hcju/isabelleps huggingface dataset:
simps_3: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d = (\<lambda>\<I>. \<lbrakk>insert (t \<cdot> \<I>) M; S\<rbrakk>\<^sub>d \<I>)"
numneg_nb: fixes t :: "Cooper.num" assumes "numbound0 t" shows "numbound0 (numneg t)"
len_join_adj1: fixes ts :: "'a tree23s" assumes "not_T ts" shows "len (join_adj ts) < len ts"
Inv_preserves_Can_1: fixes t :: "'c local.term" assumes "Can t" shows "Can (Inv t)"
LLs: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "LLs t1 t2 \<in> fmla"
is_arg_min_linorder: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "is_arg_min f P x = (P x \<and> (\<forall>y. P y \<longrightarrow> order.greater_eq (f y) (f x)))"
Re_def: fixes complex :: "complex" shows "Re complex = (case complex of Complex x1 x2 \<Rightarrow> x1)"
feasible_unitaryD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "feasible t"   and "unitary P" shows "unitary (t P)"
indets_plus_subset: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (p + q) \<subseteq> indets p \<union> indets q"
divide_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "field R"   and "x \<in> carrier R" shows "x \<oslash>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> = x"
select_convs_1: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c" shows "Chn \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = Chn"
lim_sequence_def: fixes x :: "'a" shows "lim_sequence x = projective_family.lim UNIV (Ionescu_Tulcea.CI (K' x) (\<lambda>_. M)) (\<lambda>_. M)"
preserves_reflects_arr: fixes f :: "'a" shows "C.arr (C.map f) = C.arr f"
simps_2: fixes succ :: "'c list \<Rightarrow> 'c list"   and un :: "'c list \<Rightarrow> 'b \<Rightarrow> 'b"   and memb :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and new :: "'c list"   and "have" :: "'b"   and m :: "nat" shows "relpow_impl succ un memb new have (Suc m) = (if new = [] then have else let maybe = succ new; have' = un new have; new' = filter (\<lambda>n. \<not> memb n have') maybe in relpow_impl succ un memb new' have' m)"
has_next_filter_generator: fixes g :: "('a, 'c) generator"   and s :: "'c" shows "list.has_next (local.filter_generator g) s = (list.has_next g s \<and> (let (x, s') = list.next g s in if P x then True else list.has_next (local.filter_generator g) s'))"
mat1: fixes nr :: "nat"   and ze :: "'b"   and on :: "'b" shows "mat nr nr (mat1I ze on nr)"
elims_3: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'c"   and xa :: "('a, 'b) lit" assumes "U.Struct wtFsym wtPsym arOf parOf D intF intP"   and "\<not> U.Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'c" and a :: "('a, 'b) atm" where "x = \<xi>__" and "xa = Pos a__" and "\<not> U.Struct.satA intF intP \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'c" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> U.Struct.satA intF intP \<xi> a"
assms_4: shows "lincomb l1 X + lincomb l2 W \<in> local.span X"
Rep_resumption: fixes x :: "('h, 'g, 'f, 'i) resumption" shows "Rep_resumption x \<in> resumption"
C_eq_Until_InsertDenies: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)))) = C p"
step: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "x \<in> segment f a b"   and "f x \<noteq> b" shows "f x \<in> segment f a b"
cramer_lemma: fixes k :: "'b"   and A :: "(('a, 'b) vec, 'b) vec"   and x :: "('a, 'b) vec" shows "det (\<chi>i j. if j = k then (A *v x) $ i else A $ i $ j) = x $ k * det A"
