hcju/isabelleps huggingface dataset:
sum_list_replicate: fixes n :: "nat"   and c :: "'q" shows "sum_list (replicate n c) = of_nat n * c"
N: shows "N \<in> X \<rightarrow>\<^sub>M subprob_algebra K"
c: shows "lookup (tail p) (t \<oplus> lt f) = lookup p (t \<oplus> lt f)"
abs_sum_upto_moebius_mu_over_n_less: fixes x :: "real" assumes "2 \<le> x" shows "\<bar>M x\<bar> < 1"
opfType: shows "(\<bullet>) f \<in> Hom A B \<rightarrow> Hom A C"
smap2_applicative: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b stream"   and y :: "'c stream" shows "smap2 f x y = pure f \<diamondop> x \<diamondop> y"
dec_nemp: fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>\<^sub>+" shows "Dec G u \<noteq> \<epsilon>"
assms: shows "\<alpha> \<in>\<^sub>\<circ> \<beta>"
ext_inject: fixes mbal :: "nat"   and bal :: "nat"   and inp :: "InputsOrNi"   and more :: "'a"   and mbal' :: "nat"   and bal' :: "nat"   and inp' :: "InputsOrNi"   and more' :: "'a" shows "(\<lparr>mbal = mbal, bal = bal, inp = inp, \<dots> = more\<rparr> = \<lparr>mbal = mbal', bal = bal', inp = inp', \<dots> = more'\<rparr>) = (mbal = mbal' \<and> bal = bal' \<and> inp = inp' \<and> more = more')"
Ints_cnj: fixes x :: "complex" assumes "x \<in> \<int>" shows "cnj x \<in> \<int>"
finter_mono: fixes A :: "'b fset"   and C :: "'b fset"   and B :: "'b fset"   and D :: "'b fset" assumes "A |\<subseteq>| C"   and "B |\<subseteq>| D" shows "A |\<inter>| B |\<subseteq>| C |\<inter>| D"
unique_connecting_path_def: fixes G :: "('e, 'f) Graph_scheme"   and v :: "'e"   and w :: "'e" shows "v \<leadsto>\<^bsub>G\<^esub> w \<equiv> THE xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
invh_joinR: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invh l"   and "invh r"   and "bheight r \<le> bheight l" shows "invh (joinR l x r)"
distinct_39: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> While x61 x62"
simps_4: fixes M :: "'b nmultiset multiset"   and N :: "'b nmultiset multiset" shows "(MSet M < MSet N) = less_multiset_ext\<^sub>D\<^sub>M (<) M N"
mem_simps_9: fixes A :: "'a"   and C :: "'a set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
tllist_ord_bot: fixes b :: "'f"   and xs :: "('g, 'f) tllist" shows "tllist_ord b (TNil b) xs"
MPoly_monomial_power': fixes x' :: "nat \<Rightarrow>\<^sub>0 nat"   and n :: "nat" shows "MPoly (monomial (1::'a) x') ^ (n + 1) = MPoly (monomial (1::'a) (((\<lambda>x. x + x') ^^ n) x'))"
Derivs_simps_2: fixes c :: "'a"   and s :: "'a list"   and A :: "'a list set" shows "Derivs (c # s) A = Derivs s (Deriv c A)"
obsp_def: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "obsp \<sigma> k = (case local.att (path \<sigma> k) of None \<Rightarrow> None | Some f \<Rightarrow> Some (f (\<sigma>\<^bsup>k\<^esup>)))"
Rsafe_def: fixes R :: "(('a, 'b, 'c) formula list \<times> ('a, 'b, 'c) formula list) list \<times> ('a, 'b, 'c) formula list \<times> ('a, 'b, 'c) formula list" shows "Rsafe R = ((\<forall>i\<ge>0. i < length (fst R) \<longrightarrow> Ssafe (fst R ! i)) \<and> Ssafe (snd R))"
root_unity: fixes n :: "nat" shows "FFT.root n ^ n = 1"
