hcju/isabelleps huggingface dataset:
statically_complete_calculus_axioms_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set" shows "statically_complete_calculus_axioms Bot Inf entails Red_I \<equiv> \<forall>B N. B \<in> Bot \<longrightarrow> calculus.saturated Inf Red_I N \<longrightarrow> entails N {B} \<longrightarrow> (\<exists>B'\<in>Bot. B' \<in> N)"
Plus_ZeroL: fixes r :: "'a language" shows "Plus Zero r = r"
vreal_ub_def: fixes S :: "V"   and M :: "V" shows "vreal_ub S M \<equiv> S \<subseteq>\<^sub>\<circ> \<real>\<^sub>\<circ> \<and> M \<in>\<^sub>\<circ> \<real>\<^sub>\<circ> \<and> (\<forall>x\<in>elts S. x \<le>\<^sub>\<real> M)"
emptyset_part_emptyset2: fixes P :: "'b set set" assumes "P partitions {}" shows "P = {}"
intros_40: fixes vs :: "e list"   and n :: "nat"   and j :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and i :: "nat"   and vls :: "v list" assumes "const_list vs"   and "length vs = n"   and "Lfilled j lholed (vs @ [$Return]) es" shows "\<lparr>[Local n i vls es]\<rparr> \<leadsto> \<lparr>vs\<rparr>"
scale_def: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'a"   and v :: "'b \<times> 'c" assumes "module_prod s1 s2" shows "module_prod.scale s1 s2 a v = (s1 a (fst v), s2 a (snd v))"
path_loop_shift: fixes E :: "('e \<times> 'e) set"   and u :: "'e"   and p :: "'e list"   and v :: "'e" assumes "path E u p u"   and "v \<in> set p" obtains p' :: "'e list" where "set p' = set p" and "path E v p' v"
InitConf_CarAudioSystem: shows "InitConf CarAudioSystem = {''Off'', ''CDEmpty'', ''CarAudioSystem''}"
ceq: fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "ceq = Some eq" shows "eq = (=)"
subspaceI: fixes Sa :: "('a \<Rightarrow> real) set" assumes "Sa \<subseteq> diff_fun_space"   and "0 \<in> Sa"   and "\<And>x y. \<lbrakk>x \<in> Sa; y \<in> Sa\<rbrakk> \<Longrightarrow> x + y \<in> Sa"   and "\<And>c x. x \<in> Sa \<Longrightarrow> c *\<^sub>R x \<in> Sa" shows "diff_fun_space.m1.subspace Sa"
n_annil: fixes x :: "'a" shows "0 \<cdot> x = 0"
minus_1_eq: fixes x :: "real" shows "x + - 1 * p = x - p"
simps_3: fixes v :: "trm" shows "Pluso (Aterm v) undeft = undeft"
init: fixes q :: "'state"   and v :: "'action stream" shows "reduced_run q [] v [] [] [] [] v"
\<delta>\<^sub>p_def: shows "(\<delta>\<^sub>p, p\<^sub>1) = find_closest_bf_code p\<^sub>0 (p\<^sub>2 # p\<^sub>3 # ps)"
div_mult_self2: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and b :: "'e"   and a :: "'e"   and c :: "'e" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times b c)) b = plus c (divide a b)"
subst_if: fixes x :: "nat"   and v :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp" shows "subst x v (EIf e1 e2 e3) = EIf (subst x v e1) (subst x v e2) (subst x v e3)"
val_ab: shows "valof (a * b) = valof (zerosign (of_bool (sign a \<noteq> sign b)) (IEEE.round To_nearest (valof a * valof b)))"
rev_is_rev_conv: fixes xs :: "'f list"   and ys :: "'f list" shows "(rev xs = rev ys) = (xs = ys)"
P2: shows "monom P ((p \<otimes>\<^bsub>P\<^esub> q) (deg R (p \<otimes>\<^bsub>P\<^esub> q))) (deg R (p \<otimes>\<^bsub>P\<^esub> q)) = monom P (((p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)) (deg R ((p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)))) (deg R ((p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q)))"
two_not_one: shows "(2::'a) \<noteq> (1::'a)"
