hcju/isabelleps huggingface dataset:
mkIde_def: fixes A :: "'s set" shows "mkIde A = (if A \<subseteq> Univ \<and> |A| <o \<AA> then inv_into (Collect ide) local.set A else null)"
c2dc_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex" assumes "residues_nat n" shows "residues_nat.c2dc n \<chi> = (\<lambda>x. \<chi> (x mod n))"
negligible_abs: fixes f :: "nat \<Rightarrow> real" shows "negligible (\<lambda>x. \<bar>f x\<bar>) = negligible f"
normalize_gcd: fixes a :: "'a"   and b :: "'a" shows "normalize (gcd a b) = gcd a b"
hom_lead_coeff: fixes p :: "'a poly" shows "lead_coeff (map_poly hom p) = hom (lead_coeff p)"
ltl_convert_def: fixes \<phi> :: "expr ltlc" shows "ltl_convert \<phi> = (let APs = ltlc_aps_list \<phi> in Let (map_ltlc ((the \<circ>\<circ> idx) APs) \<phi>) (Pair (IArray APs)))"
Comm: fixes P :: "ccs"   and a :: "act"   and P' :: "ccs"   and Q :: "ccs"   and Q' :: "ccs" assumes "P \<longmapsto>a \<prec> P'"   and "Q \<longmapsto>coAction a \<prec> Q'"   and "a \<noteq> \<tau>" shows "P \<parallel> Q \<longmapsto>\<tau> \<prec> P' \<parallel> Q'"
transition_def: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" shows "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l' = ((l, g, \<mu>) \<in> PTA.trans_of A \<and> (X, l') \<in> set_pmf \<mu>)"
simps_7: shows "merge_cstatus CSUCCESS CSUCCESS = CSUCCESS"
longest_prefix_def: fixes j :: "nat" shows "longest_prefix j \<equiv> THE y. (\<forall>x<y. \<psi> j x \<down>) \<and> (\<forall>x\<ge>y. \<psi> j x \<up>)"
add_decreasing: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> (0::'a)"   and "c \<le> b" shows "a + c \<le> b"
Eq_tree: fixes x :: "'a" shows "f \<diamondop> pure x = g \<diamondop> pure x"
borel_measurable_nn_integral: fixes f :: "'a \<Rightarrow> val \<Rightarrow> ennreal"   and N :: "'a measure"   and i :: "nat" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (N \<Otimes>\<^sub>M stock_measure (\<Gamma> i))" shows "(\<lambda>x. integral\<^sup>N (stock_measure (\<Gamma> i)) (f x)) \<in> borel_measurable N"
C: shows "prv (imp (lcnj \<phi>2s') (lcnj \<phi>2s))"
B: fixes B :: "bool" assumes "my_locale2 B" shows "B"
q_pderiv: shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly q x) = sgn (poly (pderiv p) x)"
pr_naturality_2: fixes g :: "'a"   and b :: "'a"   and d :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "arr g"   and "local.dom g = b"   and "cod g = d"   and "arr f"   and "local.dom f = a"   and "cod f = c" shows "prj1' c d \<cdot> ECC.prod f g = f \<cdot> prj1' a b"
indRelRTPO_impl_TRel_is_weak_reduction_correspondence_simulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_correspondence_simulation (indRelRTPO TRel) (STCal Source Target)" shows "weak_reduction_correspondence_simulation (TRel\<^sup>+) Target"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "B.in_hhom \<mu> a b \<equiv> B.arr \<mu> \<and> src\<^sub>B \<mu> = a \<and> trg\<^sub>B \<mu> = b"
rtrancl_path_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b" assumes "rtrancl_path r x xs y"   and "rtrancl_path r y ys z" shows "rtrancl_path r x (xs @ ys) z"
finite_distinct_list: fixes A :: "'a set" assumes "finite A" shows "\<exists>xs. set xs = A \<and> distinct xs"
HRB_slice_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S \<equiv> {n'. \<exists>n\<in>S. n' \<in> SDG.combine_SDG_slices sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses (SDG.sum_SDG_slice1 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n)}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "one_hom hom"   and "injective hom" shows "inj_one_hom hom"
