hcju/isabelleps huggingface dataset:
assms_2: shows "cl = Func_host (t1s _> t2s) f"
assms_2: shows "S\<lparr>ArrDom\<rparr> = A"
\<delta>_def: shows "\<delta> \<equiv> \<lambda>w. Derivs w A"
lookup_eq_valueI: fixes k :: "'a"   and v :: "'b"   and xs :: "'x" assumes "(k, v) \<in> set (fst (list_of_oalist xs))" shows "lookup xs k = v"
assms_1: shows "vs @ xs \<in> L M2 \<inter> L M1"
IH_2: fixes x :: "'a \<times> 'b"   and xa :: "('a \<times> 'b) set"   and xb :: "('a \<times> 'b) set" assumes "\<not> (infinite A \<or> A = {})"   and "x = arg_min_on snd A"   and "xa = {p \<in> A. fst p < fst x}"   and "xb = {p \<in> A. fst x < fst p}"   and "finite xb"   and "inj_on fst xb"   and "inj_on snd xb" shows "set_tree (treap_of xb) = xb"
assms: shows "\<delta> \<in> elts (\<omega>_sum \<alpha>s)"
records_def: fixes c :: "('c, 'b, 'd) Exchange.configuration_scheme" shows "cri.records c = sum (c_caps c) UNIV + timestamps (zmset_of (c_data_msg c))"
r_2: shows "(w \<circ> r) \<longlonglongrightarrow> limsup w"
assms_2: shows "\<Gamma> \<turnstile>\<^sub>c e2 : REAL"
wf\<^sub>s\<^sub>tI: fixes S :: "('d, 'c) strand_step list"   and V :: "'c set" assumes "wfrestrictedvars\<^sub>s\<^sub>t S \<subseteq> V" shows "wf\<^sub>s\<^sub>t V S"
rename_SKIP: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "rename h \<bottom> = \<bottom>"
forall_spec: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "\<forall>b. P b \<longrightarrow> Q b"   and "P a" shows "Q a"
lexord_mono': fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "\<And>x y. f x y \<longrightarrow> g x y"   and "(xs, ys) \<in> lexord {(x, y). f x y}" shows "(xs, ys) \<in> lexord {(x, y). g x y}"
xs_converg: shows "\<forall>z\<in>set (map (\<lambda>i. Cn (Suc m) (r_code_constn n) [recf.Id (Suc m) i]) [0..<Suc m]). eval z (p # cs) \<down>"
store_typing_imp_inst_length_eq: fixes s :: "s"   and \<S> :: "s_context" assumes "store_typing s \<S>" shows "length (inst s) = length (s_inst \<S>)"
not_col_permutation_4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet B A C"
dASSERT_def: shows "dASSERT \<equiv> iASSERT dRETURN"
cscc_invar_ext_axioms: shows "cscc_invar_ext G l' D'"
V_BCLIM_def: shows "V\<^bsub>BC-LIM\<^esub> \<equiv> \<Union> (range V_bclim)"
degree_smult_eq: fixes a :: "'b"   and p :: "'b poly" shows "degree (smult a p) = (if a = (0::'b) then 0 else degree p)"
ccProd_union1: fixes S' :: "Vars.var set"   and S'' :: "Vars.var set"   and S :: "Vars.var set" shows "(S' \<union> S'') G\<times> S = S' G\<times> S \<squnion> S'' G\<times> S"
uset_cong: fixes p :: "fm"   and x :: "real"   and bs :: "real list"   and U' :: "(Ferrack.num \<times> int) set"   and U :: "(Ferrack.num \<times> int) set" assumes "isrlfm p"   and "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` U' = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (U \<times> U)"   and "\<forall>(t, n)\<in>U. numbound0 t \<and> 0 < n"   and "\<forall>(t, n)\<in>U'. numbound0 t \<and> 0 < n" shows "(\<exists>(t, n)\<in>U. \<exists>(s, m)\<in>U. Ifm (x # bs) (usubst p (Add (Mul m t) (Mul n s), 2 * n * m))) = (\<exists>(t, n)\<in>U'. Ifm (x # bs) (usubst p (t, n)))"
