hcju/isabelleps huggingface dataset:
ih: assumes "valid ((x, T\<^sub>1) # \<Gamma>\<^sub>2)"   and "(x, T\<^sub>1) # \<Gamma>\<^sub>1 \<subseteq> (x, T\<^sub>1) # \<Gamma>\<^sub>2" shows "(x, T\<^sub>1) # \<Gamma>\<^sub>2 \<turnstile> t : T\<^sub>2"
gt_total: fixes y :: "'a"   and x :: "'a" shows "gt y x \<or> gt x y \<or> y = x"
res_function_euclidean_def: fixes b :: "'a"   and n :: "'a" shows "res_function_euclidean b n = (if b = (0::'a) then n else n mod b)"
gen_igba_impl_rel_sv: fixes Rl :: "('a \<times> ('b \<Rightarrow> 'c \<Rightarrow> bool)) set"   and Rm :: "('d \<times> 'e) set" assumes "single_valued Rl"   and "single_valued Rm" shows "single_valued (\<langle>Rm, Rl\<rangle>gen_igba_impl_rel_eext)"
d2_def: shows "stm2.d2 = prod_list stm2.dims2"
pdevs_apply_add_pdevs: fixes X :: "'d pdevs"   and Y :: "'d pdevs"   and n :: "nat" shows "pdevs_apply (add_pdevs X Y) n = pdevs_apply X n + pdevs_apply Y n"
intros_14: fixes A :: "('a, 'b) form"   and n :: "'a"   and G :: "('a, 'b) form list" assumes "\<stileturn> Neg (A[App n []/0]) # G"   and "news n (A # G)" shows "\<stileturn> Neg (Forall A) # G"
Rep_ge_0: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "0 \<le> Rep x"
pr_ideal_axioms_def: fixes R :: "'b set"   and I :: "'b set"   and multiplication :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "pr_ideal_axioms R I multiplication \<equiv> I \<noteq> R \<and> (\<forall>x y. x \<in> R \<longrightarrow> y \<in> R \<longrightarrow> multiplication x y \<in> I \<longrightarrow> x \<in> I \<or> y \<in> I)"
rec_disj_def: shows "rec_disj = CN rec_sign [rec_add]"
p2s_disj_hom: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "p2s (P \<squnion> Q) = p2s P \<union> p2s Q"
adds_term_def: fixes u :: "'a"   and v :: "'a" shows "local.punit.adds_term u v = (local.punit.component_of_term u = local.punit.component_of_term v \<and> local.punit.pp_of_term u adds local.punit.pp_of_term v)"
poly_subst_zero: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f 0 = 0"
bex_simps_7: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>f ` A. P x) = (\<exists>x\<in>A. P (f x))"
se_Assign_eq: fixes c :: "('c, 'd) conf"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and c' :: "('c, 'd) conf" shows "se c (Assign v e) c' = (\<exists>sv. Conf.fresh_symvar sv c \<and> fst sv = v \<and> c' = \<lparr>store = (store c)(v := snd sv), pred = insert (\<lambda>\<sigma>. \<sigma> sv = adapt_aexp e (store c) \<sigma>) (pred c)\<rparr>)"
widen_i_i_iE: fixes x :: "'a prog"   and xa :: "ty"   and xb :: "ty"   and y :: "unit" assumes "pred.eval (widen_i_i_i x xa xb) y" obtains "x \<turnstile> xa \<le> xb"
le_INF_iff: fixes u :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(u \<le> Inf (f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
system_needed_vars_def: fixes PROB :: "(('a, 'b) fmap \<times> 'c) set"   and s :: "('a, 'b) fmap" shows "system_needed_vars PROB s \<equiv> \<Union> {action_needed_vars a s |a. a \<in> PROB}"
\<alpha>'_ide_simp: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "ide a"   and "ide b"   and "ide c" shows "\<alpha>' (a, b, c) = \<a>\<^sup>-\<^sup>1[a, b, c]"
herbrand'_contra: fixes Cs :: "fterm literal set set" assumes "finite Cs"   and "\<forall>C\<in>Cs. finite C"   and "\<forall>G. \<not> eval\<^sub>c\<^sub>s HFun G Cs" shows "\<exists>T. anybranch T (\<lambda>G. closed_branch G T Cs)"
spanset_reduce_Cons: fixes ms :: "'v list"   and m :: "'v" assumes "set ms \<subseteq> V"   and "m \<in> RSpan ms" shows "RSpan (m # ms) = RSpan ms"
