hcju/isabelleps huggingface dataset:
cases: fixes a :: "('b com option \<times> 'b) list" assumes "a \<in> cptn" obtains P :: "'b com option" and s :: "'b" where "a = [(P__, sa__)]"   | P :: "'b com option" and t :: "'b" and xs :: "('b com option \<times> 'b) list" and s :: "'b" where "a = (Pa__, sb__) # (Pa__, t__) # xsa__" and "(Pa__, t__) # xsa__ \<in> cptn"   | P :: "'b com option" and s :: "'b" and Q :: "'b com option" and t :: "'b" and xs :: "('b com option \<times> 'b) list" where "a = (P, s) # (Q, t) # xs" and "(P, s) -c\<rightarrow> (Q, t)" and "(Q, t) # xs \<in> cptn"
simps: fixes A :: "'a list"   and n :: "nat"   and a :: "'a" shows "array_set (Array A) n a = Array (A[n := a])"
coeff_poly_mult_rat_main: fixes n :: "'a"   and d :: "'a"   and f :: "'a poly"   and i :: "nat" shows "coeff (poly_mult_rat_main n d f) i = coeff f i * n ^ (degree f - i) * d ^ i"
height_acomplete: fixes t :: "'b tree" assumes "acomplete t" shows "height t = nat \<lceil>log 2 (real (size1 t))\<rceil>"
tl_upt: fixes m :: "nat"   and n :: "nat" shows "tl [m..<n] = [Suc m..<n]"
product_run_embed: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and q\<^sub>0 :: "'a"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "\<iota>\<^sub>m x = Some q\<^sub>0" shows "run\<^sub>t (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w i \<in> \<upharpoonleft>\<^sub>x (run\<^sub>t (\<delta>\<^sub>m x) q\<^sub>0 w i)"
init: shows "dfa.init \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr> \<in> dfa.states \<lparr>dfa.states = {0}, init = 0, final = {}, nxt = \<lambda>q x. q\<rparr>"
neg_numeral_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus m) = uminus (numeral.numeral one plus n)) = (m = n)"
assms: shows "a \<in> runiqs \<inter> Pow (UNIV \<times> (UNIV - {{}}))"
inject_1: fixes x11 :: "nat"   and x12 :: "nat set"   and y11 :: "nat"   and y12 :: "nat set" shows "(\<langle>x11; x12\<rangle> = \<langle>y11; y12\<rangle>) = (x11 = y11 \<and> x12 = y12)"
th0: shows "fps_deriv (fps_binomial c * fps_binomial d - fps_binomial (c + d)) = fps_const (c + d) * (fps_binomial c * fps_binomial d - fps_binomial (c + d)) / (1 + fps_X)"
fv_convert_multiway: fixes \<phi> :: "formula"   and b :: "nat" assumes "safe_formula \<phi>" shows "fvi b (convert_multiway \<phi>) = fvi b \<phi>"
card_Diff_subset: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "order.greater_eq A B" shows "card (A - B) = card A - card B"
dom_S': fixes A :: "'a"   and t :: "'b"   and S :: "nat \<Rightarrow> typ" shows "Type.dom (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) \<subseteq> free_tv A \<union> free_tv t"
proper_parts_principle: fixes x :: "i"   and y :: "i" assumes "\<exists>z. PP z x"   and "\<forall>z. PP z x \<longrightarrow> P z y" shows "P x y"
axioms_1: fixes r :: "V" assumes "v11 r" shows "vsv r"
monofun_def: fixes f :: "'c \<Rightarrow> 'd" shows "monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
ZObis_pres_discrCfL: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and df :: "('test, 'atom, 'choice) cmd \<times> 'state" assumes "fst cf \<approx>01 fst df"   and "snd cf \<approx> snd df"   and "discrCf df" shows "discrCf cf"
set_ins_list: fixes x :: "'b"   and xs :: "'b list" shows "set (ins_list x xs) = set xs \<union> {x}"
