hcju/isabelleps huggingface dataset:
psimps: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "Wellfounded.accp (Smith_Impl.Smith_mxn_rel Smith_1x2 Smith_2x2 div_op) A" shows "Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op A = (if dim_row A = 0 \<or> dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A)) else if dim_row A = 1 then (1\<^sub>m 1, Smith_Impl.Smith_1xn Smith_1x2 A) else if dim_row A = 2 then Smith_Impl.Smith_2xn Smith_1x2 Smith_2x2 div_op A else if dim_col A = 1 then let (P, S) = Smith_Impl.Smith_nx1 Smith_1x2 A in (P, S, 1\<^sub>m 1) else if dim_col A = 2 then Smith_Impl.Smith_nx2 Smith_1x2 Smith_2x2 div_op A else let A1 = mat_of_row (Matrix.row A 0); A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A]); (P1, D1, Q1) = Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op A2; C = A1 * Q1 @\<^sub>r P1 * A2 * Q1; D = mat_of_rows (dim_col A) [Matrix.row C 0, Matrix.row C 1]; E = mat_of_rows (dim_col A) (map (Matrix.row C) [2..<dim_row A]); (P2, F, Q2) = Smith_Impl.Smith_2xn Smith_1x2 Smith_2x2 div_op D; H = P2 * D * Q2 @\<^sub>r E * Q2; (P_H2, H2) = reduce_column div_op H; (H2_UL, H2_UR, H2_DL, H2_DR) = split_block H2 1 1; (P3, S', Q3) = Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_row A - 1) 1) S'; P1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P1; P2' = four_block_mat P2 (0\<^sub>m 2 (dim_row A - 2)) (0\<^sub>m (dim_row A - 2) 2) (1\<^sub>m (dim_row A - 2)); P3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P3; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P3' * P_H2 * P2' * P1', S, Q1 * Q2 * Q3'))"
Diff_subset_conv: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
lift_ltl_transformer_axioms: shows "lift_ltl_transformer af_G_letter"
inter_overappr: fixes Z :: "'a set"   and n :: "'a"   and g :: "real" shows "Z \<inter> {x. x \<bullet> n = g} \<subseteq> {x. \<forall>i\<in>Basis. x \<bullet> i \<in> {y. (g, y) \<in> (\<lambda>x. (x \<bullet> n, x \<bullet> i)) ` Z}}"
con_rews: fixes A :: "A" shows "B\<cdot>A \<noteq> \<bottom>"
proper_interval_None2: fixes ao :: "'a option" shows "proper_interval ao None = (above ao \<noteq> {})"
simps: fixes i :: "'a \<times> nat"   and j :: "'a \<times> nat" shows "happened_immediately_before i j = (is_valid_event_id i \<and> is_valid_event_id j \<and> (fst i = fst j \<and> Suc (snd i) = snd j \<or> is_reception i j))"
abs_eq: shows "1 = (1::'a) \<delta>\<delta> (0::'b)"
DF_ne: fixes k :: "nat"   and i :: "nat" shows "DF k i \<noteq> {}"
ldrop_lzip: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'d llist" shows "ldrop n (lzip xs ys) = lzip (ldrop n xs) (ldrop n ys)"
extendEmpty: fixes C :: "sequent" shows "extend ( \<Empt> \<Rightarrow>* \<Empt>) C = C"
dbm_not_lt_eq: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "\<not> a \<prec> b"   and "\<not> b \<prec> a" shows "a = b"
distinct_9: fixes x2 :: "bexp"   and x41 :: "aexp"   and x42 :: "aexp" shows "bexp.Not x2 \<noteq> Less x41 x42"
l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs \<equiv> type_wf = ShadowRootClass.type_wf \<and> set_mode = set_mode \<and> set_mode_locs = set_mode_locs"
surj_iff: fixes f :: "'b \<Rightarrow> 'a" shows "top_pres ((`) f) = (f \<circ> inv f = id)"
1: fixes C1 :: "'b set"   and C2 :: "'b set" assumes "span C1 = span C2" shows "span (insert p C1) \<subseteq> span (insert p C2)"
sig_red_singleD5: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "ordered_powerprod_lin.is_le_rel top_tail"
D_ASSIGN_H: fixes x :: "Var"   and e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" assumes "CONTEXT x = high"   and "\<forall>s. noLowDPs s \<longrightarrow> Expr_good e s" shows "(G, Assign x e, HighSec) \<in> Deriv"
