hcju/isabelleps huggingface dataset:
equiv_imp_le_alt: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans u t"
set: fixes x1 :: "'g"   and x2 :: "'f" shows "b_set (make_rec x1 x2) = {x2}"
y_def: shows "y = (\<lambda>n. if n = 0 then 0 else inv\<^bsub>residue_ring (p ^ n)\<^esub> x n)"
add_mono_thms_linordered_field_1: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i < j \<and> k = l" shows "i + k < j + l"
stack_discovered: shows "set (stack s) \<subseteq> dom (discovered s)"
set_mp: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
Bernstein_coeffs_def: fixes p :: "nat"   and c :: "real"   and d :: "real"   and P :: "real poly" shows "Bernstein_coeffs p c d P = map (\<lambda>j. inverse (real (p choose j)) * coeff (reciprocal_poly p (P \<circ>\<^sub>p [:c, 1:] \<circ>\<^sub>p [:0, d - c:]) \<circ>\<^sub>p [:1, 1:]) (p - j)) [0..<p + 1]"
assms_2: shows "t\<^sub>0 \<in> U"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.uniqify (t # bq) = SkewBinomialHeapStruc.ins t bq"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "b \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "c \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "(a = inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> b \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> c) = (c = b \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> a)"
C: shows "key_sets (Hash (Pwd n)) (crypts (Log -` spied s)) = {} \<and> key_sets \<lbrace>Pwd n, X\<rbrace> (crypts (Log -` spied s)) = {} \<and> key_sets \<lbrace>X, Pwd n\<rbrace> (crypts (Log -` spied s)) = {}"
26264: fixes x :: "'a"   and y :: "'a" shows "- x \<le> - y \<sqinter> - x \<squnion> - - y"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b list" shows "set (map f v) = f ` set v"
elims: fixes x :: "'b trm"   and y :: "indices list set" assumes "pos_of x = y" obtains t :: "'b trm" where "x = t" and "y = {p. position_in p t}"
mset_mergesort_by_rel_split: fixes xs1 :: "'a list"   and xs2 :: "'a list"   and xs :: "'a list" shows "mset (fst (mergesort_by_rel_split (xs1, xs2) xs)) + mset (snd (mergesort_by_rel_split (xs1, xs2) xs)) = mset xs + mset xs1 + mset xs2"
matrix_vector_mult_def: fixes m :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "m *v x \<equiv> \<chi>i. \<Sum>j\<in>UNIV. m $ i $ j * x $ j"
cong: fixes a :: "'a fract"   and c :: "'a fract"   and b :: "'a fract"   and d :: "'a fract" assumes "a = c"   and "b = d" shows "a * b = c * d"
nth_prime_asymptotics: shows "(\<lambda>n. real (nth_prime n)) \<sim>[sequentially] (\<lambda>n. real n * ln (real n))"
abs_eq: fixes x :: "8 word" shows "uint64_of_uint8 (Abs_uint8 x) = Abs_uint64 (ucast x)"
C1_differentiable_on_const: fixes a :: "'a"   and S :: "real set" shows "(\<lambda>z. a) C1_differentiable_on S"
atom_tvar_def: fixes a :: "Syntax.tvar" shows "atom a = Rep_tvar a"
subcls1_induct: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and 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 Q :: "char list \<Rightarrow> bool" assumes "wf_prog wf_md P"   and "is_class P C"   and "Q Object"   and "\<And>C D. \<lbrakk>C \<noteq> Object; P \<turnstile> C \<prec>\<^sup>1 D; is_class P D; Q D\<rbrakk> \<Longrightarrow> Q C" shows "Q C"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (src f)"
array_grow'_def: fixes a :: "'b array"   and n :: "nat"   and x :: "'b" shows "array_grow' a n x \<equiv> Array.len a \<bind> (\<lambda>l. Array.new (l + n) x \<bind> (\<lambda>a'. blit a 0 a' 0 l \<bind> (\<lambda>_. return a')))"
exp_of_derived_img: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and K :: "'a set"   and n :: "nat" assumes "group_hom G H h"   and "K \<subseteq> carrier G" shows "(derived H ^^ n) (h ` K) = h ` (derived G ^^ n) K"
halfspace_Int_eq_1: fixes a :: "'b"   and b :: "real" shows "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
