hcju/isabelleps huggingface dataset:
fmdomI: fixes m :: "('b, 'a) fmap"   and x :: "'b"   and y :: "'a" assumes "fmlookup m x = Some y" shows "x |\<in>| fmdom m"
list_all2_Cons1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and ys :: "'d list" shows "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
inverse_unique: fixes a :: "'c"   and b :: "'c" assumes "a + b = (0::'c)" shows "- a = b"
inj_on_def: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "inj_on f A = (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
less_score_ext_def: fixes t1 :: "'a score_scheme"   and t2 :: "'a score_scheme" shows "(t1 < t2) = ((Score t2, S1 t1, S2 t1, score.more t1) < (Score t1, S1 t2, S2 t2, score.more t2))"
mpoly_eqI: fixes p :: "'a mpoly"   and q :: "'a mpoly" assumes "\<And>mon. MPoly_Type.coeff p mon = MPoly_Type.coeff q mon" shows "p = q"
bdoubleton_same: fixes x :: "'d" shows "bdoubleton x x = bsingleton x"
dim_basis: fixes A :: "'a vec set" assumes "finite A"   and "basis A" shows "local.dim = card A"
suminf_emeasure: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "disjoint_family A" shows "(\<Sum>i. emeasure M (A i)) = emeasure M (\<Union> (range A))"
simps: fixes a1 :: "'a tag_const"   and a2 :: "'a tag_const" shows "(a1 \<le> a2) = (\<exists>n m. a1 = \<tau>\<^sub>c\<^sub>s\<^sub>t n \<and> a2 = \<tau>\<^sub>c\<^sub>s\<^sub>t m \<and> n \<le> m)"
conc_abs_swap: fixes m' :: "'a nres"   and m :: "'b nres" shows "(m' \<le> \<Down> R m) = (\<Up> R m' \<le> m)"
g1: shows "\<forall>(x, y)\<in>set (step p \<tau>'). x \<in> succs p"
can_cope_with_unrestrict_effects: fixes S :: "'b literal set set"   and T :: "'b literal set set"   and v :: "bool"   and a :: "'b"   and n :: "'b literal set set" assumes "S \<turnstile> \<box>"   and "S \<subseteq> T" shows "\<exists>R\<subseteq>{make_lit v a}. (\<lambda>c. if c \<in> n then {make_lit v a} \<union> c else c) ` T \<turnstile> R"
prod_ffact_nat: fixes m :: "nat"   and n :: "nat" shows "prod ((-) m) {0..<n} = ffact n m"
inf1: fixes A :: "'h"   and k :: "'h"   and a :: "'h"   and b :: "'h" assumes "A \<equiv> k \<sqinter> a" shows "A \<sqinter> b \<equiv> k \<sqinter> (a \<sqinter> b)"
Lit: fixes dt :: "loc \<Rightarrow> ty option"   and x :: "val"   and T :: "ty"   and E :: "env" assumes "typeof dt x = Some T" shows "E,dt\<Turnstile>Lit x\<Colon>-T"
SUP_bot_conv_2: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and B :: "'h \<Rightarrow> 'g"   and A :: "'h set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup (B ` A)) = (\<forall>x\<in>A. B x = bot)"
in_I: fixes x :: "nat"   and y :: "nat" assumes "x \<in> I" shows "eval psi [x, y] = eval r_phi [j\<^sub>1, y]"
in_dom_m_lookup_iff: shows "(a' \<in># dom_m a'a) = (fmlookup a'a a' \<noteq> None)"
degree_eq_zeroE: fixes p :: "'a Polynomial.poly" assumes "Polynomial.degree p = 0" obtains a :: "'a" where "p = [:a:]"
path_circlepath: fixes z :: "complex"   and r :: "real" shows "path (circlepath z r)"
theFT_length: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list" assumes "n \<in> theFT cl" shows "n < length cl"
ax: fixes n :: "nat" assumes "0 < n"   and "Lyndon_rec (a # w) n" shows "a # w <lex rotate n (a # w)"
