hcju/isabelleps huggingface dataset:
assms_2: shows "X \<inter> range_vars \<theta> = {}"
cond_spmf_fst_inverse: fixes p :: "('d \<times> 'e) spmf" shows "map_spmf fst p \<bind> (\<lambda>x. map_spmf (Pair x) (cond_spmf_fst p x)) = p"
length_to_bl_eq: fixes w :: "'a word" shows "length (to_bl w) = LENGTH('a)"
arr_cod_iff_arr: fixes f :: "'a" shows "L'.C.arr (L'.C.cod f) = L'.C.arr f"
S_def: shows "S \<equiv> trace_core_closure core1 core2 E IA IU p q"
Y: shows "entropy b T Y = - (LINT x|(S \<Otimes>\<^sub>M T). Pxy x * log b (Py (snd x)))"
less_diff_eq: fixes a :: "complex"   and c :: "complex"   and b :: "complex" shows "(a <\<^sub>\<complex> c - b) = (a + b <\<^sub>\<complex> c)"
index: fixes r :: "nat" shows "(J ^\<^sub>m r) $$ (i, j) = (if bi = bj then (jordan_block nn la ^\<^sub>m r) $$ (li, lj) else 0)"
A31_DAcc_level1: shows "DAcc level1 sA31 = {sA41}"
cases_scheme: fixes r :: "('b, 'd, 'a, 'e, 'c, 'f) BEState_scheme" obtains es :: "'a" and ps :: "'b \<Rightarrow> 'c" and pubActs :: "'d \<times> ('b \<Rightarrow> 'e)" and more :: "'f" where "r = \<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr>"
innotin: fixes i1 :: "nat"   and i2 :: "nat"   and t :: "nat" assumes "i1 < len"   and "i2 < len"   and "t < m"   and "t \<in> Is i1"   and "t \<notin> Is i2" shows "\<exists>j<len. t = hs j"
simps_3: fixes uu :: "'c"   and pr :: "('b, 'c) Digit"   and m :: "('b, 'c) FingerTreeStruc"   and sf :: "('b, 'c) Digit" shows "toList (Deep uu pr m sf) = digitToList pr @ toList m @ digitToList sf"
intros_2: fixes A :: "'b set"   and a :: "'b" assumes "finite A" shows "finite (insert a A)"
UP_triangle_4: shows "S.UP (\<l>\<^sup>-\<^sup>1[f] \<cdot> \<r>[f]) = (S.cmp\<^sub>U\<^sub>P (trg f, f) \<cdot>\<^sub>S (UP.unit (trg f) \<star>\<^sub>S S.UP f)) \<cdot>\<^sub>S (S.UP f \<star>\<^sub>S S.inv (UP.unit (src f))) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, src f))"
bounded_byI: fixes P :: "'b \<Rightarrow> real"   and b :: "real" assumes "\<And>x. P x \<le> b" shows "bounded_by b P"
disc_eq_case_2: fixes step :: "('c, 'd) step" shows "is_Skip step = (case step of Skip uu_ \<Rightarrow> True | _ \<Rightarrow> False)"
cexp_mono1: fixes p1 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "p1 \<le>o r1"   and "Card_order q" shows "p1 ^c q \<le>o r1 ^c q"
projection_plus: fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> is_orthogonal x y"   and "closed_csubspace A"   and "closed_csubspace B" shows "projection (A +\<^sub>M B) = (\<lambda>x. projection A x + projection B x)"
llist_all2_LCons1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist"   and ys :: "'d llist" shows "llist_all2 P (LCons x xs) ys = (\<exists>y ys'. ys = LCons y ys' \<and> P x y \<and> llist_all2 P xs ys')"
assms_1: shows "\<tt>\<tt> p \<le> \<tt>\<tt> i"
Gt_lt_conv: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Gt) = lt_of_comp comp y x"
h0: shows "\<forall>x\<in>set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)). \<exists>x1\<in>set (characterize_consistent_signs_at_roots_copr p qs1). \<exists>x2\<in>set (characterize_consistent_signs_at_roots_copr p qs2). x = x1 @ x2"
map_of_map_restrict: fixes f :: "'b \<Rightarrow> 'c"   and ks :: "'b list" shows "map_of (map (\<lambda>k. (k, f k)) ks) = (Some \<circ> f) |` set ks"
"case": fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'c" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
unique: fixes b2 :: "'b" assumes "a \<sim> b2"   and "QB b2" shows "b2 = b"
UP_mapsto: shows "UP \<in> UNIV \<rightarrow> S.Univ"
