hcju/isabelleps huggingface dataset:
invmono: fixes I :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<turnstile> I \<longrightarrow> P"   and "|~ P \<and> _actrans N f \<longrightarrow> \<circle>P" shows "\<turnstile> I \<and> \<box>[N]_f \<longrightarrow> \<box>P"
horner_sum_eq_sum_funpow: fixes f :: "'c \<Rightarrow> 'a"   and a :: "'a"   and xs :: "'c list" shows "horner_sum f a xs = (\<Sum>n = 0..<length xs. ((*) a ^^ n) (f (xs ! n)))"
Ord_VsetI: fixes i :: "V"   and b :: "V" assumes "Ord i"   and "rank b \<subset>\<^sub>\<circ> i" shows "b \<in>\<^sub>\<circ> Vset i"
prod_filter_INF2: fixes J :: "'a set"   and A :: "'b filter"   and B :: "'a \<Rightarrow> 'c filter" assumes "J \<noteq> {}" shows "A \<times>\<^sub>F Inf (B ` J) = (INF i\<in>J. A \<times>\<^sub>F B i)"
sum_assoc_bij: fixes a :: "V"   and b :: "V"   and c :: "V" shows "bij_betw (restrict (sum_case (sum_case ZFC_Cardinals.Inl (\<lambda>y. ZFC_Cardinals.Inr (ZFC_Cardinals.Inl y))) (\<lambda>y. ZFC_Cardinals.Inr (ZFC_Cardinals.Inr y))) (elts (a \<Uplus> b \<Uplus> c))) (elts (a \<Uplus> b \<Uplus> c)) (elts (a \<Uplus> (b \<Uplus> c)))"
assms: shows "\<forall>n. is_zero_row_upt_k n k A"
opp_half_apartment_def: shows "folding.Y \<equiv> {x \<in> X. \<exists>C\<in>folding.\<C> - f \<turnstile> folding.\<C>. order.greater_eq C x}"
r_one: fixes x :: "'a" assumes "x \<in> carrier (Ring_Divisibility.mult_of R)" shows "x \<otimes> \<one> = x"
4: shows "e\<^sup>T * top \<le> - (d\<^sup>T * top)"
singleton_singleton: fixes a :: "'b" shows "fis_singleton {|a|}"
psubst_def: shows "psubst \<equiv> \<lambda>\<theta>. trm_rec (lookup \<theta>) (\<lambda>s t. App) (\<lambda>x s. Lam x) (\<lambda>t x s ta. To ta x) (\<lambda>t. Ret)"
that_2: fixes xs' :: "'a list" assumes "xs = z # xs'"   and "prefix (xs' @ [y]) zs" shows "thesis"
smap_empty_def: fixes k :: "'d" shows "{}. k = {}"
inf_sup_aci_8: fixes 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 x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "sup x (sup x y) = sup x y"
topeqI: fixes S :: "'a set set" assumes "\<And>m. m open\<^bsub>T\<^esub> \<Longrightarrow> m open\<^bsub>S\<^esub>"   and "\<And>m. m open\<^bsub>S\<^esub> \<Longrightarrow> m open\<^bsub>T\<^esub>" shows "T = S"
map_int_pair: fixes f :: "int \<Rightarrow> int \<Rightarrow> 'a"   and g :: "val \<Rightarrow> 'a"   and i :: "int"   and j :: "int" shows "map_int_pair f g <|IntVal i, IntVal j|> = f i j"
card_const_lt_component_le: fixes B :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "finite B" shows "card (const_lt_component ` (B - {0}) - {None}) \<le> card (component_of_term ` Keys B)"
sin_monotone_2pi_le: fixes y :: "real"   and x :: "real" assumes "- (pi / 2) \<le> y"   and "y \<le> x"   and "x \<le> pi / 2" shows "sin y \<le> sin x"
e2_v2: shows "psubst \<rho>' e2 \<longrightarrow>* v2'"
induct: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" obtains x :: "'d" where "has_bochner_integral M f x"
word_sdiv_div0: fixes a :: "'a word" shows "a sdiv 0 = 0"
Diff_insert0: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "x \<notin> A" shows "A - insert x B = A - B"
dim_7: shows "dim_row (mat_of_rows n (map gso [0..<length fs])) = length fs"
conf: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) (u @ v) = Partial_Cost_Model.config' (rTS h0) ([x, y], rev u @ h) v"
in_out_indep: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'h \<Longrightarrow> 'g" shows "$x \<bowtie> $y\<acute>"
A_7: shows "v \<notin> \<lbrace>EQ x c\<rbrace>"
new_document_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr \<noteq> new_document_ptr" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
set_bd_3: fixes x :: "('e, 'f, 'd) F1" shows "|F1set3 x| \<le>o bd_F1"
map\<^sub>1_simps_2: fixes f :: "'a" assumes "C.ide f" shows "src\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[G f] \<cdot>\<^sub>D \<r>\<^sub>D[G f]) = G.map\<^sub>0 (src\<^sub>C f)"