cgenideal_eq_genideal: fixes i :: "'a list" assumes "i \<in> carrier (K [X])" shows "PIdl\<^bsub>K [X]\<^esub> i = Idl\<^bsub>K [X]\<^esub> {i}"
frequently_at_top_elim: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "frequently P at_top"   and "\<And>i. P i \<Longrightarrow> \<exists>j>i. Q j" shows "frequently Q at_top"
order_gt_1: fixes \<G> :: "'a cyclic_group" assumes "okamoto_base \<G>" shows "1 < order \<G>"
CFG_Def_SDG_Def: fixes m :: "'node"   and V :: "'var"   and n :: "'node SDG_node" assumes "valid_node m"   and "V \<in> Def m"   and "n = CFG_node m" shows "V \<in> Def\<^bsub>SDG\<^esub> n"
w0_1: fixes f :: "'f" shows "w0 \<le> w (f, 0)"
cross7_simps_11: fixes a :: "'b"   and b :: "'b" shows "a * b = b * a"
rel_def: shows "rel = (\<lambda>q1 q2. mlt q1 \<prec>\<^sub>t mlt q2 \<or> mlt q1 = mlt q2 \<and> mnum q1 < mnum q2)"
dom_locals_eval_mono: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)" shows "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1)) \<and> (\<forall>vv. v = In2 vv \<and> normal s1 \<longrightarrow> (\<forall>s val. dom (locals (snd s)) \<subseteq> dom (locals (snd (snd vv val s)))))"
field_chain: fixes R :: "'a ring" assumes "R \<in> C" shows "field R"
lbisimulation_def: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c, 'd) sa_rec_scheme"   and B :: "('b, 'c, 'e) sa_rec_scheme" shows "lbisimulation R A B \<equiv> (sa A \<and> sa B) \<and> lsimulation R A B \<and> lsimulation (R\<inverse>) B A \<and> bisimulation R A B"
sees_method_compP: fixes P :: "'a prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal"   and f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'a \<Rightarrow> 'b" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "compP f P \<turnstile> C sees M: Ts\<rightarrow>T = map_option (f D M Ts T) m in D"
bezout_ring_imp_diagonal_admits_SNF: assumes "OFCLASS('b, bezout_ring_class)" shows "\<forall>A. isDiagonal A \<longrightarrow> (\<exists>P Q. invertible P \<and> invertible Q \<and> Smith_normal_form (P ** A ** Q))"
atLeastLessThan_empty: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le>s a" shows "signed.atLeastLessThan a b = {}"
FW_neg_cycle_detect: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" assumes "\<not> linordered_monoid.cycle_free m n" shows "\<exists>i\<le>n. linordered_monoid.fw m n n n n i i \<prec> Le (0::'d)"
in_listsI: fixes xs :: "'a list"   and A :: "'a set" assumes "\<forall>x\<in>set xs. x \<in> A" shows "xs \<in> lists A"
0: fixes k :: "'a" shows "(lookup (Poly_Mapping.single v n) k when k \<noteq> v) = (0::'b)"
monom_mult_poly_inv: fixes p :: "('c monom \<times> 'd) list"   and m :: "'c monom"   and c :: "'d" assumes "poly_inv p" shows "poly_inv (monom_mult_poly (m, c) p)"
assms_2: shows "V'' \<in> N (vs @ xs) M1 V"
gallery_in_apartment: fixes Y :: "'a set set"   and ys :: "'a set list" assumes "Y \<in> \<A>"   and "order.greater_eq Y (set ys)"   and "gallery ys" shows "SimplicialComplex.maxsimpchain Y ys"
roundup_ge_emBits: fixes emBits :: "nat"   and x :: "nat" assumes "0 < emBits"   and "0 < x" shows "emBits \<le> roundup emBits x * x"
monoidal_category_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b" shows "monoidal_category C T \<alpha> \<iota> \<equiv> ((category C \<and> product_category C C) \<and> product_category C (product_category.comp C C) \<and> binary_endofunctor C T) \<and> (natural_isomorphism (product_category.comp C (product_category.comp C C)) C (binary_endofunctor.ToTC C T) (binary_endofunctor.ToCT C T) \<alpha> \<and> equivalence_functor C C (\<lambda>f. T (partial_magma.cod C \<iota>, f))) \<and> equivalence_functor C C (\<lambda>f. T (f, partial_magma.cod C \<iota>)) \<and> monoidal_category_axioms C T \<alpha> \<iota>"