rotate_class_rotate': fixes w :: "'b list"   and u :: "'b list"   and l :: "nat" shows "(\<exists>n. rotate n w = u) = (\<exists>n. rotate n (rotate l w) = u)"
simps_3: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "gval (Gt a1 a2) s = value_gt (aval a1 s) (aval a2 s)"
countable_INT': fixes X :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "countable X"   and "X \<noteq> {}"   and "A ` X \<subseteq> M" shows "\<Inter> (A ` X) \<in> M"
slice_edge_def: fixes S :: "'node SDG_node set"   and cs :: "'edge list"   and a :: "'edge" shows "slice_edge S cs a \<equiv> (\<forall>c\<in>set cs. sourcenode c \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<and> (case kind a of Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> True | _ \<Rightarrow> sourcenode a \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>)"
axioms_1: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "FunctorM F" shows "PreFunctor F"
trigpoly_def: shows "trigpoly \<equiv> \<lambda>f. \<exists>n a b. f = (\<lambda>x. \<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))"
IH: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "higher_differentiable_on S g n"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "higher_differentiable_on S f n"
closedin_def: fixes U :: "'b topology"   and S :: "'b set" shows "closedin U S = (S \<subseteq> topspace U \<and> openin U (topspace U - S))"
dbl_dec_simps_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one (uminus (numeral.numeral one plus k)) = uminus (neg_numeral.dbl_inc plus one (numeral.numeral one plus k))"
unique_subheap: fixes \<sigma>1 :: "'a"   and \<sigma>2 :: "'a"   and \<sigma> :: "'a" assumes "\<sigma>1,\<sigma>2\<triangleright>\<sigma>" shows "\<exists>!\<sigma>2'. \<sigma>1,\<sigma>2'\<triangleright>\<sigma>"
r2f_epi: shows "\<forall>f g. f \<circ> \<F> = g \<circ> \<F> \<longrightarrow> f = g"
pot_comp_zero_min: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and v :: "'a" assumes "cmp u v \<noteq> Gt"   and "snd (rep_nat_term u) = snd (rep_nat_term v)" shows "pot_comp cmp u v \<noteq> Gt"
cases: fixes x :: "'a \<times> 'a \<times> 'a \<times> 'a" obtains a :: "'a" and b :: "'a" and c :: "'a" and d :: "'a" where "x = (a, b, c, d)"
class_of_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set"   and s :: "'d" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I" shows "direct_lim.class_of \<FF> \<rho> I U s \<equiv> equivalence.Class (Sigma I \<FF>) {(x, y). direct_lim.rel \<FF> \<rho> I x y} (U, s)"
Rep_zet_cases: fixes y :: "'b set" assumes "y \<in> zet" obtains x :: "'b zet" where "y = Rep_zet x"
inf_le1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) x"
lemma_3_5_iv_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "d2 a b * d2 b c * d2 a b \<le> d2 a c"
ys_opts: shows "h \<turnstile> map_M f xs \<rightarrow>\<^sub>r ys_opts"
cos_gt_zero_pi: fixes x :: "real" assumes "- (pi / 2) < x"   and "x < pi / 2" shows "0 < cos x"
Zero_neq_Suc: fixes m :: "nat"   and R :: "bool" assumes "0 = Suc m" shows "R"
can_def: fixes u :: "'a local.term"   and t :: "'a local.term" shows "can u t \<equiv> \<lbrace>Inv (u\<^bold>\<down>) \<^bold>\<cdot> t\<^bold>\<down>\<rbrace>"
elims: fixes x :: "char list"   and y :: "unit + char \<times> char list" assumes "string_expose x = y" obtains "x = []" and "y = Inl ()"   | c :: "char" and cs :: "char list" where "x = c # cs" and "y = Inr (c, cs)"
s_def: shows "s = (\<lambda>_. Some (Suc i))"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" obtains "seq (fst g) (fst f)" and "seq (snd g) (snd f)"