relation_of_Guard: fixes g :: "'d \<Rightarrow> bool"   and P :: "('c, 'd) action" shows "action.relation_of g `&` P = R (g \<circ> alpha_rp.more \<circ> fst \<longrightarrow> \<not> action.relation_of P\<^sup>f\<^sub>f \<turnstile> g \<circ> alpha_rp.more \<circ> fst \<and> action.relation_of P\<^sup>t\<^sub>f \<or> \<not> g \<circ> alpha_rp.more \<circ> fst \<and> (\<lambda>(A, A'). tr A' = tr A \<and> wait A'))"
ereal_le_less: fixes y :: "real"   and a :: "ereal"   and x :: "real" assumes "ereal y \<le> a"   and "x < y" shows "ereal x < a"
PL_Indis_axioms: shows "PL_Indis indis"
prod_topology_def: fixes X :: "'d topology"   and Y :: "'e topology" shows "prod_topology X Y \<equiv> topology (arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T}))"
map_poly_inj_idom_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_inj_idom_hom hom \<equiv> inj_idom_hom hom"
cos_eq_neg_periodic_intro: fixes x :: "real"   and y :: "real"   and k :: "int" assumes "x - y = 2 * real_of_int k * pi + pi \<or> x + y = 2 * real_of_int k * pi + pi" shows "cos x = - cos y"
add_im_sheaf_def: shows "add_im_sheaf \<equiv> \<lambda>V. add_str (f \<^sup>\<inverse> X V)"
poly_minus_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.poly_minus one times uminus p = semiring_0.cmult times (uminus one) p"
ide_char: fixes a :: "'a" shows "hom'.ide a = (hom'.arr a \<and> ide a)"
sup_eq_bot_iff: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bot :: "'e"   and x :: "'e"   and y :: "'e" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(sup x y = bot) = (x = bot \<and> y = bot)"
intro: fixes ops1 :: "('a, 'b, 'c) set_ops_scheme"   and ops2 :: "('d, 'e, 'f) set_ops_scheme"   and ops3 :: "('g, 'h, 'i) set_ops_scheme" assumes "StdSet ops1"   and "StdSet ops2"   and "StdSet ops3" shows "inj_image_filter_cp_loc ops1 ops2 ops3"
wf_cdecl_hides_SomeD: fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and sig :: "sig"   and newM :: "methd"   and old :: "qtname \<times> methd" assumes "wf_cdecl G (C, c)"   and "C \<noteq> Object"   and "table_of (methods c) sig = Some newM"   and "G,sig\<turnstile> (C, newM) hides old" shows "accmodi old \<le> access newM \<and> is_static old"
af_nested_prop_atoms: fixes \<phi> :: "'b ltln"   and w :: "'b set list" shows "nested_prop_atoms (af \<phi> w) \<subseteq> nested_prop_atoms \<phi>"
that: fixes g :: "'c \<Rightarrow> 'a" assumes "continuous_on U g"   and "g ` U \<subseteq> C"   and "g z = a"   and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y" shows "thesis"
upd_type: fixes t :: "('c \<times> 'd) bro"   and h :: "nat"   and x :: "'c"   and y :: "'d" shows "(t \<in> B h \<longrightarrow> upd x y t \<in> Bp h) \<and> (t \<in> U h \<longrightarrow> upd x y t \<in> T h)"
suminf_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'b)" shows "suminf f = sum f N"
finite_ReZ_segments_induct: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and P :: "(real \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> bool" assumes "finite_ReZ_segments g z"   and "\<And>g z. P (subpath 0 0 g) z"   and "\<And>s g z. \<lbrakk>s \<in> {0..<1}; s = 0 \<or> Re (g s) = Re z; \<forall>t\<in>{s<..<1}. Re (g t) = Re z; finite_ReZ_segments (subpath 0 s g) z; P (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P g z"   and "\<And>s g z. \<lbrakk>s \<in> {0..<1}; s = 0 \<or> Re (g s) = Re z; \<forall>t\<in>{s<..<1}. Re (g t) \<noteq> Re z; finite_ReZ_segments (subpath 0 s g) z; P (subpath 0 s g) z\<rbrakk> \<Longrightarrow> P g z" shows "P g z"
simps_4: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and ao :: "'a option"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.Compl_set_less_eq_aux less proper_interval ao (x # xs) (y # ys) = (if less x y then \<not> proper_interval ao (Some x) \<and> proper_intrvl.Compl_set_less_eq_aux less proper_interval (Some x) xs (y # ys) else if less y x then \<not> proper_interval ao (Some y) \<and> proper_intrvl.Compl_set_less_eq_aux less proper_interval (Some y) (x # xs) ys else \<not> proper_interval ao (Some y))"
