hcju/isabelleps huggingface dataset:
at_within_Icc_at_left: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order_topology less_eq less open"   and "less a b" shows "topological_space.at_within open b (ord.atLeastAtMost less_eq a b) = topological_space.at_within open b (ord.lessThan less b)"
mono_Det: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "(P [+] S) \<sqsubseteq> (Q [+] S)"
prems: shows "sp F (x # xs) \<Sigma> = Ok \<Sigma>'"
unroll: shows "(s, t) \<in> (unroll G l e c)\<^sup>*"
prefix_imp_take_eq: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "n \<le> length xs"   and "prefix xs ys" shows "xs \<down> n = ys \<down> n"
map_dual_def: fixes f :: "'a \<Rightarrow> 'b" shows "\<partial>\<^sub>F f = \<partial> \<circ> f \<circ> \<partial>"
is_bound_def: fixes rel :: "(nat \<times> nat) set"   and n :: "nat" shows "is_bound rel n = (\<forall>m\<in>Field rel. m < n)"
not_0_coeffs_not_Nil: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "coeffs p \<noteq> []"
coeff_reciprocal: fixes i :: "nat"   and p :: "nat"   and P :: "'b poly" assumes "i \<le> p"   and "degree P \<le> p" shows "coeff (reciprocal_poly p P) i = coeff P (p - i)"
RP_saturated_if_fair: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>) Sts"   and "fair_state_seq Sts"   and "Q_of_state (lhd Sts) = {}" shows "redundancy_criterion.saturated_upto (ground_resolution_with_selection.ord_\<Gamma> (S_Q Sts)) src.Rf (standard_redundancy_criterion.Ri (ground_resolution_with_selection.ord_\<Gamma> (S_Q Sts))) (Liminf_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts))"
\<b>_zero_exact: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "qs \<noteq> []" shows "Suc (Max (poly_deg ` fst ` set qs)) \<le> \<b> (exact k qs) 0"
\<theta>\<^sub>\<tau>': shows "\<guillemotleft>(u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] : t\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine \<Rightarrow> u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
l11_36: 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 A' :: "'a"   and D :: "'a"   and E :: "'a"   and D' :: "'a"   and C :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A' \<noteq> B"   and "D \<noteq> E"   and "D' \<noteq> E"   and "Bet A B A'"   and "Bet D E D'" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F = Tarski_neutral_dimensionless.LeA Bet Cong D' E F A' B C"
eigen_value_transpose: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and v :: "'a" shows "eigen_value (Finite_Cartesian_Product.transpose A) v = eigen_value A v"
a1: shows "s' = snd (fst (store_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
abs_eq: fixes x :: "int" shows "bit (Word.Word x) = (\<lambda>n. n < LENGTH('b) \<and> bit x n)"
simps_5: shows "OUTfromCh data5 = {data2}"
inducts: fixes P :: "'a + 'b \<Rightarrow> bool"   and sum :: "'a + 'b" assumes "\<And>a. P (Inl a)"   and "\<And>b. P (Inr b)" shows "P sum"
inj_on_app_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and S :: "('a, 'b) pre_digraph set" assumes "digraph_isomorphism hom"   and "S \<subseteq> {H. subgraph H G}" shows "inj_on (app_iso hom) S"
less_eq_matrix_def: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd" shows "(f \<preceq> g) = (\<forall>e. f e \<le> g e)"
return_env_def: fixes return :: "'i \<Rightarrow> 'h"   and x :: "'i" shows "return_env return x = EnvT (\<lambda>_. return x)"
less_bot_Bot: fixes x :: "'b bot" shows "(x < bot.Bot) = False"
inf_left_idem: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "infdist x B \<le> D"
subtopology_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
P_def: shows "P = (\<lambda>(n, k). fact k = (\<Prod>i<n. 2 ^ n - 2 ^ i))"
subset_class_def: fixes \<Omega> :: "'b set"   and M :: "'b set set" shows "subset_class \<Omega> M \<equiv> M \<subseteq> Pow \<Omega>"
word_neg_and_le: fixes ptr :: "'b word"   and n :: "nat" shows "ptr \<le> (ptr AND NOT (mask n)) + (2 ^ n - 1)"
history: shows "\<exists>hs. rev a @ h = [x, y] @ hs"
