hcju/isabelleps huggingface dataset:
at1: fixes x :: "'a" assumes "at TYPE('a)" shows "[] \<bullet> x = x"
invertible_right_inverse: fixes u :: "'a" assumes "sub.invertible u"   and "u \<in> N" shows "u \<cdot> sub.inverse u = \<one>"
bisim1CondElse: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and e1 :: "'addr expr1" assumes "P,e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,if (e) e1 else e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (Suc (length (compE2 e) + length (compE2 e1) + pc)), xcp)"
set_ins_list: fixes x :: "'b"   and xs :: "'b list" shows "list.set (ins_list x xs) = list.set xs \<union> {x}"
vrange_vempty: shows "\<R>\<^sub>\<circ> 0 = 0"
obs1: shows "Y\<^sub>1 \<in> Sols f (\<lambda>s. {t\<^sub>0--t}) S t\<^sub>0 s"
mbisim_def: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s1 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'f \<Rightarrow>f nat) option) \<times> 'c) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set"   and s2 :: "('f, 'a) locks \<times> (('a \<Rightarrow> ('d \<times> 'f \<Rightarrow>f nat) option) \<times> 'e) \<times> ('a \<Rightarrow> 'g wait_set_status option) \<times> 'a set" shows "FWbisimulation_base.mbisim bisim bisim_wait s1 s2 \<equiv> finite (dom (thr s1)) \<and> locks s1 = locks s2 \<and> wset s1 = wset s2 \<and> wset_thread_ok (wset s1) (thr s1) \<and> interrupts s1 = interrupts s2 \<and> (\<forall>t. FWbisimulation_base.tbisim bisim bisim_wait (wset s2 t = None) t (thr s1 t) (shr s1) (thr s2 t) (shr s2))"
RedCast: fixes s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and v :: "'addr val"   and U :: "ty"   and P :: "'addr J_prog"   and T :: "ty"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "P \<turnstile> U \<le> T" shows "extTA,P,t \<turnstile> \<langle>Cast T (Val v),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val v,s\<rangle>"
alpha_prod_closed: fixes x :: "'a"   and y :: "'a" shows "\<nu> (\<nu> x \<cdot> \<nu> y) = \<nu> x \<cdot> \<nu> y"
le_ref_def: fixes P :: "'b process"   and Q :: "'b process" shows "P \<le> Q \<equiv> D Q \<subseteq> D P \<and> F Q \<subseteq> F P"
Inf_UNIV: 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" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf UNIV = bot"
affected_order_greater_eq_two: fixes a :: "'b"   and f :: "'b perm" assumes "a \<in> affected f" shows "2 \<le> order f a"
Healthy_def': fixes P :: "'a upred"   and H :: "'a upred \<Rightarrow> 'a upred" shows "(P is H) = (H P = P)"
pair_pair_spmf: fixes p :: "'a spmf"   and q :: "'b spmf"   and r :: "'c spmf" shows "pair_spmf (pair_spmf p q) r = map_spmf (\<lambda>(x, y, z). ((x, y), z)) (pair_spmf p (pair_spmf q r))"
foo4: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and s :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "set p = set s \<and> (\<forall>r. r \<in> set p \<longrightarrow> x \<notin> dom (Cp r))"   and "r \<in> set s" shows "x \<notin> dom (Cp r)"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (r \<star> s) \<rho>\<sigma>.tab (s\<^sub>0 \<star> \<rho>\<sigma>.p\<^sub>0) (r\<^sub>1 \<star> \<rho>\<sigma>.p\<^sub>1) (t \<star> u) \<tau>\<mu>.tab (u\<^sub>0 \<star> \<tau>\<mu>.p\<^sub>0) (t\<^sub>1 \<star> \<tau>\<mu>.p\<^sub>1) (\<omega> \<star> \<chi>)"
FreeGroup_def: fixes S :: "'b set" shows "FreeGroup S \<equiv> {x. order.greater_eq S (fst ` set (freeword x))}"
isLb_def: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" shows "isLb R S x = (x <=* S \<and> x \<in> R)"
simps_1: fixes x :: "'v" shows "local.ground_heads (Var x) = ground_heads_var x"
3: shows "length w\<^sub>1 = length (w\<^sub>1 @ v\<^sub>1)"
h3: shows "distinct (get_signs (reduce_system p (qs, matrix, subsets, signs)))"
converse_rtranclp_into_rtranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
zero_one_infty_not_equal_2: shows "0\<^sub>h \<noteq> \<infinity>\<^sub>h"