On_def: shows "On = (\<lambda>N. if N = 1 then O1 else {x \<in> space M. \<forall>B\<ge>Nf N. real (card ({n. \<exists>l\<in>{Nf N..n}. u n x - u (n - l) x \<le> - eps N * real l} \<inter> {..<B})) < eps N * real B})"
wf_jvm_prog\<^sub>k_def: fixes P :: "jvm_prog" shows "wf_jvm_prog\<^sub>k P \<equiv> wf_prog (\<lambda>P C' (M, b, Ts, T\<^sub>r, mxs, mxl\<^sub>0, is, xt). wt_kildall P C' b Ts T\<^sub>r mxs mxl\<^sub>0 is xt) P"
assms: shows "b \<in> arcs H - (G.face_cycle_set a \<union> G.face_cycle_set a')"
bind_game_def: fixes key_gen :: "('a \<times> 'b) spmf"   and verify :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and valid_msg :: "'c \<Rightarrow> bool"   and \<A> :: "'a \<Rightarrow> ('d \<times> 'c \<times> 'e \<times> 'c \<times> 'e) spmf" shows "abstract_commitment.bind_game key_gen verify valid_msg \<A> = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A> ck \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. let b = verify vk m c d; b' = verify vk m' c d' in return_spmf (b \<and> b')))) ELSE return_spmf False"
\<R>\<^sub>2_bisim_step: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>" shows "\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle> \<leadsto> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle> \<and> \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>"
multiplicity_prime: fixes i :: "nat"   and j :: "nat" assumes "prime i"   and "j \<noteq> i" shows "multiplicity j i = 0"
d_OUT_h_minus_aux: fixes h' :: "'v \<times> 'v \<Rightarrow> ennreal"   and j :: "nat"   and x :: "'v" assumes "\<And>e. h' e \<noteq> \<top>"   and "cycles \<Delta> \<noteq> {}" shows "KIR (h_minus_aux h' j) x"
phl_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "p ; x \<le> x ; r"   and "r ; y \<le> y ; q" shows "p ; (x ; y) \<le> x ; y ; q"
simps_1: shows "SkewBinomialHeapStruc.rank_invar [] = True"
rtrancl: shows "(red B)\<^sup>*\<^sup>* p r"
bdia_def: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bdia (\<lambda>x y. y \<cdot> x) ad x y = ad (ad (x \<cdot> y))"
1: fixes m1 :: "nat"   and m2 :: "nat" assumes "m1 < m2" shows "mkdisjoint A m1 \<inter> mkdisjoint A m2 = {}"
B_setops: shows "\<forall>p\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t B. \<exists>q\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. \<exists>\<delta>. fst p = fst q \<and> snd p = snd q \<cdot>\<^sub>p \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
While_CFGExit_wf_aux: fixes prog :: "cmd" shows "CFGExit_wf sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (_Exit_)"
w0_2: shows "w0 \<subseteq> fst x \<inter> fst y"
assms: shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
scalar_left_zero: fixes nn :: "nat"   and v :: "'a list" shows "scalar_prod (vec0 nn) v = (0::'a)"
assms: shows "rhs \<equiv> let n = CARD('a) in if n = 0 then False else let xs' = remdups xs; ys' = remdups ys in length xs' + length ys' = n \<and> (\<forall>x\<in>set xs'. x \<notin> set ys') \<and> (\<forall>y\<in>set ys'. y \<notin> set xs')"
resChainFreshSimps_3: fixes xvec :: "name list"   and P :: "('a, 'b, 'c) psi" shows "xvec \<sharp>* (\<lparr>\<nu>*xvec\<rparr>P)"
redT_updTs_Some: fixes ts :: "'i \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'i"   and xw :: "'g \<times> 'h \<Rightarrow>f nat"   and tas :: "('i, 'g, 'j) new_thread_action list" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_oks ts tas" shows "redT_updTs ts tas t = \<lfloor>xw\<rfloor>"