lex_trans: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex a b"   and "Counterclockwise_2D_Arbitrary.lex b c" shows "Counterclockwise_2D_Arbitrary.lex a c"
Quotient_equiv_abs2: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'j"   and y :: "'j" assumes "Quotient R Abs Rep T"   and "R x y" shows "T y (Abs x)"
UINF_upto_expand_first: fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> 'a upred" assumes "m < n" shows "(\<Sqinter> i \<in> {m..<n} \<bullet> P i) = (P m \<or> (\<Sqinter> i \<in> {Suc m..<n} \<bullet> P i))"
selem_bottom_iff: fixes x :: "'a"   and xs :: "[:'a:]" shows "(selem\<cdot>x\<cdot>xs = \<bottom>) = (xs = \<bottom> \<or> xs \<noteq> [::] \<and> x = \<bottom>)"
pcr_spmf_gpv_def: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "pcr_spmf_gpv A = cr_spmf_gpv OO rel_gpv A (=)"
valid_refs_invD_3: fixes x :: "'e"   and m :: "'f"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state"   and y :: "'e" assumes "x \<in> mut_m.tso_store_refs m s"   and "(x reaches y) s"   and "valid_refs_inv s" shows "obj_at (\<lambda>s. True) y s"
normalized_cancels_to: fixes l :: "(bool \<times> 'd) list" shows "cancels_to l (Cancelation.normalize l)"
simps_1: fixes i :: "name"   and x :: "trm" shows "subst i x zer = zer"
marked_insertionD: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and m :: "'h"   and r :: "'g"   and f :: "'e"   and r' :: "'g"   and ws :: "('e, 'f, 'g) mem_store_action list" assumes "sys_mem_store_buffers (mutator m) s = mw_Mutate r f (Some r') # ws"   and "mut_m.marked_insertions m s" shows "marked r' s"
showsp_nat_append: fixes p :: "nat"   and n :: "nat"   and x :: "char list"   and y :: "char list" shows "showsp_nat p n (x @ y) = showsp_nat p n x @ y"
sgn2_is_pr: shows "sgn2 \<in> PrimRec1"
ind_zero_str_def: fixes V :: "'b set" shows "ind_zero_str V \<equiv> zero_str (X \<inter> V)"
trans_FD: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q"   and "Q \<sqsubseteq>\<^sub>F\<^sub>D S" shows "P \<sqsubseteq>\<^sub>F\<^sub>D S"
sorted_root_list_set_def: fixes p :: "(real \<times> real \<times> real) set" shows "sorted_root_list_set p \<equiv> sorted_list_of_set (root_set p)"
assms_3: fixes g :: "'a"   and h :: "'a" assumes "g \<in> G"   and "h \<in> G" shows "g \<cdot> h \<in> G"
dvdmE: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" assumes "poly_mod.dvdm m f g" obtains h :: "int poly" where "poly_mod.Mp m g = poly_mod.Mp m (f * h)" and "poly_mod.Mp m h = h"
imp_def: fixes \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<longrightarrow>\<^sup>b \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<longrightarrow> \<gamma>' t n"
filter_nb_def: fixes b :: "'c \<Rightarrow> bool"   and I :: "'c nres" shows "filter_nb b I \<equiv> I \<bind> (\<lambda>s. ASSUME (\<not> b s) \<bind> (\<lambda>_. RETURN s))"
fin_cut_same_snoc: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "fin_cut_same x (xs @ [y]) = (if x = y then fin_cut_same x xs else xs @ [y])"
rc: shows "\<forall>z\<in>l ` {- r..r}. 0 < f z \<bullet> f x"
frequently_def: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "frequently P F = (\<not> (\<forall>\<^sub>F x in F. \<not> P x))"
One_not_eq_j1: shows "J.One \<noteq> J.j1"
compl_mono: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq x y" shows "less_eq (uminus y) (uminus x)"
leftderives_refl: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.leftderives \<NN> \<TT> \<RR> u u"
assms_1: shows "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"
mult_map_scale_conv_mult: fixes s :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'd" shows "Poly_Mapping.map ((*) s) p = monomial s (0::'c) * p"
distinct_3: fixes x1 :: "agent"   and x3 :: "agent" shows "sharK x1 \<noteq> privK x3"
map\<^sub>0_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (\<tau>\<^sub>0 a) = G.map\<^sub>0 a"