sorted_Cons_iff: fixes x :: "'b"   and xs :: "'b list" shows "Sorted_Less.sorted (x # xs) = ((\<forall>y\<in>set xs. x < y) \<and> Sorted_Less.sorted xs)"
AE: shows "AE x in M. f x \<noteq> \<infinity>"
valid_node_def: fixes n :: "'node" shows "valid_node n \<equiv> \<exists>a. valid_edge a \<and> (n = sourcenode a \<or> n = targetnode a)"
wt_def: fixes c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "wt c s i = fst (wt_cont_eff c s i)"
S12: shows "fn_to_set (\<lambda>x y. p_a x (c_fst y) + p_b x (c_snd y)) = A \<inter> B"
simps_6: fixes F :: "'b formula"   and G :: "'b formula" shows "depth (F \<^bold>\<rightarrow> G) = Suc (max (depth F) (depth G))"
trace_consE: fixes conf :: "'a"   and conf' :: "'a"   and T :: "'a list"   and final :: "'a" assumes "trace conf (conf' # T) final" obtains "trace conf' T final" and "conf \<Rightarrow> conf'"
\<A>: shows "\<A> = (\<A>\<^sub>1, \<A>\<^sub>2)"
vars0_def: shows "st3.vars0 = {0..<Suc k} \<union> {Suc (Suc k)..<n}"
bu_eq_akzk: shows "b \<bullet> u = a ! k * z ! k"
simps_2: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "stl (sfilter P s) = sfilter P (stl (sdrop_while (Not \<circ> P) s))"
simps_3: fixes v :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S) = ik\<^sub>s\<^sub>t S"
commute: fixes a :: "'a"   and b :: "'a" shows "max_torder r a b = max_torder r b a"
preserves_cod: fixes f :: "'j" assumes "J.arr f" shows "J.cod (J.map f) = J.map (J.cod f)"
inf_uminus: fixes x :: "'a" assumes "x \<in> assertion" shows "x \<sqinter> neg_assert x = \<bottom>"
unrest_sublens: fixes x :: "'c \<Longrightarrow> 'd"   and P :: "('e, 'd) uexpr"   and y :: "'f \<Longrightarrow> 'd" assumes "x \<sharp> P"   and "y \<subseteq>\<^sub>L x" shows "y \<sharp> P"
exhaust: fixes y :: "'b Literal" obtains x1 :: "'b" where "y = Pos x1"   | x2 :: "'b" where "y = Neg x2"
homeomorphism_sym: fixes S :: "'a set"   and t :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "homeomorphism S t f g = homeomorphism t S g f"
einterval_def: fixes a :: "ereal"   and b :: "ereal" shows "einterval a b = {x. a < ereal x \<and> ereal x < b}"
isCont_divide: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "isCont f a"   and "isCont g a"   and "g a \<noteq> (0::'b)" shows "isCont (\<lambda>x. f x / g x) a"
sgn1_nz_impl_arg_pos: fixes x :: "nat" assumes "sgn1 x \<noteq> 0" shows "0 < x"
wa_intro: fixes Qi :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and addi :: "(('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)) set" assumes "while_algo (bre'_algo Qi \<delta>)"   and "wa_invar (brw_algo Qi \<delta>) = addi \<inter> {s. brw_\<alpha> s \<in> wa_invar (bre'_algo Qi \<delta>)}"   and "\<And>s s'. \<lbrakk>s \<in> addi; s \<in> wa_cond (brw_algo Qi \<delta>); brw_\<alpha> s \<in> wa_invar (bre'_algo Qi \<delta>); (s, s') \<in> wa_step (brw_algo Qi \<delta>)\<rbrakk> \<Longrightarrow> s' \<in> addi"   and "wa_initial (brw_algo Qi \<delta>) \<subseteq> addi" shows "while_algo (brw_algo Qi \<delta>)"
assms: shows "(xs, ys) \<in> local.Minimal_Solutions"
cont2cont_fun: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and y :: "'b" assumes "cont f" shows "cont (\<lambda>x. f x y)"
congruenceSubstDef_3: fixes Rel :: "(pi \<times> pi) set" shows "substClosed Rel \<equiv> {(P, Q) |P Q. \<forall>\<sigma>. (P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel}"