l1_inv3E: fixes x :: "l1_state" assumes "x \<in> l1_inv3" obtains "\<And>A B Rb Ra. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; progress x Rb = Some {xpkE, xsk}; guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; progress x Ra = Some {xpkE, xskE}; can_signal x A B\<rbrakk> \<Longrightarrow> l1_state.signals x (Commit A B \<langle>epubKF (Ra $ kE), NonceF (Rb $ sk)\<rangle>) < l1_state.signals x (Running A B \<langle>epubKF (Ra $ kE), NonceF (Rb $ sk)\<rangle>)"
le_1: fixes a :: "'a" assumes "a \<in> set as" shows "norm a \<le> 1"
assms_1: shows "b \<notin>\<^sub>\<circ> A"
subst_cls_negs: fixes AA :: "'a multiset"   and \<sigma> :: "'s" shows "negs AA \<cdot> \<sigma> = negs (AA \<cdot>am \<sigma>)"
eSwap_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and s :: "'sort"   and X :: "'a"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "igWls MOD s X" shows "eSwap MOD zs z1 z2 (OK X) = OK (igSwap MOD zs z1 z2 X)"
filter_nth_pairE: fixes i :: "nat"   and j :: "nat"   and P :: "'b \<Rightarrow> bool"   and xs :: "'b list" assumes "i < j"   and "i < length (filter P xs)"   and "j < length (filter P xs)" obtains i' :: "nat" and j' :: "nat" where "i' < j'" and "i' < length xs" and "j' < length xs" and "filter P xs ! i = xs ! i'" and "filter P xs ! j = xs ! j'"
least_mem: fixes L :: "('a, 'b) gorder_scheme"   and l :: "'a"   and A :: "'a set" assumes "least L l A" shows "l \<in> A"
supp_set_mem: fixes x :: "'b"   and L :: "'b list" assumes "x \<in> set L" shows "supp x \<subseteq> supp L"
pure_eq_iff: fixes h :: "'ap"   and f :: "('ap, 'ar, 'aq) prog" shows "(\<forall>h' x. h \<turnstile> f \<rightarrow>\<^sub>r x \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h' \<longrightarrow> h = h') = pure f h"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (x \<le>p y \<and> y \<le>p x)"
e'_3: shows "\<forall>i<n. e' i = e i"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a" shows "(case SOURCE of SOURCE \<Rightarrow> f1 | vertex.SINK \<Rightarrow> f2 | \<langle>x\<rangle> \<Rightarrow> f3 x) = f1"
assms_3: fixes s :: "'s" assumes "guard s"   and "I s" shows "lossless_spmf (body s)"
assms_3: shows "set (received_messages i) = set (received_messages j)"
implode_vend: shows "String.implode ''vend'' = STR ''vend''"
freeliftid_kernel': fixes ss :: "'w list" assumes "ss \<in> lists S"   and "sum_list ss = (0::'w)" shows "Abs_freelist ss \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
intros: fixes t :: "'a tree" assumes "\<And>t'. t' |\<in>| cont t \<Longrightarrow> tfinite t'" shows "tfinite t"
characteristic_holds: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "finite A" shows "\<sigma> \<Turnstile>\<^sub>p characteristic_formula A (\<sigma> 0)"
HEndPhase0_HInv4b_p_dblock: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2a s"   and "Inv2c_inner s p" shows "bal (dblock s' p) < mbal (dblock s' p)"
countCurrentLevelLiterals_def: fixes state :: "State" shows "countCurrentLevelLiterals state = (let cl = currentLevel (getM state) in state\<lparr>getCn := length (filter (\<lambda>l. elementLevel (opposite l) (getM state) = cl) (getC state))\<rparr>)"
exec_gpv_plus_oracle_right: fixes oracle1 :: "'i \<Rightarrow> 'j \<Rightarrow> ('l \<times> 'i) spmf"   and oracle2 :: "'i \<Rightarrow> 'k \<Rightarrow> ('m \<times> 'i) spmf"   and gpv :: "('h, 'k, 'm) gpv"   and s :: "'i" shows "exec_gpv (oracle1 \<oplus>\<^sub>O oracle2) (right_gpv gpv) s = exec_gpv oracle2 gpv s"