bisim\<^sub>\<sigma>: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "(cms\<^sub>1 ! k, subst \<sigma> (fst (mems ! k))) \<approx> (cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k)))"
inj_map: fixes f :: "'c \<Rightarrow> 'd" shows "inj (map f) = inj f"
that_1: shows "0 < (x - y) \<bullet> rot (a - b)"
Domain: fixes x :: "ZF"   and f :: "ZF" shows "Elem x (HOLZF.Domain f) = (\<exists>y. Elem (Opair x y) f)"
is_interval_convex: fixes S :: "'c set" assumes "is_interval S" shows "convex S"
is_processT2_TR: fixes P :: "'c process" shows "\<forall>s. s \<in> T P \<longrightarrow> front_tickFree s"
B1: shows "\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel"
pp_increasing: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<preceq> \<ominus> \<ominus> x"
finsum_ldistr: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "finite A" shows "sum f A * a = (\<Sum>i\<in>A. f i * a)"
Cons_eqvt: fixes p :: "perm"   and x :: "'a"   and xs :: "'a list" shows "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
simps_70: fixes x7 :: "'a OCL_Types.type"   and x6 :: "'a OCL_Types.type" shows "Bag x7 \<noteq> OrderedSet x6"
subgroup_generated_is_subset: fixes H :: "'a set" shows "additive.subgroup_generated H \<subseteq> R"
invertible_right_inverse: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> G" shows "u \<cdot> sub.inverse u = \<one>"
set_replicate_conv_if: fixes n :: "nat"   and x :: "'g" shows "set (replicate n x) = (if n = 0 then {} else {x})"
s_prod_pa_def: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" shows "R \<otimes> S = {(a, A). \<exists>B. (a, B) \<in> R \<and> (\<forall>b\<in>B. (b, A) \<in> S)}"
VAs: fixes j :: "'sz" shows "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT (x2a j)}. FVT (\<sigma> i))"
valid_def: fixes ds :: "'d set"   and P :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool"   and Q :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool" shows "valid ds P Q = (Q ds {} \<and> (\<forall>fp. P ds fp \<and> Q ds fp \<longrightarrow> Q ds (cop_F ds fp)))"
D: shows "evalDdb v1\<cdot>\<rho> = evalDdb v2\<cdot>\<rho>"
inc_LE_def: shows "inc_LE \<equiv> inv_image lex_triple inc_measure"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat option" assumes "rank x xa = y" obtains x :: "nat" and n :: "nat" where "xa__ = x" and "xa = n" and "y = (if x \<le> n \<and> \<not> sink (token_run x n) then Some (card (older_seniors x n)) else None)"
Trivial2: fixes a :: "ennreal" shows "a / (1 + a) \<le> 1"
basis_partition_2: shows "basis1 \<inter> basis2 = {}"
subst_eq_lit: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and l :: "'b literal" assumes "\<sigma> \<doteq> \<eta>" shows "subst_lit l \<sigma> = subst_lit l \<eta>"
env_C_restr_lookup: fixes r :: "C"   and \<rho> :: "'b \<Rightarrow> C \<rightarrow> 'a"   and v :: "'b" shows "(\<rho>|\<^sup>\<circ>\<^bsub>r\<^esub>) v = \<rho> v|\<^bsub>r\<^esub>"
sup_inf_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "sup_class.sup x (inf_class.inf y z) = inf_class.inf (sup_class.sup x y) (sup_class.sup x z)"
of_nat_1: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
sum_list_map_power2: fixes ls :: "real poly list" shows "0 \<le> sum_list (map power2 ls)"
fps_of_poly_simps_10: fixes c :: "'b"   and p :: "'b poly" shows "fps_of_poly (smult c p) = fps_const c * fps_of_poly p"
index_hma_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and i :: "'c"   and j :: "'b" shows "index_hma A i j \<equiv> A $h i $h j"
coord_matrix': fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and v :: "('a, 'b) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)"   and "linear (*s) (*s) f" shows "coord Y (f v) = matrix' X Y f *v coord X v"
take_bit_minus_one_eq_mask: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "take_bit n (uminus one) = mask n"