P_def: shows "P = (\<lambda>x. \<exists>N. \<forall>m>N. s m k = x)"
ftv_ty: fixes T :: "ty" shows "supp T = set (ftv T)"
arr_runit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C a)"
stl_sset: fixes x :: "'c"   and a :: "'c stream" assumes "x \<in> sset (stl a)" shows "x \<in> sset a"
CNVR: shows "(ecnvi, ecnv) \<in> igbg_impl_rel_ext Re Rv \<rightarrow> Re'"
pbernpoly_integral_conv_pbernpoly_integral_Suc: fixes n :: "nat"   and N :: "nat" assumes "1 \<le> n" shows "integral {0..real N} (\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n) = complex_of_real (pbernpoly (Suc n) (real N)) / (of_nat (Suc n) * (s + of_nat N) ^ n) - complex_of_real (bernoulli (Suc n)) / (of_nat (Suc n) * s ^ n) + of_nat n / of_nat (Suc n) * integral {0..real N} (\<lambda>x. complex_of_real (pbernpoly (Suc n) x) / (complex_of_real x + s) ^ Suc n)"
discounted_measurable: fixes X :: "'b \<Rightarrow> real"   and N :: "'b measure"   and r :: "real"   and m :: "nat" assumes "X \<in> borel_measurable N" shows "discounted_value r (\<lambda>m. X) m \<in> borel_measurable N"
wf_fdecl_def: fixes G :: "prog"   and P :: "pname" shows "wf_fdecl G P = (\<lambda>(fn, f). is_acc_type G P (type f))"
diff_eq: fixes x :: "'b upred"   and y :: "'b upred" shows "diff_upred x y = (x \<and> \<not> y)"
size_remove_neg: fixes \<phi> :: "Formula.formula" shows "size (remove_neg \<phi>) \<le> size \<phi>"
below_monic_pair: fixes f :: "'a \<rightarrow> 'b"   and g :: "'a \<rightarrow> 'c" assumes "below_monic_cfun f"   and "below_monic_cfun g" shows "below_monic_cfun (\<Lambda> x. (f\<cdot>x, g\<cdot>x))"
simps_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" shows "fold f (x # xs) = fold f xs \<circ> f x"
assms_2: shows "{p, q, r} \<subseteq> S"
valid_Act: fixes P :: "'state"   and \<alpha> :: "'act"   and x :: "('a, 'pred, 'act) formula" shows "P \<Turnstile> Act \<alpha> x = (\<exists>\<alpha>'' x' P'. Act \<alpha> x = Act \<alpha>'' x' \<and> P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
le_less: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less" shows "less_eq x y = (less x y \<or> x = y)"
neq: shows "p \<noteq> \<zero>\<^bsub>P\<^esub>"
assms_6: shows "m2 \<le>\<^sub>L 1\<^sub>m d2"
atLeastAtMost_singleton: fixes a :: "'b word" shows "signed.atLeastAtMost a a = {a}"
snd_crit_pair_below_lcs: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "snd (crit_pair p q) \<prec>\<^sub>p monomial (1::'c) (term_of_pair (lcs (lp p) (lp q), component_of_term (lt p)))"
validn_def: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and n :: "nat"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "G \<Turnstile>\<^sub>n c : A = ( \<Turnstile>\<^sub>n G   \<longrightarrow>  \<Turnstile>\<^sub>n c : A )"
0: shows "convergent_subseq s l l = residue (p ^ l) (acc_point s l)"
c_def: shows "c \<equiv> 2 * complex_of_real pi * \<i>"
refl: fixes x :: "'a" shows "x \<le>\<cdot> x"
plane_fin_exceptionalVertex_def: fixes g :: "graph"   and v :: "nat" assumes "Invariants.inv g"   and "final g"   and "v \<in> \<V> g" shows "exceptionalVertex g v = (|filter (\<lambda>f. 5 \<le> |vertices f|) (facesAt g v)| \<noteq> 0)"