eventually_division_filter_tagged_division: fixes s :: "'a set" shows "\<forall>\<^sub>F p in division_filter s. p tagged_division_of s"
O_simps_2: fixes \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and trn :: "('a, 'b, 'c) trans"   and g :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and tr :: "('a, 'b, 'c) trans list" assumes "\<not> \<gamma> trn" shows "BD_Security.O \<gamma> g (trn # tr) = BD_Security.O \<gamma> g tr"
erH_rHom: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B" shows "erH R S X A B Y h \<in> pHom R S X, A B Y"
widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and T :: "ty"   and C :: "char list" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C))"
snd_set_split: fixes ps :: "'a"   and pth :: "('b \<times> 'a) list" assumes "ps \<in> snd ` set pth" obtains pth1 :: "('b \<times> 'a) list" and pth2 :: "('b \<times> 'a) list" and e :: "'b \<times> 'a" where "pth = pth1 @ [e] @ pth2" and "snd e = ps" and "ps \<notin> snd ` set pth1"
min_gallery_least_length: fixes C :: "'a set"   and D :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<noteq> D" shows "domain.min_gallery (C # (ARG_MIN length Cs. domain.gallery (C # Cs @ [D])) @ [D])"
ipd_not_self: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" shows "x \<noteq> ipd x"
po_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "po_on P A = (irreflp_on P A \<and> transp_on P A)"
inf1D1: fixes A :: "'g \<Rightarrow> bool"   and B :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "(A \<sqinter> B) x" shows "A x"
valid_append_valid_suff: fixes as1 :: "'a list"   and as2 :: "'a list"   and PROB :: "'a set" assumes "as1 @ as2 \<in> valid_plans PROB" shows "as2 \<in> valid_plans PROB"
free_hull_hull: fixes G :: "'b list set" shows "\<langle>\<langle>G\<rangle>\<^sub>F\<rangle> = \<langle>G\<rangle>\<^sub>F"
load_word_mem_10_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and address :: "word32" assumes "low_equal s1 s2" shows "load_word_mem s1 address 10 = load_word_mem s2 address 10"
Domainp_rel_fun_le: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "Domainp (rel_fun A B) \<le> pred_fun (Domainp A) (Domainp B)"
face_cycle_sets_def: shows "GM.face_cycle_sets = GM.face_cycle_set ` arcs G"
b_c: shows "Partial_Cost_Model.config'_rand BIT (type0 init x y) v = type0 init x y"
if_true: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "ev\<^sub>B mem b" shows "eval\<^sub>w_simplep (Stmt.If b t e) mem t mem"
comm_ring_1: shows "OFCLASS('a bit0, comm_ring_1_class)"
NSconvergentI: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S L" shows "NSconvergent X"
mset_set_empty_iff: fixes A :: "'a set" shows "(mset_set A = {#}) = (A = {} \<or> infinite A)"
tuple_simps_1: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "arr (ECC.tuple f g)"
distr_mult_real: fixes c :: "real"   and M :: "real measure"   and f :: "real \<Rightarrow> ennreal" assumes "c \<noteq> 0"   and "has_density M lborel f" shows "has_density (distr M borel ((*) c)) lborel (\<lambda>x. f (x / c) * ennreal (inverse \<bar>c\<bar>))"
elims_3: fixes x :: "'b literal" assumes "\<not> positive_literal x" obtains xa :: "'b equation" where "x = Neg xa"
goto_decision: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and X :: "final_decision"   and rest :: "'c rule list" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision X" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>Rule m (Goto chain) # rest, Undecided\<rangle> \<Rightarrow> Decision X"
IH: fixes a :: "(real mpoly \<times> nat) list"   and b :: "(real mpoly \<times> nat) list"   and L' :: "(real mpoly \<times> nat) list" assumes "(a, b) = chophelper L L'" shows "chop L = b"
\<alpha>_remove_le: fixes G :: "nat set \<times> nat set set"   and u :: "nat" shows "\<alpha> (G -- u) \<le> \<alpha> G"