insert_absorb: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "insert a A = A"
distinct_3: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PUnit \<noteq> PApp x31 x32"
simps: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and a1 :: "('e, 'd, 'f) com list"   and a2 :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and a3 :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>a1,a2 \<Down> a3 = ((\<exists>s. a1 = [] \<and> a2 = [] \<and> a3 = s) \<or> (\<exists>nrms css s abrs. a1 = [] \<and> a2 = (nrms, abrs) # css \<and> a3 = Normal s \<and> \<Gamma>\<turnstile>nrms,css \<Down> Normal s) \<or> (\<exists>abrs css s nrms. a1 = [] \<and> a2 = (nrms, abrs) # css \<and> a3 = Abrupt s \<and> \<Gamma>\<turnstile>abrs,css \<Down> Normal s) \<or> (\<exists>nrms css f abrs. a1 = [] \<and> a2 = (nrms, abrs) # css \<and> a3 = Fault f \<and> \<Gamma>\<turnstile>nrms,css \<Down> Fault f) \<or> (\<exists>nrms css abrs. a1 = [] \<and> a2 = (nrms, abrs) # css \<and> a3 = Stuck \<and> \<Gamma>\<turnstile>nrms,css \<Down> Stuck) \<or> (\<exists>c s cs css. a1 = c # cs \<and> a2 = css \<and> a3 = s \<and> \<Gamma>\<turnstile>c \<down> s \<and> (\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> t)))"
S_supt: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "s \<rhd> t" shows "S s t"
nprv_clear5_3: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<phi>4 :: "'c"   and \<phi>5 :: "'c"   and \<psi> :: "'c"   and \<phi>3 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>4, \<phi>5} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<phi>5 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2, \<phi>3, \<phi>4, \<phi>5} \<psi>"
csum_cinfinite_bound: fixes p :: "('d \<times> 'd) set"   and r :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p \<le>o r"   and "q \<le>o r"   and "Card_order p"   and "Card_order q"   and "Cinfinite r" shows "p +c q \<le>o r"
dvd_Lcm_int: fixes m :: "int"   and M :: "int set" assumes "m \<in> M" shows "m dvd Lcm M"
comp_Abs_inverse: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "type_definition Rep Abs A"   and "f \<circ> Abs = g" shows "g \<circ> Rep = f"
simps_7: fixes bs :: "int list"   and c :: "int"   and a :: "Cooper.num" shows "Inum bs (Mul c a) = c * Inum bs a"
assms_3: shows "h \<turnstile> g x \<rightarrow>\<^sub>e e"
hom_one: shows "int.lift f \<one>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub> = \<one>\<^bsub>int_group\<^esub>"
Smith_mxn_pinduct_carrier_less_2: fixes A :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat m n"   and "n \<le> 2 \<or> m \<le> 2" shows "fst (Smith_mxn A) \<in> carrier_mat m m \<and> fst (snd (Smith_mxn A)) \<in> carrier_mat m n \<and> snd (snd (Smith_mxn A)) \<in> carrier_mat n n"
rel_sum_simps_4: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a2 :: "'c"   and b2 :: "'d" shows "rel_sum R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
simps_3: fixes y :: "int"   and x :: "int" shows "(Num_arc y \<le> Num_arc x) = (y \<le> x)"
run_one_step_basic_unop_f_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x251 :: "t"   and x252 :: "unop_f"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Unop_f x251 x252) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
support: shows "pmf (sds R10) c = 0 \<or> pmf (sds R10) b = 0"
Zm_Module: fixes R :: "('e, 'f) Ring_scheme"   and e :: "'g" assumes "Ring R" shows "R module Zm R e"