bf_imp_def: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "bf_imp a b = bf_ite a b bf_True"
fold_update_conv: fixes g :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and m :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "fold (\<lambda>n m. m(n \<mapsto> g n)) xs m x = (if x \<in> set xs then Some (g x) else m x)"
n1i1: shows "n1 \<noteq> pickFT cl \<or> i1 \<noteq> 0"
AObjOp: shows "A \<in> obj\<^bsub>Op C\<^esub>"
cases: fixes x :: "'b ext" obtains "x = ext.Bot"   | uu_ :: "'b" where "x = ext.Val uu_"   | "x = ext.Top"
le_imp_less_or_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x < y \<or> x = y"
Suc_n_not_le_n: fixes n :: "nat" shows "\<not> Suc (n::nat) \<le> n"
distinct_47: fixes x3 :: "db"   and x4 :: "db" shows "DBAbsN x3 \<noteq> DBAbsV x4"
summarize_pdevs_list_def: fixes p :: "nat"   and I :: "nat \<Rightarrow> 'a list \<Rightarrow> bool"   and d :: "nat"   and xs :: "'a pdevs list" shows "summarize_pdevs_list p I d xs = map2 (summarize_pdevs p (\<lambda>i _. I i (pdevs_applys xs i))) [d..<d + length xs] xs"
arr_lunit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C a)"
measurable_restrict_mono: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and A :: "'d set"   and N :: "'e measure"   and B :: "'d set" assumes "f \<in> restrict_space M A \<rightarrow>\<^sub>M N"   and "B \<subseteq> A" shows "f \<in> restrict_space M B \<rightarrow>\<^sub>M N"
as: shows "as \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
induct: fixes x1 :: "('b, 'c, 'd) Tree"   and x2 :: "('b, 'c, 'd) Tree"   and P :: "('b, 'c, 'd) Tree \<Rightarrow> ('b, 'c, 'd) Tree \<Rightarrow> bool" assumes "(x1, x2) \<in> Tree_wf"   and "\<And>t tset. t \<in> set_bset tset \<Longrightarrow> P t (tConj tset)"   and "\<And>t. P t (tNot t)"   and "\<And>t \<alpha>. P t (tAct \<alpha> t)" shows "P x1 x2"
length_list_update: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" shows "length (xs[i := x]) = length xs"
vcg_rem_frame: fixes m :: "'a nres"   and m' :: "'a nres" assumes "m \<le> m'" shows "VCG_FRAME m \<le> m'"
betw_c_in_path: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and ab :: "'a set" assumes "[[a b c]]"   and "path ab a b" shows "c \<in> ab"
dist_stereographic: fixes M1 :: "riemann_sphere"   and M2 :: "riemann_sphere" shows "dist_riemann_sphere' M1 M2 = dist_fs (stereographic M1) (stereographic M2)"
ter: shows "TER (\<Squnion> Y) \<inter> ({x} \<union> set p) = {}"
iT_Plus_neg_empty_iff: fixes I :: "nat set"   and k :: "nat" shows "(I \<oplus>- k = {}) = (I = {} \<or> finite I \<and> Max I < k)"
distinct_3: fixes x1 :: "nat"   and x31 :: "'b"   and x32 :: "('a, 'b) form list" shows "At x1 \<noteq> Modal x31 x32"
MakeFaceFinal_edges_sym: fixes f :: "face"   and g :: "graph" assumes "f \<in> \<F> g"   and "minGraphProps g" shows "edges_sym (makeFaceFinal f g)"
funion_finsert_right: fixes A :: "'x fset"   and a :: "'x"   and B :: "'x fset" shows "A |\<union>| finsert a B = finsert a (A |\<union>| B)"
monad_altc_optionT': assumes "monad_altc Monad_Overloading.return Monad_Overloading.bind altc" shows "monad_altc Monad_Overloading.return Monad_Overloading.bind altc"
inf_sup_ord_4: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and y :: "'e"   and x :: "'e" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