crossproduct_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and w :: "'b"   and y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(plus (times w y) (times x z) = plus (times w z) (times x y)) = (w = x \<or> y = z)"
fs_int_mu_d_Z: fixes j :: "nat"   and ii :: "nat" assumes "j \<le> ii"   and "ii < length fs'" shows "rat_of_int (fs'.d fs' (Suc j)) * fs'.gs.\<mu> ii j \<in> \<int>"
inner_minus_left: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (uminus x) y = - inner x y"
simps_2: fixes n :: "nat"   and xs :: "'a list" shows "repeat (Suc n) xs = xs @ repeat n xs"
convergent_def: fixes X :: "nat \<Rightarrow> 'd" shows "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
check_field_access_def: fixes G :: "prog"   and accC :: "qtname"   and statDeclC :: "qtname"   and fn :: "vname"   and stat :: "bool"   and a' :: "val"   and s :: "state" shows "check_field_access G accC statDeclC fn stat a' s = (let oref = if stat then Inr statDeclC else Inl (the_Addr a'); dynC = case oref of Inl a \<Rightarrow> obj_class (the (globs (snd s) oref)) | Inr C \<Rightarrow> C; f = the (table_of (DeclConcepts.fields G dynC) (fn, statDeclC)) in abupd (error_if (\<not> G \<turnstile>Field fn (statDeclC, f) in dynC dyn_accessible_from accC) AccessViolation) s)"
Morera_triangle: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "continuous_on S f"   and "open S"   and "\<And>a b c. convex hull {a, b, c} \<subseteq> S \<longrightarrow> contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0" shows "f analytic_on S"
wf_jvm_prog_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" shows "wf_jvm_prog P \<equiv> \<exists>\<Phi>. wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"
col_trivial_1: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Col Bet A A B"
path_in_verts: fixes G :: "nat graph_rec"   and l :: "nat list"   and v :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) l v" shows "set l \<subseteq> set (g_V G)"
le_add_same_cancel2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "(a \<subseteq># b + a) = ({#} \<subseteq># b)"
cases: fixes a :: "(('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent" assumes "a \<in> edges" obtains c :: "'form" and "is" :: "nat list" where "a = edge_at ca__ isa__" and "ca__ \<in> set conclusions" and "isa__ \<in> it_paths (it' ca__)"   | c :: "'form" and "is" :: "nat list" and n :: "nat" and s :: "'subst" and ants :: "('form, 'rule, 'subst, 'var) itnode rose_tree list" where "a = hyp_edge_at c is n s" and "c \<in> set conclusions" and "is \<in> it_paths (it' c)" and "tree_at (it' c) is = HNode n s ants"
eventually_inf_principal: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and s :: "'b set" shows "eventually P (inf F (principal s)) = (\<forall>\<^sub>F x in F. x \<in> s \<longrightarrow> P x)"
minus_minus: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> (\<ominus>\<^bsub>G\<^esub> x) = x"
dom_fmlookup: fixes m :: "('a, 'b) fmap" shows "dom (fmlookup m) = fmdom' m"
ereal_tendsto_simps2_1: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" shows "((ereal \<circ> f) \<longlongrightarrow> ereal a) F = (f \<longlongrightarrow> a) F"
replicate_stutter: fixes n :: "nat"   and v :: "nat \<Rightarrow> 'b" shows "replicate n (v 0) \<frown> v \<approx> v"
carrier_eq: shows "alts = {x. Pareto (permute_profile \<sigma> R) x x}"
min_bot2: fixes bot :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.order_bot bot less_eq less" shows "ord.min less_eq x bot = bot"
lexord_eq_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "lexord_eq_fusion g1 g2 s1 s2 = lexordp_eq (unstream g1 s1) (unstream g2 s2)"
claim: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "cs\<^bsup>\<pi>\<^esup> i = cs\<^bsup>\<pi>'\<^esup> i'"   and "cs\<^bsup>\<pi>\<^esup> j = cs\<^bsup>\<pi>'\<^esup> j'"   and "butlast (cs\<^bsup>\<pi>\<^esup> i) = butlast (cs\<^bsup>\<pi>\<^esup> j)"   and "\<pi> i \<noteq> local.return"   and "i < j" shows "i' < j'"
zero_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_monoid_diff plus minus zero" shows "minus zero a = zero"