ide_char: fixes a :: "'c local.term set" shows "\<F>\<^sub>SC.ide a = (\<F>\<^sub>SC.arr a \<and> ide a)"
path_entry_append: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and l :: "nat list"   and v :: "nat"   and w :: "nat" assumes "cfg_doms G"   and "path_entry E l v"   and "(v, w) \<in> E" shows "path_entry E (v # l) w"
rotate_back: fixes n :: "nat"   and u :: "'b list" obtains m :: "nat" where "rotate m (rotate n u) = u"
induct: fixes x1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and x2 :: "'d"   and x3 :: "('c \<Rightarrow> real) set"   and x4 :: "'d"   and x5 :: "('c \<Rightarrow> real) set"   and P :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> ('c \<Rightarrow> real) set \<Rightarrow> 'd \<Rightarrow> ('c \<Rightarrow> real) set \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>x4, x5\<rangle>"   and "\<And>A l Z. P A l Z l Z"   and "\<And>A l Z l' Z' l'' Z''. \<lbrakk>A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z'\<rangle>; P A l Z l' Z'; A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l'', Z''\<rangle>\<rbrakk> \<Longrightarrow> P A l Z l'' Z''" shows "P x1 x2 x3 x4 x5"
option2list_simps_1: fixes x :: "'c" shows "option2list (Some x) = [x]"
move_stability_wdr: fixes ts :: "traffic"   and c :: "cars"   and n :: "nat"   and ts' :: "traffic"   and v :: "view" shows "ts \<^bold>\<midarrow>wdr( c, n ) \<^bold>\<rightarrow> ts' \<longrightarrow> move ts ts' v = v"
group_l_invI: assumes "\<And>x. x \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>) \<Longrightarrow> \<exists>y\<in>carrier (G\<lparr>carrier := generate G {a}\<rparr>). y \<otimes>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> x = \<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>" shows "Group.group (G\<lparr>carrier := generate G {a}\<rparr>)"
p_antitone: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "- x \<le> - y"
gen_entries_def: fixes kvts :: "(('a \<times> 'b) \<times> ('a, 'b) rbt) list"   and t :: "('a, 'b) rbt" shows "gen_entries kvts t = entries t @ concat (map (\<lambda>(kv, t). kv # entries t) kvts)"
Gmult_flincomb_comm: fixes g :: "'g"   and vs :: "'w list"   and as :: "'f list" assumes "g \<in> H"   and "set vs \<subseteq> W" shows "g *\<star> as \<bullet>\<sharp>\<star> vs = as \<bullet>\<sharp>\<star> map ((*\<star>) g) vs"
double_ruler: fixes w :: "'b list"   and r :: "'b list"   and w' :: "'b list"   and r' :: "'b list" assumes "w \<bowtie> r"   and "w' \<bowtie> r'"   and "\<not> r \<bowtie> r'" shows "w \<and>\<^sub>p w' \<le>p r \<and>\<^sub>p r'"
finitely_many_lists_for_multiset: fixes m :: "'a multiset" shows "finite {l. mset l = m}"
Nsetn_sub: fixes n :: "nat" shows "{j. j \<le> n} \<subseteq> {j. j \<le> Suc n}"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> x"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> x" shows "b = c"
const_list_no_progress: fixes es :: "e list"   and s :: "s"   and vs :: "v list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list" assumes "const_list es" shows "\<not> \<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
x0_hb_bound8: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real"   and x :: "real"   and i :: "nat" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p"   and "x\<^sub>1 \<le> x"   and "i < k" shows "x\<^sub>0 < bs ! i * x - hb * x / ln x powr (1 + e)"
word_arith_nat_defs_7: fixes a :: "'a word"   and b :: "'a word" shows "a mod b = word_of_nat (unat a mod unat b)"