MkArr_in_hom: fixes A :: "'a"   and B :: "'a"   and f :: "'a set" assumes "A \<in> Collect B.obj"   and "B \<in> Collect B.obj"   and "f \<in> Hom A B" shows "\<guillemotleft>MkArr A B f : MkIde A \<rightarrow> MkIde B\<guillemotright>"
INF_mono: fixes B :: "'b set"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd \<times> 'e"   and g :: "'b \<Rightarrow> 'd \<times> 'e" assumes "\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. less_eq_prod (f n) (g m)" shows "less_eq_prod (Inf_prod (f ` A)) (Inf_prod (g ` B))"
nsum_eqTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A \<and> g j \<in> carrier A \<and> f j = g j) \<longrightarrow> \<Sigma>\<^sub>e A f n = \<Sigma>\<^sub>e A g n"
exhaust: fixes y :: "('a, 'b) BinomialHeapStruc.BinomialTree" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "nat" and x4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "y = BinomialHeapStruc.Node x1 x2 x3 x4"
ncoplanar_perm_6: 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"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B A C D"
assms_4: shows "\<beta> \<in>\<^sub>\<circ> \<gamma>"
DERIV_D: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c" assumes "(f has_field_derivative D) (at x)" shows "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'c\<rightarrow> D"
finite_measure_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets N"   and "B \<subseteq> A" shows "Sigma_Algebra.measure N (A - B) = Sigma_Algebra.measure N A - Sigma_Algebra.measure N B"
distinct_1: fixes x2 :: "nat" shows "INIT \<noteq> WAIT x2"
ortho: fixes x :: "'a vec" assumes "x \<in> set S" shows "v \<bullet> x = (0::'a)"
build_local_iwf: fixes t :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" assumes "tfinite t"   and "local.wf t" shows "\<exists>it. local_iwf it (fst (tree.root t))"
assms_3: shows "Sigma_Algebra.measure M t = Sigma_Algebra.measure M (space M)"
composition_of_substs_lit: fixes l :: "'b literal"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list" shows "subst_lit (subst_lit l \<sigma>) \<eta> = subst_lit l (\<sigma> \<lozenge> \<eta>)"
outer_measure_of_attain: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<subseteq> space M" shows "\<exists>E\<in>sets M. A \<subseteq> E \<and> outer_measure_of M A = emeasure M E"
inverse_le_imp_le_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq (inverse a) (inverse b)"   and "less b zero" shows "less_eq b a"
try_spmf_bind_out: fixes p :: "'a spmf"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'b spmf" assumes "lossless_spmf p" shows "p \<bind> (\<lambda>x. TRY f x ELSE q) = TRY p \<bind> f ELSE q"
compute_monom_mult_poly_mapping: fixes c :: "'c"   and t :: "'a"   and xs :: "('a \<times> 'b, 'c) fmap" shows "pprod.monom_mult c t (Pm_fmap xs) = Pm_fmap (if c = (0::'c) then fmempty else pprod.shift_map_keys t ((*) c) xs)"
nth_root_nat_0: fixes k :: "nat" shows "nth_root_nat k 0 = 0"
an_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "an_sds agents alts sds \<equiv> anonymous_sds agents alts sds \<and> neutral_sds agents alts sds"
9: fixes p :: "nat \<times> 'b" assumes "p \<in> sset (gtrace (sdrop n (fromN 1 ||| r)) (gtarget (stake n (fromN 1 ||| r)) (0, p__)))"   and "gaccepting A p" shows "even (f p)"
bind_ASSERT_eq_if: fixes \<Phi> :: "bool"   and m :: "'e nres" shows "ASSERT \<Phi> \<bind> (\<lambda>_. m) = (if \<Phi> then m else FAIL)"
L_delta: fixes a :: "'b"   and s :: "bool \<times> (bool \<times> 'b) rexp" shows "L_a (delta_a a s) = Deriv a (L_a s)"
expandAction': fixes R :: "pi"   and S :: "pi set"   and Rs :: "residual" assumes "(R, S) \<in> sumComposeSet"   and "R \<longmapsto> Rs" shows "\<exists>P\<in>S. P \<longmapsto> Rs"
t\<^sub>p_def: fixes s :: "'a list"   and q :: "'a"   and a :: "nat \<times> nat list" shows "t\<^sub>p s q a = (let (mf, sws) = a in index (swaps sws s) q + length sws)"
chain_boundary_add: fixes p :: "nat"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "chain_boundary p (a + b) = chain_boundary p a + chain_boundary p b"