independentD: fixes s :: "'b set"   and t :: "'b set"   and u :: "'b \<Rightarrow> 'a"   and v :: "'b" assumes "independent s"   and "finite t"   and "t \<subseteq> s"   and "(\<Sum>v\<in>t. u v *s v) = (0::'b)"   and "v \<in> t" shows "u v = (0::'a)"
elims: fixes x :: "'b equation"   and y :: "'b trm set" assumes "subterms_of_eq x = y" obtains t :: "'b trm" and s :: "'b trm" where "x = Eq t s" and "y = subterms_of t \<union> subterms_of s"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "g \<cdot>\<^sub>C f = C.dom f"
that: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "f i adds\<^sub>t f j" shows "thesis"
indoc_append: fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" shows "indoc a (u @ v) = (indoc a u \<or> a \<notin> set u \<and> Ind {a} (set u) \<and> indoc a v)"
lvars_tableau_ge_start: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat" shows "\<forall>var\<in>lvars (Tableau (preprocess' cs start)). start \<le> var"
aext_twice: fixes P :: "('b, 'e) uexpr"   and a :: "'e \<Longrightarrow> 'd"   and b :: "'d \<Longrightarrow> 'c" shows "(P \<oplus>\<^sub>p a) \<oplus>\<^sub>p b = P \<oplus>\<^sub>p (a ;\<^sub>L b)"
set_img: fixes f :: "'s" assumes "arr f" shows "local.set (img f) = Img f"
degree_npolyhCN: fixes c :: "poly"   and n :: "nat"   and p :: "poly"   and n0 :: "nat" assumes "isnpolyh (CN c n p) n0" shows "Reflected_Multivariate_Polynomial.degree c = 0"
irreflp_on_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<subseteq> B"   and "irreflp_on P B" shows "irreflp_on P A"
is_red_indI2: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "p \<noteq> 0"   and "is_red F (tail p)" shows "is_red F p"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
filterlim_tendsto_neg_mult_at_bot_iff: fixes f :: "'a \<Rightarrow> real"   and c :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "c < 0" shows "(LIM x F. f x * g x :> at_bot) = filterlim g at_top F"
hyps_6: assumes "Dup q i ((p # ps, a) # branch) xs" shows "A, Suc n \<turnstile> omit_branch xs ((p # ps, a) # branch)"
set_sum_alt: fixes I :: "'b set"   and S :: "'b \<Rightarrow> 'c set" assumes "finite I" shows "sum S I = {sum s I |s. \<forall>i\<in>I. s i \<in> S i}"
monomials_add_disjoint: fixes a :: "'a mpoly"   and b :: "'a mpoly" assumes "monomials a \<inter> monomials b = {}" shows "monomials (a + b) = monomials a \<union> monomials b"
Posix_Atom: fixes c :: "'b" shows "[c] \<in> Atom c \<rightarrow> Atm c"
fair_run_axioms_def: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" shows "fair_run_axioms s\<^sub>0 lstep s \<equiv> df_system.is_fair_run s\<^sub>0 lstep s"
rep_eq: fixes x :: "'a fract" shows "quot_of_fract x = normalize_quot (rep_fract x)"
hyps_1: fixes s :: "('a, 'b) xstate"   and n :: "nat"   and t :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile> \<langle>mark_guards f c1,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c1,s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' = Fault f \<longrightarrow> t' = t) \<and> (isFault t' \<longrightarrow> isFault t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
arcs_set_elem2: fixes i :: "nat"   and k :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "i \<noteq> k"   and "k \<in> set (j # xs)" shows "\<exists>l. (l, k) \<in> set (arcs i j xs)"
