hcju/isabelleps huggingface dataset:
INF_eq_const: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and I :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and x :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Inf (f ` I) = x"
hmemrec: fixes G :: "(hf \<Rightarrow> 'a) \<Rightarrow> hf \<Rightarrow> 'a"   and a :: "hf" shows "hmemrec G a = G (ecut (hmemrec G) a) a"
special_soundness_def: shows "\<Sigma>_OR.special_soundness \<equiv> \<forall>h e e' a z z'. h \<in> valid_pub_OR \<longrightarrow> e \<in> challenge_space \<longrightarrow> e' \<in> challenge_space \<longrightarrow> e \<noteq> e' \<longrightarrow> check_OR h a e z \<longrightarrow> check_OR h a e' z' \<longrightarrow> lossless_spmf (\<A>ss_OR h (a, e, z) (a, e', z')) \<and> (\<forall>w'\<in>set_spmf (\<A>ss_OR h (a, e, z) (a, e', z')). (h, w') \<in> Rel_OR)"
P: shows "sets (tree_sigma M \<Otimes>\<^sub>M tree_sigma M) = sets (sigma (space (tree_sigma M) \<times> space (tree_sigma M)) {A \<times> B |A B. A \<in> trees_cyl ` trees (sets M) \<and> B \<in> trees_cyl ` trees (sets M)})"
I2: shows "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
completeD: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "complete A r"   and "x \<in> A"   and "y \<in> A"   and "x \<noteq> y" shows "x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
old: fixes q :: "int poly"   and ri :: "root_info" assumes "(q, ri) \<in> set old" shows "root_info.l_r ri l r \<noteq> 0"
sh: shows "sh \<unlhd>\<^sub>s sh(C \<mapsto> (sfs, Error))"
R\<^sub>C_of_simp: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and c\<^sub>1\<^sub>C :: "'Com\<^sub>C"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>1\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>2\<^sub>C :: "'Com\<^sub>C"   and mem\<^sub>2\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val" assumes "secure_refinement \<R>\<^sub>A \<R> P" shows "((\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> R\<^sub>C_of \<R>\<^sub>A \<R> P) = ((\<exists>c\<^sub>1\<^sub>A c\<^sub>2\<^sub>A. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<and> (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C\<rangle>\<^sub>A) \<in> \<R>\<^sub>A) \<and> conc.low_mds_eq mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C \<and> (\<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>2\<^sub>C, mds\<^sub>C, mem\<^sub>2\<^sub>C\<rangle>\<^sub>C) \<in> P)"
h_3: shows "nonstop (code tp) (bl2wc (<lm>)) stp = 0"
multiplicity_prod_prime_powers: fixes S :: "'a set"   and p :: "'a"   and f :: "'a \<Rightarrow> nat" assumes "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> prime x"   and "prime p" shows "multiplicity p (\<Prod>p\<in>S. p ^ f p) = (if p \<in> S then f p else 0)"
igWlsAbsIsInBar_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igWlsAbsIsInBar MOD \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> isInBar (us, s)"
Bseq_bdd_above: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X" shows "bdd_above (range X)"
jensens_inequality: fixes M :: "'a pmf"   and X :: "'a \<Rightarrow> real"   and I :: "real set"   and a :: "real"   and b :: "real"   and q :: "real \<Rightarrow> real" assumes "integrable (measure_pmf M) X"   and "AE x in measure_pmf M. X x \<in> I"   and "I = {a<..<b} \<or> I = {a<..} \<or> I = {..<b} \<or> I = UNIV"   and "integrable (measure_pmf M) (\<lambda>x. q (X x))"   and "convex_on I q" shows "q (measure_pmf.expectation M X) \<le> measure_pmf.expectation M (\<lambda>x. q (X x))"
subring_props_1: fixes K :: "'a set" assumes "subfield K R" shows "K \<subseteq> carrier R"
iT_Mult_iWeakSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "0 < k" shows "(P t1. t1 \<B> t2 I \<otimes> k. Q t2) = (P (t1 * k). t1 \<B> t2 I. Q (t2 * k))"
prod_ineqs1: fixes m :: "nat"   and k :: "nat" assumes "0 < m * k" shows "0 < m \<and> 0 < k"
length_list_update: fixes xs :: "'c list"   and i :: "nat"   and x :: "'c" shows "length (xs[i := x]) = length xs"
left_commute: fixes b :: "'e"   and a :: "'e"   and c :: "'e" shows "b \<cdot> (a \<cdot> c) = a \<cdot> (b \<cdot> c)"
sign_changes_Cons_Cons_same: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "(0::'a) < x * y" shows "sign_changes (x # y # xs) = sign_changes (y # xs)"
assms_1: fixes s :: "'a set" shows "r \<in> g s"
null_sets_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "\<Omega> \<in> sets M" shows "(A \<in> null_sets (restrict_space M \<Omega>)) = (A \<subseteq> \<Omega> \<and> A \<in> null_sets M)"
quotient_web_sel_2: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "weight (quotient_web \<Gamma> f) x = (if x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<or> x \<in> TER\<^bsub>\<Gamma>\<^esub> f \<inter> B \<Gamma> then 0 else weight \<Gamma> x)"