sep_map_general_def: fixes p :: "32 word"   and obj :: "cdl_object"   and gs :: "nat option set" shows "sep_map_general p obj gs \<equiv> \<lambda>s. sep_heap s = [p \<mapsto> obj] \<and> sep_ghost_state s p = gs"
abs_eq: fixes x :: "'b" shows "Var x = abs_trm (PVar x)"
assms_2: shows "x [^]\<^bsub>R\<^esub> CARD('a) ^ m' = x"
sq_norm_poly_eq_0: fixes p :: "'d poly" shows "(\<parallel>p\<parallel>\<^sup>2 = (0::'d)) = (p = 0)"
cnf_form_of_cnf: fixes S :: "'a literal list list" shows "cnf (form_of_cnf S) = set (map set S)"
simps_11: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm"   and q :: "fm" shows "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
hyps: shows "snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<k]) = foldl Gauss_Jordan_column_k (0, A) [0..<k]"
set_sum: fixes H :: "'a set"   and K :: "'a set" assumes "H \<subseteq> carrier A"   and "K \<subseteq> carrier A" shows "H \<minusplus> K = {x. \<exists>h\<in>H. \<exists>k\<in>K. x = h \<plusminus> k}"
elims: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b \<times> 'b"   and y :: "'a \<times> 'a" assumes "vec_map x xa = y" obtains f :: "'b \<Rightarrow> 'a" and x :: "'b" and y :: "'b" where "xa__ = f" and "xa = (x, y)" and "ya__ = (f x, f y)"
must_syncI: fixes t :: "'t"   and x :: "'x"   and m :: "'m" assumes "\<exists>taa x' m' s. t \<turnstile> \<langle>x, m\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t taa" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>"
rim_2: shows "r ` (W \<union> T) \<subseteq> T"
exhaust: fixes y :: "('b, 'a) Op" obtains x11 :: "'a" and x12 :: "'b" and x13 :: "'b" where "y = Dec x11 x12 x13"   | x21 :: "'a" and x22 :: "'b" where "y = Inc x21 x22"
homeomorphism_cont1: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphism S T f g" shows "continuous_on S f"
assms_3: shows "set Procs \<subseteq> dom \<Gamma>"
length_0_conv: fixes xs :: "'d list" shows "(length xs = 0) = (xs = [])"
pathfinish_def: fixes g :: "real \<Rightarrow> 'a" shows "pathfinish g = g 1"
hyps: fixes I :: "nat \<Rightarrow> 'a set" assumes "\<And>i. i \<le> n \<Longrightarrow> ideal (I i) R" shows "canonical_proj_ext I n \<in> ring_hom R (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
intros_4: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> equality_axioms" shows "H \<turnstile> A"
help'''_in: fixes ram :: "'c"   and b :: "'c list"   and xs :: "'c list" assumes "ram \<in> set b" shows "fst (splitAtRec ram xs b) = xs @ fst (splitAtRec ram [] b)"
Sigma_empty2: fixes A :: "'b set" shows "A \<times> {} = {}"
l10: fixes S :: "'a set"   and n :: "nat"   and g :: "'a \<Rightarrow> int" assumes "card S = n"   and "\<And>x. x \<in> S \<Longrightarrow> [g x = a] (mod int p)" shows "[prod g S = a ^ n] (mod int p)"
vcg_pre: fixes coms :: "'a \<Rightarrow> ('b, 'c, 'd, 'e) com"   and p :: "'a"   and aft :: "'e \<Rightarrow> 'c set"   and Q :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool"   and c :: "('b, 'c, 'd, 'e) com"   and R :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool"   and P :: "('b, 'c, 'a, 'd, 'e) system_state \<Rightarrow> bool" assumes "valid_syn coms p aft Q c R"   and "\<And>s. P s \<Longrightarrow> Q s" shows "valid_syn coms p aft P c R"
coeff_pCons': fixes c :: "'a"   and p :: "'a poly"   and n :: "nat" shows "poly.coeff (pCons c p) n = (if n = 0 then c else poly.coeff p (n - 1))"