remove: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A = gcd a (Gcd\<^sub>f\<^sub>i\<^sub>n (A - {a}))"
Sup_mono: fixes A :: "('b \<times> 'c) set"   and B :: "('b \<times> 'c) set" assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. less_eq_prod a b" shows "less_eq_prod (Sup_prod A) (Sup_prod B)"
boundedBySuc: fixes sigma :: "'b"   and subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and n :: "nat" assumes "sigma \<in> subs Gamma"   and "boundedBy (Suc n) (tree subs Gamma)" shows "boundedBy n (tree subs sigma)"
cod_simp: fixes f :: "'a" assumes "R.arr f" shows "R.cod f = cod f"
ring_is_ag: fixes R :: "('d, 'e) Ring_scheme" assumes "Ring R" shows "aGroup R"
add_right_mono: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "\<alpha> \<le> \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<alpha> + \<gamma> \<le> \<beta> + \<gamma>"
poly_induct3: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and Q :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "\<And>p q. \<lbrakk>q \<in> carrier (UP R); p \<in> carrier (UP R); Q p; Q q\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>UP R\<^esub> q)"   and "\<And>a n. a \<in> carrier R \<Longrightarrow> Q (monom (UP R) a n)" shows "Q p"
IH: fixes ys :: "'a list"   and zs :: "'a list" assumes "list_emb P (xs @ ys) zs" shows "\<exists>us vs. zs = us @ vs \<and> list_emb P xs us \<and> list_emb P ys vs"
R2_H2_commute: shows "R2 \<circ> H2 = H2 \<circ> R2"
insert_inc2: fixes a :: "'c"   and A :: "'c set" shows "a \<in> insert a A"
updEnv_overwrite: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "rho [x \<leftarrow> X]_xs [x \<leftarrow> X']_xs = rho [x \<leftarrow> X']_xs"
simps_3: fixes x21 :: "'d"   and x22 :: "'d list" shows "x21 # x22 \<noteq> []"
simps: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "get_min \<langle>l, a, r\<rangle> = a"
Collect_cong: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "\<lbrace>\<acute>P\<rbrace> = \<lbrace>\<acute>Q\<rbrace>"
collapse_4: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "is_Insert stateful_strand_step" shows "insert\<langle>the_elem_term stateful_strand_step,the_set_term stateful_strand_step\<rangle> = stateful_strand_step"
ta_hb_consistent_LNil: fixes P :: "'a prog"   and obs :: "('b \<times> ('c, 'b) obs_event action) list" shows "ta_hb_consistent P obs LNil = True"
object_ptr_kinds_M_eq_h3: fixes ptrs :: "(_) object_ptr list" shows "h3 \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs = h' \<turnstile> object_ptr_kinds_M \<rightarrow>\<^sub>r ptrs"
south_def: shows "south = proj2_abs (vector [0, - 1, 1])"
elims_2: fixes x :: "'c regular \<times> 'c dense Filters.filter"   and xa :: "'c regular \<times> 'c dense Filters.filter" assumes "stone_phi.pairs_less_eq x xa" obtains xb :: "'c regular" and y :: "'c dense Filters.filter" and z :: "'c regular" and w :: "'c dense Filters.filter" where "x = (xb, y)" and "xa = (z, w)" and "xb \<le> z \<and> w \<le> y"
h_def: shows "h = (\<lambda>i. hd (m i))"
c2: fixes dsn :: "nat"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "0 < dsn" shows "P (dsn, kno, val, hops, nhip, pre)"
pseudo_proj_True_def: fixes n :: "nat" shows "pseudo_proj_True n = (\<lambda>w. stake n w @- sconst True)"