l2_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Nr :: "msg"   and gny :: "msg" shows "l2_step3 Ra A B Nr gny \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xnx, xni, xgnx} \<and> sklvl1.guessed_frame Ra xgny = Some gny \<and> sklvl1.guessed_frame Ra xnr = Some Nr \<and> sklvl1.guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx))) \<and> Confid B A Nr \<in> chan s \<and> Insec B A \<langle>gny, hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr)\<rangle> \<in> chan s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}), chan := {Insec A B (hmac \<langle>Number 1, gny, Exp Gen (NonceF (Ra $ nx)), Messages.Agent A, Messages.Agent B\<rangle> (hmac (NonceF (Ra $ ni)) Nr))} \<union> chan s, secret := {x. x = Exp gny (NonceF (Ra $ nx)) \<and> Ra = test} \<union> secret s, signalsInit := if can_signal s A B then addSignal (signalsInit s) (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsInit s, signalsInit2 := if can_signal s A B then addSignal (signalsInit2 s) (Commit A B (Exp gny (NonceF (Ra $ nx)))) else signalsInit2 s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) else signalsResp s, signalsResp2 := if can_signal s A B then addSignal (signalsResp2 s) (Running A B (Exp gny (NonceF (Ra $ nx)))) else signalsResp2 s\<rparr>}"
pwd_spy_sep_map_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_spy_sep_map n of (s, S, A, B, C, D) \<Rightarrow> pwd_spy_sep_map_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
same_pmod_iff_pdivides: fixes K :: "'a list set"   and a :: "'a list list"   and b :: "'a list list"   and q :: "'a list list" assumes "subfield K (K [X])"   and "a \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "b \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "(UP.pmod a q = UP.pmod b q) = q pdivides\<^bsub>K [X]\<^esub> (a \<ominus>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> b)"
ord_isom_Pre1: fixes E :: "('e, 'f) Order_scheme"   and a :: "'a"   and f :: "'a \<Rightarrow> 'e" assumes "Worder E"   and "a \<in> carrier D"   and "ExPre D a"   and "ord_isom D E f" shows "ExPre E (f a)"
larger_is_join1: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.cpo below"   and "below y x" shows "cpo.join below x y = x"
converseD: fixes a :: "'a"   and b :: "'c"   and r :: "('c \<times> 'a) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
winding_number_comp: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "open s"   and "f holomorphic_on s"   and "path_image \<gamma> \<subseteq> s"   and "valid_path \<gamma>"   and "z \<notin> path_image (f \<circ> \<gamma>)" shows "winding_number (f \<circ> \<gamma>) z = 1 / (complex_of_real (2 * pi) * \<i>) * contour_integral \<gamma> (\<lambda>w. deriv f w / (f w - z))"
sp_fundef_same_arities: fixes F1 :: "'c \<Rightarrow> 'd fundef option"   and F2 :: "'c \<Rightarrow> 'e fundef option" assumes "all_same_arities F1 F2" shows "sp_fundef F1 = sp_fundef F2"
star_ext: fixes x :: "'a" shows "x \<le> x\<^sup>\<star>"
defined: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c spmf"   and x :: "'d"   and A :: "'c set"   and p :: "'d spmf" assumes "y \<in> set_spmf (f x)"   and "y \<in> A"   and "x \<in> set_spmf p" shows "Some ` A \<inter> (\<Union>x\<in>set_pmf p. set_pmf (case x of None \<Rightarrow> return_pmf None | Some x \<Rightarrow> f x)) \<noteq> {}"
finite_representation: fixes basis :: "'b set"   and v :: "'b" shows "finite {b. vs1.representation basis v b \<noteq> (0::'a)}"
inject_3: fixes x31 :: "'b local.term"   and x32 :: "'b local.term"   and y31 :: "'b local.term"   and y32 :: "'b local.term" shows "(x31 \<^bold>\<star> x32 = y31 \<^bold>\<star> y32) = (x31 = y31 \<and> x32 = y32)"
eta_nt: fixes f :: "'a \<Rightarrow> 'b" shows "\<P> f \<circ> \<eta> = \<eta> \<circ> id f"
code_ent_kl_log: shows "code_rate enc X - \<H>(X) = KL_div L fi (\<lambda>i. 1 / (b powr real (cw_len i) * \<K>)) + log b (1 / \<K>)"
oppositeIdempotency: fixes literal :: "Literal" shows "opposite (opposite literal) = literal"
diff_single_eq_union: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<in># M" shows "(M - \<LM> x  \<RM> = N) = (M = add_mset x N)"
INF_emeasure_decseq: fixes A :: "nat \<Rightarrow> 'd set"   and M :: "'d measure" assumes "range A \<subseteq> sets M"   and "decseq A"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<Sqinter>n. emeasure M (A n)) = emeasure M (\<Inter> (range A))"