measure_le_0_iff: fixes M :: "'c measure"   and X :: "'c set" shows "(Sigma_Algebra.measure M X \<le> 0) = (Sigma_Algebra.measure M X = 0)"
le_less_linear: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "x \<le>s y \<or> y <s x"
sqmatrix_mult_0_left: fixes A :: "('c, 'd) sqmatrix" shows "0 \<cdot> A = 0"
norm_EM_remainder_le_strong_int: fixes Y :: "real set"   and n :: "nat"   and D :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and C :: "real" assumes "finite Y"   and "\<forall>x. \<bar>pbernpoly n x\<bar> \<le> D"   and "continuous_on {a..} f"   and "continuous_on {a..} g"   and "continuous_on {a..} g'"   and "(g \<longlongrightarrow> C) at_top"   and "\<And>x. a \<le> x \<Longrightarrow> norm (f x) \<le> g' x"   and "\<And>x. x \<in> {a..} - Y \<Longrightarrow> (g has_real_derivative g' x) (at x)" shows "\<forall>x. a \<le> real_of_int x \<longrightarrow> norm (EM_remainder n f x) \<le> D / fact n * (C - g (real_of_int x))"
compact_imp_bounded: fixes U :: "'a set" assumes "compact U" shows "bounded U"
f: shows "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
fiff: fixes x :: "V"   and y :: "V"   and i :: "nat" assumes "f {x, y} = i"   and "x \<in> elts \<alpha>"   and "y \<in> elts \<alpha>"   and "x < y" shows "\<pi> x < \<pi> y \<and> i = 0 \<or> \<pi> y < \<pi> x \<and> i = 1"
pp_def: shows "pp = (\<lambda>x. x ^ degree p / poly p x)"
span_eq: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" shows "(vec.span S = vec.span T) = (S \<subseteq> vec.span T \<and> T \<subseteq> vec.span S)"
case_optionE: fixes P :: "bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'d option" assumes "case x of \<bottom> \<Rightarrow> P | \<lfloor>y\<rfloor> \<Rightarrow> Q y" obtains "x = \<bottom>" and "P"   | y :: "'d" where "x = \<lfloor>y\<rfloor>" and "Q y"
proper_designI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "design point_set block_collection"   and "int (size block_collection) \<noteq> 0" shows "proper_design point_set block_collection"
moebius_rotation_dilatation_def: fixes a :: "complex" shows "moebius_rotation_dilatation a = moebius_similarity a 0"
fps_lr_inverse_one_minus_fps_X_2: shows "fps_right_inverse (1 - fps_X) (1::'a) = Abs_fps (\<lambda>n. 1::'a)"
map_index_nth: fixes i :: "nat"   and l :: "'a list"   and f :: "'a \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat" assumes "i < length l" shows "map_index f l n ! i = f (l ! i) (n + i)"
Suc_0: shows "\<chi> (Suc 0) = 1"
FunctorCodDom_1: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "dom\<^bsub>CatCod F\<^esub> (F ## f) = F @@ dom\<^bsub>CatDom F\<^esub> f"
all_nth_in_listset: fixes x :: "'c list"   and As :: "'c set list" assumes "length x = length As"   and "\<And>i. i < length As \<Longrightarrow> x ! i \<in> As ! i" shows "x \<in> listset As"
simps_1: fixes x :: "Preliminaries.var" shows "nv2T (Var x) = {x}"
det_echelon_form_of_euclidean: fixes A :: "(('c, 'd) vec, 'd) vec" shows "det A = (1::'c) div fst (echelon_form_of_det A euclid_ext2) * (\<Prod>i\<in>UNIV. snd (echelon_form_of_det A euclid_ext2) $ i $ i)"
Suc_4: shows "(Fun.swap a b id \<circ> p) a = a"