prime_irreducible: fixes p :: "'a" assumes "prime (Ring_Divisibility.mult_of R) p" shows "irreducible (Ring_Divisibility.mult_of R) p"
extremum_uniqueI: fixes a :: "'a multiset" assumes "a \<subseteq># {#}" shows "a = {#}"
equal_poly_def: fixes p :: "'a poly"   and q :: "'a poly" shows "equal_class.equal p q = equal_class.equal (coeffs p) (coeffs q)"
IsoInvId1: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "ciso\<^bsub>C\<^esub> f" shows "Cinv\<^bsub>C\<^esub> f ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)"
prems_1: fixes B :: "'a tensor" assumes "B \<in> set (B__ # Bs)" shows "Tensor.order B = 1"
preserves_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<F>C.D f : \<F>C.D a \<rightarrow>\<^sub>S \<F>C.D b\<guillemotright>"
idx_iteratei_aux_simps_2: fixes c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and get :: "'d \<Rightarrow> nat \<Rightarrow> 'e"   and sz :: "nat"   and i :: "nat"   and l :: "'d"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<not> c \<sigma>" shows "idx_iteratei_aux get sz i l c f \<sigma> = \<sigma>"
right_idem: fixes a :: "'a"   and b :: "'a" shows "inf (inf a b) b = inf a b"
valid_regions_intv_distinct: fixes X :: "'c set"   and I :: "'c \<Rightarrow> Regions.intv"   and r :: "('c \<times> 'c) set"   and I' :: "'c \<Rightarrow> Regions.intv"   and r' :: "('c \<times> 'c) set"   and u :: "'c \<Rightarrow> real"   and x :: "'c" assumes "Regions.valid_region X k I r"   and "Regions.valid_region X k I' r'"   and "u \<in> Regions.region X I r"   and "u \<in> Regions.region X I' r'"   and "x \<in> X" shows "I x = I' x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b tree \<Rightarrow> 'b tree \<Rightarrow> nat \<Rightarrow> 'a"   and tree :: "'b tree" shows "P (case tree of ET \<Rightarrow> f1 | MKT x xa xb xc \<Rightarrow> f2 x xa xb xc) = ((tree = ET \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24. tree = MKT x21 x22 x23 x24 \<longrightarrow> P (f2 x21 x22 x23 x24)))"
node_props_formaldef_def: fixes P :: "('v, 'a) TopoS_Params" shows "node_props_formaldef P \<equiv> \<lambda>i. if i \<in> dom (node_properties P) then the (node_properties P i) else \<bottom>"
3: shows "\<Phi>.map yx = S.mkArr (HomC.set (F (fst yx), snd yx)) (HomD.set (fst yx, G (snd yx))) (inD \<circ> \<phi> (fst yx) \<circ> HomC.\<psi> (F (fst yx), snd yx))"
Cod_preserves_Diag: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t" shows "monoidal_language.Diag C (monoidal_language.Cod C t)"
mkarr_memb_2: fixes t :: "'c local.term"   and f :: "'c local.term set" assumes "t \<in> f"   and "arr f" shows "mkarr t = f"
invar: shows "\<forall>i. enat i < llength TNs \<longrightarrow> lgc_invar TNs (enat i)"
initial_lower_bound_def: fixes l :: "rat" shows "local.initial_lower_bound l = (if l \<le> 1 then l else rat_of_int (root_rat_floor n l))"
lookup_insert: fixes k :: "'b"   and v :: "'a"   and xs :: "('b, 'a) oalist_ntm"   and k' :: "'b" shows "OAlist_lookup_ntm (OAlist_insert_ntm (k, v) xs) k' = (if k = k' then v else OAlist_lookup_ntm xs k')"
RestrictedP_Mem_D: fixes H :: "fm set"   and s :: "tm"   and k :: "tm"   and t :: "tm"   and a :: "tm"   and A :: "fm" assumes "H \<turnstile> RestrictedP s k t"   and "H \<turnstile> a IN t"   and "insert (a IN s) H \<turnstile> A" shows "H \<turnstile> A"
r_totalizer_def: fixes d :: "nat" shows "r_totalizer d \<equiv> Cn 2 (r_lifz (r_constn 1 d) (Cn 2 r_phi [recf.Id 2 0, Cn 2 r_take [Cn 2 r_delay [recf.Id 2 0, recf.Id 2 1], recf.Id 2 1]])) [Cn 2 r_delay [recf.Id 2 0, recf.Id 2 1], recf.Id 2 0, recf.Id 2 1]"