in_bounds_def: fixes i :: "int"   and k :: "int" shows "(i in_bounds k) = (0 \<le> i \<and> i < k)"
intros: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "encoding.indRelLp Enc (TargetTerm (Enc S)) (SourceTerm S)"
i_shrink_eq_i_last_message_hold_shrink_last: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "0 < k" shows "f \<div> k = f \<longmapsto> k \<div>\<^sub>l k"
integral_fst': fixes f :: "'b \<times> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) f" shows "LINT x|M2. LINT y|M1. f (x, y) = integral\<^sup>L (M2 \<Otimes>\<^sub>M M1) f"
induct: fixes P :: "modality \<Rightarrow> safety \<Rightarrow> 'a Regex.regex \<Rightarrow> bool"   and a0 :: "modality"   and a1 :: "safety"   and a2 :: "'a Regex.regex" assumes "\<And>m uu_ n. P m uu_ (regex.Skip n)"   and "\<And>m g \<phi>. P m g (regex.Test \<phi>)"   and "\<And>m g r s. \<lbrakk>P m g r; P m g s\<rbrakk> \<Longrightarrow> P m g (regex.Plus r s)"   and "\<And>g r s. \<lbrakk>P Futu g s; P Futu Lax r\<rbrakk> \<Longrightarrow> P Futu g (regex.Times r s)"   and "\<And>g r s. \<lbrakk>P Past g r; P Past Lax s\<rbrakk> \<Longrightarrow> P Past g (regex.Times r s)"   and "\<And>m g r. P m g r \<Longrightarrow> P m g (regex.Star r)" shows "P a0 a1 a2"
C\<^sub>0s: fixes x :: "i"   and y :: "i" shows "(E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)) \<and> (E x \<^bold>\<leftarrow> E (dom x)) \<and> (E y \<^bold>\<leftarrow> E (cod y))"
H_1: fixes n :: "nat"   and m :: "nat" shows "AE x in M. u (n + m) x \<le> u n x + u m ((T ^^ n) x)"
alphaAbs_qAbs_iff_alphaAbs_all_distinct_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_all_distinct_qAFresh xs x X xs' x' X'"
b_least_less_impl_nz: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "b_least f x < x" shows "f x (b_least f x) \<noteq> 0"
inj_on: fixes A :: "'b set" shows "inj_on (nth_least A) {i. enat i < esize A}"
xs_def: shows "xs = coeffs ([:1::'a, - (1::'a):] * g)"
simps_9: fixes a :: "MIR.num" shows "zsplit0 (Floor a) = (let (i', a') = zsplit0 a in (i', Floor a'))"
final_Accessible_dfa: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.final (dfa.Accessible_dfa M) = dfa.final M \<inter> dfa.accessible M"
peval_closed_1: fixes R :: "('c, 'd) ring_scheme"   and xs :: "'c list"   and e :: "pexpr" assumes "field R"   and "cring.in_carrier R xs" shows "field.peval R xs e \<in> carrier R"
assms_3: shows "p \<in># image_mset set_mset P"
inj_compose: fixes x :: "'a"   and y :: "'a" assumes "is_inj x"   and "is_inj y" shows "is_inj (x ; y)"
scaleRat_leq1: fixes a :: "'a"   and b :: "'a"   and k :: "rat" assumes "a \<le> b"   and "0 < k" shows "k *R a \<le> k *R b"
cases: fixes x :: "'a list \<times> 'a list" obtains "x = ([], [])"   | l :: "'a" where "x = ([], [l__])"   | l :: "'a" and v :: "'a" and va :: "'a list" where "x = ([], la__ # v # va)"   | f :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = (f__ # fs__, l # ls)"   | f :: "'a" and fs :: "'a list" where "x = (f # fs, [])"
ivl_disj_un_1: fixes l :: "'b" shows "{l} \<union> {l<..} = {l..}"
simps_1: fixes h :: "'b \<Rightarrow> int"   and e :: "'b" shows "remove h e Tip = Tip"