tranclD2: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R\<^sup>* \<and> (z, y) \<in> R"
older_seniors_finite: fixes x :: "nat"   and n :: "nat" shows "finite (older_seniors x n)"
lSup_finite_gen_prefixes: fixes zs :: "'c llist"   and xs :: "'c llist" assumes "zs \<sqsubseteq> xs"   and "lfinite zs" shows "lSup {ys. ys \<sqsubseteq> xs \<and> zs \<sqsubseteq> ys \<and> lfinite ys} = xs"
support_def: fixes z :: "'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "support z A f = {x \<in> A. f x \<noteq> z}"
Act_eq_iff_perm_renaming: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c, 'd) formula"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c, 'd) formula" shows "(Act f1 \<alpha>1 x1 = Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1))"
gkernel_def: fixes G :: "('e, 'f) Group_scheme"   and F :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" shows "gker\<^bsub>G,F\<^esub> f = {x \<in> carrier G. f x = \<one>\<^bsub>F\<^esub>}"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "B.dom (\<tau> f) = \<tau> (A.dom f)"
deriv: fixes x :: "real"   and s :: "real set" assumes "x0 \<le> x" shows "(f has_real_derivative f' x) (at x within s)"
trans_fm_inject: fixes e :: "name list"   and A :: "fm"   and B :: "fm" shows "(trans_fm e A = trans_fm e B) = (A = B)"
p53: shows "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. \<exists>rb\<in>blocksRead s p d. block rb = disk s d p"
divergent_def: fixes P :: "'a"   and Cal :: "'a processCalculus" shows "P \<longmapsto>Cal\<omega> \<equiv> \<forall>P'. P \<longmapsto>Cal* P' \<longrightarrow> Ex (step P' Cal)"
simps_6: fixes f1 :: "'b"   and f2 :: "'b" shows "rec_T1 f1 f2 B = f2"
swapidseq_identity_even: fixes n :: "nat" assumes "swapidseq n id" shows "even n"
smult_closed: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and a :: "'e"   and x :: "'g" assumes "module R M"   and "a \<in> carrier R"   and "x \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"
funs\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) strand_step list" shows "funs\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map funs\<^sub>s\<^sub>t\<^sub>p S))"
reflexive: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c" assumes "equivalence S E"   and "a \<in> S" shows "(a, a) \<in> E"
nlistToTree_def: fixes xs :: "('c, 'd) Node list" shows "nlistToTree xs = lconsNlist xs Empty"
lappend_eq_lappend_conv: fixes xs :: "'b llist"   and us :: "'b llist"   and ys :: "'b llist"   and vs :: "'b llist" assumes "Coinductive_List.llength xs = Coinductive_List.llength us" shows "(xs @@ ys = us @@ vs) = (xs = us \<and> (lfinite xs \<longrightarrow> ys = vs))"
tick_count_strict_is_fun: fixes r :: "'b run"   and c :: "clock"   and n :: "nat" shows "tick_count_strict r c n = #\<^sub>< r c n"
Cres': shows "Cres' = map (\<lambda>(F, Tm). ((F, C), Tm)) fs @ Dres'"
rref_upt_condition1_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c"   and j :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "i < j" shows "is_zero_row_upt_k j k A"
less_poly_def: fixes x :: "'b poly"   and y :: "'b poly" shows "(x < y) = pos_poly (y - x)"
invf_closed1: fixes K :: "('e, 'f) Ring_scheme"   and x :: "'e" assumes "Corps K"   and "x \<in> carrier K - {\<zero>\<^bsub>K\<^esub>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<in> carrier K - {\<zero>\<^bsub>K\<^esub>}"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<lbrace>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<rbrace> = \<alpha> (\<lbrace>t\<rbrace>, \<lbrace>u\<rbrace>, \<lbrace>v\<rbrace>)"