sel_12: fixes x31 :: "'c"   and x32 :: "'a"   and x33 :: "'b" shows "new_var (Extension x31 x32 x33) = x32"
iIN_Min: fixes n :: "nat"   and d :: "nat" shows "iMin [n\<dots>,d] = n"
rBang: fixes B :: "('a, 'b, 'c) boundOutput"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> ROut M B"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) M B"   and "guarded P" shows "Prop C \<Psi> (!P) M B"
sets_lim: shows "sets local.lim = sets (Pi\<^sub>M I M)"
assms_5: shows "b2 = p / (2::'a) + m + q / ((2::'a) * a)"
horner_sum_simps_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.comm_semiring_0 plus zero times" shows "comm_semiring_0.horner_sum plus zero times f a [] = zero"
butterfly_scaleR_left: fixes r :: "real"   and \<psi> :: "'c"   and \<phi> :: "'b" shows "butterfly (r *\<^sub>R \<psi>) \<phi> = complex_of_real r *\<^sub>C butterfly \<psi> \<phi>"
f_Exec_Stream_Acc_Output_eq_last_message_hold: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c = map output_fun (f_Exec_Comp_Stream trans_fun (xs \<odot> k) c) \<longmapsto> k \<div>\<^sub>l k"
d2_def: fixes a :: "'a"   and b :: "'a" shows "d2 a b = (a r\<rightarrow> b) * (b r\<rightarrow> a)"
pre: shows "wf_since_aux \<sigma> n R pos \<phi> I \<psi> aux ne"
11: shows "\<ominus>\<^bsub>K\<^esub> (inv\<^bsub>K\<^esub> a v) \<in> carrier K"
termMOD_igSubstIGVar2: shows "igSubstIGVar2 termMOD"
getFr_def: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list" shows "getFr xs ts \<phi>s = getFresh (set xs \<union> \<Union> (FvarsT ` set ts) \<union> \<Union> (Fvars ` set \<phi>s))"
D: shows "\<forall>X\<in>A. X \<inter> next_events P xs = {}"
tpart2_def: fixes V :: "Preliminaries.var set" shows "tpart2 V \<equiv> \<Union>\<sigma>. part2 (V \<inter> tpOfV -` {\<sigma>})"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> [a, b]\<^sub>\<circ>"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> [a, b]\<^sub>\<circ>" shows "b = c"
minf: shows "- \<infinity> = Abs_Ainteg (0, - 1)"
bind_spmf_cong: fixes p :: "'b spmf"   and q :: "'b spmf"   and f :: "'b \<Rightarrow> 'c spmf"   and g :: "'b \<Rightarrow> 'c spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
alignUp_unfold: fixes w :: "'b word"   and n :: "nat" shows "alignUp w n = w + mask n AND NOT (mask n)"
Skip: fixes G :: "prog"   and s :: "st"   and n :: "nat" shows "G\<turnstile>Norm s \<midarrow>Skip\<midarrow>n\<rightarrow> Norm s"
distinct_101: fixes x8 :: "int"   and x101 :: "nat"   and x102 :: "msg" shows "Pub_AgrK x8 \<noteq> Crypt x101 x102"
Derives1_sentence1: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "Derives1 a i r b" shows "is_sentence a"
hmset_of_enat_numeral: fixes n :: "num" shows "hmset_of_enat (numeral n) = numeral n"
addSubsetSetI_2: fixes t :: "'b set"   and ts :: "'b set set"   and s :: "'b set" assumes "t \<in> ts" shows "s \<inter> t \<in> addSubsetSet s ts"
cet_pow_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : REAL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c e1 ^\<^sub>c e2 : REAL"
in_notinI: fixes r :: "('a \<times> 'a) set"   and j :: "'a"   and i :: "'a" assumes "wo_rel r"   and "(j, i) \<notin> r \<or> j = i"   and "i \<in> Field r"   and "j \<in> Field r" shows "(i, j) \<in> r"
assms_2: shows "N2 \<subseteq>\<^sub>N N"
insertion_one: fixes f :: "nat \<Rightarrow> 'b" shows "insertion f 1 = (1::'b)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "rev (x # xs) = rev xs \<cdot> [x]"
