hcju/isabelleps huggingface dataset:
reachable_ab: fixes as :: "'b list"   and r :: "'b rexp" shows "reachable_b as r = hom_ab ` reachable_a as r"
impl_lasso_loc_axioms: shows "impl_lasso_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G"
euler_genus_def: shows "H.euler_genus \<equiv> (int (2 * card H.sccs) - int (card H.isolated_verts) - H.euler_char) div 2"
374: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - (x \<sqinter> y) = - (x \<sqinter> y)"
bal_split_left_1: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ (a, b) # rs) t)" shows "bal (Node ls a)"
sorted_spvec_cons1: fixes a :: "nat \<times> 'a"   and as :: "(nat \<times> 'a) list" assumes "sorted_spvec (a # as)" shows "sorted_spvec as"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" shows "New_max.filterQueryNeg V Q = Set.filter (\<lambda>(A, uu_). A \<subseteq> V) Q"
simps_3: fixes F :: "'b formula"   and G :: "'b formula" shows "to_mini_formula (F \<^bold>\<rightarrow> G) = to_mini_formula F \<^bold>\<rightarrow> to_mini_formula G"
n_Suc_3: fixes i :: "nat" assumes "1 < i"   and "i < Suc n" shows "(deriv ^^ i) f z = 0"
rel_refl_strong: fixes x :: "'c \<times> 'd"   and R1a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> fsts x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> snds x \<Longrightarrow> R2a z2 z2" shows "rel_prod R1a R2a x x"
decr_grading_p_zero: fixes d :: "'c \<Rightarrow> nat"   and n :: "nat" shows "Quasi_PM_Power_Products.punit.decr_grading_p d n 0 = 0"
SeqAuto_select: fixes SA :: "('d, 'e, 'f) seqauto" shows "SeqAuto (States SA) (InitState SA) (Labels SA) (Delta SA)"
body3_def: fixes c_o :: "'c_o"   and c_a :: "'c_a"   and s :: "'s"   and j\<^sub>s :: "nat" shows "body3 c_o c_a s j\<^sub>s = exec_gpv_stop (stop_oracle c_o) (\<A> c_a) (Inr j\<^sub>s, s) \<bind> (\<lambda>(uu_, idgs, uu_). (case idgs of Inl (g, s') \<Rightarrow> eval c_o c_a s' g | Inr x \<Rightarrow> return_spmf False) \<bind> return_spmf)"
nth_drop: fixes n :: "nat"   and xs :: "'b list"   and i :: "nat" assumes "n \<le> length xs" shows "xs \<up> n ! i = xs ! (n + i)"
sub_code_1: shows "Code_Numeral.sub num.One num.One = 0"
indVfbasis: shows "set (concat indVfbasis) \<subseteq> indV \<and> indV = R_scalar_mult.RSpan UNIV (\<currency>\<currency>) (concat indVfbasis) \<and> R_scalar_mult.R_lin_independent UNIV (\<currency>\<currency>) (concat indVfbasis)"
perm_pi_simp_2: fixes pi :: "(atom \<times> atom) list"   and x :: "'a" shows "rev pi \<bullet> pi \<bullet> x = x"
merge_sum: fixes rha :: "'c \<Rightarrow> 'd"   and boa :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and rhb :: "'e \<Rightarrow> 'f"   and bob :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and mb :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option" assumes "merge_on UNIV rha boa ma"   and "merge_on UNIV rhb bob mb" shows "merge_on UNIV (map_sum rha rhb) (rel_sum boa bob) (merge_sum ma mb)"
cinner_simps_6: fixes x :: "'b"   and r :: "complex"   and y :: "'b" shows "\<langle>x, r *\<^sub>C y\<rangle> = r * \<langle>x, y\<rangle>"
IH: assumes "\<forall>i\<in>I. subgroup (S i) G"   and "\<forall>i\<in>I. finite (S i)"   and "pairwise (\<lambda>x y. coprime (card (S x)) (card (S y))) I" shows "card (IDirProds G S I) = (\<Prod>i\<in>I. card (S i))"
relto_absorb_1: fixes E :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" shows "(E\<^sup>* O R O E\<^sup>*) O E\<^sup>* = E\<^sup>* O R O E\<^sup>*"