cr_2: fixes a :: "real" shows "nr a = card {x. x \<le> a \<and> poly (real_of_int_poly p) x = 0}"
gu_condition_imply_secure_aux: fixes P :: "'a process"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and I :: "('b \<times> 'b) set"   and U :: "'b set"   and xs :: "'a list"   and ys :: "'a list"   and xs' :: "'a list"   and u :: "'b" assumes "view_partition P D R"   and "weakly_future_consistent P I D R"   and "weakly_step_consistent P D R"   and "locally_respects P I D R"   and "U \<subseteq> range D"   and "U \<noteq> {}"   and "xs @ ys \<in> traces P"   and "\<And>u. u \<in> unaffected_domains I D U [] \<Longrightarrow> (xs, xs') \<in> R u"   and "u \<in> unaffected_domains I D U ys" shows "(xs @ ys, xs' @ ipurge_tr_aux I D U ys) \<in> R u"
eq: fixes z :: "int"   and n :: "nat"   and f :: "nat \<Rightarrow> int" shows "z ^ Suc n * prod f {0..n} = (\<Prod>x = 0..n. z * f x)"
shleg_poly_def: fixes n :: "nat" shows "shleg_poly n = gen_shleg_poly n n div [:fact n:]"
singular_chain_add: fixes p :: "nat"   and X :: "'c topology"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X a"   and "singular_chain p X b" shows "singular_chain p X (a + b)"
insert_iff: fixes a :: "'c"   and b :: "'c"   and A :: "'c set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>t {\<lambda>s. P (s[a/x])} x ::= a {P}"
cases: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and D :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('a, 'c, 'd \<Rightarrow> 'e) generat"   and a2 :: "('b, 'c, 'd \<Rightarrow> 'f) generat" assumes "eq_\<I>_generat A \<I> D a1 a2" obtains x :: "'a" and y :: "'b" where "a1 = Pure x" and "a2 = Pure y" and "A x y"   | out :: "'c" and c :: "'d \<Rightarrow> 'e" and c' :: "'d \<Rightarrow> 'f" where "a1 = IO out c" and "a2 = IO out c'" and "out \<in> outs_\<I> \<I>" and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> D (c input) (c' input)"
BrnFT_notIn_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n" shows "BrnFT cl dl \<notin> UNpart1 cl dl P"
jump_not_infinity: fixes a :: "real"   and f :: "real \<Rightarrow> real" assumes "isCont f a" shows "jump f a = 0"
OK_def: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd program" shows "OK I F = (\<forall>i\<in>I. \<forall>j\<in>I - {i}. Acts (F i) \<subseteq> AllowedActs (F j))"
simps: fixes a1 :: "any_unop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'a OCL_Types.type" shows "any_unop_type a1 a2 a3 = ((\<exists>\<tau>. a1 = OclAsSetOp \<and> a2 = \<tau> \<and> a3 = Set (to_required_type \<tau>) \<and> \<tau> \<le> OclAny[?]) \<or> (\<exists>\<tau>. a1 = OclIsNewOp \<and> a2 = \<tau> \<and> a3 = Boolean[1] \<and> \<tau> \<le> OclAny[?]) \<or> (\<exists>\<tau>. a1 = OclIsUndefinedOp \<and> a2 = \<tau> \<and> a3 = Boolean[1] \<and> \<tau> \<le> OclAny[?]) \<or> (\<exists>\<tau>. a1 = OclIsInvalidOp \<and> a2 = \<tau> \<and> a3 = Boolean[1] \<and> \<tau> \<le> OclAny[?]) \<or> (\<exists>\<tau>. a1 = OclLocaleOp \<and> a2 = \<tau> \<and> a3 = String[1] \<and> \<tau> \<le> OclAny[?]) \<or> (\<exists>\<tau>. a1 = ToStringOp \<and> a2 = \<tau> \<and> a3 = String[1] \<and> \<tau> \<le> OclAny[?]))"