0: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp \<and> f \<bullet> x = \<zero> \<and> val_Zp (pderiv f \<bullet> a) < val_Zp (a \<ominus> x) \<and> val_Zp (pderiv f \<bullet> a) \<noteq> val_Zp (a \<ominus> x)" shows "x = \<alpha>"
lm01: fixes P :: "('a \<times> 'b) set" shows "card P = card (P\<inverse>)"
commute: fixes a :: "'g"   and b :: "'g" shows "inf a b = inf b a"
affine_hull_open_segment: fixes a :: "'a"   and b :: "'a" shows "affine hull open_segment a b = (if a = b then {} else affine hull {a, b})"
inv_f_f: fixes x :: "'b" shows "inv to_fract (to_fract x) = x"
1: shows "ia \<in> {u. Pi (Suc i) ^ u dvd godel_code' nl (Suc 0)}"
dagger_denest: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> ; y)\<^sup>\<star> ; x\<^sup>\<star>"
cnf2bhv_greater_llength: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "\<langle>c \<and> t\<rangle> < n" shows "the_enat (llength (\<pi>\<^bsub>c\<^esub>inf_llist t)) - 1 < \<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n"
MGFn_free_wt_da_NormalConformI: fixes G :: "prog"   and t :: "Term.term"   and A :: "state triple set"   and n :: "nat" assumes "\<forall>T L C B. \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T \<longrightarrow> G,A\<turnstile>{Normal ((\<lambda>Y' s' s. s' = s \<and> normal s) \<and>. G\<turnstile>init\<le>n) \<and>. (\<lambda>s. s\<Colon>\<preceq>(G, L)) \<and>. (\<lambda>s. \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile> dom (locals (snd s)) \<guillemotright>t\<guillemotright> B)} t\<succ> {\<lambda>Y s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y, s')}" shows "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}"
considers_def: fixes f :: "nat \<Rightarrow> nat \<times> (nat \<times> form) list"   and n :: "nat"   and nf :: "nat \<times> form" shows "considers f n nf = (case snd (f n) of [] \<Rightarrow> False | x # xs \<Rightarrow> x = nf)"
set_intros_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 \<in> set (x21 # x22)"
elims_3: fixes x :: "('a \<times> 'c) list \<times> 'o"   and xa :: "('a \<times> 'c) list \<times> 'o" assumes "\<not> oalist_eq_raw x xa" obtains xs :: "('a \<times> 'c) list" and ox :: "'o" and ys :: "('a \<times> 'c) list \<times> 'o" where "x = (xs, ox)" and "xa = ys" and "xs \<noteq> sort_oalist_aux ox ys"
bs': shows "length bs = length vs \<and> v' = bs \<bullet>\<cdot> vs"
tensor_prod_of_id_2_y: shows "Quantum.Id 2 \<Otimes> Z * X = M4"
simps_1: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" shows "noDenyAll (x # xs) = (\<not> member DenyAll x \<and> noDenyAll xs)"
exec_move_Sync1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (sync\<^bsub>V\<^esub> (e) e') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
aligned_sub_aligned: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word"   and m :: "nat" assumes "is_aligned x n"   and "is_aligned y m"   and "m \<le> n" shows "is_aligned (x - y) m"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'c \<Rightarrow> 'a"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
rel_simps_46: shows "(0::'a) \<le> (0::'a)"
va: fixes \<nu> :: "variable \<Rightarrow> real" assumes "Uvariation \<nu> (F 0) {RVar x, DVar x}" shows "term_sem I \<theta> \<nu> = term_sem J \<eta> \<nu>"
one_less_inverse_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 x :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less one (inverse x) = (less zero x \<and> less x one)"
ipd_in_pdt: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "x \<noteq> IFC_def.return prob" shows "(x, IFC_def.ipd prob x) \<in> IFC_def.pdt prob"
simps_4: shows "TernaryUnknown \<noteq> TernaryTrue"
hs_delete_impl: shows "imp_set_delete is_hashset hs_delete"
not_int_def: fixes k :: "int" shows "NOT k = - k - 1"
trimmed_map_ms: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and C :: "'a ms" assumes "is_lifting f basis basis'" shows "trimmed (map_ms f C) = trimmed C"