tp_commute: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and b :: "'c" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<cdot>\<^sub>r\<^bsub>R\<^esub> b = b \<cdot>\<^sub>r\<^bsub>R\<^esub> a"
Left_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> R" shows "g +| I = additive.Class g"
paths_disjoint: fixes xs :: "'a list"   and ys :: "'a list"   and v :: "'a" assumes "xs \<in> paths"   and "ys \<in> paths"   and "xs \<noteq> ys"   and "v \<in> set xs"   and "v \<in> set ys" shows "v = v0 \<or> v = v1"
sign_simps_41: fixes am :: "'a"   and bm :: "'a"   and ck :: "'a" shows "am - (bm - ck) = am + ck - bm"
conc_star_comm: fixes A :: "'a list set" shows "A @@ star A = star A @@ A"
valid_equiv_PL: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<longleftrightarrow>. q) = ({p} \<turnstile> q \<and> {q} \<turnstile> p)"
simple_function_compose1: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "simple_function M f" shows "simple_function M (\<lambda>x. g (f x))"
not_less_Least: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and k :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
simps_8: fixes e1 :: "exp"   and C :: "ctx"   and e3 :: "exp"   and e :: "exp" shows "plug (CIf2 e1 C e3) e = EIf e1 (plug C e) e3"
that: fixes c' :: "real"   and d' :: "real" assumes "{a..b} \<inter> g -` {c..d} = {c'..d'}"   and "c' \<le> d'"   and "g c' = c"   and "g d' = d" shows "thesis"
of_bl_rep_False: fixes n :: "nat"   and bs :: "bool list" shows "of_bl (replicate n False @ bs) = of_bl bs"
hd_filter_prop: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" assumes "filter P xs \<noteq> []" shows "P (hd (filter P xs))"
cases: fixes a1 :: "Arity"   and a2 :: "Value"   and a3 :: "Value" assumes "eq a1 a2 a3" obtains v :: "Value" where "a1 = 0" and "a2 = v" and "a3 = v"   | n :: "Arity" and v1 :: "Value" and v2 :: "Value" where "a1 = inc\<cdot>n" and "a2 = v1" and "a3 = v2" and "\<And>v. eq n (v1 \<down>Fn v) (v2 \<down>Fn v)"
same_arities: shows "all_same_arities (Fubx_get F2) (Fubx_get (Fubx_add F2 f (rewrite_fundef_body fd pc instr)))"
wf_update_until: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'muaux"   and ne :: "nat"   and n :: "nat"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and I :: "\<I>"   and pos :: "bool" assumes "wf_until_aux \<sigma> V R args \<phi> \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<psi>) rel2"   and "fv \<phi> \<subseteq> fv \<psi>"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>"   and "args_R args = fv \<psi>"   and "args_pos args = pos" shows "wf_until_aux \<sigma> V R args \<phi> \<psi> (add_new_muaux args rel1 rel2 (\<tau> \<sigma> (ne + length_muaux args aux)) aux) ne \<and> length_muaux args (add_new_muaux args rel1 rel2 (\<tau> \<sigma> (ne + length_muaux args aux)) aux) = Suc (length_muaux args aux)"
PPs_closed_single: fixes x :: "'c"   and X :: "'c set"   and e :: "nat" assumes "x \<in> X" shows "monomial e x \<in> .[X]"
is_map_from_is_homomorphism: fixes U :: "'a set set"   and V :: "'a set set" assumes "is_zariski_open U"   and "is_zariski_open V"   and "V \<subseteq> U" shows "Set_Theory.map (sheaf_spec_morphisms U V) (\<O> U) (\<O> V)"
path_connected_space_topspace_empty: fixes X :: "'b topology" assumes "topspace X = {}" shows "path_connected_space X"
Int_insert_left_if1: fixes a :: "'d"   and C :: "'d set"   and B :: "'d set" assumes "a \<in> C" shows "insert a B \<inter> C = insert a (B \<inter> C)"
le_swap: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c \<le> a + b" shows "c \<le> b + a"
induct: fixes P :: "('a \<times> 'b) list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "('a \<times> 'b) list"   and a1 :: "'a list" assumes "\<And>xs. P [] xs"   and "\<And>v va. P (v # va) []"   and "\<And>x y rs z zs. \<lbrakk>x = z \<Longrightarrow> P rs (z # zs); \<lbrakk>x \<noteq> z; x < z\<rbrakk> \<Longrightarrow> P rs (z # zs); \<lbrakk>x \<noteq> z; \<not> x < z\<rbrakk> \<Longrightarrow> P ((x, y) # rs) zs\<rbrakk> \<Longrightarrow> P ((x, y) # rs) (z # zs)" shows "P a0 a1"