S_intv_cl: fixes a :: "'a"   and b :: "'a"   and S :: "'a set" assumes "a \<in> A"   and "b \<in> A"   and "S \<subseteq> interval r a b" shows "S \<subseteq> A"
surjective: fixes r :: "('c, 'd) state_scheme" shows "r = \<lparr>counter = counter r, discovered = discovered r, finished = finished r, pending = pending r, stack = stack r, tree_edges = tree_edges r, back_edges = back_edges r, cross_edges = cross_edges r, \<dots> = state.more r\<rparr>"
hp: shows "h p = finsum A (h \<circ> (\<lambda>t. [fst t] \<otimes>\<^bsub>K [X]\<^esub> X [^]\<^bsub>K [X]\<^esub> snd t)) (set (dense_repr p))"
degree_reflect_poly_eq: fixes p :: "'b poly" assumes "coeff p 0 \<noteq> (0::'b)" shows "degree (reflect_poly p) = degree p"
param_optci_rule: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and bdd :: "bdd"   and p :: "nat option"   and bdd' :: "bdd"   and bddi :: "bddi" assumes "mi.param_opt_impl i t e bdd = Some (p, bdd')" shows "<is_bdd_impl bdd bddi> param_optci i t e bddi <\<lambda>r. case r of (pi, bddi') \<Rightarrow> is_bdd_impl bdd' bddi' * \<up> (pi = p)>\<^sub>t"
Cons1_5: shows "\<forall>i<length ys'. ((x # xs) ! i, ys' ! i) \<in> ns"
C_eq_All_untilSortedQ: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)) = C p"
gj_fst: shows "fst (gauss_jordan M (1\<^sub>m n)) = 1\<^sub>m n"
distinct_fst_appendD: fixes kxs :: "('b \<times> 'c) list"   and kxs' :: "('b \<times> 'c) list" assumes "distinct_fst (kxs @ kxs')" shows "distinct_fst kxs \<and> distinct_fst kxs'"
setI: shows "set I = {LI dir s' (lhs (eq_for_lvar (\<T> s') x\<^sub>i))} \<union> {LI dir s' x |x. x \<in> rvars_eq (eq_for_lvar (\<T> s') x\<^sub>i) \<and> coeff (rhs (eq_for_lvar (\<T> s') x\<^sub>i)) x < 0} \<union> {UI dir s' x |x. x \<in> rvars_eq (eq_for_lvar (\<T> s') x\<^sub>i) \<and> 0 < coeff (rhs (eq_for_lvar (\<T> s') x\<^sub>i)) x}"
A: fixes a :: "('b, 'c) OrdSum"   and b :: "('b, 'c) OrdSum" shows "(a l\<rightarrow> b) * a = a * (a r\<rightarrow> b)"
tail_in_verts: fixes e :: "nat \<times> nat" assumes "e \<in> g_E G" shows "tail G e \<in> set (g_V G)"
assms: fixes U :: "'a set" assumes "openin (top_of_set S) U"   and "U \<noteq> {}" shows "T \<inter> U \<noteq> {}"
inv: fixes a :: "'a" assumes "a \<in> H" shows "inv a \<in> H"
preListpreSet: fixes xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and l :: "'a \<Rightarrow> nat"   and C :: "acom"   and s :: "char list \<Rightarrow> int" assumes "preSet (set xs) C l s" shows "preList xs C l s"
monom_add: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
finite_UNIV_bit0_def: shows "finite_UNIV = Phantom('a bit0) True"
MkIde_Dom: fixes a :: "('a E.term, 'a) arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
append_Nil: fixes ys :: "'a list" shows "[] @ ys = ys"
groupF_fst: shows "groupF f (x # xs) = (x # filter (\<lambda>y. f x = f y) xs) # groupF f (filter (\<lambda>y. f x \<noteq> f y) xs)"
simps: fixes a :: "'a set \<Rightarrow> 'a set" shows "CK_nf_pos a = (a = id \<or> a = I \<or> a = K \<or> a = I \<circ> K \<or> a = K \<circ> I \<or> a = I \<circ> K \<circ> I \<or> a = K \<circ> I \<circ> K)"
Afix_nonrec_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallArityAnalysis.Afix_nonrec cccExp x e\<cdot>i = fup\<cdot>(CoCallArityAnalysis.Aexp cccExp e)\<cdot>(ABind_nonrec x e\<cdot>i) \<squnion> fst i"