simps_2: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "insert_sort_by cmp (x # xs) = insert_sort_insert_by cmp x (insert_sort_by cmp xs)"
cod_simps_1: shows "\<mu>\<nu>_\<pi>.cod.apex = \<mu>\<nu>_\<pi>.apex"
preorder_on_axioms: shows "Order_Predicates.preorder_on carrier le"
AX: shows "inv_into X f ` f ` X \<subseteq> X"
s_def: shows "s = a \<otimes> j \<oplus> b \<otimes> i"
elem_def: fixes S :: "('a, 'b) eq_object_scheme"   and x :: "'a"   and A :: "'a set" shows "(x .\<in>\<^bsub>S\<^esub> A) = (\<exists>y\<in>A. x .=\<^bsub>S\<^esub> y)"
in_sccs_verts_conv_reachable: fixes S :: "'a set" shows "(S \<in> H.sccs_verts) = (S \<noteq> {} \<and> (\<forall>u\<in>S. \<forall>v\<in>S. u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v) \<and> (\<forall>u\<in>S. \<forall>v. v \<notin> S \<longrightarrow> \<not> u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v \<or> \<not> v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> u))"
ordLeq_reflexive: fixes r :: "('b \<times> 'b) set" assumes "Well_order r" shows "r \<le>o r"
bidel_arc_diff_face_axioms_def: fixes M :: "'c pre_map"   and a :: "'c" shows "bidel_arc_diff_face_axioms M a \<equiv> pre_digraph_map.face_cycle_set M (edge_rev M a) \<noteq> pre_digraph_map.face_cycle_set M a"
Order_od_fm_fun: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N" shows "Order (od_fm_fun R M N f H)"
multiplicative_dirichlet_prod: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (dirichlet_prod f g)"
uint64_nat_rel_def: shows "uint64_nat_rel = br nat_of_uint64 (\<lambda>_. True)"
inv: fixes x :: "'a" assumes "P x" shows "\<exists>y. f y = x"
silent_move_no_slice_edge: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and rs :: "'edge list"   and cs :: "'edge list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"   and "tl ms = targetnodes rs"   and "length rs = length cs"   and "\<forall>i<length cs. call_of_return_node (tl ms ! i) (sourcenode (cs ! i))" shows "\<not> slice_edge S cs a"
dghm_HomDom: shows "\<KK>\<lparr>HomDom\<rparr> = \<AA>"
sets_range: fixes A :: "'d \<Rightarrow> 'c set"   and I :: "'d set"   and M :: "'c measure"   and i :: "'d" assumes "A ` I \<subseteq> sets M"   and "i \<in> I" shows "A i \<in> sets M"
G_is_essentially_surjective: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "equivalence_of_categories C D F G \<eta> \<epsilon>" shows "essentially_surjective_functor C D G"
iso_cancel_right: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "CC.iso g"   and "CC.comp f g = CC.comp f' g"   and "CC.seq f g"   and "CC.iso g" shows "f = f'"
assms: shows "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px"
upRuleCharacterise: fixes Ps :: "'a list"   and C :: "'b sequent" assumes "(Ps, C) \<in> upRules" shows "\<exists>F Fs. C = ( \<Empt> \<Rightarrow>* Compound F Fs) \<or> C = ( \<LM> Compound F Fs  \<RM> \<Rightarrow>* Em)"
S3: shows "c_fst (c_pair 4 (c_pair (sch_to_nat sch1) (sch_to_nat sch2))) = 4"
mem_simps_9: fixes A :: "'d"   and C :: "'d set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
set_less_eq_aux_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "ord.set_less_eq_aux less_eq A {} = (A = {})"