simps: fixes a :: "'b formula" shows "AX0p (a::'b formula) = ((\<exists>(F::'b formula) G::'b formula. a = F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F) \<or> (\<exists>(F::'b formula) (G::'b formula) H::'b formula. a = (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H))"
eq_reflection: fixes x :: "'f"   and y :: "'f" assumes "x = y" shows "x \<equiv> y"
exhaust: fixes y :: "pratt" obtains x1 :: "nat" where "y = Prime x1"   | x21 :: "nat" and x22 :: "nat" and x23 :: "nat" where "y = Triple x21 x22 x23"
intros: fixes v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 \<in> carrier_vec n"
IH: shows "H \<otimes>\<^bsup>n\<^esup> = Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n))"
Writer_def: shows "Writer \<equiv> \<Lambda> w a. writer_abs\<cdot>(:up\<cdot>w, up\<cdot>a:)"
good_item_simps_38: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X" shows "Abs xsa xa X $[y1 \<and> y2]_ys = Abs xsa (xa @xsa[y1 \<and> y2]_ys) (X #[y1 \<and> y2]_ys)"
getEdges_impl: shows "(uncurry0 getEdges_impl, uncurry0 (getEdges_SPEC csuper_E)) \<in> unit_assn\<^sup>k \<rightarrow>\<^sub>a list_assn (nat_assn \<times>\<^sub>a int_assn \<times>\<^sub>a nat_assn)"
inf_sup_ord_4: fixes y :: "'d"   and x :: "'d" shows "y \<le> x \<sqinter> y"
A_to_prodag_aHom: fixes A :: "('d, 'e) aGroup_scheme"   and I :: "'f set"   and B :: "'f \<Rightarrow> ('g, 'h) aGroup_scheme"   and S :: "'f \<Rightarrow> 'd \<Rightarrow> 'g" assumes "aGroup A"   and "\<forall>k\<in>I. aGroup (B k)"   and "\<forall>k\<in>I. S k \<in> aHom A (B k)" shows "A_to_prodag A I S B \<in> aHom A (a\<Pi>\<^bsub>I\<^esub> B)"
i_set_defs_7: shows "iMODb2_set \<equiv> {[ r, mod m, c ] |r m c. 2 \<le> m \<and> 1 \<le> c}"
Bernstein_Poly_01_span1: fixes P :: "real poly"   and p :: "nat" assumes "degree P \<le> p" shows "P \<in> poly_vs.span {Bernstein_Poly_01 x p |x. x \<le> p}"
in_lang_INTERSECT: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and n :: "nat"   and w :: "'c list"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'d rexp list" assumes "project \<Sigma> project"   and "alphabet.wf_word \<Sigma> n w" shows "(w \<in> project.lang \<Sigma> project lookup n (INTERSECT xs)) = (\<forall>r\<in>set xs. w \<in> project.lang \<Sigma> project lookup n r)"
simps_1: fixes d :: "nat"   and rA :: "'a \<times> 'a mat" shows "local.sub3 d 0 rA = rA"
act3: fixes X :: "'a set"   and p :: "'b" shows "\<alpha> (\<Squnion> X) p = (\<Squnion>x\<in>X. \<alpha> x p)"
i_Exec_Stream_Init_eq_i_Exec_Stream_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and input :: "nat \<Rightarrow> 'e"   and c :: "'d" shows "i_Exec_Comp_Stream_Init trans_fun input c = [c] \<frown> i_Exec_Comp_Stream trans_fun input c"
null_def: fixes xs :: "'b list" shows "List.null xs = (xs = [])"
characters_def: fixes G :: "('c, 'd) monoid_scheme" shows "characters G = {\<chi>. character G \<chi>}"
lim: shows "(f \<longlongrightarrow> 0) at_top"
dgrad_p_set_le_def: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and G :: "('a \<Rightarrow>\<^sub>0 'c) set" shows "punit.dgrad_p_set_le d F G = dgrad_set_le d (local.punit.pp_of_term ` Keys F) (local.punit.pp_of_term ` Keys G)"
lm040: fixes a :: "('c \<times> 'd set) set" assumes "a \<in> allocationsUniverse" shows "a \<in> allAllocations (Domain a) (\<Union> (Range a))"
exp_cf2_pos: fixes x :: "real" shows "0 < exp_cf2 x"
abN: shows "a\<^sup>2 + N * b\<^sup>2 = 0"
dgrad_p_set_le_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and G :: "('t \<Rightarrow>\<^sub>0 'd) set"   and d :: "'a \<Rightarrow> nat" assumes "F \<subseteq> G" shows "dgrad_p_set_le d F G"