rel_gpv''_map_gpv2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and gpv :: "('a, 'c, 'e) gpv"   and f :: "'g \<Rightarrow> 'b"   and g :: "'h \<Rightarrow> 'd"   and gpv' :: "('g, 'h, 'f) gpv" shows "rel_gpv'' A C R gpv (map_gpv f g gpv') = rel_gpv'' (\<lambda>a b. A a (f b)) (\<lambda>c d. C c (g d)) R gpv gpv'"
plus_absorb2: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "g \<in> o[F](f)" shows "L F (\<lambda>x. f x + g x) = L F f"
minus_eq: fixes x :: "'e"   and y :: "'e" shows "x \<ominus>\<^bsub>N\<^esub> y = x \<oplus>\<^bsub>N\<^esub> \<ominus>\<^bsub>N\<^esub> y"
sq_norm_vec_le_linf_norm: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "\<parallel>v\<parallel>\<^sup>2 \<le> of_nat n * \<parallel>v\<parallel>\<^sub>\<infinity>\<^sup>2"
supt_supteq_set_conv: shows "{\<rhd>} = {\<unrhd>} - Id"
downset_set_prop: shows "prod.downset_set = \<Union> \<circ> (`) prod.downset"
cod_in_codomains: fixes f :: "'a \<times> 'a" assumes "CC.codomains f \<noteq> {}" shows "CC.cod f \<in> CC.codomains f"
lift_nth: fixes i :: "nat"   and xs :: "('b com option \<times> 'b) list"   and Q :: "'b com" assumes "i < length xs" shows "map (lift Q) xs ! i = lift Q (xs ! i)"
strictify_cancel: fixes f :: "'b \<rightarrow> 'a" assumes "f\<cdot>\<bottom> = \<bottom>" shows "strictify\<cdot>f = f"
angle_actrans_def: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<langle>P\<rangle>_v \<equiv> \<not> _actrans (\<not> P) v"
fg: shows "f = g \<or> (\<exists>a b. (a, b) \<in> c\<^sup>+ \<and> f = Abs a \<and> g = Abs b)"
accepts_trace_simulation: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list \<times> value list) list"   and f :: "nat \<Rightarrow> nat"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option" assumes "accepts_trace e1 s1 r1 t"   and "trace_simulation f e1 s1 r1 e2 s2 r2 t" shows "accepts_trace e2 s2 r2 t"
reflpD: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "reflp r" shows "r x x"
leadsTo: shows "F \<in> A \<longmapsto> B"
distinct_3: shows "Newton \<noteq> Neville_Aitken"
Tr: shows "Tr \<equiv> \<lambda>(s\<^sub>1, s\<^sub>2). Tr\<^sub>1 s\<^sub>1 \<inter> Tr\<^sub>2 s\<^sub>2"
intro: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "Matrix_Kernel.kernel nr nc A"
length_take: fixes n :: "nat"   and xs :: "'d list" shows "length (take n xs) = min (length xs) n"
is_valid_problem_strips_initial_of_dom: fixes \<Pi> :: "'a strips_problem" assumes "is_valid_problem_strips \<Pi>" shows "dom (\<Pi>\<^sub>I) = set (\<Pi>\<^sub>\<V>)"
eq_inorder_Nil: fixes t :: "'c tree" shows "(inorder t = []) = (t = \<langle>\<rangle>)"
lprefix_not_lnullD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "\<not> lnull xs" shows "\<not> lnull ys"
spmf_of_pmf_return_pmf: fixes x :: "'c" shows "spmf_of_pmf (return_pmf x) = return_spmf x"
upper_asymptotic_density_union: fixes A :: "nat set"   and B :: "nat set" shows "upper_asymptotic_density (A \<union> B) \<le> upper_asymptotic_density A + upper_asymptotic_density B"
distinct_5: fixes x1 :: "'b"   and x41 :: "'b formula"   and x42 :: "'b formula" shows "Atom x1 \<noteq> x41 \<^bold>\<and> x42"
null_sets_subset: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M"   and "A \<in> sets M"   and "A \<subseteq> B" shows "A \<in> null_sets M"
iTILL_D: fixes x :: "nat"   and n :: "nat" assumes "x \<in> [\<dots>n]" shows "x \<le> n"
bij_to_nat: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.to_nat Rep) UNIV {0..<CARD('a)}"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> sup_class.sup a b"
provRules_def: shows "provRules \<equiv> {(xa, x). provRulesp xa x}"
Subgroup_RSubmodule: fixes N :: "'v set" assumes "RModule FG (\<cdot>) N \<and> N \<subseteq> V" shows "VecGroup.Subgroup N"