apsnd_def': fixes f :: "'b \<Rightarrow> 'c" shows "apsnd f = (\<lambda>(k, v). (k, f v))"
stricts_1: shows "EMB('e)\<cdot>\<bottom> = \<bottom>"
ih_15: shows "InvariantGetReasonIsReason (getReason state') (getF state') (getM state') (set (getQ state'))"
F'_iter_def: shows "F'_iter = while (\<lambda>A. F' A \<noteq> A) F' (XD_smallest ds X, dual (XH_largest ds X))"
chine_eq_apex: shows "chine = \<mu>\<nu>.apex"
rev_finite_subset: fixes B :: "'b set"   and A :: "'b set" assumes "finite (B::'b set)"   and "(A::'b set) \<subseteq> (B::'b set)" shows "finite (A::'b set)"
primitive_root_unityD_1: fixes k :: "nat"   and x :: "'a" assumes "primitive_root_unity k x" shows "k \<noteq> 0"
assms_1: fixes x :: "'a" assumes "P x" shows "P (f x)"
fin_digraph_subgraph: fixes H :: "('a, 'b) pre_digraph" assumes "subgraph H G" shows "fin_digraph H"
coeffs_monom: fixes a :: "'a"   and n :: "nat" shows "coeffs (Polynomial.monom a n) = (if a = (0::'a) then [] else replicate n (0::'a) @ [a])"
is_nextElem_nth2: fixes ls :: "'a list"   and x :: "'a"   and y :: "'a" assumes "\<exists>i j. i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i mod |ls| = j" shows "is_nextElem ls x y"
repNEG_INF: fixes r :: "real" assumes "r \<le> real_of_int (sint NEG_INF)" shows "NEG_INF \<equiv>\<^sub>E r"
ub_exp_nonneg: fixes prec :: "nat"   and x :: "float" shows "0 \<le> real_of_float (ub_exp prec x)"
cast_env: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and v :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and v' :: "val"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E' :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>\<lparr>C\<rparr>Val v,(h, l)\<rangle> \<rightarrow> \<langle>Val v',(h, l)\<rangle>" shows "P,E' \<turnstile> \<langle>\<lparr>C\<rparr>Val v,(h, l)\<rangle> \<rightarrow> \<langle>Val v',(h, l)\<rangle>"
\<sigma>: fixes v :: "'a" assumes "v \<in> winning_region p" shows "\<sigma>' v = \<sigma> v"
prems_2: shows "wf\<^sub>t\<^sub>r\<^sub>m (Fun f' T')"
lucas_lehmer_correct: fixes p :: "nat" shows "prime (2 ^ p - 1) = (prime p \<and> (p = 2 \<or> 2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2)))"
hyperUn_comm: fixes A :: "'f set option"   and B :: "'f set option" shows "A \<squnion> B = B \<squnion> A"
less_imp_not_less: fixes x :: "'c Word.word"   and y :: "'c Word.word" assumes "x <s y" shows "(\<not> y <s x) = True"
h3b_a: shows "\<forall>x. List.member (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets))) x \<longrightarrow> x < length subsets"
intS: shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) absolutely_integrable_on S"
set_bd_2: fixes x :: "('o, 'p, 'n) F" shows "|F'set4 x| \<le>o F'bd"
k_is_map: shows "is_left_adjoint t\<^sub>0"
subspace_diff: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "csubspace S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
set_4_atLeast0: shows "{0..<4} = {0, 1, 2, 3}"
le_infI2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and b :: "'e"   and x :: "'e"   and a :: "'e" assumes "class.semilattice_inf inf less_eq less"   and "less_eq b x" shows "less_eq (inf a b) x"
Diff_Un: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A - (B \<union> C) = (A - B) \<inter> (A - C)"