abs_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a set \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'a set"   and x :: "'a" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
fps_div_fps_X_nth': fixes f :: "'a fps"   and k :: "nat" assumes "inverse (1::'a) = (1::'a)" shows "fps_nth (f / fps_X) k = fps_nth f (Suc k)"
map_spmf_bind_spmf: fixes f :: "'e \<Rightarrow> 'd"   and p :: "'f spmf"   and g :: "'f \<Rightarrow> 'e spmf" shows "map_spmf f (p \<bind> g) = p \<bind> map_spmf f \<circ> g"
is_weakly_dom_component_safe_def: shows "Shadow_DOM_DOM_Components.is_weakly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"
cscc_invar_outer_Dnode: fixes it :: "'v set"   and l :: "'v set list"   and D :: "'v set" assumes "cscc_outer_invar it (l, D)" shows "cscc_invar_ext G l D"
step: fixes y :: "'var \<Rightarrow> 'value"   and z :: "'var \<Rightarrow> 'value" assumes "P y z" shows "P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>1 \<^bold>\<rbrakk>\<^bsub>y\<^esub>) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>2 \<^bold>\<rbrakk>\<^bsub>z\<^esub>)"
SUP_ereal_add_directed: fixes I :: "'a set"   and f :: "'a \<Rightarrow> ereal"   and g :: "'a \<Rightarrow> ereal" assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"   and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g i"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<exists>k\<in>I. f i + g j \<le> f k + g k" shows "(SUP i\<in>I. f i + g i) = Sup (f ` I) + Sup (g ` I)"
OclIncludes_charn1: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> Set('b)"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> \<delta> X"   and "\<tau> \<Turnstile> \<upsilon> x" shows "\<tau> \<Turnstile> X->including\<^sub>S\<^sub>e\<^sub>t(x)->includes\<^sub>S\<^sub>e\<^sub>t(x)"
hyps_3: shows "mds\<^sub>1 \<sqsubseteq> mds\<^sub>2''"
select_convs_6: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "marked \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = marked"
l1_inv1_def: shows "l1_inv1 \<equiv> {s. \<forall>Ra A B K. 0 < l1_state.signals s (Commit A B \<langle>epubKF (Ra $ kE), K\<rangle>) \<longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xpkE, xskE, xsk} \<and> guessed_frame Ra xsk = Some K}"
generate_into_G: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> generate (G \<inter> H)" shows "a \<in> G"
bounded_closed_segment: fixes a :: "'b"   and b :: "'b" shows "bounded (closed_segment a b)"
elims: fixes x :: "ereal"   and xa :: "ereal"   and y :: "ereal" assumes "x + xa = y" obtains r :: "real" and p :: "real" where "x = ereal r__" and "xa = ereal p__" and "y = ereal (r__ + p__)"   | a :: "ereal" where "x = \<infinity>" and "xa = a__" and "y = \<infinity>"   | a :: "ereal" where "x = a" and "xa = \<infinity>" and "y = \<infinity>"   | r :: "real" where "x = ereal r" and "xa = - \<infinity>" and "y = - \<infinity>"   | p :: "real" where "x = - \<infinity>" and "xa = ereal p" and "y = - \<infinity>"   | "x = - \<infinity>" and "xa = - \<infinity>" and "y = - \<infinity>"
u_K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace>, servTicket\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "\<exists>Ta. Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace> \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
list_emb_imp_reflclp_mulex_on: fixes xs :: "'b list"   and A :: "'b set"   and ys :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "xs \<in> lists A"   and "ys \<in> lists A"   and "list_emb P xs ys" shows "(mulex_on P A)\<^sup>=\<^sup>= (mset xs) (mset ys)"
prime_mn_reverse: fixes n :: "nat"   and f :: "recf"   and m :: "nat"   and P :: "bool" assumes "primerec (Mn n f) m" shows "P"
Gromov_extension_isometry_composition: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "isometry f"   and "isometry g" shows "Gromov_extension (g \<circ> f) = Gromov_extension g \<circ> Gromov_extension f"