thin_triangles_implies_hyperbolic: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and delta :: "real" assumes "class.geodesic_space dist uniformity open"   and "\<And>x y z w Gxy Gyz Gxz. \<lbrakk>geodesic_segment_between Gxy x y; geodesic_segment_between Gxz x z; geodesic_segment_between Gyz y z; w \<in> Gxy\<rbrakk> \<Longrightarrow> infdist w (Gxz \<union> Gyz) \<le> delta" shows "Gromov_hyperbolic_subset (4 * delta) UNIV"
equivalence_is_left_adjoint: fixes f :: "'a" assumes "B.equivalence_map f" shows "B.is_left_adjoint f"
simps_1: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm"   and at :: "('e, 'f) atm" shows "substL \<pi> (Pos at) = Pos (substA \<pi> at)"
inverse_mat_base_case_2_R: shows "inverts_mat (mat_of_rows_list 3 [[1, 1, 1], [0, 1, 0], [1, 0, - 1]]) (mat_of_rows_list 3 [[1 / 2, - 1 / 2, 1 / 2], [0, 1, 0], [1 / 2, - 1 / 2, - 1 / 2]])"
simps_2: fixes v :: "'b"   and first :: "indices"   and "next" :: "indices list"   and S :: "'b trm" shows "subterm (Var v) (first # next) S = False"
4: shows "coreflexive (prim_P w v e\<^sup>T * (w \<sqinter> - prim_EP w v e)\<^sup>T)"
not_nofail_inres: fixes S :: "'c nres"   and x :: "'c" assumes "\<not> nofail S" shows "inres S x"
measurable_lift_RealVal: fixes f :: "real \<Rightarrow> real" assumes "f \<in> borel_measurable borel" shows "lift_RealVal f \<in> stock_measure REAL \<rightarrow>\<^sub>M stock_measure REAL"
finite_subset_Union: fixes A :: "'c set"   and \<B> :: "'c set set" assumes "finite A"   and "A \<subseteq> \<Union> \<B>" obtains \<F> :: "'c set set" where "finite \<F>" and "\<F> \<subseteq> \<B>" and "A \<subseteq> \<Union> \<F>"
healthy_intros_9: fixes f :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy (wp f)"   and "healthy (wp g)" shows "healthy (wp (f \<Sqinter> g))"
ctvalid_def: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" shows "(C \<Turnstile>\<^sub>t {P} c {Q}) = (|\<Turnstile>\<^sub>t C \<longrightarrow> \<Turnstile>\<^sub>t {P} c {Q})"
simps_5: fixes \<iota> :: "nat"   and b :: "'a"   and c :: "('a, 'b) MWLsCom" shows "PPc (while\<^bsub>\<iota>\<^esub> b do c od) = [\<iota>] @ PPc c"
sees_method_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and M :: "char list"   and TS :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'b"   and D' :: "char list" assumes "P \<turnstile> C sees M: TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M: TS'\<rightarrow>T' = m' in D'" shows "TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
si_neq: fixes x :: "'c event"   and s :: "'c event list"   and X :: "'c event set"   and y :: "'c event"   and t :: "'c event list" shows "setinterleaving (x # s, X, y # t) = (if x \<in> X then if y \<in> X then if x = y then {x # u |u. u setinterleaves ((s, t), X)} else {} else {y # u |u. u setinterleaves ((x # s, t), X)} else if y \<notin> X then {x # u |u. u setinterleaves ((s, y # t), X)} \<union> {y # u |u. u setinterleaves ((x # s, t), X)} else {x # u |u. u setinterleaves ((s, y # t), X)})"
measure_of_st_vec'_def: fixes x :: "'a st_vec"   and I :: "'a set" shows "measure_of_st_vec' x I = ennreal (sum (($h) (st_vec x)) I)"
nhds_def: fixes "open" :: "'a set \<Rightarrow> bool"   and a :: "'a" assumes "class.topological_space open" shows "topological_space.nhds open a = Inf (principal ` {S. open S \<and> a \<in> S})"
simps_6: fixes x21 :: "char list"   and x22 :: "aexp" shows "x21 ::= x22 \<noteq> SKIP"
bot_prod_def: shows "bot_prod = (\<bottom>, \<bottom>)"
matrix_matrix_mult_def: fixes m :: "(('b, 'e) vec, 'd) vec"   and m' :: "(('b, 'c) vec, 'e) vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $ i $ k * m' $ k $ j"
join_raw_def: fixes f :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list"   and ys :: "('b \<times> 'c) list" shows "join_raw f xs ys = foldr (\<lambda>(k, v). map_default k v (\<lambda>v'. f k (v', v))) ys xs"