mod_plus_right: fixes a :: "nat"   and x :: "nat"   and m :: "nat"   and b :: "nat" shows "((a + x) mod m = (b + x) mod m) = (a mod m = b mod m)"
ESem_Lam: fixes x :: "Vars.var"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue" shows "\<N>\<lbrakk> Lam [x]. e \<rbrakk>\<^bsub>\<rho>\<^esub> = C_case\<cdot>((\<Lambda> f r. CFn\<cdot>(\<Lambda> v. f\<cdot>v|\<^bsub>r\<^esub>))\<cdot>(\<Lambda> v. \<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>(x := v)\<^esub>))"
r_nf_t_prim: shows "prim_recfn 3 r_nf_t"
happens_before_committed_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "happens_before_committed P E J = (\<forall>n. happens_before P (justifying_exec (J n)) |` committed (J n) = inv_imageP (happens_before P E) (action_translation (J n)) |` committed (J n))"
Graph7: fixes M :: "node list"   and E :: "(nat \<times> nat) list"   and T :: "nat"   and a :: "node" assumes "Proper_Edges (M, E)" shows "Proper_Edges (M[T := a], E)"
boundedByUn: fixes N :: "nat"   and A :: "(nat \<times> 'b) set"   and B :: "(nat \<times> 'b) set" shows "boundedBy N (A \<union> B) = (boundedBy N A \<and> boundedBy N B)"
discI_2: fixes literal :: "'a literal"   and x21 :: "char list"   and x22 :: "'a list" assumes "literal = Neg x21 x22" shows "\<not> sign literal"
simps_6: fixes n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" shows "local.wf n (Times r s) = (local.wf n r \<and> local.wf n s)"
inv'_2: shows "length (map of_int_hom.vec_hom fs') = m"
simps_1: fixes F :: "'a \<Rightarrow> 'b" shows "lupd [] [] F = F"
fminus_fempty: fixes A :: "'a fset" shows "A |-| {||} = A"
CP_subseteq_TOTAL: shows "CP \<subseteq> TOTAL"
analz_LtKeys_Tag: fixes implem :: "chan \<Rightarrow> msg"   and NI :: "msg set"   and K :: "msg set" assumes "semivalid_implem implem"   and "NI \<subseteq> payload"   and "K \<subseteq> range LtK" shows "analz (NI \<union> K \<union> Tags) \<subseteq> analz NI \<union> K \<union> Tags"
substInp_def2: fixes ys :: "'varSort"   and Y :: "('f, 'g, 'varSort, 'var, 'h) qTerm set"   and y :: "'var" shows "substInp ys Y y = lift (subst ys Y y)"
pr_conv_1_to_2_def: shows "pr_conv_1_to_2 = (\<lambda>f x y. f (c_pair x y))"
plus_\<I>_sel_1: fixes \<I>l :: "('y, 'aa) \<I>"   and \<I>r :: "('z, 'ab) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
split_block_5: fixes A :: "'a mat"   and sr1 :: "nat"   and sc1 :: "nat"   and A1 :: "'a mat"   and A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat"   and sr2 :: "nat"   and sc2 :: "nat" assumes "split_block A sr1 sc1 = (A1, A2, A3, A4)"   and "dim_row A = sr1 + sr2"   and "dim_col A = sc1 + sc2" shows "A = four_block_mat A1 A2 A3 A4"
maxsimp_def: fixes x :: "'a set" shows "folding.chamber x \<equiv> x \<in> X \<and> (\<forall>z\<in>X. order.greater_eq z x \<longrightarrow> z = x)"
is_fair_run_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b" assumes "df_system s\<^sub>0 lstep" shows "df_system.is_fair_run s\<^sub>0 lstep s \<equiv> \<exists>l. df_system.is_lrun s\<^sub>0 lstep l s \<and> df_system.is_lfair lstep l s"
ereal_uminus_lessThan: fixes a :: "ereal" shows "uminus ` {..<a} = {- a<..}"
t_find_closest_cnt: fixes p :: "int \<times> int"   and \<delta> :: "real"   and ps :: "(int \<times> int) list" shows "t_find_closest p \<delta> ps \<le> 1 + length (filter (\<lambda>q. real_of_int (snd q - snd p) \<le> \<delta>) ps)"