v_w_2: shows "(v, w) \<in> edges_of_walk ys"
bacc_stepp_bacc_step_eq: fixes \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set" shows "bacc_stepp (\<lambda>x. x \<in> \<delta>) Q = (\<lambda>x. x \<in> bacc_step \<delta> Q)"
qfree_or: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" assumes "qfree \<phi>\<^sub>1"   and "qfree \<phi>\<^sub>2" shows "qfree (or \<phi>\<^sub>1 \<phi>\<^sub>2)"
set_pmf_bind_spmf: fixes M :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "set_pmf (M \<bind> f) = set_pmf M \<bind> map_option_set (set_pmf \<circ> f)"
rPar1: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and \<Psi>\<^sub>Q :: "'b"   and A\<^sub>Q :: "name list" assumes "cP = P \<parallel> Q"   and "cRs = \<alpha> \<prec> P' \<parallel> Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* C"   and "A\<^sub>Q \<sharp>* P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)" shows "Prop"
cr_float_def: shows "IEEE.cr_float \<equiv> \<lambda>x y. x = Rep_float y"
rotate1_hd_tl: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "rotate1 xs = tl xs @ [hd xs]"
fold_stores_points_to: fixes ws :: "('a, 'b, 'c) mem_store_action list"   and s :: "'d process_name \<Rightarrow> ('a, 'd, 'b, 'c) Model.local_state"   and r :: "'c"   and obj :: "('a, 'b, 'c) object"   and f :: "'a"   and r' :: "'c" assumes "heap (fold_stores ws (s sys)) r = Some obj"   and "obj_fields obj f = Some r'" shows "(r points_to r') s \<or> (\<exists>w\<in>set ws. r' \<in> store_refs w)"
RPDs_safe: fixes g :: "safety"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Past g (from_mregex mr \<phi>s)"   and "ms \<in> RPDs mr" shows "safe_regex Past g (from_mregex ms \<phi>s)"
floatI: fixes m :: "int"   and e :: "int"   and x :: "real" assumes "real_of_int m * 2 powr real_of_int e = x" shows "x \<in> float"
split_option_ex: fixes P :: "'k option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
numeral_index: fixes w :: "num"   and i :: "'b" shows "numeral w $ i = numeral w"
cone_smult_basis: fixes Vs :: "'a vec set"   and l :: "'a vec \<Rightarrow> 'a" assumes "Vs \<subseteq> carrier_vec (n + 1)"   and "l ` Vs \<subseteq> {x. (0::'a) < x}" shows "next_dim.cone {l v \<cdot>\<^sub>v v |v. v \<in> Vs} = next_dim.cone Vs"
not_loose: shows "\<not> loose_bvar (subst_term [((idn, ty), t)] B) lev"
rspp_def: fixes m :: "'node"   and S :: "'node SDG_node set"   and xs :: "'var list"   and f :: "'var \<Rightarrow> 'val option"   and g :: "'var \<Rightarrow> 'val option" shows "rspp m S xs f g \<equiv> map_merge f Map.empty(ParamDefs m [:=] map g xs) (\<lambda>i. Actual_out (m, i) \<in> S) (ParamDefs m)"
cod_trg_eq_dom_trg: shows "assoc_fgh.cod.trg = assoc_fgh.dtrg"
not_neg_one_le_neg_numeral_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(\<not> less_eq (uminus one) (uminus (numeral.numeral one plus m))) = (m \<noteq> num.One)"
carrier_ccTTree: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "carrier (ccTTree S G) = S"
equation_3: fixes xa :: "'a" shows "eq_o_i xa = Predicate.bind (Predicate.single xa) Predicate.single"
simps_3: fixes v :: "('c, 'd) Term.term"   and S :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S) = ik\<^sub>s\<^sub>t S"
inst_def: fixes \<phi> :: "'fmla"   and t :: "'trm" shows "inst \<phi> t = subst \<phi> t xx"