rel_sel: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a list"   and b :: "'b list" shows "list_all2 R a b = ((a = []) = (b = []) \<and> (a \<noteq> [] \<longrightarrow> b \<noteq> [] \<longrightarrow> R (lead_coeff a) (lead_coeff b) \<and> list_all2 R (tl a) (tl b)))"
lagrange: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H__\<rparr>)" shows "card (rcosets\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> H) * card H = order (G\<lparr>carrier := H__\<rparr>)"
replication_number_value: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B> rep x = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
update_convs_2: fixes M' :: "nat \<Rightarrow> nat"   and I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "M_update M' \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = \<lparr>vars.I = I, M = M' M, N = N, S = S, \<dots> = more\<rparr>"
LFP_fixed_point: fixes f :: "'a \<Rightarrow> 'a" assumes "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "LFP\<^bsub>L\<^esub> f \<in> fps L f"
eq_iff: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (le x y \<and> le y x)"
AuthReq: fixes evsAReq :: "event list"   and KM :: "nat"   and Transaction :: "msg"   and M :: "agent"   and C :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and HOD :: "msg"   and OIData :: "msg"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and Chall_M :: "nat"   and k :: "nat"   and P_I :: "msg"   and HPIData :: "msg"   and encPANData :: "msg"   and P :: "agent"   and EKj :: "nat" assumes "evsAReq \<in> set_pur"   and "Key KM \<notin> used evsAReq"   and "KM \<in> symKeys"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "HOD = Hash \<lbrace>Number OrderDesc, Number PurchAmt\<rbrace>"   and "OIData = \<lbrace>Number LID_M, Number XID, Nonce Chall_C, HOD, Nonce Chall_M\<rbrace>"   and "CardSecret k \<noteq> 0 \<longrightarrow> P_I = \<lbrace>sign (priSK C) \<lbrace>HPIData, Hash OIData\<rbrace>, encPANData\<rbrace>"   and "Gets M \<lbrace>P_I, OIData, HPIData\<rbrace> \<in> set evsAReq"   and "Says M C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Nonce Chall_M, cert P EKj onlyEnc (priSK RCA)\<rbrace>) \<in> set evsAReq"   and "Notes M \<lbrace>Number LID_M, Agent P, Transaction\<rbrace> \<in> set evsAReq" shows "Says M P (EncB (priSK M) KM (pubEK P) \<lbrace>Number LID_M, Number XID, Hash OIData, HOD\<rbrace> P_I) # evsAReq \<in> set_pur"
arr_MkArr: fixes A :: "'a E.term"   and B :: "'a E.term"   and f :: "'a" assumes "A \<in> IDE"   and "B \<in> IDE"   and "f \<in> HOM A B" shows "arr (MkArr A B f)"
lexordp_append_leftD: fixes xs :: "'a list"   and us :: "'a list"   and vs :: "'a list" assumes "lexordp (xs @ us) (xs @ vs)"   and "\<forall>a. \<not> a < a" shows "lexordp us vs"
filter_join_False: fixes A :: "'a set"   and m :: "('a, 'b) mapping" assumes "finite A" shows "filter_join False A m = Finite_Set.fold Mapping.delete m A"
least_unique: fixes k :: "'b"   and A :: "'b set" assumes "k \<in> A"   and "k \<le> least A" shows "k = least A"
mono: shows "monotone (\<le>) ord (\<lambda>x. f x (eSuc x))"
left_neutral: fixes a :: "'d" shows "(1::'d) * a = a"
core_case: assumes "c = 3"   and "a * \<^bold>|rev x\<^bold>| \<le> b * \<^bold>|rev y\<^bold>|"   and "rev y \<noteq> \<epsilon>"   and "rev x \<noteq> \<epsilon>"   and "b * \<^bold>|rev y\<^bold>| < \<^bold>|rev z\<^bold>| + \<^bold>|rev y\<^bold>|"   and "a * \<^bold>|rev x\<^bold>| < \<^bold>|rev z\<^bold>| + \<^bold>|rev x\<^bold>|" shows "rev y \<cdot> rev x = rev x \<cdot> rev y"
simps: fixes x :: "'a extT" shows "- x = (if x = extT.Val bot then extT.Top else extT.Val bot)"
le_sup_iff: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x + y \<le> z) = (x \<le> z \<and> y \<le> z)"
add_zero: fixes a :: "'a" assumes "a \<in> S" shows "z \<oplus>\<^sub>o\<^sub>w a = a"
f1: shows "(x\<^sup>\<infinity>)\<^sup>\<star> \<cdot> x\<^sup>\<star> = x\<^sup>\<infinity>"
subgroup_indicator_pos: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" assumes "finite_comm_group G"   and "subgroup H G"   and "a \<in> carrier G" shows "0 < subgroup_indicator G H a"
Quotient3_rep_reflp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and a :: "'f" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep a)"
ex1_Collect_singleton: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<exists>!x. P x"   and "P x" obtains "Collect P = {x}"
dlex_fun_strict_def: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "dlex_fun_strict s t = strict dlex_fun s t"
