hcju/isabelleps huggingface dataset:
bounded_countable_web_web_of_network: assumes "\<And>x. \<not> edge \<Delta> x (source \<Delta>)"   and "\<And>y. \<not> edge \<Delta> (sink \<Delta>) y"   and "\<And>x y. edge \<Delta> x y \<Longrightarrow> (\<exists>z. edge \<Delta> y z) \<or> (\<exists>z. edge \<Delta> z x)"   and "\<not> edge \<Delta> (source \<Delta>) (sink \<Delta>)"   and "\<And>x. \<not> edge \<Delta> x x" shows "bounded_countable_web (web_of_network \<Delta>)"
E'_to_E: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and u :: "'c"   and v :: "'c"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd" assumes "g_rename_precond G f"   and "(u, v) \<in> g_E (fr_rename_ext ecnv f G)" shows "(g_rename_precond.fi G f u, g_rename_precond.fi G f v) \<in> g_E G"
filter_def: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and xs :: "('b, 'c) oalist_ntm" shows "OAlist_filter_ntm P xs = OAlist_ntm (filter_raw P (list_of_oalist_ntm xs))"
list_less_def: fixes xs :: "'b list"   and ys :: "'b list" shows "list_less xs ys = ((xs, ys) \<in> lexord {(u, v). u < v})"
left_total_cr_spmf_gpv: shows "left_total cr_spmf_gpv"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and sum :: "'c + 'd" shows "P (case sum of Sum_Type.Inl x \<Rightarrow> f1 x | Sum_Type.Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Sum_Type.Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Sum_Type.Inr x2 \<and> \<not> P (f2 x2))))"
lookup_base_Nil: fixes v :: "'b list" shows "lookup_base [] v [] = hd v"
ws_def: shows "ws = [countVertices g'..<countVertices g' + n]"
gState_inv_to\<^sub>I: fixes prog :: "program"   and g :: "gState" shows "gState_inv prog g = gState_inv prog (to\<^sub>I g)"
vquot_dbfm_eq: fixes V :: "name set"   and A :: "dbfm"   and W :: "name set" assumes "atom ` V \<inter> supp A = atom ` W \<inter> supp A" shows "vquot_dbfm V A = vquot_dbfm W A"
connected_ivt_component: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and k :: "'a"   and a :: "real" assumes "connected S"   and "x \<in> S"   and "y \<in> S"   and "x \<bullet> k \<le> a"   and "a \<le> y \<bullet> k" shows "\<exists>z\<in>S. z \<bullet> k = a"
build_def: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "build n seed = (let bound = of_nat n * 2 in gen_remove bound n (gen_build bound n ({}, seed)))"
continuous_at_Arg: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont Arg z"
simps_2: fixes s :: "'b set"   and t :: "'b set"   and ts :: "'b set list" shows "partList4 s (t # ts) = (if s = {} then t # ts else if s \<inter> t = {} then t # partList4 s ts else if t - s = {} then t # partList4 (s - t) ts else t \<inter> s # (t - s) # partList4 (s - t) ts)"
find_consistent_signs_at_roots_copr: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "\<And>q. q \<in> set qs \<Longrightarrow> coprime p q" shows "set (find_consistent_signs_at_roots p qs) = set (characterize_consistent_signs_at_roots_copr p qs)"
sublist_imp_len_filter_le: fixes l' :: "'d list"   and l :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "subseq l' l" shows "length (filter P l') \<le> length (filter P l)"
vec_inj: fixes x :: "'a"   and y :: "'a" shows "(Finite_Cartesian_Product.vec x = Finite_Cartesian_Product.vec y) = (x = y)"
MOST_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "\<forall>\<^sub>\<infinity>x. Q x"
closure_convex_hull: fixes S :: "'b set" assumes "compact S" shows "closure (convex hull S) = convex hull S"
le_SUP_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less_eq x (Sup (f ` A)) = (\<forall>y. less y x \<longrightarrow> (\<exists>i\<in>A. less y (f i)))"