no_hyp_change_no_cons: fixes x :: "nat"   and z :: "nat"   and i :: "nat"   and j :: "nat" assumes "0 < x"   and "\<not> (s (e_snoc (local.adverse z i j \<triangleright> x) 0) \<noteq> s (local.adverse z i j \<triangleright> x) \<or> s (e_snoc (local.adverse z i j \<triangleright> x) 1) \<noteq> s (local.adverse z i j \<triangleright> x))" shows "\<not> learn_cons \<phi> {local.adverse 0 i j, local.adverse 1 i j} s"
S: shows "(\<Union>s\<in>S. set_pmf (K s)) \<subseteq> S \<union> N"
uminus_def: fixes uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linorder_stone_algebra_expansion uminus less_eq less bot top inf sup" shows "uminus x = (if x = bot then top else bot)"
get_all_matching_src_ips: fixes m :: "'a common_primitive match_expr"   and iface :: "iface"   and a :: "action" assumes "normalized_nnf_match m" shows "get_all_matching_src_ips iface m \<subseteq> {ip. \<forall>p. matches (common_matcher, in_doubt_allow) m a (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>)}"
Kas_never_says_newer_timestamp: fixes evs :: "event list"   and T :: "nat"   and X :: "msg" assumes "CT evs \<le> T"   and "Number T \<in> parts {X}"   and "evs \<in> kerbV" shows "\<forall>A. Says Kas A X \<notin> set evs"
row_submatrix_UNIV: fixes i :: "nat"   and A :: "'a mat"   and I :: "nat set" assumes "i < card {i. i < dim_row A \<and> i \<in> I}" shows "Matrix.row (submatrix A I UNIV) i = Matrix.row A (pick I i)"
inorder2_def: fixes t :: "'b tree"   and xs :: "'b list" shows "Root_Balanced_Tree.inorder2 t xs = Time_Monad.val (inorder2_tm t xs)"
WhileTrue: fixes b :: "bexp"   and s1 :: "char list \<Rightarrow> int"   and c :: "com"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int"   and z :: "nat" assumes "bval b s1"   and "(c, s1) \<Rightarrow> x \<Down> s2"   and "(WHILE b DO c, s2) \<Rightarrow> y \<Down> s3"   and "1 + x + y = z" shows "(WHILE b DO c, s1) \<Rightarrow> z \<Down> s3"
splitset_fun_finite: fixes xs :: "'c list" shows "finite (splitset_fun xs)"
assms_2: shows "g field_differentiable at (f a) within f ` S"
AssumeH_4: fixes A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and H :: "fmla set" shows "insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> A4"
setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t_cons: fixes x :: "'d \<times> ('e, 'f) stateful_strand_step"   and S :: "('d \<times> ('e, 'f) stateful_strand_step) list" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (x # S) = setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p x \<union> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t S"
fnt_carrier_exists_util_fun: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "finite carrier"   and "rational_preference carrier relation" shows "\<exists>u. ordinal_utility carrier relation u"
labeled_static_ref: assumes "statically_complete_calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) no_labels.Red_I_\<G> no_labels.Red_F_\<G>_empty" shows "statically_complete_calculus Bot_FL Inf_FL (\<Turnstile>\<inter>\<G>L) Red_I Red_F"
step_preserves_sconf: 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 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
sum_flatten1: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "finite A"   and "finite B" shows "\<Sum> {\<Sum> {f x y |y. y \<in> B} |x. x \<in> A} = \<Sum> {f x y |x y. x \<in> A \<and> y \<in> B}"
power_abs: 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 sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times a n) = power.power one times (abs a) n"
ladder_shift_n_at: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and N :: "nat" assumes "index < length L" shows "ladder_shift_n N L ! index = (fst (L ! index) - N, snd (L ! index))"
circ_mult_sub: fixes x :: "'a"   and y :: "'a" shows "Rf (x * (y * x)\<^sup>\<circ>) y \<le> (x * y)\<^sup>\<circ>"
1: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "Span.arr (SPN \<mu>)"