starP2_star_n: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'c" shows "(*p2* P) (star_n X) (star_n Y) = (\<forall>\<^sub>F n in \<U>. P (X n) (Y n))"
bisim1CondThen: fixes P :: "'b prog"   and h :: "'heap"   and e1 :: "'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 e2 :: "'addr expr1" assumes "P,e1,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,if (e) e1 else e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (length (compE2 e) + pc), xcp)"
assms: fixes n :: "nat" assumes "n < array_length a \<Longrightarrow> P (Suc n)" shows "P n"
ccINF_empty: 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 f :: "'b \<Rightarrow> 'a" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` {}) = top"
cases_scheme: fixes r :: "('b, 'c) routing_rule_scheme" obtains routing_match :: "'b prefix_match" and metric :: "nat" and routing_action :: "'b routing_action" and more :: "'c" where "r = \<lparr>routing_match = routing_match, metric = metric, routing_action = routing_action, \<dots> = more\<rparr>"
wt_subst_subst_upd: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and x :: "'var"   and t :: "('fun, 'var) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "\<Gamma> (Var x) = \<Gamma> t" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<theta>(x := t))"
ih_20: shows "getSATFlag state' = TRUE \<longrightarrow> satisfiable F0'"
chine_composite'_1: shows "Chn (trg f \<star> trg f \<star> f) = \<langle>trg.chine \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom (trg f))) \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom f))]] \<lbrakk>trg.cod.leg0, trg.cod.leg1 \<cdot> \<p>\<^sub>1[trg.cod.leg0, f.cod.leg1]\<rbrakk> \<langle>trg.chine \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom f))] \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom (trg f))) \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom f))]] \<lbrakk>trg.cod.leg0, f.cod.leg1\<rbrakk> f.chine \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom f))] \<cdot> \<p>\<^sub>0[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom (trg f))) \<cdot> \<p>\<^sub>1[Leg0 (Dom (local.dom (trg f))), Leg1 (Dom (local.dom f))]]\<rangle>\<rangle>"
fmsubset_drop: fixes a :: "'a"   and m :: "('a, 'b) fmap" shows "fmdrop a m \<subseteq>\<^sub>f m"
convex_is_geodesic: fixes S :: "'b set" assumes "convex S" shows "geodesic_subset S"
shuffle_to_top_def: fixes Y :: "'a set" shows "Pdds'.shuffle_to_top Y = (\<lambda>da. Pd_shuffle_to_top (insert d ds') Y da - Pdds'.dX Y da \<times> UNIV \<union> (Domain (Pd_shuffle_to_top (insert d ds') Y da) \<union> Pdds'.dX Y da) \<times> Pdds'.dX Y da)"
circ_decompose_9: fixes x :: "'b"   and y :: "'b" shows "(x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>+"
prv_neg_fls: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla"   and "bprv \<phi>"   and "bprv (B.neg \<phi>)" shows "bprv fls"
r_neg: fixes x :: "'c" assumes "x \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> \<ominus>\<^bsub>V\<^esub> x = \<zero>\<^bsub>V\<^esub>"
Diag_Diagonalize_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Cod C (monoidal_language.Diagonalize C t) = monoidal_language.Diagonalize C (monoidal_language.Cod C t)"
yun_1: shows "p = smult c (\<Prod>(a, i)\<in>set pis. a ^ Suc i)"
assms_2: shows "a \<notin> set u\<^sub>1"