absorb_iff1: fixes b :: "'e multiset"   and a :: "'e multiset" shows "(b \<subseteq># a) = (a \<union># b = a)"
arc_imp_path: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "path g"
smalls_pscope: fixes \<pi>' :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and c' :: "com"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option" assumes "small_steps \<pi>' (c, s) (Some (c', s'))" shows "small_steps \<pi> (PScope \<pi>' c, s) (Some (PScope \<pi>' c', s'))"
Ptl_0: shows "ltl P $ 0 = lhd (ltl P)"
right: fixes P :: "'b"   and Cal :: "'b processCalculus"   and P' :: "'b"   and Q :: "'b"   and Rel :: "('b \<times> 'b) set" assumes "P \<longmapsto>Cal* P'"   and "(Q, P) \<in> Rel" shows "(P', Q) \<in> cSim_cs Rel Cal"
l_inv: fixes x :: "'a" assumes "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
assms_2: shows "w\<^sub>2 \<preceq>\<^sub>F w\<^sub>3"
17: shows "e\<^sup>T * top * e\<^sup>T \<le> - F"
ctxt_subst_identity: fixes X :: "tyvrs"   and \<Gamma> :: "binding list"   and U :: "ty" assumes "X \<sharp> \<Gamma>" shows "\<Gamma>[X \<mapsto> U]\<^sub>e = \<Gamma>"
\<alpha>_char: fixes i :: "nat" assumes "3 \<le> i"   and "i < k + 2" shows "\<alpha> i = f (i - 1) ^ (Suc (length (coeffs (F (i - 2)))) - length (coeffs (F (i - 1))))"
always_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<box>F \<equiv> %s. \<forall>n. F (s |\<^sub>s n)"
exec_n_exec: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and n :: "nat"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list" assumes "P \<turnstile> c \<rightarrow>^n c'" shows "P \<turnstile> c \<rightarrow>* c'"
filterlim_uminus_at_top_at_bot: shows "filterlim uminus at_top at_bot"
prefs_wf'_1: fixes i :: "'a" assumes "i \<in> agents'" shows "total_preorder_on alts' (R' i)"
extend_basis_raw_def: shows "extend_basis \<equiv> vector_space.extend_basis (*\<^sub>R)"
card_face_cycle_sets_conv: shows "card face_cycle_sets = length (lists_fcs (remdups (snd G)))"
da_elim_cases_1: fixes Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>Skip\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
eval_finalsId: fixes es :: "char list exp list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "finals es" shows "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es,s\<rangle>"
assms_1: shows "\<Gamma> \<oplus> A = \<Gamma>' \<oplus> B"
swap: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and B :: "'f set"   and A :: "'e set" shows "(\<Sum>i\<in>A. sum (g i) B) = (\<Sum>j\<in>B. \<Sum>i\<in>A. g i j)"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [e2]))"
coboundedI2: fixes c :: "'c multiset"   and b :: "'c multiset"   and a :: "'c multiset" assumes "c \<subseteq># b" shows "c \<subseteq># a \<union># b"
theta_lessE: fixes m :: "nat"   and n :: "nat" assumes "theta m < theta n"   and "1 \<le> m" obtains p :: "nat" where "p \<in> {m<..n}" and "prime p"
group_ringD: fixes G :: "'a set" assumes "Group G" shows "aezfun_setspan G = {x. supp (aezfun x) \<subseteq> G}"
536: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> - - y) \<sqinter> y = (x \<squnion> top) \<sqinter> y"
basis: shows "\<BB>\<^sub>F {u, v} = {hd (Dec \<BB>\<^sub>F {u, v} u), hd (Dec \<BB>\<^sub>F {u, v} v)}"
PreFunctorFunctor: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" assumes "Functor F" shows "PreFunctor F"
map_of_SomeD: fixes xs :: "('e \<times> 'd) list"   and k :: "'e"   and y :: "'d" assumes "map_of xs k = Some y" shows "(k, y) \<in> set xs"
rel_mono: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R \<le> Ra" shows "list_all2 R \<le> list_all2 Ra"
conv_concat_2: fixes ss1 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss2 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" assumes "ss1 \<in> conv ars"   and "ss2 \<in> conv ars"   and "lst_conv ss1 = fst ss2" shows "lst_conv (fst ss1, snd ss1 @ snd ss2) = lst_conv ss2"