unfoldr_rbt_keys_generator: fixes t :: "('b, 'c) rbt" shows "list.unfoldr rbt_keys_generator (rbt_init t) = RBT_Impl.keys t"
z_property_def: fixes bullet :: "'b \<Rightarrow> 'b"   and R :: "('b \<times> 'b) set" shows "z_property bullet R \<equiv> \<forall>a b. (a, b) \<in> R \<longrightarrow> (b, bullet a) \<in> R\<^sup>* \<and> (bullet a, bullet b) \<in> R\<^sup>*"
imf: shows "f ` ({x. norm x = 1 \<and> b \<bullet> x \<noteq> 1} \<inter> T) = {x. b \<bullet> x = 0} \<inter> T"
max_spec_def: fixes G :: "prog"   and S :: "qtname"   and rt :: "ref_ty"   and sig :: "sig" shows "max_spec G S rt sig = {m \<in> appl_methds G S rt sig. \<forall>m'\<in>appl_methds G S rt sig. more_spec G m' m \<longrightarrow> m' = m}"
r_leap_Suc: fixes t :: "nat"   and i :: "nat"   and x :: "nat" shows "eval r_leap [Suc t, i, x] = eval r_step [the (eval r_leap [t, i, x])]"
insort_left_comm: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "insort x (insort y xs) = insort y (insort x xs)"
comp_converter_parallel2: fixes conv1l :: "('a, 'c, 'i, 'k) converter"   and conv1r :: "('b, 'd, 'j, 'l) converter"   and conv2l :: "('i, 'k, 'e, 'g) converter"   and conv2r :: "('j, 'l, 'f, 'h) converter" shows "(conv1l |\<^sub>= conv1r) \<odot> (conv2l |\<^sub>= conv2r) = conv1l \<odot> conv2l |\<^sub>= conv1r \<odot> conv2r"
no_disc: shows "\<not> has_disc_negated disc2 False m"
minus_closed: fixes a :: "'ag"   and b :: "'ag" assumes "a \<in> U"   and "b \<in> U" shows "a -\<^sub>o\<^sub>w b \<in> U"
tmsubst_nb: fixes m :: "nat"   and t :: "tm"   and a :: "tm" assumes "tmbound m t" shows "tmbound m (tmsubst m t a)"
ridmap_rhom: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "carrier (Vr K v) \<subseteq> carrier (Vr K v')" shows "ridmap (Vr K v) \<in> rHom (Vr K v) (Vr K v')"
STA_struct_block: fixes A :: "'b set"   and n :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and ps' :: "('c, 'b) fm list" assumes "infinite UNIV"   and "finite A"   and "A, n \<turnstile> (ps, a) # branch"   and "set ps \<subseteq> set ps'" shows "A \<turnstile> (ps', a) # branch"
connected_space_imp_connected_component_of: fixes X :: "'b topology"   and a :: "'b"   and b :: "'b" assumes "connected_space X"   and "a \<in> topspace X"   and "b \<in> topspace X" shows "connected_component_of X a b"
emBits_roundup_cancel2: fixes emBits :: "nat" assumes "emBits mod 8 \<noteq> 0" shows "roundup emBits 8 * 8 - (8 - emBits mod 8) = emBits"
sum_constant: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and y :: "'a"   and A :: "'b set" assumes "class.semiring_1 one times plus zero" shows "comm_monoid_add.sum plus zero (\<lambda>x. y) A = times (semiring_1.of_nat one plus zero (card A)) y"
p: assumes "S = (x \<leftrightarrow> y) \<bullet> T"   and "x \<noteq> y"   and "atom y \<sharp> T"   and "atom x \<sharp> S" shows "(x \<leftrightarrow> y) \<bullet> f x T = f y S"
mem_CD_on_Cd: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and x :: "'d"   and ds :: "'e set"   and X :: "'d set" assumes "Contracts Xd Xh Pd Ch" shows "(x \<in> Contracts.CD_on Pd ds X) = (x \<in> Contracts.Cd Pd (Xd x) X \<and> Xd x \<in> ds)"
lsided_bres: fixes x :: "'b \<times> 'c" shows "prod.lsd x = less_eq_prod x (prod.bres top_prod x)"
update_inf_different: fixes u :: "'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "u \<le> - y" shows "(x[y\<longmapsto>z]) \<sqinter> u = x \<sqinter> u"
is_nnf_NotD: fixes F :: "'c formula" assumes "is_nnf (\<^bold>\<not> F)" shows "(\<exists>k. F = Atom k) \<or> F = \<bottom>"