image_set_diff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "inj f" shows "f ` (A - B) = f ` A - f ` B"
ha: fixes x :: "'a"   and s :: "'s" assumes "x \<in> supp (p s)" shows "healthy (wp (a x))"
is_num_normalize_8: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
a_rcos_module_imp: fixes x :: "'a"   and x' :: "'a" assumes "x \<in> carrier R"   and "x' \<in> a_kernel R S h +> x" shows "x' \<oplus> \<ominus> x \<in> a_kernel R S h"
RN: fixes \<phi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<phi> in v]" shows "[\<^bold>\<box>\<phi> in v]"
C_2: fixes n :: "nat" shows "B (Suc n) \<le> D \<or> B (Suc n) \<le> C * B n"
assms_2: shows "subgroup H (G\<lparr>carrier := K\<rparr>)"
intros_1: shows "local.S (return_spmf (Void, None, Map.empty)) (return_spmf ((False, ()), Map.empty, Void))"
not_le: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.linorder less_eq less" shows "(\<not> less_eq x y) = less y x"
lasso_of_prpl_simps_1: fixes prpl :: "'e list \<times> 'e list" shows "lasso_reach (lasso_of_prpl prpl) = fst prpl"
dist_bounded: fixes f :: "'d \<Rightarrow>\<^sub>C 'c"   and x :: "'d"   and g :: "'d \<Rightarrow>\<^sub>C 'c" shows "dist (apply_bcontfun f x) (apply_bcontfun g x) \<le> dist f g"
Min_gr_iff: fixes A :: "'c DBMEntry set"   and x :: "'c DBMEntry" assumes "finite A"   and "A \<noteq> {}" shows "(x \<prec> linordered_monoid.Min A) = (\<forall>a\<in>A. x \<prec> a)"
inner_zero_right: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x zero = 0"
deadlock_free_v2_def: fixes P :: "'b process" shows "deadlock_free_v2 P \<equiv> DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F P"
orthogonal_mat_def: fixes A :: "'b mat" shows "orthogonal_mat A \<equiv> let B = transpose_mat A * A in diagonal_mat B \<and> (\<forall>i<dim_col A. B $$ (i, i) \<noteq> (0::'b))"
arbitrary_intersection_of_complement: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "(arbitrary intersection_of P) S = (arbitrary union_of (\<lambda>S. P (- S))) (- S)"
ran_def: fixes m :: "'f \<Rightarrow> 'e option" shows "ran m = {b. \<exists>a. m a = Some b}"
relation_of_R: fixes x :: "('c, 'd) action" shows "action.relation_of x is R healthy"
axioms_2: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring_hom_ring R S h" shows "ring S"
is_SDG_path_ics_SDG_path: 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 n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" shows "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"
sort_append: fixes xs :: "'b list"   and ys :: "'b list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> x \<le> y" shows "sort (xs @ ys) = sort xs @ sort ys"
Subst_all_cls_remains: fixes X :: "'b list set"   and xrhs :: "'b trm set"   and ES :: "('b list set \<times> 'b trm set) set"   and Y :: "'b list set"   and yrhs :: "'b trm set" assumes "(X, xrhs) \<in> ES" shows "\<exists>xrhs'. (X, xrhs') \<in> Subst_all ES Y yrhs"
mk_linear_orders_termination: fixes B :: "'a set"   and C :: "'a set" assumes "B \<subseteq> A" shows "local.mk_linear_orders_dom (C, B)"
bij_betw_push_forward: fixes p :: "'a" assumes "p \<in> src.carrier" shows "bij_betw push_forward (src.tangent_space p) (dest.tangent_space (f p))"