ordermap_inc_eq: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set"   and \<pi> :: "'a \<Rightarrow> 'b"   and s :: "('b \<times> 'b) set" assumes "x \<in> A"   and "small A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; (x, y) \<in> r\<rbrakk> \<Longrightarrow> (\<pi> x, \<pi> y) \<in> s"   and "wf r"   and "total_on A r"   and "wf s" shows "ordermap (\<pi> ` A) s (\<pi> x) = ordermap A r x"
R1I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat option" assumes "recfn 1 r"   and "Partial_Recursive.total r"   and "\<And>x. eval r [x] = f x" shows "f \<in> \<R>"
fls_integral_conv_fps_zeroth_integral: fixes a :: "'a fls" assumes "0 \<le> fls_subdegree a" shows "fls_integral a = fps_to_fls (fps_integral0 (fls_regpart a))"
special_points_on_x_axis''_1: shows "0\<^sub>h \<in> circline_set x_axis"
B': shows "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"
le_iff_sup: fixes y :: "'c"   and x :: "'c" shows "(y \<le> x) = (x \<sqinter> y = y)"
prv_exi_inst_same: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var" shows "prv (imp \<phi> (exi x \<phi>))"
reds: shows "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
simps_2: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" shows "inf_concat_simple f (Suc n) = (let (i, j) = inf_concat_simple f n in if Suc j < f i then (i, Suc j) else (Suc i, 0))"
of_bool_conj: fixes P :: "bool"   and Q :: "bool" shows "of_bool (P \<and> Q) = of_bool P * of_bool Q"
unbounded: fixes x :: "ordinal" shows "\<exists>y\<in>A. x < y"
eq: shows "(\<lambda>v. Blinfun (\<lambda>v'. sterm_sem I \<theta>\<^sub>1 v * frechet I \<theta>\<^sub>2 v v' + frechet I \<theta>\<^sub>1 v v' * sterm_sem I \<theta>\<^sub>2 v)) = blin_frechet (good_interp I) (simple_term (Times \<theta>\<^sub>1 \<theta>\<^sub>2))"
subst_with_ax2: fixes M :: "trm"   and b :: "coname"   and x :: "name"   and a :: "coname" shows "M{b:=x.Ax x a} \<longrightarrow>\<^sub>a* M[b\<turnstile>c>a]"
one_imp_carrier: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R"   and "\<one>\<^bsub>R\<^esub> \<in> I" shows "I = carrier R"
assms: shows "a \<noteq> b \<or> b \<noteq> c \<or> a \<noteq> c"
allAllocationsBridgingLemma: fixes N :: "'a set"   and G :: "'b list" assumes "0 < card N"   and "distinct G" shows "allAllocations N (set G) = set (allAllocationsAlg N G)"
rep_eq: fixes x :: "'a sq_mtx" shows "to_vec x\<^sup>-\<^sup>1 = matrix_inv (to_vec x)"
lprod_2_2: fixes a :: "'a"   and b :: "'a"   and R :: "('a \<times> 'a) set"   and c :: "'a" assumes "(a, b) \<in> R" shows "([a, c], [c, b]) \<in> lprod R"
ustar_sub_unfoldl: fixes P :: "'b hrel" shows "P\<^sup>\<star> \<sqsubseteq> II \<sqinter> (P ;; P\<^sup>\<star>)"
init_reds_sync_unit_throw: fixes e\<^sub>0 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e' :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>0,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"   and "init_exp_of e\<^sub>0 = \<lfloor>unit\<rfloor>" shows "P \<turnstile> \<langle>init_switch e\<^sub>0 e',s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
noDAID: fixes p :: "('b, 'c) Combinators list" assumes "noDenyAll p" shows "noDenyAll (insertDenies p)"
max_fun_diff_le_eq: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "(s.max_fun_diff f g, x) \<in> s"   and "f \<noteq> g"   and "f \<in> FINFUNC"   and "g \<in> FINFUNC"   and "x \<noteq> s.max_fun_diff f g" shows "f x = g x"
list_se_match_1: fixes l1 :: "'a list"   and l2 :: "'a list"   and a :: "'a" assumes "l1 \<noteq> []" shows "(l1 @ l2 = [a]) = (l1 = [a] \<and> l2 = [])"
jipsen1r: fixes x :: "'a"   and y :: "'a" shows "x \<le> inv_resr (inv_resl y x) y"
strict_trans1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ordering less_eq less"   and "less_eq a b"   and "less b c" shows "less a c"
sel_1: fixes x1 :: "('c, 'd) hd" shows "head (Hd x1) = x1"
ring_iso_sym: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme" assumes "ring R"   and "R \<simeq> S" shows "S \<simeq> R"