vec_of_list_carrier: fixes x :: "'c list" shows "vec_of_list x \<in> carrier_vec (length x)"
smooth_compat_restrict_chartD2: fixes c1 :: "('c, 'd) chart"   and U :: "'c set"   and k :: "enat"   and c2 :: "('c, 'd) chart" assumes "domain c1 \<subseteq> U"   and "open U"   and "k-smooth_compat (restrict_chart U c2) c1" shows "k-smooth_compat c2 c1"
list_rel_induct: fixes l :: "'a list"   and l' :: "'b list"   and R :: "('a \<times> 'b) set"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "(l, l') \<in> \<langle>R\<rangle>list_rel"   and "P [] []"   and "\<And>x x' l l'. \<lbrakk>(x, x') \<in> R; (l, l') \<in> \<langle>R\<rangle>list_rel; P l l'\<rbrakk> \<Longrightarrow> P (x # l) (x' # l')" shows "P l l'"
nn_integral_embed_measure: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> ennreal"   and M :: "'a measure" assumes "inj f"   and "g \<in> borel_measurable (embed_measure M f)" shows "integral\<^sup>N (embed_measure M f) g = \<integral>\<^sup>+ x. g (f x) \<partial>M"
line_extension_of_combine_set: fixes u :: "'a"   and Us :: "'a list" assumes "u \<in> carrier R" shows "line_extension K u {combine Ks Us |Ks. set Ks \<subseteq> K} = {combine Ks (u # Us) |Ks. set Ks \<subseteq> K}"
leg1_in_hom_2: shows "\<guillemotleft>F g : F g \<Rightarrow>\<^sub>D F g\<guillemotright>"
dist_le_additive_strength: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion"   and n :: "nat"   and x :: "'b" assumes "isometry f"   and "Gromov_extension f xi = xi"   and "0 \<le> additive_strength f xi"   and "1 \<le> n" shows "dist x ((f ^^ n) x) \<le> dist x (f x) + real n * additive_strength f xi + real_of_int (\<lceil>log 2 (real n)\<rceil> * 16) * deltaG TYPE('b)"
round_of_nat: fixes n :: "nat" shows "round (of_nat n) = int n"
g_bounds1: obtains c3 :: "real" where "\<And>x i. \<lbrakk>x\<^sub>1 \<le> x; i < k\<rbrakk> \<Longrightarrow> c3 * g x \<le> g_approx i x" and "0 < c3"
subst_term_list_obtain: fixes T :: "'c list"   and P :: "'c \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> bool"   and S :: "('d, 'f) Term.term list"   and \<delta> :: "'e \<Rightarrow> ('d, 'f) Term.term" assumes "\<forall>i<length T. \<exists>s. P (T ! i) s \<and> S ! i = s \<cdot> \<delta>"   and "length T = length S" shows "\<exists>U. length T = length U \<and> (\<forall>i<length T. P (T ! i) (U ! i)) \<and> S = U \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>"
lcp_len': fixes u :: "'b list"   and v :: "'b list" assumes "\<not> u \<le>p v" shows "\<^bold>|u \<and>\<^sub>p v\<^bold>| < \<^bold>|u\<^bold>|"
smember_def: fixes x :: "'b"   and s :: "'b stream" shows "smember x s = (x \<in> sset s)"
cong_diff_mod: fixes xa :: "'a"   and xb :: "'a"   and c :: "'a" assumes "xa \<noteq> xb"   and "c dvd xa - xb"   and "xa = xa mod c" shows "xb \<noteq> xb mod c"
lsumr_rsuml_parallel_converter2: fixes conv1 :: "('h, 'k, 'n, 'q) converter"   and conv2 :: "('i, 'l, 'o, 'r) converter"   and conv3 :: "('j, 'm, 'p, 's) converter" shows "map_converter id id lsumr rsuml (conv1 |\<^sub>= conv2 |\<^sub>= conv3) = map_converter lsumr rsuml id id ((conv1 |\<^sub>= conv2) |\<^sub>= conv3)"
rep_eq: fixes x :: "'c upred"   and xa :: "'d \<Longrightarrow> 'c" shows "\<lbrakk>x \<restriction>\<^sub>v xa\<rbrakk>\<^sub>e = (\<lambda>b. \<exists>b'. \<lbrakk>x\<rbrakk>\<^sub>e (b' \<triangleleft>\<^bsub>xa\<^esub> b))"