card_setsum_partition: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "sum card ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) = card A"
index_map_vec_2: fixes f :: "'c \<Rightarrow> 'b"   and v :: "'c Matrix.vec" shows "dim_vec (map_vec f v) = dim_vec v"
S''_def: shows "S'' = {(p, i). prime p \<and> 1 < i \<and> real (p ^ Suc i) \<le> x}"
simps: fixes g :: "nat list \<Rightarrow> nat"   and xs :: "nat list" shows "UF.Sigma g xs = (if last xs = 0 then g xs else UF.Sigma g (butlast xs @ [last xs - 1]) + g xs)"
map_def: shows "dnelist.map = map_fun id (map_fun list_of_dnelist Abs_dnelist) (\<lambda>f xs. remdups (map f xs))"
compact_lemma_general: fixes basis :: "'b set"   and proj :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "nat \<Rightarrow> 'd"   and unproj :: "('b \<Rightarrow> 'c) \<Rightarrow> 'd" assumes "finite basis"   and "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. proj x k) ` range f)"   and "\<And>e k. k \<in> basis \<Longrightarrow> proj (unproj e) k = e k"   and "\<And>x. unproj (proj x) = x" shows "\<forall>d\<subseteq>basis. \<exists>l r. strict_mono r \<and> (\<forall>e>0. \<forall>\<^sub>F n in sequentially. \<forall>i\<in>d. dist (proj (f (r n)) i) (proj l i) < e)"
extended_Gromov_distance_commute: fixes x :: "'b Gromov_completion"   and y :: "'b Gromov_completion" shows "extended_Gromov_distance x y = extended_Gromov_distance y x"
fin_mono: fixes A :: "'c fset"   and B :: "'c fset"   and x :: "'c" assumes "A |\<subseteq>| B" shows "x |\<in>| A \<longrightarrow> x |\<in>| B"
f_expand_length: fixes xs :: "'d message_af list"   and k :: "nat" shows "length (xs \<odot> k) = length xs * k"
uminus_eq_minus: fixes x :: "'a" shows "- x = top - x"
assms_1: shows "(\<box>(\<lambda>s. R s \<longrightarrow> R' s)) \<sigma>"
simps_5: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I (Exists x \<phi>) = {uu_. \<exists>v r. uu_ = v \<and> repv v x r \<in> fml_sem I \<phi>}"
env_delete_def: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b" shows "env_delete x m = m(x := \<bottom>)"
invar_merge: fixes h1 :: "'b hp option"   and h2 :: "'b hp option" assumes "invar h1"   and "invar h2" shows "invar (merge h1 h2)"
pos_add_strict: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.strict_ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less b c" shows "less b (plus a c)"
Eps_case_prod_eq: fixes x :: "'c"   and y :: "'d" shows "(SOME (x', y'). x = x' \<and> y = y') = (x, y)"
distrib_right: fixes a :: "'a hrel"   and b :: "'a hrel"   and c :: "'a hrel" shows "a \<sqinter> b ;; c = (a ;; c) \<sqinter> (b ;; c)"
refines_reachable_strengthening: fixes R :: "('e \<times> 'f) set"   and pi :: "'g \<Rightarrow> 'h"   and Sa :: "('e, 'h) spec"   and Sc :: "('f, 'g) spec" assumes "refines R pi Sa Sc" shows "refines (R \<inter> reach Sa \<times> reach Sc) pi Sa Sc"
lprod_3_7: fixes z' :: "'a"   and z :: "'a"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "(z', z) \<in> R" shows "([x, z', y], [x, y, z]) \<in> lprod R"
inject: fixes x1 :: "'b"   and y1 :: "'b" shows "(bot.Value x1 = bot.Value y1) = (x1 = y1)"
id_comp: fixes g :: "'a \<Rightarrow> 'b" shows "id \<circ> g = g"
abs_eq: fixes x :: "'a" shows "blinfun_inner_left x = Blinfun (\<lambda>y. y \<bullet> x)"
map2_val_compat'I: fixes rep_key_order :: "'c \<Rightarrow> 'a key_order"   and f :: "('a \<times> 'b) list \<times> 'c \<Rightarrow> ('a \<times> 'd) list \<times> 'c" assumes "\<And>zs. oalist_raw.oalist_inv rep_key_order zs \<Longrightarrow> oalist_raw.oalist_inv rep_key_order (f zs)"   and "\<And>zs. oalist_raw.oalist_inv rep_key_order zs \<Longrightarrow> snd (f zs) = snd zs"   and "\<And>zs. oalist_raw.oalist_inv rep_key_order zs \<Longrightarrow> fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs)" shows "oalist_raw.map2_val_compat' rep_key_order f"