gtrE: fixes c :: "'f multiset"   and e :: "'g"   and c' :: "'h multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'h \<times> 'h multiset) set" assumes "(c, e, c') \<in> gtr T" obtains s :: "'f" and ce :: "'f multiset" and s' :: "'h" and ce' :: "'h multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> T"
cmp_rfn2: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g"   and "l \<le> s * r" shows "G \<guillemotright> cmp_rfn G r f s g l "
induced_def: shows "induced = (\<lambda>A\<in>Partition. THE b. \<exists>a\<in>A. b = \<alpha> a)"
nprv_exiE: fixes F :: "'fmla set"   and x :: "'var"   and \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "nprv F (exi x \<phi>)"   and "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "x \<in> var"   and "\<psi> \<in> fmla"   and "x \<notin> \<Union> (Fvars ` F)"   and "x \<notin> Fvars \<psi>" shows "nprv F \<psi>"
set_less_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" shows "ord.set_less less_eq A B = (ord.set_less_eq less_eq A B \<and> \<not> ord.set_less_eq less_eq B A)"
assms: shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w"
trail_def: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a" shows "H.trail a u p v \<equiv> H.awalk a u p v \<and> distinct p"
eqpoll_iff_card_of_ordIso: fixes A :: "'a set"   and B :: "'b set" shows "(A \<approx> B) = ordIso2 (card_of A) (card_of B)"
compl: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "'c set" assumes "Dynkin_system \<Omega> M"   and "A \<in> M" shows "\<Omega> - A \<in> M"
assms_1: shows "sKS m \<notin> specKeysSecrets P"
irreducible_dvd_imp_factor: fixes 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 a :: "'a"   and b :: "'a"   and G :: "'a multiset" assumes "class.ufd times one plus zero minus uminus"   and "dvd.dvd times a b"   and "comm_semiring_1.irreducible times one zero a"   and "comm_semiring_1.mset_factors times one zero G b" shows "\<exists>g\<in>#G. dvd.dvd times a g \<and> dvd.dvd times g a"
restrict_def: fixes S :: "'c set"   and s :: "'c \<Rightarrow> 'd" shows "restrict S s = (\<lambda>x. if x \<in> S then Some (s x) else None)"
summand_eq: fixes j :: "nat" assumes "j \<in> {0..<dim_vec Ts}" shows "row A i $ j * map_vec (\<lambda>T. lookup T is) Ts $ j = lookup (A $$ (i, j) \<cdot> Ts $ j) is"
simps: fixes a :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "(a \<in> oreachable A S U) = ((\<exists>s. a = s \<and> s \<in> init A) \<or> (\<exists>s aa s'. a = s' \<and> s \<in> oreachable A S U \<and> (s, aa, s') \<in> automaton.trans A \<and> S (fst s) (fst s') aa) \<or> (\<exists>s \<sigma>'. a = (\<sigma>', snd s) \<and> s \<in> oreachable A S U \<and> U (fst s) \<sigma>'))"
vec_nth_Basis: fixes x :: "(real, 'e) vec"   and i :: "'e" assumes "x \<in> Basis" shows "x $ i = (if x = axis i 1 then 1 else 0)"
functor_axioms: shows "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F"
inf_concat_step: fixes n :: "nat \<Rightarrow> nat"   and k :: "nat"   and i :: "nat"   and j :: "nat"   and m :: "nat" assumes "inf_concat n k = (i, j)"   and "Suc (j + m) = n i" shows "inf_concat n (k + Suc m) = (LEAST i'. i < i' \<and> 0 < n i', 0)"
causal_Write: shows "causal_program_history (Write volatile a sop A L R W # is\<^sub>s\<^sub>b) sb"
7: shows "(left`?`x \<rightarrow> right`!`x \<rightarrow> P) \<le> (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY)"
negate_wf_dfa: fixes A :: "nat bdd list \<times> bool list"   and l :: "nat" shows "wf_dfa (negate_dfa A) l = wf_dfa A l"
eval_const: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and c :: "real"   and x :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "eval (const_expansion c) x = c"
nop_v11_vrange: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_v11 A n f" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