abelian_groupI: fixes R :: "('a, 'b) ring_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"   and "\<zero>\<^bsub>R\<^esub> \<in> carrier R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"   and "\<And>x y. \<lbrakk>x \<in> carrier R; y \<in> carrier R\<rbrakk> \<Longrightarrow> x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"   and "\<And>x. x \<in> carrier R \<Longrightarrow> \<exists>y\<in>carrier R. y \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>" shows "abelian_group R"
sel_2: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "ltl (lzip xs ys) = lzip (ltl xs) (ltl ys)"
not_symKeys_pubK: fixes b :: "keymode"   and A :: "agent" shows "publicKey b A \<notin> symKeys"
Pbij_inverse_Rng: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_Dom (Pbij_inverse \<beta>) = Pbij_Rng \<beta>"
simps_4: fixes r :: "mregex"   and s :: "mregex" shows "LPD (MPlus r s) = LPD r \<union> LPD s"
bind_UNION: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'c set" shows "Set.bind A f = \<Union> (f ` A)"
assms_1: shows "\<forall>x. P x \<longrightarrow> P (f x)"
le_cases: fixes x :: "'b word"   and y :: "'b word" obtains "x \<le>s y"   | "y \<le>s x"
ex_disj_distrib: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<exists>x. P x \<or> Q x) = ((\<exists>x. P x) \<or> (\<exists>x. Q x))"
fMin_le: fixes x :: "'b"   and A :: "'b fset" assumes "x |\<in>| A" shows "fMin A \<le> x"
Lim_null_comparison: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> real"   and net :: "'b filter" assumes "\<forall>\<^sub>F x in net. norm (f x) \<le> g x"   and "(g \<longlongrightarrow> 0) net" shows "(f \<longlongrightarrow> (0::'c)) net"
minus_closed: fixes x :: "int"   and y :: "int" assumes "x \<in> carrier INTEG"   and "y \<in> carrier INTEG" shows "x \<ominus>\<^bsub>INTEG\<^esub> y \<in> carrier INTEG"
normalize_coeff: fixes p :: "'c list" shows "S.coeff p = S.coeff (S.normalize p)"
take_bit_int_greater_self_iff: fixes k :: "int"   and n :: "nat" shows "(k < take_bit n k) = (k < 0)"
relcompp_exists: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and r' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'a"   and y :: "'c" shows "(r OO r') x y = (\<exists>u. r x u \<and> r' u y)"
d_def: shows "d \<equiv> \<lambda>z w. if w = z then deriv f z else (f w - f z) / (w - z)"
x_coord_back_img: fixes x :: "real" shows "finite ({0..1} \<inter> x_coord -` {x})"
dual_Join: fixes A :: "'b set" shows "dual (\<Squnion>A) = \<Sqinter>dual ` A"
assms_2: fixes U :: "'a set" assumes "U \<subseteq> T" shows "closedin (top_of_set S) (S \<inter> f -` U) = closedin (top_of_set T) U"
s_Unroll_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Unroll e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e' :: "Syntax.term" where "e'a__ = Unroll e'" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   | "e = Roll e'a__" and "\<pi>' = \<pi>" and "value e'a__"
chine_assoc_props_2: shows "cods.Prj\<^sub>1 \<cdot> cods.chine_assoc = cods.Prj\<^sub>1\<^sub>1"
simps_9: fixes b :: "nat"   and \<phi>s :: "formula list" shows "fvi b (Ands \<phi>s) = (let xs = map (fvi b) \<phi>s in \<Union>x\<in>set xs. x)"
connected_component_maximal: fixes x :: "'c"   and T :: "'c set"   and S :: "'c set" assumes "x \<in> T"   and "connected T"   and "T \<subseteq> S" shows "T \<subseteq> connected_component_set S x"