w: shows "rep_nat_term w = (fst (rep_nat_term u), snd (rep_nat_term v))"
islimpt_def: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and S :: "'a set" assumes "class.topological_space open" shows "topological_space.islimpt open x S = (\<forall>T. x \<in> T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x))"
quasicomm_var: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>) = (y\<^sup>\<star> \<cdot> x \<le> x \<cdot> (x + y)\<^sup>\<star>)"
one_reducible_lm_3: fixes n :: "nat" shows "one_reducible_to (nat_to_ce_set n) univ_ce"
rtrancl_trancl_reflcl: fixes r :: "('c \<times> 'c) set" shows "r\<^sup>* = (r\<^sup>+)\<^sup>="
RelStay: fixes R :: "pi"   and S :: "pi" assumes "(R \<parallel> !R, S) \<in> Rel'" shows "(!R, S) \<in> Rel'"
assms_2: shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> B"
less_supI1: fixes x :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "x \<subset># a" shows "x \<subset># a \<union># b"
insert_ops_ref_older: fixes pre :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and suf :: "('a \<times> 'a option) list" assumes "insert_ops (pre @ [(oid, Some ref)] @ suf)" shows "ref < oid"
subst_def: shows "subst \<equiv> \<lambda>x0 x1 x2. projl (subst_subst_heap_sumC (Inl (x0, x1, x2)))"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
insert_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "order.greater_eq D C" shows "order.greater_eq (insert a D) (insert a C)"
bisimCoinduct: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto>[(X \<union> bisim)] S"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> (S, R) \<in> X" shows "P \<sim> Q"
ide_dom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C f" shows "partial_magma.ide C (partial_magma.dom C f)"
cod_eqI: fixes b :: "'b"   and f :: "'b" assumes "J.ide b"   and "J.seq b f" shows "J.cod f = b"
subspace_scale: fixes S :: "'a set"   and x :: "'a"   and c :: "complex" assumes "csubspace S"   and "x \<in> S" shows "c *\<^sub>C x \<in> S"
span_subset_spanI: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> More_Modules.ideal B" shows "More_Modules.ideal A \<subseteq> More_Modules.ideal B"
1: shows "\<forall>(t, s)\<in>set (unlabel D). (fv t \<union> fv s) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A = {}"
smult_prod_extract1: fixes a :: "'a"   and A :: "'a tensor"   and B :: "'a tensor" shows "a \<cdot> (A \<otimes> B) = a \<cdot> A \<otimes> B"
hom_2: shows "homotopic_with_canon (\<lambda>x. True) T T (h \<circ> k) id"
transferTauFrame: fixes \<Psi>\<^sub>F :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>G :: "'b" assumes "\<Psi>\<^sub>F \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>P\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>G, \<Psi>\<^sub>G \<otimes> \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>F \<sharp>* P"   and "A\<^sub>G \<sharp>* P"   and "A\<^sub>P \<sharp>* A\<^sub>F"   and "A\<^sub>P \<sharp>* A\<^sub>G"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>F"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>G" shows "\<Psi>\<^sub>G \<rhd> P \<longmapsto> \<tau> \<prec> P'"
all_in_listAppend: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and ys :: "('b, 'c) Combinators list" assumes "all_in_list xs l"   and "all_in_list ys l" shows "all_in_list (xs @ ys) l"
infinite_cball: fixes r :: "real"   and a :: "'a" assumes "0 < r" shows "infinite (cball a r)"
idempotent_bot_closed: shows "comp_inf.idempotent bot"
build_ip_partition_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "build_ip_partition c rs = map (\<lambda>xs. wordinterval_sort (wordinterval_compress (foldr wordinterval_union xs Empty_WordInterval))) (groupWIs3 c rs)"
not_below2not_eq: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.po below"   and "\<not> below x y" shows "x \<noteq> y"
snd_sum_list: fixes xs :: "('c \<times> 'b) list" shows "snd (sum_list xs) = sum_list (map snd xs)"