ring_is_ag: fixes R :: "('h, 'i) Ring_scheme" assumes "Ring R" shows "aGroup R"
assms_2: shows "y \<subseteq>\<^sub>L x"
finite_maxsimp: fixes x :: "'a set" assumes "chamber x" shows "finite x"
assms_2: shows "AE x in M. g x \<le> f x"
pivot_update_precond_2: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "x\<^sub>i \<notin> lvars (\<T> (pivot x\<^sub>i x\<^sub>j s))"
map_lsumr_plus_oracle: fixes oracle1 :: "'h \<Rightarrow> 'i \<Rightarrow> ('l \<times> 'h) spmf"   and oracle2 :: "'h \<Rightarrow> 'j \<Rightarrow> ('m \<times> 'h) spmf"   and oracle3 :: "'h \<Rightarrow> 'k \<Rightarrow> ('n \<times> 'h) spmf" shows "map_fun id (map_fun lsumr (map_spmf (map_prod rsuml id))) ((oracle1 \<oplus>\<^sub>O oracle2) \<oplus>\<^sub>O oracle3) = oracle1 \<oplus>\<^sub>O oracle2 \<oplus>\<^sub>O oracle3"
set_minus_one: fixes v :: "'a"   and v' :: "'a"   and vs :: "'a list" assumes "v \<noteq> v'"   and "v' \<in> set vs" shows "(set vs - {v'} \<subseteq> {v}) = (set vs = {v'} \<or> set vs = {v, v'})"
bounded_solution_set_strips'_def: fixes \<Pi> :: "('c \<times> 'd) strips_problem"   and k :: "nat" shows "bounded_solution_set_strips' \<Pi> k \<equiv> {\<pi>. STRIPS_Semantics.is_serial_solution_for_problem \<Pi> \<pi> \<and> length \<pi> = k}"
b_pos: shows "\<forall>\<^sub>F x in at_top. 0 < hd basis x"
emeasure_eq_measure: fixes n :: "nat"   and A :: "real set" shows "emeasure (M n) A = ennreal (Mn.prob n A)"
simps_2: fixes l1 :: "value"   and v :: "'b" shows "AExp_Lexorder.less_aexp_aux (L l1) (V v) = True"
awalkE: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" obtains "set (pawalk_verts u p) \<subseteq> pverts (subdivide G e w)" and "set p \<subseteq> parcs (subdivide G e w)" and "pcas u p v" and "hd (pawalk_verts u p) = u" and "last (pawalk_verts u p) = v"
upset_prop2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "y \<in> \<up> x"
top_on_acom_narrow: fixes C1 :: "'a st option acom"   and X :: "char list set"   and C2 :: "'a st option acom" assumes "top_on\<^sub>c C1 X"   and "strip C1 = strip C2"   and "top_on\<^sub>c C2 X" shows "top_on\<^sub>c (C1 \<triangle> C2) X"
returns_result_def: fixes h :: "'d"   and p :: "('d, 'e, 'f) prog"   and r :: "'f" shows "h \<turnstile> p \<rightarrow>\<^sub>r r = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (r', x) \<Rightarrow> r = r')"
wf_subst_nil: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t Var"
two_sides_cases: fixes PO :: "'p"   and A :: "'p"   and B :: "'p"   and P :: "'p" assumes "\<not> Col PO A B"   and "PO P OS A B" shows "PO A TS P B \<or> PO B TS P A"
opt_map_def: fixes f :: "'d \<Rightarrow> 'f option"   and g :: "'f \<Rightarrow> 'e option" shows "f |> g \<equiv> \<lambda>s. case f s of None \<Rightarrow> None | Some x \<Rightarrow> g x"
snd_mono: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" assumes "x \<le> y" shows "snd x \<le> snd y"
fps_mult_fps_X_nonzero_1: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "fps_X * f \<noteq> 0"
minorM_def: fixes M :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" shows "minorM M i j = (\<chi>k l. if k = i \<and> l = j then 1::'d else if k = i \<or> l = j then 0::'d else M $ k $ l)"