simps: fixes i :: "nat"   and j :: "nat" shows "gs2.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs' ! i \<bullet> gs2.gso j / \<parallel>gs2.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
stl_def: fixes stream :: "'b stream" shows "stl stream = (case stream of x1 ## x2 \<Rightarrow> x2)"
Quotient_hmultiset: shows "Quotient (eq_onp no_elem) Hereditary_Multiset.Abs_hmultiset Hereditary_Multiset.Rep_hmultiset cr_hmultiset"
map_ident: fixes t :: "'c stream" shows "smap (\<lambda>x. x) t = t"
thetaParWL1_converse_Wretr: shows "thetaParWL1\<inverse> \<subseteq> Wretr (thetaParWL1\<inverse> \<union> Wbis)"
distinct_permutations_of_list_impl: fixes xs :: "'a list" shows "distinct (permutations_of_list_impl xs)"
Diff_disjoint: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> (B - A) = {}"
\<K>_power: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and k :: "nat" assumes "source_code M b fi X enc dec" shows "source_code.\<K> M b X enc ^ k = (\<Sum>w | length w = k \<and> set w \<subseteq> source_code.L M X. 1 / b ^ foldr (\<lambda>x. (+) (code_word_length enc x)) w 0)"
cdens_ctxt_invar_insert: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and e :: "expr"   and t' :: "pdf_type"   and f :: "cexpr" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "\<Gamma> \<turnstile> e : t'"   and "free_vars e \<subseteq> set vs \<union> set vs'"   and "dens_ctxt_\<alpha> ([], vs @ vs', \<Gamma>, CReal 1) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>x xa. ennreal (eval_cexpr f x xa))"   and "is_density_expr ([], vs @ vs', \<Gamma>, CReal 1) t' f" shows "cdens_ctxt_invar (shift_vars vs) (map Suc vs') (case_nat t' \<Gamma>) (map_vars Suc \<delta> *\<^sub>c f)"
not_nonzero_iff: fixes x :: "nat list" shows "(\<not> nonzero x) = (x = zeroes (length x))"
INV1: fixes I :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "|~ I \<and> _actrans N f \<longrightarrow> \<circle>I" shows "\<turnstile> I \<and> \<box>[N]_f \<longrightarrow> \<box>I"
simE_2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<longmapsto> a<x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel"
distinct_56: fixes x7 :: "CTypeId" shows "CClassT x7 \<noteq> NullT"
C_def: shows "C = (\<lambda>n. \<Union> (A ` {..n}))"
assms_1: shows "eval c t t' n [\<phi>]\<^sub>b"
tendsto_neg_powr: fixes s :: "real"   and f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "s < 0"   and "filterlim f at_top F" shows "((\<lambda>x. f x powr s) \<longlongrightarrow> 0) F"
Rep_unit_induct: fixes y :: "bool"   and P :: "bool \<Rightarrow> bool" assumes "y \<in> {True}"   and "\<And>x. P (Rep_unit x)" shows "P y"
infinite_growing: fixes X :: "'c DBMEntry set" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. x \<prec> y" shows "infinite X"
abs_eq: fixes x :: "'b poly fract" assumes "eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'b)) x x" shows "quot_of_ratfps (Abs_ratfps x) = quot_of_fract x"
is_inf_def: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and i :: "'a" assumes "dpo le" shows "dpo.is_inf le x y i = (le i x \<and> le i y \<and> (\<forall>z. le z x \<and> le z y \<longrightarrow> le z i))"
assms_4: fixes g1 :: "'t \<Rightarrow>\<^sub>0 'b"   and g2 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "g1 \<in> G"   and "g2 \<in> G"   and "g1 \<noteq> 0"   and "g2 \<noteq> 0"   and "spoly g1 g2 \<noteq> 0" shows "spoly_rep d m G g1 g2"
"1_2": shows "q \<in> transition A (w !! n) p"