left_commute: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "linordered_monoid.max b (linordered_monoid.max a c) = linordered_monoid.max a (linordered_monoid.max b c)"
cases: fixes a1 :: "'a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a"   and a2 :: "'a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" assumes "a1 \<lessapprox> a2" obtains ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" and ve' :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" and ds :: "'a \<aproc> set list" and ds' :: "'a \<aproc> set list" and proc :: "'a \<aproc>" and b :: "'a" where "a1 = (proc, ds, ve, b)" and "a2 = (proc, ds', ve', b)" and "ve \<lessapprox> ve'" and "ds \<lessapprox> ds'"
neg_divide_le_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less_eq (divide b c) a = less_eq (times a c) b"
sub1_preserves_dimensions_2: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub1 mf q k l (r, A) = (r', A')" shows "dim_col A' = dim_col A"
BrnFT_Int_BrnNFT_1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "length cl = length dl" shows "BrnFT cl dl \<inter> BrnNFT cl dl = {}"
same_cs_implies_same_resulting_cs: fixes c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and d :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration" assumes "cs c i = cs d i"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'"   and "regular_event ev" shows "cs c' i = cs d' i"
fst_set_fst_gb_red: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "fst ` set (fst (gb_red gs bs ps sps data)) = set (gb_red_aux (gs @ bs) sps)"
space_density: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> ennreal" shows "space (density M f) = space M"
ANDLEFT1_elim: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "x:M \<in> ANDLEFT1 (B AND C) (\<parallel>(B)\<parallel>)" obtains x' :: "name" and M' :: "trm" where "M = AndL1 x'.M' x" and "fin (AndL1 x'.M' x) x" and "x':M' \<in> \<parallel>(B)\<parallel>"
trancl_eq: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" assumes "wf_sub_rel subs" shows "subs\<^sup>+ = subs"
parallel_resource1_wiring_def: shows "parallel_resource1_wiring = swap_lassocr"
ok: shows "term_ok \<Theta> (Abs T (subst_bvs1' s 1 (map (\<lambda>(x, y). Fv x y) vs)))"
addfirststut: fixes s :: "nat \<Rightarrow> 'a" assumes "first s = second s" shows "s \<approx> tail s"
subspace_dim: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and X :: "'d set" assumes "vectorspace K V"   and "subspace K X V"   and "vectorspace.fin_dim K V"   and "vectorspace.fin_dim K (V\<lparr>carrier := X\<rparr>)" shows "vectorspace.dim K (V\<lparr>carrier := X\<rparr>) \<le> vectorspace.dim K V"
length_update_matchF_base: fixes I :: "nat"   and mr :: "\<I>"   and mrs :: "mregex"   and nt :: "mregex list"   and entry :: "'b option list set list"   and st :: "'a" shows "length (fst (update_matchF_base I mr mrs nt entry st)) = Suc 0"
cross3_simps_11: fixes a :: "'c"   and b :: "'c" shows "a * b = b * a"
returns_result_eq: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and y :: "'ar"   and y' :: "'ar" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
simps_4: fixes x51 :: "bexp"   and x52 :: "com"   and y51 :: "bexp"   and y52 :: "com" shows "(WHILE x51 DO x52 = WHILE y51 DO y52) = (x51 = y51 \<and> x52 = y52)"
empty_eff_exec_eq: fixes a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and s :: "('c, 'd) fmap" assumes "fmdom' (snd a) = {}" shows "state_succ s a = s"
eval_nembed_true: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "P s" shows "\<guillemotleft> \<N> P \<guillemotright> s = 0"
ss_1: shows "ss \<in> lists (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
prefixeq_antisym: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys"   and "prefix ys xs" shows "xs = ys"