Cassoc: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g"   and "g \<approx>>\<^bsub>C\<^esub> h" shows "f ;;\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h = f ;;\<^bsub>C\<^esub> (g ;;\<^bsub>C\<^esub> h)"
vwalk_wf_digraph_consI: fixes p :: "'a list"   and a :: "'a" assumes "vwalk p G"   and "a \<rightarrow> hd p" shows "vwalk (a # p) G"
rhss_def: fixes rhs :: "'b trm set" shows "rhss rhs \<equiv> {uu_. \<exists>X r. uu_ = X \<and> Trn X r \<in> rhs}"
distinct_swap: fixes i :: "nat"   and xs :: "'b list"   and j :: "nat" assumes "i < length xs"   and "j < length xs" shows "distinct (xs[i := xs ! j, j := xs ! i]) = distinct xs"
proj2_incident_left_abs: fixes v :: "(real, 3) vec"   and l :: "proj2_line" assumes "v \<noteq> 0" shows "proj2_incident (proj2_abs v) l = (v \<bullet> proj2_line_rep l = 0)"
eq: shows "\<forall>bs. wf_bs bs p \<longrightarrow> Ipoly bs p = (0::'a)"
finite_dimension_def: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and E :: "'a set" assumes "ring R" shows "ring.finite_dimension R K E = (\<exists>n. ring.dimension R n K E)"
suffix_drop: fixes n :: "nat"   and as :: "'b list" shows "drop n as \<le>s as"
dcd_3: shows "l cd\<^bsup>path \<sigma>\<^esup>\<rightarrow> k"
sym_lr_def: shows "sym_lr = (\<lambda>(rm, rp). (dual (F (dual rp, undual rm)), F (rm, rp)))"
adjoint_pair_antipar_1: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.adjoint_pair f g" shows "S.ide f"
mix: fixes p :: "'a pmf"   and a :: "real"   and y :: "'a" assumes "set_pmf p \<subseteq> A"   and "a \<in> {0<..<1}"   and "y \<in> A"   and "P p" shows "P (mix_pmf a p (return_pmf y))"
fls_shift_uminus: fixes m :: "int"   and f :: "'a fls" shows "fls_shift m (- f) = - fls_shift m f"
vimage_Int: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" shows "f -` (A \<inter> B) = f -` A \<inter> f -` B"
refine: fixes R_Iec :: "('b \<times> unit) set" shows "(impl_blue_dfs, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
absorb_iff2: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (max a b = b)"
domain_unique: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b"   and a' :: "'b" assumes "category C"   and "a \<in> partial_magma.domains C f"   and "a' \<in> partial_magma.domains C f" shows "a = a'"
lt_ge_tt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tt p \<preceq> punit.lt p"
Union_upper: fixes B :: "'a set"   and A :: "'a set set" assumes "B \<in> A" shows "B \<subseteq> \<Union> A"
powr_of_int: fixes z :: "complex"   and n :: "int" assumes "z \<noteq> 0" shows "z powr of_int n = (if 0 \<le> n then z ^ nat n else inverse (z ^ nat (- n)))"
induced_wellformed: fixes G :: "nat set \<times> nat set set"   and V :: "nat set" assumes "uwellformed G"   and "V \<subseteq> uverts G" shows "uwellformed (induced_subgraph V G)"
hyps_2: shows "atom ` domA \<Delta> \<sharp>* L"
that: fixes gs :: "'a list" assumes "set gs \<subseteq> carrier G"   and "cycle gs"   and "DirProds (\<lambda>g. G\<lparr>carrier := generate G {g}\<rparr>) (set gs) \<cong> G"   and "successively (dvd) (map ord gs)"   and "card (set gs) \<le> card gen"   and "compl_fam (\<lambda>g. generate G {g}) (set gs)"   and "\<one> \<notin> set gs" shows "thesis"
P2_security: fixes D :: "bool \<times> (nat \<times> nat) \<times> bool \<times> bool \<Rightarrow> bool spmf"   and m2 :: "bool"   and HCP_ad :: "real"   and m1 :: "bool \<times> bool" assumes "\<forall>a. lossless_spmf (D a)"   and "\<forall>b\<^sub>\<sigma>. etp_rsa.HCP_adv etp.\<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad" shows "etp.OT_12.adv_P2 m1 m2 D \<le> 2 * HCP_ad"