assms_1: shows "\<beta> < \<omega> \<up> \<delta>"
prv_exi_PPf_iff_isTrue: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)) = isTrue (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>))"
assms_5: shows "z t0 = flow t0 x0 t0"
prime_elem_power_dvd_cases: fixes p :: "'a"   and c :: "nat"   and m :: "'a"   and n :: "'a"   and a :: "nat"   and b :: "nat" assumes "p ^ c dvd m * n"   and "a + b = Suc c"   and "prime_elem p" shows "p ^ a dvd m \<or> p ^ b dvd n"
tiTable_st_call_ok: fixes req :: "nat \<Rightarrow> reqType list"   and x :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and k :: "nat"   and t :: "nat" assumes "tiTable_SampleT req x stop lose (fin_inf_append [init_state] st) b ack i1 vc st"   and "ts lose"   and "\<forall>m\<le>k. ack (Suc (Suc (t + m))) = [connection_ok]"   and "st (Suc t) = call" shows "st (Suc (Suc t)) = connection_ok"
lfilter_lappend_lfinite: fixes xs :: "'a llist"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a llist" assumes "lfinite xs" shows "lfilter P (xs @\<^sub>l ys) = lfilter P xs @\<^sub>l lfilter P ys"
delta_interpretation: fixes \<delta> :: "'a"   and def :: "'a" assumes "(0::'a) < \<delta>"   and "\<delta> \<le> def" shows "SN_strict_mono_ordered_semiring_1 def (delta_gt \<delta>) delta_mono"
dom_eqI': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and f :: "'b" assumes "category C"   and "a \<in> partial_magma.domains C f" shows "a = partial_magma.dom C f"
ct_suffix_exists: fixes x1 :: "ct list" shows "\<exists>ts_c. ct_suffix x1 (to_ct_list ts_c)"
Y_last: shows "\<forall>x\<in>Y. x $ n = (1::'a)"
intros_2: fixes s :: "Store"   and l :: "Location"   and k :: "Location" assumes "s\<turnstile> l reachable_from (s@@k)"   and "ref k \<noteq> nullV" shows "s\<turnstile> l reachable_from ref k"
bit_word_eqI: fixes a :: "'a word"   and b :: "'a word" assumes "\<And>n. n < LENGTH('a) \<Longrightarrow> bit a n = bit b n" shows "a = b"
AE_imp: shows "AE x in M. P x \<longrightarrow> Q x"
attach_map_converter: fixes f :: "'h \<Rightarrow> 'l"   and g :: "'m \<Rightarrow> 'i"   and h :: "'n \<Rightarrow> 'j"   and k :: "'k \<Rightarrow> 'o"   and conv :: "('l, 'm, 'n, 'o) converter"   and res :: "('j, 'k) resource" shows "map_converter f g h k conv \<rhd> res = map_resource f g (conv \<rhd> map_resource h k res)"
Union_paths_carrier: fixes t :: "'a ttree" shows "\<Union> (set ` paths t) = carrier t"
dist_diff_2: fixes a :: "'a"   and b :: "'a" shows "dist (a - b) a = norm b"
minimality_offending_flows_mono_edges_graph: fixes V :: "'a set"   and E :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set"   and F :: "('a \<times> 'a) set"   and nP :: "'a \<Rightarrow> 'b" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "F \<in> set_offending_flows \<lparr>nodes = V, edges = E'\<rparr> nP" shows "\<forall>(e1, e2)\<in>F. \<not> sinvar (add_edge e1 e2 (delete_edges \<lparr>nodes = V, edges = E\<rparr> F)) nP"
g_def: shows "g \<equiv> \<lambda>x. 1 / of_nat k * (\<Sum>m<k. f m * unity_root k (- int x * int m))"
dd: shows "{y + k |y k. y \<in> path_image \<gamma> \<and> k \<in> ball 0 dd} \<subseteq> U"
\<psi>T_W: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "\<psi> T ` indV \<subseteq> W"