sign_def: fixes x :: "'b" shows "sign x \<equiv> if (0::'b) < x then 1 else if x = (0::'b) then 0 else - 1"
IH: fixes k :: "nat" shows "list.set (acc_lengths k bs) \<subseteq> list.set (acc_lengths k bs2)"
subdiv_step_axioms: shows "subdiv_step G rev_G H rev_H u v w uv uw vw"
as: shows "as \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
nth_map: fixes n :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
JJ12: shows "JJ1 \<in> lift P F ` Q \<and> JJ2 \<in> lift P F ` Q \<and> JJ1 \<noteq> JJ2"
elims: fixes x :: "'a list"   and y :: "String.literal + 'a" assumes "fromSingleton x = y" obtains "x = []" and "y = Inl STR ''Expected list of length 1''"   | xa :: "'a" where "x = [xa__]" and "y = Inr xa__"   | xa :: "'a" and ya :: "'a" and ys :: "'a list" where "x = xa # ya # ys" and "y = Inl STR ''Expected list of length 1''"
sb\<^sub>h: shows "sb\<^sub>h = rs @ Write\<^sub>s\<^sub>b volatile a (D, f) v A L R W # rws"
IH: assumes "distinct (map fst xs)" shows "{#the (map_of xs i). i \<in># mset (map fst xs)#} = mset (map snd xs)"
expr_typing_varE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and x :: "nat"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> Var x : t" obtains "t = \<Gamma> x"
F_leb: fixes m :: "nat" shows "F m \<in> sets lebesgue"
i: fixes j :: "nat" assumes "i \<le> j" shows "(as j, as (Suc j)) \<in> NS - S"
distinct_1: fixes x1 :: "Preliminaries.var"   and x21 :: "'e"   and x22 :: "'e trm list" shows "Var x1 \<noteq> Fn x21 x22"
ffold_empty: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "ffold f z {||} = z"
fundamental_theorem_of_calculus_interior: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b" assumes "a \<le> b"   and "continuous_on {a..b} f"   and "\<And>x. x \<in> {a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)" shows "(f' has_integral f b - f a) {a..b}"
Apply_def: fixes f :: "'a \<Rightarrow> 'a" shows "Apply f \<equiv> \<lambda>ab P s. P (f s)"
norm_unsigned_result: fixes xs :: "bit list" shows "norm_unsigned xs = [] \<or> bv_msb (norm_unsigned xs) = \<one>"
algebraicI: fixes p :: "'a poly"   and x :: "'a" assumes "\<And>i. poly.coeff p i \<in> \<int>"   and "p \<noteq> 0"   and "poly p x = (0::'a)" shows "algebraic x"
splitted_def: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" shows "Id.S.splitted p = (size (Id.S.roots p) = degree p)"
bij_to_def: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" shows "bij_to f A B = (surj_to f A B \<and> inj_on f A)"
split_head_repeat_4: fixes j :: "nat"   and list2 :: "cell list"   and list1 :: "cell list" shows "(Oc \<up> j @ list2 = Bk # list1) = (j = 0 \<and> Bk # list1 = list2)"
WeakF_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "WF(F)_v \<equiv> \<diamond>\<box>Enabled \<langle>F\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>F\<rangle>_v"
coeff_add: fixes p :: "'b mpoly"   and q :: "'b mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (p + q) m = MPoly_Type.coeff p m + MPoly_Type.coeff q m"
Quotient3_rel_abs: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and r :: "'a"   and s :: "'a" assumes "Quotient3 R Abs Rep"   and "R r s" shows "Abs r = Abs s"
measurable_extract_bool: shows "Measurable.pred (stock_measure BOOL) extract_bool"
collapse: fixes "thm" :: "thm" shows "Thm (concl thm) = thm"
assms: shows "(\<lambda>n. ereal (root n (norm (f n)))) \<longlonglongrightarrow> l"
lambda_one: shows "(\<lambda>x. x) = (\<cdot>) (1::'c)"