factoringlemma_withconstant: fixes c :: "real"   and S0 :: "'c list set"   and A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)" assumes "1 \<le> c"   and "\<forall>e\<in>S0. distinct e"   and "\<forall>e\<in>S0. 0 < length e"   and "List_Factoring.pairwise A"   and "\<forall>s0\<in>S0. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. \<forall>(x, y)\<in>{(x, y). x \<in> set s0 \<and> y \<in> set s0 \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy s0 {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy s0 {x, y}, is))) (Lxy qs {x, y}) \<le> c * real (T\<^sub>p_opt (Lxy s0 {x, y}) (Lxy qs {x, y})) + b"   and "\<And>is s q. \<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"   and "\<And>init qs x. \<lbrakk>distinct init; set qs \<subseteq> set init; x < length qs\<rbrakk> \<Longrightarrow> finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))" shows "\<forall>s0\<in>S0. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. T\<^sub>p_on_rand' A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs \<le> c * real (T\<^sub>p_opt s0 qs) + b"
CauchyI': fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
B_meas: fixes N :: "nat"   and K :: "real"   and e :: "real" shows "B N K e \<in> sets M"
sup_ccINF: fixes B :: "'d set"   and a :: "'a"   and f :: "'d \<Rightarrow> 'a" assumes "countable B" shows "sup a (Inf (f ` B)) = (INF b\<in>B. sup a (f b))"
distinct_5: fixes x1 :: "('c, 'd) Term.term"   and x41 :: "'d list"   and x42 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "send\<langle>x1\<rangle>\<^sub>s\<^sub>t \<noteq> \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t"
lb: shows "\<forall>p\<in>set ps. \<forall>x. poly p x = 0 \<longrightarrow> lb < x"
union_disjoint: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "F g (A \<union> B) = F g A \<^bold>* F g B"
measurable_smap: fixes X :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'d measure" assumes "X \<in> N \<rightarrow>\<^sub>M M" shows "smap X \<in> stream_space N \<rightarrow>\<^sub>M stream_space M"
a0i_trans_def: shows "a0i_trans \<equiv> ((\<Union>A B. \<Union> (range (a0i_running A B))) \<union> (\<Union>A B. \<Union> (range (a0i_commit A B))))\<^sup>="
insert_UNIV: fixes x :: "'d" shows "insert x UNIV = UNIV"
assms: shows "SubnetS s1 s2 \<in> init (pnet np p)"
sdom_dom_succs: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "dom_sl G"   and "cfg_doms.strict_dom G i j"   and "j \<in> cfg_doms.succs G k" shows "cfg_doms.dom G i k"
lcm_proj1_iff_int: fixes m :: "int"   and n :: "int" shows "(lcm m n = \<bar>m\<bar>) = (n dvd m)"
select_convs_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
assms_12: shows "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"
lupd_def2: fixes vn :: "lname"   and v :: "val"   and g :: "oref \<Rightarrow> obj option"   and l :: "lname \<Rightarrow> val option" shows "lupd(vn\<mapsto>v) (st g l) = st g (l(vn \<mapsto> v))"
s\<^sub>1: shows "s\<^sub>1 = (h\<^sub>1, l\<^sub>1)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "category C"   and "replete_set_category S"   and "functor C S F" shows "set_valued_functor C S F"
distinct_list_find_indices: fixes i :: "nat"   and Vs :: "'b list"   and x :: "'b" assumes "i < length Vs"   and "Vs ! i = x"   and "distinct Vs" shows "find_indices x Vs = [i]"
real_cond_exp_F_meas: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f"   and "f \<in> borel_measurable F" shows "AEeq M (real_cond_exp M F f) f"
try_spmf_def: fixes p :: "'d spmf"   and q :: "'d spmf" shows "TRY p ELSE q = p \<bind> (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
assms_2: shows "is2 \<lhd> nths ds (- A)"
lcm_int_int_eq: fixes m :: "nat"   and n :: "nat" shows "lcm (int m) (int n) = int (lcm m n)"
xfA: fixes x :: "'a"   and j :: "nat" assumes "x \<in> A j" shows "xf j x < e / real n"