bot2E: fixes x :: "'c"   and y :: "'d"   and P :: "bool" assumes "\<bottom> x y" shows "P"
prems_5: shows "last (msgs (S t i) cid) = Marker"
scaleR_matrix_vector_assoc: fixes k :: "real"   and A :: "((real, 'b) vec, 'a) vec"   and v :: "(real, 'b) vec" shows "k *\<^sub>R (A *v v) = k *\<^sub>R A *v v"
UN_extend_simps_8: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set set" shows "(\<Union>y\<in>A. \<Union> (B ` y)) = \<Union> (B ` \<Union> A)"
continuous_on_cong_simp: fixes s :: "'d set"   and t :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "s = t"   and "\<And>x. x \<in> t =simp=> f x = g x" shows "continuous_on s f = continuous_on t g"
bidirected_digraph_axioms: shows "bidirected_digraph (with_proj (list_digraph G_list)) (edge_rev (to_map' (snd G_list) xss))"
sub_num_simps_8: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and k :: "num"   and l :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit1 k) (num.Bit0 l) = neg_numeral.dbl_inc plus one (neg_numeral.sub minus plus one k l)"
getParts_complete: fixes rs :: "'b simple_rule list" shows "wordinterval_list_to_set (getParts rs) = UNIV"
y1_prop: shows "q < y1 \<and> (\<forall>a\<in>set neq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f \<noteq> 0)"
Diagonalize_DOM: fixes f :: "'c local.term set" assumes "arr f" shows "\<^bold>\<lfloor>DOM f\<^bold>\<rfloor> = Dom \<^bold>\<lfloor>rep f\<^bold>\<rfloor>"
simps_6: fixes v :: "'c match_expr"   and rs :: "'c rule list" shows "process_ret (Rule v Reject # rs) = Rule v Reject # process_ret rs"
fmeasurable_cbox: fixes a :: "'c"   and b :: "'c" shows "cbox a b \<in> fmeasurable lborel"
basic_trans_rules_31: fixes A :: "'y set"   and B :: "'y set"   and c :: "'y" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
kat_4_equiv_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(r x \<cdot> y = r x \<cdot> (y \<cdot> r z)) = (r x \<cdot> y \<le> y \<cdot> r z)"
dom_comp: fixes g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.seq g f" shows "S.VV.dom (S.VV.comp g f) = S.VV.dom f"
cases: fixes x :: "'d list" obtains "x = []"   | xa :: "'d" where "x = [xa__]"   | xa :: "'d" and y :: "'d" and xs :: "'d list" where "x = xa # y # xs"
last_op_greatest: fixes op_list :: "('b \<times> 'b option) list"   and oid :: "'b"   and oper :: "'b option"   and x :: "'b" assumes "insert_ops (op_list @ [(oid, oper)])"   and "x \<in> set (map fst op_list)" shows "x < oid"
Busemann_function_Lipschitz: fixes xi :: "'b Gromov_completion"   and x :: "'b"   and y :: "'b"   and x' :: "'b"   and y' :: "'b" shows "\<bar>Busemann_function_at xi x y - Busemann_function_at xi x' y'\<bar> \<le> dist x x' + dist y y'"
hinsert_eq_sup: fixes a :: "hf"   and b :: "hf" shows "b \<triangleleft> a = b \<squnion> \<lbrace>a\<rbrace>"
simp_2: fixes res :: "(nat \<times> 'i list) list" shows "dist_degree_factorize_main_i p ff_ops (arith_ops_record.zero ff_ops) (arith_ops_record.one ff_ops) (degree_i V) V W d res = (if V = [arith_ops_record.one ff_ops] then res else if degree_i V < d + d then (degree_i V, V) # res else let w = power_poly_f_mod_i ff_ops (\<lambda>f. mod_field_poly_i ff_ops f V) W (nat p); d = Suc d; gd = gcd_poly_i ff_ops (minus_poly_i ff_ops w [arith_ops_record.zero ff_ops, arith_ops_record.one ff_ops]) V in if gd = [arith_ops_record.one ff_ops] then dist_degree_factorize_main_i p ff_ops (arith_ops_record.zero ff_ops) (arith_ops_record.one ff_ops) (degree_i V) V w d res else let v' = div_field_poly_i ff_ops V gd in dist_degree_factorize_main_i p ff_ops (arith_ops_record.zero ff_ops) (arith_ops_record.one ff_ops) (degree_i v') v' (mod_field_poly_i ff_ops w v') d ((d, gd) # res))"