height_balL: fixes m :: "nat"   and l :: "('b \<times> nat) tree"   and r :: "('b \<times> nat) tree"   and a :: "'b" assumes "HBT m"   and "HBT.hbt m l"   and "HBT.hbt m r"   and "height l = height r + m + 1" shows "height (HBT.balL m l a r) \<in> {height r + m + 1, height r + m + 2}"
monotonicity: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> R\<^sup>*" shows "(a\<^sup>\<bullet>, b\<^sup>\<bullet>) \<in> R\<^sup>*"
add_right_mono: fixes a :: "'h multiset"   and b :: "'h multiset"   and c :: "'h multiset" assumes "a \<subseteq># b" shows "a + c \<subseteq># b + c"
is_type_array: fixes P :: "'b prog"   and A :: "ty" shows "is_type P (A\<lfloor>\<rceil>) = (case ground_type A of NT \<Rightarrow> False | Class C \<Rightarrow> is_class P C | _ \<Rightarrow> True)"
gp_int: shows "integrable M (\<lambda>x. f x * enn2real (nn_cond_exp M F gp x))"
additive_mu_G: shows "Measure_Space.additive generator \<mu>G"
coerce_idem: fixes x :: "'c" assumes "DEFL('c) \<sqsubseteq> DEFL('d)" shows "COERCE('d, 'e)\<cdot>(COERCE('c, 'd)\<cdot>x) = COERCE('c, 'e)\<cdot>x"
composition_helper: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q"   and "pres x p" shows "p \<cdot> (q \<cdot> x)\<^sup>\<dagger> \<cdot> !q \<cdot> p = p \<cdot> (q \<cdot> x)\<^sup>\<dagger> \<cdot> !q"
interpret_floatarith_inner_floatariths: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and vs :: "real list"   and x :: "'a"   and y :: "'a" assumes "length xs = DIM('a)"   and "length ys = DIM('a)"   and "einterpret xs vs = x"   and "einterpret ys vs = y" shows "interpret_floatarith (inner_floatariths xs ys) vs = x \<bullet> y"
AbstForm_trans_fm: fixes i :: "name"   and e :: "(name, hf) finfun"   and A :: "fm" shows "AbstForm (q_Var i) 0 \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<lbrakk>quot_dbfm (trans_fm [i] A)\<rbrakk>e"
focus_coeffs_subset_Polys': fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and Y :: "'b set"   and X :: "'b set" assumes "p \<in> P[Y]" shows "range (lookup (focus X p)) \<subseteq> P[Y - X]"
distinct_29: fixes x51 :: "'a rexp"   and x52 :: "'a rexp"   and x6 :: "'a rexp" shows "Times x51 x52 \<noteq> Star x6"
lambda_interpret_1: fixes v :: "i"   and a :: "\<kappa>"   and b :: "\<kappa>"   and R :: "\<Pi>\<^sub>2" assumes "[a \<^bold>= b in v]" shows "(\<^bold>\<lambda>x. \<lparr>R,x\<^sup>P,a\<rparr>) = (\<^bold>\<lambda>x. \<lparr>R,x\<^sup>P,b\<rparr>)"
BinOp: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e1 :: "expr"   and T1 :: "ty"   and e2 :: "expr"   and T2 :: "ty"   and binop :: "binop"   and T :: "ty" assumes "E,dt\<Turnstile>e1\<Colon>-T1"   and "E,dt\<Turnstile>e2\<Colon>-T2"   and "wt_binop (prg E) binop T1 T2"   and "T = PrimT (binop_type binop)" shows "E,dt\<Turnstile>BinOp binop e1 e2\<Colon>-T"
index_of_zero_def: shows "index_of_zero = sch_to_nat Base_zero"
k'_2: shows "i - j cd\<^bsup>\<pi> \<guillemotleft> j\<^esup>\<rightarrow> k + j - j"
eval_arith_domain: fixes xs :: "dynamic list"   and op :: "op" assumes "length xs = ar op" shows "\<exists>y. eval op xs = y"
pick_def: fixes X :: "'c set" shows "pick X \<equiv> SOME x. x \<in> X"
comp_null_2: fixes f :: "'a \<times> 'a" shows "CC.comp f CC.null = CC.null"