omega_1: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<omega> \<cdot> y \<le> x\<^sup>\<omega>"
sgnx_able_poly_1: fixes p :: "real poly"   and a :: "real" shows "poly p sgnx_able at_right a"
carrier_quotient_ring_iff: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and X :: "('c \<times> 'c) set" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit" shows "(X \<in> S \<^sup>\<inverse> R \<^bsub>addition multiplication zero\<^esub>) = quotient_ring.valid_frac S R addition multiplication zero X"
orbit_inv_eq: fixes f :: "'e \<Rightarrow> 'e"   and x :: "'e" assumes "permutation f" shows "orbit (inv f) x = orbit f x"
primify_contains: fixes x :: "'d + 'd" shows "x \<in> primify x"
dist_0_norm: fixes x :: "'b" shows "dist (0::'b) x = \<parallel>x\<parallel>"
pred_inject_1: fixes P :: "'d \<Rightarrow> bool" shows "list_all P []"
nb_le_less: fixes b :: "contour"   and c :: "nat"   and b' :: "contour" shows "(nb b c \<le> b') = (b < b')"
inj2: shows "inj_on H (ball (x, y) (r / 2))"
execute_parallel_operator_def: fixes s :: "'c \<Rightarrow> bool option"   and ops :: "'c strips_operator list" shows "execute_parallel_operator s ops \<equiv> foldl (++) s (map (map_of \<circ> effect_to_assignments) ops)"
fps_of_ratfps_eq_iff: fixes p :: "'b ratfps"   and q :: "'b ratfps" shows "(fps_of_ratfps p = fps_of_ratfps q) = (p = q)"
simps_3: fixes v :: "('b \<times> nat) tree"   and va :: "'b \<times> nat"   and vb :: "('b \<times> nat) tree" shows "combine \<langle>v, va, vb\<rangle> \<langle>\<rangle> = \<langle>v, va, vb\<rangle>"
oexp_def: fixes a :: "V"   and b :: "V" shows "a \<up> b \<equiv> transrec (\<lambda>f x. if x = 0 then 1 else if Limit x then if a = 0 then 0 else \<Squnion> (f ` elts x) else f (\<Squnion> (elts x)) * a) b"
Infs_def: fixes X :: "'a set" shows "Sup_lattice.Infs Inf (\<lambda>x y. y \<le> x) X = \<Sqinter> {y. \<forall>x\<in>X. x \<le> y}"
up'_\<beta>: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and l :: "nat"   and lm :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < length b"   and "l + level b = lm"   and "b \<in> sparsegrid' dm"   and "p \<in> sparsegrid' dm" shows "snd (up' d l b \<alpha>) p = (if p \<in> lgrid b {d} lm then \<Sum>p'\<in>lgrid p {d} lm - {p}. \<alpha> p' * l2_\<phi> (p' ! d) (p ! d) else \<alpha> p)"
at_top_def: shows "at_top = (INF k. principal {k..})"
rel_nabla_d_closed: fixes x :: "('a \<times> 'a) set" shows "rdom (rel_nabla x) = rel_nabla x"
cases_scheme: fixes r :: "('d, 'c, 'e) ioa_scheme" obtains asig :: "'c signature" and start :: "'d set" and trans :: "('d \<times> 'c \<times> 'd) set" and more :: "'e" where "r = \<lparr>ioa.asig = asig, start = start, trans = trans, \<dots> = more\<rparr>"
le_g: shows "\<forall>i. f (g (Suc i)) \<le> f (g i)"
to_sas_plus_list_of_transformed_sas_plus_problem_operators_structure: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and ops' :: "('a \<times> 'b) strips_operator list"   and op :: "('a, 'b) sas_plus_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')" shows "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+) \<and> (\<exists>op'\<in>set ops'. op' = \<phi>\<^sub>O \<Psi> op)"
Image_Int_subset: fixes R :: "('b \<times> 'a) set"   and A :: "'b set"   and B :: "'b set" shows "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
plus_ordinal_def: shows "(+) = (\<lambda>x. ordinal_rec x (\<lambda>p. oSuc))"
Arccos_body_lemma: fixes z :: "complex" shows "z + \<i> * csqrt (1 - z\<^sup>2) \<noteq> 0"