nn_integral_PInf_AE: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "integral\<^sup>N M f \<noteq> \<infinity>" shows "AE x in M. f x \<noteq> \<infinity>"
T_splay_simps_8: fixes b :: "'a"   and x :: "'a"   and c :: "'a"   and AB :: "'a tree"   and C :: "'a tree"   and D :: "'a tree" assumes "b < x"   and "x < c" shows "T_splay x \<langle>AB, b, \<langle>C, c, D\<rangle>\<rangle> = (if C = \<langle>\<rangle> then 1 else T_splay x C + 1)"
cong1: shows "[a ^ (p - 1) = 1] (mod p)"
winning_regionI: fixes v :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> V\<^bsub>G'\<^esub>"   and "G'.strategy p \<sigma>"   and "G'.winning_strategy p \<sigma> v" shows "v \<in> G'.winning_region p"
simps_2: fixes x :: "'b"   and xl :: "'b bintree"   and xr :: "'b bintree" shows "t_sorted (Branch x xl xr) = ((\<forall>y\<in>t_set xl. y \<le> x) \<and> (\<forall>y\<in>t_set xr. x < y) \<and> t_sorted xl \<and> t_sorted xr)"
is_unifier_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and AA :: "('a, 'b) Term.term set" shows "is_unifier \<sigma> AA = (card (subst_atms AA \<sigma>) \<le> 1)"
simps_1: fixes s :: "'b" shows "tFirst (tInit s) = s"
brc_step_invar: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<Sigma> :: "('a, unit) HashMap.hashmap \<times> 'a list \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule set" assumes "rqrm_invar rqrm"   and "\<Sigma> \<in> brc_invar_add"   and "brc_\<alpha> \<Sigma> \<in> br'_invar \<delta>"   and "brc_cond \<Sigma>" shows "brc_step rqrm \<Sigma> \<in> brc_invar_add"
antisym_conv: fixes y :: "('a, 'b) Term.term"   and x :: "('a, 'b) Term.term" assumes "y \<sqsubseteq> x" shows "(x \<sqsubseteq> y) = (x = y)"
equal_oalist_tc_def: fixes xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'b) oalist_tc" shows "equal_class.equal xs ys = (list_of_oalist_tc xs = list_of_oalist_tc ys)"
generalise_output_context_check_def: fixes v :: "value"   and r :: "nat"   and s1 :: "nat"   and s2 :: "nat"   and e1 :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and e2 :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "generalise_output_context_check v r s1 s2 e1 e2 = (\<forall>c1 c2 t. obtains s1 c1 (tm e1) 0 <> t \<and> obtains s2 c2 (tm e2) 0 <> t \<longrightarrow> c2 $ r = Some v)"
add_aform_def: fixes x :: "'a \<times> 'a pdevs"   and y :: "'a \<times> 'a pdevs" shows "add_aform x y = (fst x + fst y, add_pdevs (snd x) (snd y))"
i_append_eq_i_append_conv_if_aux: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "length xs \<le> length ys" shows "(xs \<frown> f = ys \<frown> g) = (xs = ys \<down> length xs \<and> f = ys \<up> length xs \<frown> g)"
assms_1: shows "[a] \<preceq>\<^sub>F w"
empty_tree_empty_V: assumes "V\<^bsub>T\<^esub> = {}" shows "V = {}"
assms_4: shows "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"
card_Diff_singleton: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) = card A - 1"
dens: shows "has_density M (stock_measure INTEG) \<delta>"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e list" shows "map g (map f v) = map (g \<circ> f) v"
countable_insert: fixes A :: "'b set"   and a :: "'b" assumes "countable A" shows "countable (insert a A)"
map\<^sub>0_in_hom_2: fixes a :: "'a" assumes "C.obj a" shows "\<guillemotleft>\<tau>\<^sub>0 a : \<tau>\<^sub>0 a \<Rightarrow>\<^sub>D \<tau>\<^sub>0 a\<guillemotright>"
