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
G1: shows "s |\<^sub>s n \<approx> t |\<^sub>s m |\<^sub>s nextnat (t |\<^sub>s m) - 1"
prio_emb_pos_increase: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and prio :: "dir list \<Rightarrow> nat" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s"   and "prio_emb_step prio t s \<noteq> s"   and "\<And>p q dp dq. prio (q @ [dq]) < prio (p @ [dp]) \<Longrightarrow> take (length p) q \<noteq> p" shows "prio (prio_emb_pos prio t s) \<le> prio (prio_emb_pos prio (prio_emb_step prio t s) s)"
e'_1: fixes x :: "float" assumes "x \<in> U" shows "0 < e' x"
extend_vars: fixes ls :: "lit set" shows "vars_of_cnf cnf \<subseteq> var ` extend ls"
RELATESI: fixes R :: "('b \<times> 'c) set" shows "RELATES R"
lderivs_Inter: fixes xs :: "'a list"   and r :: "'b rexp"   and s :: "'b rexp" shows "lderivs xs (rexp.Inter r s) = rexp.Inter (lderivs xs r) (lderivs xs s)"
fine_division_exists_real: fixes g :: "real \<Rightarrow> real set"   and a :: "real"   and b :: "real" assumes "gauge g" obtains p :: "(real \<times> real set) set" where "p tagged_division_of {a..b}" and "g fine p"
maybe_map_def: shows "maybe_map \<equiv> \<Lambda> a. \<mu> uu. maybe_abs oo ssum_map\<cdot>ID\<cdot>(u_map\<cdot>a) oo maybe_rep"
subprob_space_imp_sigma_finite: fixes M :: "'c measure" assumes "subprob_space M" shows "sigma_finite_measure M"
power2_eq_square: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.monoid_mult one times" shows "power.power one times a 2 = times a a"
the_flag_Some: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and x :: "f" assumes "the (flag rt ip) = x"   and "ip \<in> kD rt" shows "flag rt ip = Some x"
swap_cols_rows_carrier_1: fixes k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_row (swap_cols_rows k l A) = dim_row A"
sin_60: shows "sin (pi / 3) = sqrt 3 / 2"
mset_set_set_mset_subseteq: fixes A :: "'c multiset" shows "mset_set (set_mset A) \<subseteq># A"
Red_I_of_Inf_to_N: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and \<iota> :: "'a inference"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "\<iota> \<in> Inf"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> Red_I N"
normal_axioms_def: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" shows "normal_axioms H G \<equiv> \<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H"
cond1: shows "inv_u * - Q $$ (0, 1) * (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q) $$ (0, 0) = b"
abs_eq_iff: fixes x :: "int"   and y :: "int" shows "(Word.Word x = Word.Word y) = (take_bit LENGTH('k) x = take_bit LENGTH('k) y)"
remove: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "\<Prod>\<^sub># A = x * \<Prod>\<^sub># (A - {#x#})"
I3: fixes v :: "'V"   and m2 :: "'m2"   and v' :: "'V"   and s :: "'s3" assumes "(v, m2) \<in> map_to_set (m1.\<alpha> g)"   and "(v', s) \<in> map_to_set (m2.\<alpha> m2)" shows "s3.invar s"
terminal_of_token_def: fixes t :: "symbol \<times> character list" shows "terminal_of_token t = fst t"
of_real_poly_eq_0_iff: fixes p :: "real poly" shows "(map_poly of_real p = 0) = (p = 0)"
division: fixes d :: "'b set set"   and a :: "'b"   and b :: "'b" assumes "d division_of cbox a b" shows "F g d = g (cbox a b)"
distinct_append_swap: fixes xs :: "'c list"   and ys :: "'c list" assumes "distinct (xs @ ys)" shows "distinct (ys @ xs)"
list_of_oalist_empty: fixes ko :: "'o" shows "list_of_oalist (local.empty ko) = ([], ko)"
Seq_le: fixes C1 :: "'a acom"   and C2 :: "'a acom"   and C :: "'a acom" shows "(C1;; C2 \<le> C) = (\<exists>C1' C2'. C = C1';; C2' \<and> C1 \<le> C1' \<and> C2 \<le> C2')"
mkM_simps_1: fixes T :: "'s Trace set" shows "worlds (SPR.mkM T) = T"
diff_closed: fixes U :: "'b set"   and V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "Subspace.subspace U V"   and "vectorspace V"   and "x \<in> U"   and "y \<in> U" shows "x - y \<in> U"
per_union_memI2: fixes x :: "'b"   and a :: "'b"   and R :: "('b \<times> 'b) set"   and b :: "'b"   and y :: "'b" assumes "(x, a) \<in> R"   and "(b, y) \<in> R" shows "(x, y) \<in> per_union R a b"
wfactors_unique: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and a :: "'c"   and as' :: "'c list" assumes "primeness_condition_monoid G"   and "wfactors G as a"   and "wfactors G as' a"   and "a \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set as' \<subseteq> carrier G" shows "essentially_equal G as as'"
strict_trans1: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "suffix a b"   and "strict_suffix b c" shows "strict_suffix a c"
limit_ordinalI: fixes z :: "ordinal" assumes "0 < z"   and "\<forall>x<z. oSuc x < z" shows "limit_ordinal z"
EnsuresI: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A - B Co A \<union> B"   and "F \<in> transient (A - B)" shows "F \<in> A Ensures B"
simps_5: fixes et :: "('var, 'val, 'ret, 'pname) edge_kind" shows "pred et [] = False"
weak_group_morphism_ker: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set" assumes "weak_group_morphism f H (add_monoid R)" shows "kernel (add_monoid R) (image_group f (add_monoid R)) f = H"
H_assign_iff: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a"   and Q :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool" shows "PRE P (v ::= e) POST Q = (\<forall>s. P s \<longrightarrow> Q (s(v := e s)))"
Trg_Dom_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Trg (Dom t) = Trg t \<and> Trg (Cod t) = Trg t"
preorder_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "class.preorder less_eq less \<equiv> (\<forall>x y. less x y = Restricted_Predicates.strict less_eq x y) \<and> (\<forall>x. less_eq x x) \<and> (\<forall>x y z. less_eq x y \<longrightarrow> less_eq y z \<longrightarrow> less_eq x z)"
embed_embedS_iso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "embed r r' f = (embedS r r' f \<or> iso r r' f)"
suntil_mono: fixes P :: "'a stream \<Rightarrow> bool"   and Q1 :: "'a stream \<Rightarrow> bool"   and Q2 :: "'a stream \<Rightarrow> bool"   and R1 :: "'a stream \<Rightarrow> bool"   and R2 :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "\<And>\<omega>. \<lbrakk>P \<omega>; Q1 \<omega>\<rbrakk> \<Longrightarrow> Q2 \<omega>"   and "\<And>\<omega>. \<lbrakk>P \<omega>; R1 \<omega>\<rbrakk> \<Longrightarrow> R2 \<omega>"   and "(Q1 suntil R1) \<omega>"   and "alw P \<omega>" shows "(Q2 suntil R2) \<omega>"
rep_eq: shows "coeff 0 = (\<lambda>_. 0::'a)"
borel_closed: fixes A :: "'b set" assumes "closed A" shows "A \<in> sets borel"