Inter_greatest: fixes A :: "'c set set"   and C :: "'c set" assumes "\<And>X. s2p A X \<Longrightarrow> C \<subseteq> X" shows "C \<subseteq> \<Inter> A"
Abs_bset_eqvt: fixes A :: "'a set"   and p :: "perm" assumes "|A| <o natLeq +c |UNIV|" shows "p \<bullet> Abs_bset A = Abs_bset (p \<bullet> A)"
select_convs_2: fixes dwa_cond :: "'a \<Rightarrow> bool"   and dwa_step :: "'a \<Rightarrow> 'a"   and dwa_initial :: "'a"   and dwa_invar :: "'a set"   and more :: "'b" shows "dwa_step \<lparr>dwa_cond = dwa_cond, dwa_step = dwa_step, dwa_initial = dwa_initial, dwa_invar = dwa_invar, \<dots> = more\<rparr> = dwa_step"
CIInterp2Mor: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "ZFAxioms T"   and "Sig iS\<^bsub>CI T\<^esub> \<triangleright> Vx : A \<turnstile> e : B" shows "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IMor [A,e,B]\<^bsub>T\<^esub>"
cos_theta_div_2: shows "cos (\<theta> / 2) = sqrt (real (N - M) / real N)"
poly_minus_rightE: fixes p :: "'c poly"   and x :: "'c"   and y :: "'c" obtains c :: "'c" where "poly p (x - y) = poly p x - c * y"
updSizeAux: fixes h :: "('a \<times> 'b) list"   and a :: "'a"   and obj1 :: "'b"   and n :: "nat"   and obj :: "'b" assumes "|h[a\<mapsto>obj1]| = n"   and "h\<down>a = Some obj" shows "|h| = n"
3: shows "\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow>\<langle>\<alpha>\<rangle> Q' \<and> R P' Q'))"
that: fixes B :: "'b set" assumes "S \<subseteq> B"   and "B \<subseteq> V"   and "independent B"   and "V \<subseteq> span B" shows "thesis"
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "f (f x y) z = f x (f y z)"
StrongEq_L_sym: fixes \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> x \<triangleq> y" shows "\<tau> \<Turnstile> y \<triangleq> x"
asIMOD_igFreshIGAbs1: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igFreshIGAbs1 (asIMOD SEM)"
set_pmf_bind_spmf: fixes M :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf" shows "set_pmf (M \<bind> f) = set_pmf M \<bind> map_option_set (set_pmf \<circ> f)"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) R local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) R"
Rep_dense: fixes x :: "'b dense" shows "Rep_dense x \<in> dense_elements"
CondF\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>1 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
measure_wf: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and m :: "'c \<Rightarrow> 'b" assumes "wf r"   and "r' = {(x, y). (m x, m y) \<in> r}" shows "wf r'"
SETfun: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF"   and B :: "ZF" assumes "\<forall>x. x |\<in>| A \<longrightarrow> f x |\<in>| B" shows "isZFfun (ZFfun A B f)"
tail_value_check: assumes "s \<in> verts G" shows "\<mu> c s s = ereal 0"
vbrelation_vrestriction: fixes r :: "V"   and A :: "V" shows "vbrelation (r \<restriction>\<^sub>\<circ> A)"
intros_8: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>"
assms_4: shows "list_all (\<lambda>Q. index_free Q i) Qs"
build_rqrm_correct_1: fixes \<delta> :: "('a, 'b) ta_rule dlist" assumes "ls.invar \<delta>" shows "rqrm_invar (build_rqrm \<delta>)"
simps_1: fixes ok :: "nat"   and f :: "'b list"   and f' :: "'b list"   and r :: "'b list"   and r' :: "'b list" shows "st_inv (Rev ok f f' r r') = (length f + 1 = length r \<and> length f' = length r' \<and> ok \<le> length f')"
Sup_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "\<Squnion> A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
cont: fixes z :: "'a \<times> 'b" shows "isCont ((\<lambda>x. setdist {x} (- U)) \<circ> fst) z"
lift_set_Diff_distrib: fixes i :: "nat"   and A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and B :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" shows "lift_set i (A - B) = lift_set i A - lift_set i B"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "map \<alpha> S T" shows "fiber_relation \<alpha> S T"
true_cls_mset_image_mset: fixes I :: "'b set"   and f :: "'c \<Rightarrow> 'b literal multiset"   and A :: "'c multiset" shows "(I \<Turnstile>m image_mset f A) = (\<forall>x\<in>#A. I \<Turnstile> f x)"