actions_ok_thread_oksD: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('a, 'b, 'e, 'c, 'd, 'f) thread_action" assumes "actions_ok s t ta" shows "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
simps_2: fixes A :: "'b mat"   and I :: "nat" shows "swap_row_to_front A (Suc I) = swap_row_to_front (swaprows I (Suc I) A) I"
DIM_ge_Suc0: shows "Suc 0 \<le> DIM('a)"
distinct6_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points" shows "distinct6 A B C D E F \<equiv> A \<noteq> B \<and> A \<noteq> C \<and> A \<noteq> D \<and> A \<noteq> E \<and> A \<noteq> F \<and> B \<noteq> C \<and> B \<noteq> D \<and> B \<noteq> E \<and> B \<noteq> F \<and> C \<noteq> D \<and> C \<noteq> E \<and> C \<noteq> F \<and> D \<noteq> E \<and> D \<noteq> F \<and> E \<noteq> F"
top_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'b" shows "Semilat.top r T = (\<forall>x. x \<sqsubseteq>\<^bsub>r\<^esub> T)"
h3: shows "List.member l1 m \<longrightarrow> (\<exists>n<length l1. (l1 @ l2) ! n = m)"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 g) (tab\<^sub>0 f)"
empty_subsetI: fixes A :: "'b set" shows "{} \<subseteq> (A::'b set)"
nop_v11_axioms: shows "nop_v11 A (2\<^sub>\<nat>) f"
C0: fixes \<psi> :: "'a" shows "is_orthogonal \<psi> (C *\<^sub>V \<psi>)"
nb\<^sub>3: fixes op\<^sub>1 :: "'a strips_operator"   and op\<^sub>2 :: "'a strips_operator" assumes "op\<^sub>1 \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"   and "op\<^sub>2 \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"   and "index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2" shows "\<not> are_operators_interfering op\<^sub>1 op\<^sub>2"
complex_add_cnj: fixes z :: "complex" shows "z + cnj z = cor (2 * Re z)"
Rep_process: fixes x :: "'a process" shows "Rep_process x \<in> {p. is_process p}"
term_variants_pred_refl_inv: fixes P :: "'a \<Rightarrow> 'a list"   and s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "term_variants_pred P s t"   and "\<forall>f. \<forall>g\<in>set (P f). f = g" shows "s = t"
set_tag_name_locs_def: shows "set_tag_name_locs = (\<lambda>element_ptr. all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name_update))"
fls_plus_nth: fixes f :: "'a fls"   and g :: "'a fls"   and n :: "int" shows "fls_nth (f + g) n = fls_nth f n + fls_nth g n"
iti': shows "map_iterator_rev_linord (reverse_iterateoi m) (\<alpha> m)"
distinct_39: fixes x4 :: "nat"   and x6 :: "msg" shows "Pan x4 \<noteq> Hash x6"
DynPDG_path_cdep: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>c\<^sub>d n'" shows "n -as\<rightarrow>\<^sub>d* n'"
substmap: shows "ExprEvalL arglist m1 = ExprEvalL arglist m2"
arithmetic_simps_49: fixes a :: "'a" shows "(0::'a) + a = a"
function_smult_eval: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and b :: "'a" assumes "a \<in> carrier R"   and "f \<in> carrier (function_ring (carrier R) R)"   and "b \<in> carrier R" shows "(a \<odot>\<^bsub>function_ring (carrier R) R\<^esub> f) b = a \<otimes> f b"
assms_1: fixes p :: "real poly" assumes "p \<in> set ps" shows "p \<noteq> 0"
D_2: shows "\<forall>x y. x < y \<longrightarrow> (\<exists>d\<in>D. x \<le> d \<and> d < y)"
prod_cases4: fixes y :: "'a \<times> 'b \<times> 'c \<times> 'd" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" where "y = (a, b, c, d)"
fmp: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "\<turnstile> F"   and "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> G"
eql\<^sub>2: shows "override_on (l\<^sub>2 ++ l\<^sub>3) l\<^sub>2 (set pns) = l\<^sub>2"