f6: shows "3 \<le> rk {A, A', C', C, \<beta>}"
distinct_sort: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "distinct (FWNormalisationCore.sort xs l) = distinct xs"
eucl_rel_poly_unique: fixes x :: "'b poly"   and y :: "'b poly"   and q1 :: "'b poly"   and r1 :: "'b poly"   and q2 :: "'b poly"   and r2 :: "'b poly" assumes "eucl_rel_poly x y (q1, r1)"   and "eucl_rel_poly x y (q2, r2)" shows "q1 = q2 \<and> r1 = r2"
eq_empty: shows "{xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A} = {}"
simps: fixes N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "N_of_state (N, P, Q) = N"
is_Pointwise_Output_Shrink_def: fixes pointwise_shrink :: "'d list \<Rightarrow> 'd"   and output_fun :: "'d \<Rightarrow> 'e message_af" shows "is_Pointwise_Output_Shrink pointwise_shrink output_fun \<equiv> \<forall>cs. output_fun (pointwise_shrink cs) = last_message (map output_fun cs)"
WT_resourceI: fixes \<I> :: "('c, 'd) \<I>"   and res :: "('c, 'd) resource" assumes "\<And>q r res'. \<lbrakk>q \<in> outs_\<I> \<I>; (r, res') \<in> set_spmf (run_resource res q)\<rbrakk> \<Longrightarrow> r \<in> responses_\<I> \<I> q \<and> \<I> \<turnstile>res res' \<surd>" shows "\<I> \<turnstile>res res \<surd>"
2: shows "(x \<otimes>\<^bsub>F\<^esub> (y \<otimes>\<^bsub>F\<^esub> z)) a = x a \<otimes> y a \<otimes> z a"
delete_edges_wf: fixes G :: "'b graph"   and E :: "('b \<times> 'b) set" assumes "wf_graph G" shows "wf_graph (delete_edges G E)"
not_extreme: fixes p :: "alt \<Rightarrow> real"   and b :: "alt" assumes "\<not> extreme p b" shows "\<exists>a c. distinct [a, b, c] \<and> \<not> p a < p b \<and> \<not> p b < p c"
assms: shows "\<turnstile> (\<Gamma> @ VarB x Q # \<Delta>) ok"
is_zero_cases: fixes x :: "('a, 'b) IEEE.float" assumes "IEEE.is_zero x" obtains "x = 0"   | "x = - 0"
subset_fSuc: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and f :: "nat \<Rightarrow> 'a set"   and u :: "nat"   and v :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "\<And>u. f u \<subseteq> f (Suc u)"   and "u \<le> v" shows "f u \<subseteq> f v"
aff_dim_convex_Int_open: fixes S :: "'a set"   and T :: "'a set" assumes "convex S"   and "open T"   and "S \<inter> T \<noteq> {}" shows "aff_dim (S \<inter> T) = aff_dim S"
deg: fixes u :: "'a" assumes "u \<in> verts G" shows "in_degree G u = out_degree G u"
dvd_elements_mult_matrix_left_right: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and P :: "'b mat"   and Q :: "'b mat"   and x :: "'b" assumes "A \<in> carrier_mat m n"   and "P \<in> carrier_mat m m"   and "Q \<in> carrier_mat n n"   and "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd A $$ (i, j)" shows "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd (P * A * Q) $$ (i, j)"
simps_1: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and p :: "real mpoly" shows "linear_substitution var a b (Eq p) = (let d = MPoly_Type.degree p var in Eq (\<Sum>i = 0..<d + 1. isolate_variable_sparse p var i * a ^ i * b ^ (d - i)))"
kernel_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" shows "kernel G H h = {x \<in> carrier G. h x = \<one>\<^bsub>H\<^esub>}"
degree_div: fixes q :: "'b poly"   and p :: "'b poly" assumes "q dvd p" shows "degree (p div q) = degree p - degree q"
distinct_297: fixes x151 :: "floatarith"   and x152 :: "nat"   and x17 :: "nat" shows "Power x151 x152 \<noteq> Var x17"
SC_to_LSC: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "\<Gamma> + image_mset \<^bold>\<not> \<Delta> \<Rightarrow>\<^sub>n"
assms_3: shows "b \<in> obj\<^bsub>B\<^esub>"
refl: shows "refl_on {x \<in> X. \<exists>d. I x = Intv d} r"
deriv_cmult: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and c :: "'a" assumes "f field_differentiable at z" shows "deriv (\<lambda>w. c * f w) z = c * deriv f z"
roifex_var_not_in_subtree_2: fixes b :: "'a ifex"   and v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" assumes "ro_ifex b"   and "b = IF v t e" shows "v \<notin> ifex_var_set e"