D_append: fixes es :: "expr list"   and es' :: "expr list"   and A :: "char list set option" shows "\<D>s (es @ es') A = (\<D>s es A \<and> \<D>s es' (A \<squnion> \<A>s es))"
isSPath_distinct: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "isSimplePath u p v" shows "distinct p"
reduce_le_complex: fixes b :: "complex"   and a :: "complex"   and c :: "complex" assumes "0 \<le> b"   and "a + b \<le> c" shows "a \<le> c"
rep_eq: fixes x :: "('d, 'c) lgenerator'" shows "lgenerator (ltake_trans' x) = ltake_trans (lgenerator x)"
existence_ivl_cong: fixes Y :: "('a \<Rightarrow>\<^sub>L 'a) set"   and x0a :: "'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "UNIV = Y"   and "Y.existence_ivl0 x0a = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> Y.vareq x0a t o\<^sub>L x = g t x" shows "Y.mvar.existence_ivl x0a t0 x0 = ll_on_open.existence_ivl S g Y t0 x0"
pred_def: fixes P :: "'d \<Rightarrow> bool"   and M :: "'d measure" shows "Measurable.pred M P = ({x \<in> space M. P x} \<in> sets M)"
map_simp_ide: fixes a :: "'a" assumes "A.ide a" shows "\<rho>_\<sigma>\<tau>.map a = B (\<sigma>\<tau>.map a) (\<rho> a)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) \<nu>"
cost_simps_2: fixes a :: "'a"   and c :: "nat" shows "result (a, c) = a"
same_fw_behaviour_one_def: fixes ip1 :: "'a word"   and ip2 :: "'a word"   and c :: "'b parts_connection_scheme"   and rs :: "'a simple_rule list" shows "same_fw_behaviour_one ip1 ip2 c rs \<equiv> \<forall>d s. runFw ip1 d c rs = runFw ip2 d c rs \<and> runFw s ip1 c rs = runFw s ip2 c rs"
hensel_result: shows "hensel_binary C D H = (D', H')"
return_bind: fixes x :: "'o"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog" shows "Heap_Error_Monad.bind (return x) g = g x"
skip_fun_im: fixes i :: "nat"   and n :: "nat" assumes "i \<in> {j. j \<le> Suc n}" shows "skip i ` {j. j \<le> n} = {j. j \<le> Suc n} - {i}"
sum_every_odd_summand_is_zero: fixes n :: "nat" assumes "1 \<le> n" shows "\<forall>f. (\<forall>i. i < 2 ^ (n + 1) \<and> odd i \<longrightarrow> f i = 0) \<longrightarrow> sum f {0..<2 ^ (n + 1)} = (\<Sum>k = 0..<2 ^ n. f (2 * k))"
reachable_def: shows "reachable = reaches s\<^sub>0"
finite_linorder_max_induct: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite A"   and "P {}"   and "\<And>b A. \<lbrakk>finite A; \<forall>a\<in>A. a < b; P A\<rbrakk> \<Longrightarrow> P (insert b A)" shows "P A"
max_word_max: fixes n :: "'b word" shows "n \<le> - 1"
allFinite: assumes "w1 \<noteq> NEG_INF \<and> w1 \<noteq> POS_INF \<and> w2 \<noteq> NEG_INF \<and> w2 \<noteq> POS_INF" shows "P w1 w2"
has_bochner_integralI_AE: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and g :: "'a \<Rightarrow> 'b" assumes "has_bochner_integral M f x"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "has_bochner_integral M g x"
finite_sets_supp: fixes S :: "'a set" assumes "finite S" shows "finite (supp S)"
scalar_mult_add: fixes y :: "'a"   and a :: "'a matrix"   and b :: "'a matrix" shows "scalar_mult y (a + b) = scalar_mult y a + scalar_mult y b"
assms_1: shows "f \<longlonglongrightarrow> x"
inj_on_mult: fixes x :: "nat"   and q :: "nat" assumes "coprime x q" shows "inj_on (\<lambda>b. x * b mod q) {..<q}"
powr_nat_conv_powr: fixes x :: "real"   and y :: "real" assumes "0 < x" shows "powr_nat x y = x powr y"
simps_3: fixes p :: "'a pfm"   and q :: "'a pfm" shows "reduce (p \<^bold>\<or>\<^sub>! q) = (reduce p \<^bold>\<or>\<^sub>! reduce q)"
n_subid': fixes x :: "'a" shows "!x \<le> (1::'a)"
nrows_eq_card_rows: fixes A :: "(('b, 'c) vec, 'd) vec" shows "nrows_iarray (matrix_to_iarray A) = CARD('d)"