roundup_help2: fixes emBits :: "nat" assumes "176 + sLen \<le> emBits" shows "roundup emBits 8 * 8 - emBits \<le> roundup emBits 8 * 8 - 160 - sLen - 16"
heap_is_wellformed_impl: shows "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = local.CD.a_heap_is_wellformed"
cring_sub_monom_bound: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "f = monom P a n"   and "g \<in> carrier P" shows "deg R (Cring_Poly.compose R f g) \<le> n * deg R g"
lspace_min: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f \<in> lspace (lebesgue_on S) (ennreal p)"   and "g \<in> lspace (lebesgue_on S) (ennreal p)"   and "0 < p" shows "(\<lambda>x. min (f x) (g x)) \<in> lspace (lebesgue_on S) (ennreal p)"
StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_vs_StrongEq'': fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and y :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>'e\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and H :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" assumes "WFF \<tau>"   and "\<tau> \<Turnstile> \<upsilon> x"   and "\<tau> \<Turnstile> \<upsilon> y"   and "\<And>x. x \<in> ran (heap (fst \<tau>)) \<or> x \<in> ran (heap (snd \<tau>)) \<Longrightarrow> oid_of (H x) = oid_of x"   and "\<tau> \<Turnstile> H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(x) and H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(y) or H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(x) and H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(y)" shows "(\<tau> \<Turnstile> StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y) = (\<tau> \<Turnstile> x \<triangleq> y)"
map_fun_def: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'e" shows "map_fun f g h = g \<circ> h \<circ> f"
select_convs_3: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "tail \<lparr>pre_digraph.verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = tail"
word_l_diffs_1: fixes w :: "'b word"   and z :: "'b word"   and x :: "'b word" assumes "w < z - x"   and "x \<le> z" shows "w + x < z"
contains_pred_def: fixes A :: "'b set"   and x :: "'b" shows "Predicate_Compile.contains_pred A x = (if x \<in> A then Predicate.single () else bot)"
ereal_leq_imp_neg_leq: fixes x :: "ereal"   and y :: "ereal" assumes "x \<le> y" shows "- y \<le> - x"
measurable_on_Pair: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'c" assumes "f measurable_on S"   and "g measurable_on S" shows "(\<lambda>x. (f x, g x)) measurable_on S"
all_croots_def: fixes n :: "nat"   and x :: "complex" shows "all_croots n x = (if n = 0 then [] else if algebraic x then let p = min_int_poly x; q = poly_nth_root n p in Let (complex_roots_of_int_poly q) (filter (\<lambda>y. y ^ n = x)) else SOME ys. set ys = {y. y ^ n = x})"
A: assumes "D x \<in> sources I D u (x # xs)" shows "ipurge_tr_rev_t_aux \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr> \<in> ipurge_tr_rev_t_set \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr>"
domain_eq: shows "Domainp (pcr_pos (=)) = (\<le>) (0::'a)"
lookup_not_edom: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" assumes "x \<notin> edom m" shows "m x = \<bottom>"