restrict_empty_left: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list" shows "[] \<langle>f\<rangle> ls = mbot"
le_inf_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> y \<sqinter> z) = (x \<le> y \<and> x \<le> z)"
distinct_4: shows "TernaryUnknown \<noteq> TernaryTrue"
fwi_fw_upd_mono: fixes k :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "k \<le> n"   and "i \<le> n"   and "j \<le> n" shows "fwi m n k i j i j \<le> fw_upd m k i j i j"
num_params_tmf_polys2: fixes z :: "float interval list" shows "num_params (snd (tmf_polys z)) \<le> Suc 0"
graph_homomorphism_def2: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'c) labeled_graph"   and f :: "('b \<times> 'c) set" shows "graph_homomorphism G\<^sub>1 G\<^sub>2 f = (vertices G\<^sub>1 = Domain f \<and> univalent f \<and> graph G\<^sub>1 \<and> graph G\<^sub>2 \<and> graph_union (map_graph f G\<^sub>1) G\<^sub>2 = G\<^sub>2)"
spmf_of_pmf_bind: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf" shows "spmf_of_pmf (p \<bind> f) = p \<bind> (\<lambda>x. spmf_of_pmf (f x))"
Fun_corestr: fixes f :: "'s" assumes "arr f" shows "Fun (corestr f) = Fun f"
ennreal_of_nat_neq_top: fixes i :: "nat" shows "of_nat i \<noteq> \<top>"
set_sort_oalist_aux: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "set (sort_oalist_aux ko xs) = set (fst xs)"
ideal_generated_subset: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "class.ring plus zero minus uminus times"   and "S \<subseteq> T" shows "ring.ideal_generated plus zero uminus times S \<subseteq> ring.ideal_generated plus zero uminus times T"
simps_6: fixes name :: "String.literal"   and args :: "expr list"   and prog :: "program"   and g :: "gState\<^sub>I"   and l :: "pState" shows "evalEffect (EERun name args) prog g l = runProc name args prog g l"
Uint32_def: shows "Uint32 \<equiv> map_fun int_of_integer Abs_uint32 word_of_int"
plus_iarray_def: fixes A :: "'c iarray"   and B :: "'c iarray" shows "A + B = IArray.of_fun (\<lambda>n. A !! n + B !! n) (IArray.length A)"
prv_dsj_com_imp3: fixes \<phi>1 :: "fm"   and \<phi>2 :: "fm"   and \<phi>3 :: "fm" assumes "\<phi>1 \<in> UNIV"   and "\<phi>2 \<in> UNIV"   and "\<phi>3 \<in> UNIV" shows "{} \<turnstile> \<phi>1 OR \<phi>2 OR \<phi>3 IMP \<phi>2 OR \<phi>1 OR \<phi>3"
tranclp_trancl_eq: fixes r :: "('c \<times> 'c) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>+\<^sup>+ = (\<lambda>x xa. (x, xa) \<in> r\<^sup>+)"
tendsto_unique: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "F \<noteq> bot"   and "(f \<longlongrightarrow> a) F"   and "(f \<longlongrightarrow> b) F" shows "a = b"
emeasure_pair_measure_Times: fixes A :: "'d set"   and N :: "'d measure"   and B :: "'e set"   and M :: "'e pmf" assumes "A \<in> sets N"   and "B \<in> measure_pmf.events M" shows "emeasure (N \<Otimes>\<^sub>M measure_pmf M) (A \<times> B) = emeasure N A * emeasure (measure_pmf M) B"
simps_9: shows "- \<infinity> * - \<infinity> = \<infinity>"
circle_diam_def: fixes l :: "real"   and r :: "real" shows "circle_diam l r = {x. cmod (x - complex_of_real l - (complex_of_real r - complex_of_real l) / 2) < (r - l) / 2}"
sum_comp_morphism: fixes h :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "h (0::'c) = (0::'b)"   and "\<And>x y. h (x + y) = h x + h y" shows "sum (h \<circ> g) A = h (sum g A)"
select_convs_1: fixes es :: "'a"   and ps :: "'b \<Rightarrow> 'e"   and pubActs :: "'c \<times> ('b \<Rightarrow> 'd)"   and more :: "'f" shows "es \<lparr>es = es, ps = ps, pubActs = pubActs, \<dots> = more\<rparr> = es"
mono: fixes c :: "'a word"   and a :: "'a word"   and d :: "'a word"   and b :: "'a word" assumes "c \<le>s a"   and "d \<le>s b" shows "signed.max c d \<le>s signed.max a b"
subcls_C_D: shows "G\<turnstile>C\<prec>\<^sub>C D"
smallest_genby: fixes T :: "'w set"   and w :: "'w" assumes "T \<in> Pow S"   and "w \<in> \<langle>T\<rangle>" shows "order.greater_eq T (reduced_letter_set S w)"