fold_graph_determ: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'a set"   and x :: "'b"   and y :: "'b" assumes "comp_fun_commute f"   and "fold_graph f z A x"   and "fold_graph f z A y" shows "y = x"
C_eq_Sets_mr: fixes p :: "((int \<times> int) set set, int) Combinators list"   and s :: "((int \<times> int) set set, int) Combinators list"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "set p = set s"   and "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy1_strong s"   and "wellformed_policy3 p"   and "wellformed_policy3 s"   and "allNetsDistinct p" shows "applied_rule_rev C x p = applied_rule_rev C x s"
combine_common_factor: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and e :: "'m"   and b :: "'m"   and c :: "'m" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
coActionSimps_2: fixes a :: "act" shows "(coAction a = \<tau>) = (a = \<tau>)"
less_fset_def: fixes xs :: "'a fset"   and ys :: "'a fset" shows "local.psubset_fset xs ys \<equiv> local.subset_fset xs ys \<and> xs \<noteq> ys"
typeof_lit_is_type: fixes v :: "val"   and T :: "ty"   and P :: "'a prog" assumes "typeof v = \<lfloor>T\<rfloor>" shows "is_type P T"
domains_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.seq g f" shows "CC.domains (CC.comp g f) = CC.domains f"
subcategory_axioms: shows "subcategory VxV.comp (\<lambda>\<mu>\<nu>. arr (fst \<mu>\<nu>) \<and> arr (snd \<mu>\<nu>) \<and> src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>))"
enf: shows "\<forall>m\<in>set M. \<exists>P. \<forall>G. c_sinvar m G = (\<forall>e\<in>edges G. P e)"
IH: fixes x :: "'a" assumes "\<And>z. z \<in> set xs \<Longrightarrow> f z = f x" shows "g (fold merge xs x # ys) = g (x # xs @ ys)"
suminf_le: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n \<le> g n"   and "summable f"   and "summable g" shows "suminf f \<le> suminf g"
underS_notIn: fixes a :: "'a"   and r :: "('a \<times> 'a) set" shows "a \<notin> underS r a"
simps_19: fixes v :: "nat" shows "not (NClosed v) = NOT (NClosed v)"
hyps: assumes "poly_inv p" shows "poly_inv (poly_subst \<sigma> p)"
is_tdghmI: fixes \<alpha> :: "V"   and \<NN> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<NN>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "vcard \<NN> = 5\<^sub>\<nat>"   and "vsv (\<NN>\<lparr>NTMap\<rparr>)"   and "\<D>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"   and "\<NN>\<lparr>NTDom\<rparr> = \<FF>"   and "\<NN>\<lparr>NTCod\<rparr> = \<GG>"   and "\<NN>\<lparr>NTDGDom\<rparr> = \<AA>"   and "\<NN>\<lparr>NTDGCod\<rparr> = \<BB>"   and "\<And>a. a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr> \<Longrightarrow> \<NN>\<lparr>NTMap\<rparr>\<lparr>a\<rparr> : \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<mapsto>\<^bsub>\<BB>\<^esub> \<GG>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
comp_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>B c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>B f : a \<rightarrow>\<^sub>B c\<guillemotright>"
length_remove1: fixes x :: "'a"   and xs :: "'a list" shows "length (remove1 x xs) = (if x \<in> set xs then length xs - 1 else length xs)"
1: shows "(\<Sum>i<n. if i = a then f i else (0::'b)) = (\<Sum>i<n. if i = a then g i else (0::'b))"
prems: shows "\<Gamma> \<turnstile> App Lam [x].t s : T"
hyps: shows "changes_itv_der c d2 (pderiv p) = 0"
simps_2: fixes w :: "'a"   and ws :: "'a list"   and r :: "'b rexp" shows "lderivs (w # ws) r = lderivs ws (lderiv w r)"
False: shows "\<not> (\<forall>k. b ((c ^^ k) s))"