assms_2: shows "\<forall>t\<in>closure. \<forall>(a, b)\<in>TI. \<forall>s\<in>set \<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle>. s \<in> closure"
dvd_const: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q"   and "q \<noteq> 0"   and "degree q = 0" shows "degree p = 0"
of_complex_to_complex: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "of_complex (to_complex z) = z"
put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and setter :: "('m \<Rightarrow> 'm) \<Rightarrow> ('c, 'a, 'b, 'j option) RShadowRoot_scheme \<Rightarrow> ('c, 'a, 'b, 'j option) RShadowRoot_scheme"   and v :: "'m" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr setter v"
rel_restrict_notR_1: fixes x :: "'a"   and y :: "'a"   and A :: "('a \<times> 'a) set"   and R :: "'a set" assumes "(x, y) \<in> rel_restrict A R" shows "x \<notin> R"
bag_finite: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set"   and t :: "'e" assumes "TreeDecomposition G T bag"   and "t \<in> V\<^bsub>T\<^esub>" shows "finite (bag t)"
l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs \<equiv> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_known_ptrs known_ptr known_ptrs \<and> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes get_child_nodes_locs get_parent get_parent_locs"
Under_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Under r A \<subseteq> Field r"
vsv_vdomain_vrange_vsingleton: fixes a :: "V"   and b :: "V" assumes "\<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = ZFC_in_HOL.set {b}" shows "\<BB>\<lparr>Dom\<rparr> = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
last_def: fixes s :: "nat \<Rightarrow> 'b" shows "Sequence.last s \<equiv> LEAST i. \<forall>j\<ge>i. s j = s i"
SN_imp_WN: fixes A :: "('b \<times> 'b) set" assumes "SN A" shows "WN A"
leq_path_sh: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and C' :: "char list"   and C'' :: "char list" assumes "Subobjs P C (Cs @ [C'])"   and "P \<turnstile> C' \<prec>\<^sub>S C''" shows "P,C \<turnstile> (Cs @ [C']) \<sqsubset>\<^sup>1 [C'']"
kcomp_distr: fixes f :: "'b \<Rightarrow> 'd set"   and g :: "'b \<Rightarrow> 'd set"   and h :: "'d \<Rightarrow> 'c set" shows "(f \<squnion> g) \<circ>\<^sub>K h = f \<circ>\<^sub>K h \<squnion> g \<circ>\<^sub>K h"
I_minWeightBasis_empty: shows "I_minWeightBasis ({}, carrier)"
IH_2: fixes t :: "'a bro"   and a :: "'a"   and t' :: "'a bro" assumes "t \<in> U h"   and "split_min t = Some (a, t')" shows "t' \<in> Um h"
index_Suc: fixes \<Gamma> :: "nat list"   and v :: "nat"   and i :: "nat" shows "index \<Gamma> v (Suc i) = Suc (index \<Gamma> v i)"
fun_when: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and P :: "bool" assumes "f (0::'d) = (0::'c)" shows "f (a when P) = (f a when P)"
ltl_prop_equiv_min_dnf: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>P \<psi> = (min_dnf \<phi> = min_dnf \<psi>)"
suminf_emeasure: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range A \<subseteq> sets M"   and "disjoint_family A" shows "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union> (range A))"
cong_scalar_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and c :: "'d"   and a :: "'d"   and d :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times d b) (times d c) a"
that: fixes m :: "'node" assumes "q \<in> phiUses g m"   and "m \<in> set (predecessors g (defNode g p))" shows "thesis"
ideD_1: fixes a :: "'c local.term set" assumes "\<F>\<^sub>SC.ide a" shows "\<F>\<^sub>SC.arr a"
ltln_strong_weak_1: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n F\<^sub>n \<psi> and\<^sub>n \<phi> W\<^sub>n \<psi>"
row_zero: fixes i :: "nat"   and nr :: "nat"   and nc :: "nat" assumes "i < nr" shows "row (0\<^sub>m nr nc) i = 0\<^sub>v nc"