composite_cell_in_hom: fixes w :: "'a"   and u :: "'a"   and \<theta> :: "'a" assumes "ide w"   and "\<guillemotleft>w : src u \<rightarrow> src f\<guillemotright>"   and "\<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright>" shows "\<guillemotleft>(r' \<star> \<theta>) \<cdot> \<a>[r', f, w] \<cdot> ((\<phi> \<star> f) \<cdot> \<rho> \<star> w) : g \<star> w \<Rightarrow> r' \<star> u\<guillemotright>"
min_auxlist_done: shows "min (length auxlist) (length done) = length done"
Ap: shows "countable A \<and> A \<subseteq> sets (restr_to_subalg M F) \<and> \<Union> A = space (restr_to_subalg M F) \<and> (\<forall>a\<in>A. emeasure (restr_to_subalg M F) a \<noteq> \<infinity>)"
r_ifeq_else_diverg: fixes a :: "nat"   and b :: "nat"   and v :: "nat" shows "eval r_ifeq_else_diverg [a, b, v] = (if a = b then Some v else None)"
simps_3: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "reduce' r (p \<^bold>\<or>\<^sub>! q) = (reduce' r p \<^bold>\<or>\<^sub>! reduce' r q)"
nat_induct_at_least: fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "m \<le> n"   and "P m"   and "\<And>n. \<lbrakk>m \<le> n; P n\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P n"
sorted_filter: fixes f :: "'e \<Rightarrow> 'd Word.word"   and xs :: "'e list"   and P :: "'e \<Rightarrow> bool" assumes "signed.sorted (map f xs)" shows "signed.sorted (map f (filter P xs))"
lm065: fixes pair :: "'d \<times> 'c set" assumes "finite (snd pair)" shows "finite (omega pair)"
less_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "xs < ys \<equiv> xs \<noteq> [] \<and> ys \<noteq> [] \<longrightarrow> last xs < hd ys"
BuildSeq_append: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf" assumes "BuildSeq B C s1 k1 y1"   and "BuildSeq B C s2 k2 y2" shows "BuildSeq B C (seq_append (succ k1) s1 s2) (succ (k1 @+ k2)) y2"
inv_into_Field_embed_bij_betw: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "embed r r' f"   and "bij_betw f (Field r) (Field r')" shows "embed r' r (inv_into (Field r) f)"
lNact_notActive: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and k :: "nat" assumes "dynamic_component.lNAct active c t n \<le> k"   and "k < n" shows "\<not> active c (t k)"
ex_cos_eq: fixes \<alpha> :: "real"   and a :: "real" assumes "- pi / 2 \<le> \<alpha> \<and> \<alpha> \<le> pi / 2"   and "0 \<le> a"   and "a < 1" shows "\<exists>\<alpha>'\<ge>- pi / 2. \<alpha>' \<le> pi / 2 \<and> \<alpha>' \<noteq> \<alpha> \<and> cos (\<alpha> - \<alpha>') = a"
cont_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "('c, 'd, 'a) cmd"   and s :: "'b"   and i :: "nat" assumes "PL cval" shows "PL.cont aval tval cval c s i = fst (snd (PL.wt_cont_eff aval tval cval c s i))"
pow_eq_div2: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier (G\<lparr>carrier := H\<rparr>)"   and "x [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> m = x [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> n" shows "x [^]\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> (m - n) = \<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>"
quality_increases_rreq_rrep_props': fixes \<sigma> :: "'a \<Rightarrow> state"   and \<sigma>' :: "'a \<Rightarrow> state"   and sn :: "nat"   and ip :: "nat"   and sip :: "'a"   and hops :: "nat" assumes "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"   and "1 \<le> sn"   and "ip \<in> kD (rt (\<sigma> sip)) \<and> sn \<le> nsqn (rt (\<sigma> sip)) ip \<and> (nsqn (rt (\<sigma> sip)) ip = sn \<longrightarrow> the (dhops (rt (\<sigma> sip)) ip) \<le> hops \<or> the (flag (rt (\<sigma> sip)) ip) = Aodv_Basic.inv)" shows "ip \<in> kD (rt (\<sigma>' sip)) \<and> sn \<le> nsqn (rt (\<sigma>' sip)) ip \<and> (nsqn (rt (\<sigma>' sip)) ip = sn \<longrightarrow> the (dhops (rt (\<sigma>' sip)) ip) \<le> hops \<or> the (flag (rt (\<sigma>' sip)) ip) = Aodv_Basic.inv)"