pq_ge_0: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "0 < int p * int q"
morphism_presheaves_of_rings_axioms_def: fixes is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> (\<forall>U. is_open U \<longrightarrow> ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)) \<and> (\<forall>U V. is_open U \<longrightarrow> is_open V \<longrightarrow> V \<subseteq> U \<longrightarrow> (\<forall>x. x \<in> \<FF> U \<longrightarrow> (\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x))"
lhopital_left_at_top_at_bot: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and g :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "filterlim f at_top (at_left a)"   and "filterlim g at_bot (at_left a)"   and "\<forall>\<^sub>F x in at_left a. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_left a. (g has_real_derivative g' x) (at x)"   and "LIM x at_left a. f' x / g' x :> at_bot" shows "LIM x at_left a. f x / g x :> at_bot"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "\<Squnion> (C ` A) = \<Squnion> (D ` B)"
spr_jviewInit_def: shows "spr_jviewInit \<equiv> \<lambda>a. tInit"
converse_matchrel_subset_matchless: shows "matchrel\<inverse> \<subseteq> matchless"
distinct_sorted_intersection: fixes xs :: "'c list"   and ys :: "'c list" assumes "distinct xs"   and "distinct ys"   and "sorted xs"   and "sorted ys" shows "distinct (intersection xs ys) \<and> sorted (intersection xs ys)"
lhs_def: shows "lhs \<equiv> module_ow (Collect (Domainp B))"
iprev_atLeast: fixes n :: "nat"   and t :: "nat" assumes "n < t" shows "iprev t {n..} = t - Suc 0"
foundation6: fixes \<tau> :: "'b state \<times> 'b state"   and P :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P" shows "\<tau> \<Turnstile> \<delta> P"
finite_measure_mono_AE: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b pmf" assumes "AE x\<in>A in measure_pmf M. x \<in> B"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A \<le> measure_pmf.prob M B"
vmc_path_axioms: shows "vmc_path G P v0 p \<sigma>"
g: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "(cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst (g2 \<sigma>) mem\<^sub>h\<rangle> \<and> (cms\<^sub>1' ! k, subst (g1 \<sigma>) mem\<^sub>1') \<approx> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst (g2 \<sigma>) mem\<^sub>h\<rangle>"
DSourceOfDSource: fixes Z :: "CSet"   and i :: "AbstrLevelsID"   and S :: "CSet"   and C :: "CSet" assumes "Z \<in> DSources i S"   and "S \<in> DSources i C" shows "Z \<in> Sources i C"
rcoset_reps: shows "Supgroup.is_rcoset_replist G hs"
monic_degree_one_roots: fixes a :: "'a" assumes "a \<in> carrier R" shows "roots [\<one>, \<ominus> a] = {#a#}"
prod_of_pdevs_def: fixes x :: "'b pdevs"   and y :: "'c pdevs" shows "prod_of_pdevs x y = binop_pdevs Pair x y"
Pi_closed: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and t :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "ct \<in> Pi S K"   and "s \<in> S"   and "t \<in> set_pmf (ct s)" shows "t \<in> S"
A: shows "Gromov_extension (inv f) \<circ> Gromov_extension f = id"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) \<chi>.A.map D \<chi>"
entailment_def: fixes \<Gamma> :: "'a formula set"   and F :: "'a formula" shows "\<Gamma> \<TTurnstile> F \<equiv> \<forall>\<A>. (\<forall>G\<in>\<Gamma>. \<A> \<Turnstile> G) \<longrightarrow> \<A> \<Turnstile> F"
homh: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f" assumes "group_hom G H h" shows "h \<in> hom G H"