inf_add_inf: shows "\<infinity>\<^sub>h +\<^sub>h \<infinity>\<^sub>h = \<infinity>\<^sub>h"
integrable_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integrable (restrict_space M \<Omega>) f = integrable M (\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x)"
image_Union: fixes f :: "'c \<Rightarrow> 'b"   and S :: "'c set set" shows "f ` \<Union> S = \<Union> ((`) f ` S)"
geodesic_segment_distance: fixes G :: "'c set"   and x :: "'c"   and y :: "'c"   and w :: "'c" assumes "geodesic_segment_between G x y" shows "dist w x + dist w y \<le> dist x y + 2 * infdist w G"
leg0_simps_3: shows "trg p\<^sub>0 = src (k\<^sup>* \<star> h)"
mem_le_imp_MIN_le: fixes s :: "nat set"   and k :: "nat" assumes "\<exists>x. x \<in> s \<and> x \<le> k" shows "Inf s \<le> k"
f2r_ad_fun_hom: fixes f :: "'a \<Rightarrow> 'b set" shows "\<R> (kad f) = ad_rel (\<R> f)"
sign_simps_3: fixes bf :: "'d"   and af :: "'d"   and cf :: "'d" shows "bf + (af + cf) = af + (bf + cf)"
vimage_Suc_insert_Suc: fixes n :: "nat"   and A :: "nat set" shows "Suc -` insert (Suc n) A = insert n (Suc -` A)"
Map_dom: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "Map (local.dom f) = \<lbrakk>local.Dom f\<rbrakk>\<^sub>B"
lepoll_iff: fixes A :: "'c set"   and B :: "'d set" shows "(A \<lesssim> B) = (\<exists>g. A \<subseteq> g ` B)"
assms_2: fixes fp :: "'x set" assumes "P ds fp"   and "Q ds fp" shows "Q ds (cop_F ds fp)"
norm_Pair_le: fixes x :: "'b"   and y :: "'c" shows "norm (x, y) \<le> norm x + norm y"
length_n_lists_elem: fixes ys :: "'a list"   and n :: "nat"   and xs :: "'a list" assumes "ys \<in> set (List.n_lists n xs)" shows "length ys = n"
codomains_char': fixes f :: "cospan_shape.Arr" shows "partial_magma.codomains cospan_shape.comp f = (if f = cospan_shape.Null then {} else {cospan_shape.Cod f})"
proots_within_def: fixes p :: "'c poly"   and s :: "'c set" shows "proots_within p s = {x \<in> s. poly p x = (0::'c)}"
const_OclNot: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "const X" shows "const (not X)"
induct: fixes x1 :: "cmd"   and x2 :: "nat"   and x3 :: "cmd"   and P :: "cmd \<Rightarrow> nat \<Rightarrow> cmd \<Rightarrow> bool" assumes "labels x1 x2 x3"   and "\<And>c. P c 0 c"   and "\<And>V e. P (V:=e) 1 Skip"   and "\<And>c\<^sub>1 l c c\<^sub>2. \<lbrakk>labels c\<^sub>1 l c; P c\<^sub>1 l c\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) l (c;; c\<^sub>2)"   and "\<And>c\<^sub>2 l c c\<^sub>1. \<lbrakk>labels c\<^sub>2 l c; P c\<^sub>2 l c\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (l + #:c\<^sub>1) c"   and "\<And>c\<^sub>1 l c b c\<^sub>2. \<lbrakk>labels c\<^sub>1 l c; P c\<^sub>1 l c\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (l + 1) c"   and "\<And>c\<^sub>2 l c b c\<^sub>1. \<lbrakk>labels c\<^sub>2 l c; P c\<^sub>2 l c\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (l + #:c\<^sub>1 + 1) c"   and "\<And>c' l c b. \<lbrakk>labels c' l c; P c' l c\<rbrakk> \<Longrightarrow> P (while (b) c') (l + 2) (c;; while (b) c')"   and "\<And>b c'. P (while (b) c') 1 Skip" shows "P x1 x2 x3"
Object_def: shows "Object = \<lparr>pid = java_lang, tid = Object'\<rparr>"
assoc_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a"   and "partial_magma.ide C b"   and "partial_magma.ide C c" shows "partial_magma.in_hom C (\<alpha> (a, b, c)) (T (T (a, b), c)) (T (a, T (b, c)))"
inject_2: fixes x2 :: "name"   and x1 :: "name"   and y2 :: "name"   and y1 :: "name" shows "(x2[x1] = y2[y1]) = (x2 = y2 \<and> x1 = y1)"