valid_foldl_2: shows "lin_ts_mmuaux (foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss_list) = tss_list'"
drh: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool" shows "RELATES (br \<alpha> invar)"
Group_GroupHomSet: fixes H :: "'a set" assumes "AbGroup H" shows "Group (GroupHomSet UNIV H)"
induct: fixes x :: "'node SDG_node"   and n :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> bool" assumes "x \<in> sum_SDG_slice1 n"   and "valid_SDG_node n \<Longrightarrow> P n"   and "\<And>n'' n'. \<lbrakk>n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'; n' \<in> sum_SDG_slice1 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' V n'. \<lbrakk>n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'; n' \<in> sum_SDG_slice1 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p n'. \<lbrakk>n'' s-p\<rightarrow>\<^bsub>call\<^esub> n'; n' \<in> sum_SDG_slice1 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p V n'. \<lbrakk>n'' s-p:V\<rightarrow>\<^bsub>in\<^esub> n'; n' \<in> sum_SDG_slice1 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p n'. \<lbrakk>n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'; n' \<in> sum_SDG_slice1 n; P n'\<rbrakk> \<Longrightarrow> P n''" shows "P x"
arcs_H: shows "arcs (with_proj H) = {uw, swap_in (parcs H) uw, vw, swap_in (parcs H) vw} \<union> arcs (with_proj I') - {uv, swap_in (parcs I') uv}"
g_bnd_def: fixes B :: "rat"   and fs :: "int vec list" shows "g_bnd B fs = (\<forall>i<m. \<parallel>gram_schmidt_fs.gso n (map of_int_hom.vec_hom fs) i\<parallel>\<^sup>2 \<le> B)"
accp_wfPD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "wfP r" shows "Wellfounded.accp r x"
map_of_zip_eq_SomeD: fixes xs :: "'a list"   and ys :: "'b list"   and x :: "'a"   and y :: "'b" assumes "length xs = length ys"   and "map_of (zip xs ys) x = Some y" shows "y \<in> set ys"
gfinprod_distrib: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "finite {x \<in> A. f x \<noteq> \<one>}"   and "finite {x \<in> A. g x \<noteq> \<one>}"   and "f \<in> A \<rightarrow> carrier G"   and "g \<in> A \<rightarrow> carrier G" shows "gfinprod G (\<lambda>i. f i \<otimes> g i) A = gfinprod G f A \<otimes> gfinprod G g A"
IHP_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "f a (f b c) = f (f a b) c"
hoareT_sound2_partR: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}" shows "\<Turnstile>\<^sub>3\<^sub>' {P} c { Q}"
Suc_div_le_mono: fixes m :: "nat"   and n :: "nat" shows "m div n \<le> Suc m div n"
Kelly_eval_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a set"   and A :: "'a set" shows "Kelly R B A = (\<forall>a\<in>A. \<forall>b\<in>B. R b a)"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "F (A g f) = B (F g) (F f)"
Id_on_def: fixes A :: "'a set" shows "s2r A = (\<Union>x\<in>A. {(x, x)})"
gt_def: fixes c :: "int"   and t :: "MIR.num" shows "gt c t = (if c = 0 then Gt t else if 0 < c then Gt (CN 0 c t) else Lt (CN 0 (- c) (Neg t)))"
measure_PiM_emb: fixes J :: "'a set"   and X :: "'a \<Rightarrow> 'b set" assumes "J \<subseteq> I"   and "finite J"   and "\<And>i. i \<in> J \<Longrightarrow> X i \<in> M.events i" shows "P.prob (emb I J (Pi\<^sub>E J X)) = (\<Prod>i\<in>J. M.prob i (X i))"