UN_upper: fixes a :: "'a"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "a \<in> A" shows "order.greater_eq (\<Union> (B ` A)) (B a)"
a: shows "a \<in> T \<rightarrow> S \<rightarrow> carrier_vec n"
Suc: fixes ds :: "nat list"   and a :: "nat" shows "drop n (digit_encode ds a) = digit_encode (drop n ds) (a div prod_list (take n ds))"
assms_2: fixes r :: "real" assumes "0 < r"   and "R < ereal r" shows "\<exists>z. norm z = r \<and> \<not> summable (\<lambda>n. norm (f n * z ^ n))"
continuous: fixes F :: "'d filter"   and g :: "'d \<Rightarrow> 'a" assumes "continuous F g" shows "continuous F (\<lambda>x. f (g x))"
f4: shows "\<forall>x0. (\<exists>v2. x0 = Suc v2) = (x0 = Suc (nnb x0))"
orthonormal_basis_subspace: fixes S :: "'a set" assumes "subspace S" obtains B :: "'a set" where "B \<subseteq> S" and "pairwise orthogonal B" and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B" and "card B = dim S" and "span B = S"
ring_simprules_8: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
sin_ge_zero_iff: fixes x :: "real" assumes "0 \<le> x"   and "x < 2 * pi" shows "(0 \<le> sin x) = (x \<le> pi)"
simps_3: fixes a :: "'c local.term"   and b :: "'c local.term" shows "(a \<^bold>\<otimes> b)\<^bold>\<down> = (if Diag (a \<^bold>\<otimes> b) then a \<^bold>\<otimes> b else (\<^bold>\<lfloor>a\<^bold>\<rfloor> \<^bold>\<Down> \<^bold>\<lfloor>b\<^bold>\<rfloor>) \<^bold>\<cdot> (a\<^bold>\<down> \<^bold>\<otimes> b\<^bold>\<down>))"
bounded_linear_imp_differentiable: fixes f :: "'b \<Rightarrow> 'c"   and net :: "'b filter" assumes "bounded_linear f" shows "f differentiable net"
Abs_BinomialHeap_inverse: fixes y :: "('c, 'd) BinomialHeapStruc.BinomialTree list" assumes "y \<in> {q. BinomialHeapStruc.invar q}" shows "Rep_BinomialHeap (Abs_BinomialHeap y) = y"
angle_eq_pi_imp_dist_additive: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "angle a b c = pi" shows "dist a c = dist a b + dist b c"
lhd_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lhd (xs $ ys) = (if lnull xs then lhd ys else lhd xs)"
proj2_line_abs_abs_mult: fixes v :: "(real, 3) vec"   and w :: "(real, 3) vec" assumes "proj2_line_abs v = proj2_line_abs w"   and "w \<noteq> 0" shows "\<exists>k. v = k *\<^sub>R w"
set_of_mkt_bal_l: fixes l :: "'b tree"   and r :: "'b tree"   and n :: "'b" assumes "avl l"   and "avl r" shows "set_of (mkt_bal_l n l r) = Set.insert n (set_of l \<union> set_of r)"
lossless_iff_set_pmf_None: fixes p :: "'h spmf" shows "lossless_spmf p = (None \<notin> set_pmf p)"
Lim_at_imp_Lim_at_within: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and x :: "'c"   and S :: "'c set" assumes "f \<midarrow>x\<rightarrow> l" shows "(f \<longlongrightarrow> l) (at x within S)"
refine_list_1: fixes R :: "('c \<times> 'd) set" shows "([], []) \<in> \<langle>R\<rangle>list_rel"
2: fixes j :: "nat" assumes "length sl \<le> j" shows "fst (\<pi> !! j) = toSink (last sl)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "remdups (x # xs) = (if x \<in> set xs then remdups xs else x # remdups xs)"
simps: fixes a :: "trm" shows "valuep a = ((\<exists>T t. a = (\<lambda>:T. t)) \<or> (\<exists>T t. a = (\<lambda><:T. t)) \<or> (\<exists>fs. a = Rcd fs \<and> (\<forall>(l, t)\<in>set fs. valuep t)))"
evA: fixes S :: "'b set" assumes "open S"   and "l \<in> S" shows "\<forall>\<^sub>F i in sequentially. A i \<subseteq> S"
callee_lcl_def: fixes C :: "qtname"   and sig :: "sig"   and m :: "methd" shows "callee_lcl C sig m = (\<lambda>k. case k of EName (VNam v) \<Rightarrow> (table_of (lcls (mbody m))(pars m [\<mapsto>] parTs sig)) v | EName Res \<Rightarrow> Some (resTy m) | This \<Rightarrow> if is_static m then None else Some (Class C))"
finite_imp_closed: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.t1_space open"   and "finite S" shows "topological_space.closed open S"
fmempty_lookup: fixes x :: "'f" shows "fmlookup fmempty x = None"
False: shows "lex_comp' (PP_oalist xs) (PP_oalist ys) \<noteq> Eq"
at_infinity_def: shows "at_infinity = (INF r. principal {x. r \<le> norm x})"