sse_cond_def: shows "sse_cond = {(\<Sigma>, W). W \<noteq> {}}"
rb_sp_Cons: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv1 :: "'d \<times> nat"   and re :: "('d \<times> nat) edge"   and res :: "('d \<times> nat) edge list"   and rv2 :: "'d \<times> nat" assumes "RedBlack prb" shows "SubRel.subpath (red prb) rv1 (re # res) rv2 (subs prb) = (sub_rel_of (red prb) (subs prb) \<and> (rv1 = src re \<or> (rv1, src re) \<in> subs prb) \<and> re \<in> edges (red prb) \<and> SubRel.subpath (red prb) (tgt re) res rv2 (subs prb))"
advantage\<^sub>n_def: fixes \<A> :: "nat \<Rightarrow> (bool, 'b \<times> (unit + 'cipher + 'plain \<times> 'plain), 'ekey + 'plain option + 'cipher) gpv"   and \<eta> :: "nat" shows "advantage\<^sub>n \<A> \<eta> = \<bar>spmf (ind_cca2\<^sub>n \<A> \<eta>) True - 1 / 2\<bar>"
foldSetD_closed: fixes A :: "'b set"   and z :: "'a"   and e :: "'a" assumes "(A, z) \<in> foldSetD D (\<cdot>) e" shows "z \<in> D"
field_npowf_exp_zero: fixes a :: "'a" assumes "a \<in> carrier K"   and "a \<noteq> \<zero>" shows "a\<^bsub>K\<^esub>\<^bsup>0\<^esup> = 1\<^sub>r"
simps_5: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "rev pi \<bullet> pi \<bullet> x = x"
that: fixes F :: "'a set" assumes "F facet_of S"   and "C \<subseteq> F" shows "thesis"
int_inc_zero: shows "[0] \<cdot> \<one> = \<zero>"
Transset_Sup: fixes X :: "V set" assumes "\<And>x. x \<in> X \<Longrightarrow> Transset x" shows "Transset (\<Squnion> X)"
the_map_option: fixes x :: "'b option"   and f :: "'b \<Rightarrow> 'c" assumes "\<not> Option.is_none x" shows "the (map_option f x) = f (the x)"
dy_fake_chan_mono_bad: fixes bad :: "agent set"   and bad' :: "agent set"   and I :: "msg set"   and C :: "chan set" assumes "bad \<subseteq> bad'" shows "dy_fake_chan bad I C \<subseteq> dy_fake_chan bad' I C"
of_rat_complex: fixes q :: "rat" shows "of_rat q = Complex (real_of_rat q) 0"
S_def: shows "S \<equiv> \<lambda>n. \<Sum>k | k \<in> {1..n} \<and> coprime k n. f k n"
pmatch_list_nomatch: fixes vs :: "v list"   and ps :: "pat list"   and cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and s :: "v store_v list" assumes "length vs = length ps" shows "fold2 (\<lambda>p v m. case m of No_match \<Rightarrow> No_match | Match_type_error \<Rightarrow> Match_type_error | Match env \<Rightarrow> pmatch_single cenv s p v env) Match_type_error ps vs No_match = No_match"
propa_ss1: shows "\<forall>(q, \<tau>)\<in>set (step (hd w) (ss ! hd w)). ss' ! q = hd w # ss ! hd w \<squnion>\<^bsub>f\<^esub> ss ! q"
distinct_remdups: fixes xs :: "'a list" shows "distinct (remdups xs)"
Delta_source_States: fixes T :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd"   and A :: "('d, 'e, 'f) seqauto" assumes "T \<in> Delta A" shows "source T \<in> States A"
a_subset: shows "(\<Inter>i\<le>n. I (Suc i)) \<subseteq> carrier R"
dp: shows "nmu (dports m) \<longrightarrow> nmu (dports m1) \<or> nmu (dports m2)"
G_stable_1: fixes w'' :: "nat \<Rightarrow> 'a set"   and \<psi> :: "'a ltl"   and wa :: "'a set list"   and w'b :: "'a set list" assumes "w'' \<Turnstile> G \<psi> = ((wa @ w'b) \<frown> w'') \<Turnstile> G \<psi>" shows "w'' \<Turnstile> G \<psi> = (w'b \<frown> w'') \<Turnstile> G \<psi>"
to_hma\<^sub>m_row_def: fixes A :: "'d mat"   and i :: "'f" shows "to_hma\<^sub>m_row A i = (\<chi>j. A $$ (mod_type_class.to_nat i, mod_type_class.to_nat j))"
sympartp_reflclp_equivp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "sympartp r\<^sup>=\<^sup>= = equivpartp r"