FGModIso_composite_left: fixes G :: "'c set"   and smult :: "('d, 'c) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and V :: "'e set"   and smult' :: "('d, 'c) aezfun \<Rightarrow> 'h \<Rightarrow> 'h"   and T :: "'e \<Rightarrow> 'h"   and W :: "'h set"   and smult'' :: "('d, 'c) aezfun \<Rightarrow> 'i \<Rightarrow> 'i"   and S :: "'h \<Rightarrow> 'i"   and X :: "'i set" assumes "FGModuleIso G smult V smult' T W"   and "FGModuleIso G smult' W smult'' S X" shows "FGModuleIso G smult V smult'' (S \<circ> T) X"
intros_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> \<Rightarrow> Fault f"
enum_finite_2_def: shows "enum = [a\<^sub>1, a\<^sub>2]"
simps_14: fixes x61 :: "nat"   and x62 :: "recf" shows "Mn x61 x62 \<noteq> Z"
inv_atI: fixes n :: "nat"   and p :: "'b mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'b mat"   and m :: "nat" assumes "\<And>i. i < n \<Longrightarrow> p A i m" shows "inv_at n p A m"
group_ring_spanning_set: fixes G :: "'a set" assumes "FinGroup G" shows "\<exists>gs. distinct gs \<and> set gs = G \<and> (\<forall>f\<in>aezfun_setspan G. \<exists>bs. length bs = length gs \<and> f = sum_list (map2 (\<lambda>b g. b \<delta>\<delta> (0::'a) * ((1::'b) \<delta>\<delta> g)) bs gs))"
split_max: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and i :: "'c"   and j :: "'c" assumes "class.linorder less_eq less" shows "P (ord.max less_eq i j) = ((less_eq i j \<longrightarrow> P j) \<and> (\<not> less_eq i j \<longrightarrow> P i))"
chain': shows "Complete_Partial_Order.chain (\<le>) (measure_spmf ` Y)"
powers_6: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b * (a * c) = a * (b * c)"
vdash_properties_10: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<phi> in v]" shows "[\<psi> in v]"
redT: shows "s -t\<triangleright>ta\<rightarrow> s'"
run_alt_writerT: fixes m :: "('b, 'c, 'a) writerT"   and m' :: "('b, 'c, 'a) writerT" shows "run_writer (alt m m') = alt (run_writer m) (run_writer m')"
scaleC_minus1_left: fixes x :: "'d" shows "- 1 *\<^sub>C x = - x"
assms_4: shows "maximal_literal (subst_lit L1 \<sigma>) (subst_cl C \<sigma>)"
ideal_rb_aux: fixes z :: "nat" shows "ideal (set (map rep_list (fst (fst (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)))))) = ideal (set fs)"
bounded_iff: fixes A :: "('x \<Rightarrow>\<^sub>0 nat) set"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite A"   and "A \<noteq> {}" shows "(x \<preceq> ordered_powerprod_lin.Min A) = (\<forall>a\<in>A. x \<preceq> a)"
IntD1: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A \<inter> B" shows "c \<in> A"
fst_ts_tuple_rel_before: shows "\<forall>t\<in>fst ` ts_tuple_rel (set (linearize data_in)). \<forall>t'\<in>fst ` ts_tuple_rel (set move). t \<le> t'"
smult_power: fixes a :: "'a"   and p :: "'a poly"   and n :: "nat" shows "Polynomial.smult a p ^ n = Polynomial.smult (a ^ n) (p ^ n)"
root_1: assumes "0 < degree p" shows "((\<lambda>r. Re (winding_number (poly p \<circ> part_circlepath z0 r 0 pi) 0)) \<longlongrightarrow> real (degree p) / 2) at_top"
orthogonal: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs"   and "i < length fs"   and "j < length fs"   and "i \<noteq> j" shows "gram_schmidt_fs.gso n fs i \<bullet> gram_schmidt_fs.gso n fs j = (0::'a)"
intros_1: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and n :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and n' :: "nat" assumes "((\<sigma>, n), \<sigma>', n') \<in> cp"   and "path \<sigma> n \<in> dom local.att" shows "((\<sigma>, n), \<sigma>', n') \<in> cop"
monom_altdef: fixes c :: "'a"   and n :: "nat" shows "monom c n = smult c ([:0::'a, 1::'a:] ^ n)"
hd_Cons_tl: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"