wfactors_mult_single: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and fb :: "'c list"   and b :: "'c" assumes "monoid_cancel G"   and "irreducible G a"   and "wfactors G fb b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set fb \<subseteq> carrier G" shows "wfactors G (a # fb) (a \<otimes>\<^bsub>G\<^esub> b)"
mod_minus_minus: fixes a :: "'a"   and b :: "'a" shows "- a mod - b = - (a mod b)"
case_rews_3: fixes f1 :: "'b"   and f2 :: "'c \<rightarrow> ['c] \<rightarrow> 'b"   and a :: "'c"   and list :: "['c]" shows "list_case\<cdot>f1\<cdot>f2\<cdot>(a : list) = f2\<cdot>a\<cdot>list"
Int_insert_right: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A \<inter> insert a B = (if a \<in> A then insert a (A \<inter> B) else A \<inter> B)"
rep_eq_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "(rep_nat x = rep_nat y) = (x = y)"
beukers_aux_ln_conv_integral: fixes x :: "real"   and y :: "real" assumes "x \<in> {0<..<1}"   and "y \<in> {0<..<1}" shows "- ln (x * y) / (1 - x * y) = LBINT z=0..1. 1 / (1 - (1 - x * y) * z)"
oalist_abstract3_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalistx :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_listx :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and list_of_oalisty :: "'e \<Rightarrow> ('b \<times> 'f) list \<times> 'a"   and oalist_of_listy :: "('b \<times> 'f) list \<times> 'a \<Rightarrow> 'e"   and list_of_oalistz :: "'g \<Rightarrow> ('b \<times> 'h) list \<times> 'a"   and oalist_of_listz :: "('b \<times> 'h) list \<times> 'a \<Rightarrow> 'g" shows "oalist_abstract3 rep_key_order list_of_oalistx oalist_of_listx list_of_oalisty oalist_of_listy list_of_oalistz oalist_of_listz \<equiv> oalist_abstract rep_key_order list_of_oalistx oalist_of_listx \<and> oalist_abstract rep_key_order list_of_oalisty oalist_of_listy \<and> oalist_abstract rep_key_order list_of_oalistz oalist_of_listz"
monotone_llength: shows "monotone (\<sqsubseteq>) (\<le>) llength"
cosh_arcosh: fixes x :: "real" assumes "1 \<le> x" shows "cosh (arcosh x) = x"
A: shows "\<integral>\<^sup>+ x. ennreal \<bar>subcocycle_lim u x\<bar> \<partial>M < \<infinity>"
focus_minus: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "focus X (p - q) = focus X p - focus X q"
Binp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsOpS delta"   and "\<And>i. barOf delta i = None \<and> binp i = None \<or> (\<exists>us s A. barOf delta i = Some (us, s) \<and> binp i = Some A \<and> wlsAbs (us, s) A)" shows "wlsBinp delta binp"
\<R>\<^sub>2_weak_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option" shows "weak_bisim (\<R>\<^sub>2 \<Gamma>') (\<R> \<Gamma>')"
ring_times_x_0: fixes R :: "('c, 'd) Ring_scheme"   and x :: "'c" assumes "Ring R"   and "x \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
pseudo_divmod_impl: fixes f :: "'b poly"   and g :: "'b poly" shows "pseudo_divmod f g = map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))"
rewrite_negated_src_ports_not_has_disc_negated: fixes m :: "'c common_primitive match_expr" assumes "normalized_nnf_match m" shows "\<not> has_disc_negated is_Src_Ports False (rewrite_negated_src_ports m)"
cod_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.arr f" shows "A_B.cod (A_B.cod f) = A_B.cod f"
as_needed_asses_submap_exec_xii': fixes v :: "'a"   and f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "v \<notin> fmdom' f"   and "v \<in> fmdom' g" shows "fmlookup (f ++ g) v = fmlookup g v"
funpow_neq_less_funpow_dist1: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "y \<in> orbit f x"   and "m < funpow_dist1 f x y"   and "n < funpow_dist1 f x y"   and "m \<noteq> n" shows "(f ^^ m) x \<noteq> (f ^^ n) x"