RSubmodule_inner_dirsum: fixes Ns :: "'m set list" assumes "\<forall>N\<in>set Ns. RSubmodule N" shows "RSubmodule (\<Oplus>N\<leftarrow>Ns. N)"
induct: fixes P :: "'b heap list \<Rightarrow> bool"   and a0 :: "'b heap list" assumes "\<And>h1 h2 hs. P hs \<Longrightarrow> P (h1 # h2 # hs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
assms_4: shows "src\<^sub>C g = trg\<^sub>C f"
While_P_def: fixes vars :: "nat set"   and M0 :: "complex mat"   and M1 :: "complex mat"   and C :: "com" shows "While_P vars M0 M1 C = While (\<lambda>n. if n = 0 then mat_extension dims vars M0 else if n = 1 then mat_extension dims vars M1 else undefined) C"
exhaust: fixes y :: "'b ref" obtains "y = Null"   | x2 :: "'b" where "y = Ref x2"
f_nonneg: fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "0 \<le> f x y n"
leD: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "\<not> y < x"
inf_absorb1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y" shows "inf x y = x"
degree_power_eq: fixes p :: "'b poly"   and n :: "nat" assumes "p \<noteq> 0" shows "degree (p ^ n) = n * degree p"
delete_no_failure: fixes i :: "'c"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "i \<in> fst ` set xs" shows "\<exists>xs'. delete xs i = Some xs'"
analz_Un_implAuthSet_aux: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implAuthSet_aux G \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
Afix_restr_subst': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and x :: "Vars.var"   and y :: "Vars.var"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "\<And>x' e a. (x', e) \<in> set \<Gamma> \<Longrightarrow> \<A>\<^bsub>a\<^esub> e[x::=y] f|` S = \<A>\<^bsub>a\<^esub> e f|` S"   and "x \<notin> S"   and "y \<notin> S"   and "domA \<Gamma> \<subseteq> S" shows "Afix \<Gamma>[x::h=y]\<cdot>ae f|` S = Afix \<Gamma>\<cdot>(ae f|` S) f|` S"
contains_predE: fixes A :: "'c set"   and x :: "'c"   and y :: "unit" assumes "pred.eval (Predicate_Compile.contains_pred A x) y" obtains "Predicate_Compile.contains A x"
mult_smult_right: fixes p :: "'d poly"   and a :: "'d"   and q :: "'d poly" shows "p * smult a q = smult a (p * q)"
satisfies_tableau_satisfies_tableau: fixes v1 :: "nat \<Rightarrow> 'a"   and t :: "(nat \<times> linear_poly) list"   and v2 :: "nat \<Rightarrow> 'a" assumes "v1 \<Turnstile>\<^sub>t t"   and "\<forall>var\<in>tvars t. v1 var = v2 var" shows "v2 \<Turnstile>\<^sub>t t"
prems: shows "ccw'.sortedP 0 (u # v # w # y # ys)"
integrable_subintervals_imp_measurable: fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>a b. f integrable_on cbox a b" shows "f measurable_on UNIV"
fv_assignment_rhs_subset_fv_st: fixes S :: "('d, 'c) strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (assignment_rhs\<^sub>s\<^sub>t S) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(hom x = hom y) = (x = y)"
power_eq_if: fixes p :: "'a"   and m :: "nat" shows "p ^ m = (if m = zero_class.zero then 1::'a else p \<cdot> p ^ (m - 1))"
affine_independent_iff_card: fixes s :: "'a set" shows "(\<not> affine_dependent s) = (finite s \<and> aff_dim s = int (card s) - 1)"
carrier_matI: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "dim_row A = nr"   and "dim_col A = nc" shows "A \<in> carrier_mat nr nc"
iEx_singleton: fixes t :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<diamond> t' {t}. P t') = P t"
uminus_closed: fixes A :: "'a set"   and x :: "'a" assumes "ring_closed A"   and "x \<in> A" shows "- x \<in> A"
eq_divide_eq_numeral_1: fixes w :: "num"   and b :: "'c"   and c :: "'c" shows "(numeral w = b / c) = (if c \<noteq> (0::'c) then numeral w * c = b else numeral w = (0::'c))"
nth_default_eq_dflt_iff: fixes dflt :: "'f"   and xs :: "'f list"   and k :: "nat" shows "(nth_default dflt xs k = dflt) = (k < length xs \<longrightarrow> xs ! k = dflt)"