cs_path_swap_le: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.eq_up_to \<pi> n \<pi>'"   and "k \<le> n" shows "IFC_def.cs prob \<pi> k = IFC_def.cs prob \<pi>' k"
monom_zero: shows "MPoly_Type.monom 0 (0::'a) = 0"
simps_3: fixes d :: "nat" shows "intv_const (Greater d) = d"
l: shows "trace (A + B) = (\<Sum>i = 0..<n. A $$ (i, i)) + (\<Sum>i = 0..<n. B $$ (i, i))"
not_coprimeI: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b"   and "\<not> dvd.dvd times c one" shows "\<not> algebraic_semidom.coprime times one a b"
independent_length_le_dimension: fixes n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "dimension n K E"   and "local.independent K Us"   and "set Us \<subseteq> E" shows "length Us \<le> n"
coeff_Poly: fixes xs :: "'b list"   and i :: "nat" shows "coeff (Poly xs) i = nth_default (0::'b) xs i"
bidirected_digraphI: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd" assumes "wf_digraph G"   and "\<And>a. a \<notin> arcs G \<Longrightarrow> arev a = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> arev a \<noteq> a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> arev (arev a) = a"   and "\<And>a. a \<in> arcs G \<Longrightarrow> tail G (arev a) = head G a" shows "bidirected_digraph G arev"
aff_dim_hyperplane: fixes a :: "'b"   and r :: "real" assumes "a \<noteq> (0::'b)" shows "aff_dim {x. a \<bullet> x = r} = int (DIM('b) - 1)"
poly_prod_list: fixes ps :: "'a poly list"   and x :: "'a" shows "poly (prod_list ps) x = (\<Prod>p\<leftarrow>ps. poly p x)"
commute: fixes a :: "'m"   and b :: "'m" shows "sup a b = sup b a"
ltype_simps_1: fixes cf :: "CAttId"   and a :: "nat" shows "ltype (objLoc cf a) = rtype (att cf)"
assms_2: shows "\<forall>k>n. k < n' \<longrightarrow> \<not> P k"
existence_ivl_reverse: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "t0 \<in> existence_ivl t (flow t0 x0 t)"
eq1: shows "(\<Sum>hU\<in>set ps - set (ps\<^sub>+). Hilbert_fun (cone hU) d) = card {h. (h, {}) \<in> set ps \<and> poly_deg h = d}"
unshared: assumes "s = False" shows "insert_weights s m2 (\<lambda>i. w1 (i + count_weights s m1)) = insert_weights s m2 (\<lambda>i. w2 (i + count_weights s m1))"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_uminus" shows "vrat_uminus\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vrat_uminus"
fdemodalisation22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(\<langle>x| y \<le> r z) = (r y \<cdot> (x \<cdot> ar z) = (zero_class.zero::'a))"
cong_iff_lin: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and m :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = (\<exists>k. b = plus a (times m k))"
frequently_at_top: fixes P :: "'b \<Rightarrow> bool" shows "frequently P at_top = (\<forall>t0. \<exists>t>t0. P t)"
tm_wf_change_termi: fixes tp :: "(action \<times> nat) list" assumes "tm_wf0 tp" shows "list_all (\<lambda>(acn, st). st \<le> Suc (length tp div 2)) (adjust0 tp)"
Max_torder_above_iff: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" assumes "torder_on A r"   and "finite B"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "r x (Max_torder r B) = (\<exists>a\<in>B. r x a)"
pareto_losersE: fixes x :: "'alt pmf" assumes "x \<in> pareto_losers (SD \<circ> R)" obtains y :: "'alt pmf" where "y \<in> lotteries_on alts" and "x \<prec>[Pareto (SD \<circ> R)] y"
isomorphic_to_zero_right: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd" assumes "FGModule G smult V"   and "\<exists>T. FGModuleIso G smult V smult' T 0" shows "V = 0"
Cons: fixes x :: "'a"   and y :: "'b"   and z :: "'c"   and xs :: "'a list"   and ys :: "'b list"   and zs :: "'c list" assumes "P x y z"   and "Q xs ys zs"   and "list_all3 P xs ys zs" shows "Q (x # xs) (y # ys) (z # zs)"
absorb1: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) = a"
nullable_nderivs: fixes r :: "'a rexp"   and w :: "'a list" shows "nullable (foldl (\<lambda>r a. nderiv a r) r w) = (w \<in> lang r)"
reflpI: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
mono2: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> Y"   and "a \<sqsubseteq> b"   and "a \<in> Y"   and "b \<in> Y" shows "f x a \<le> f x b"
simps_12: fixes f :: "'b \<Rightarrow> 'a" shows "mslmap f MSLNil = MSLNil"
simps_9: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Until \<phi> I \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
rep_eq: fixes x :: "('a, 'b) IEEE.float" shows "fraction x = (case Rep_float x of (uu_, uua_, xa) \<Rightarrow> unat xa)"
a: shows "\<forall>x\<in>S. \<exists>y. x = f y"
normalized_tableau_preprocess': fixes cs :: "('a \<times> QDelta ns_constraint) list" shows "\<triangle> (Tableau (preprocess' cs (start_fresh_variable cs)))"
normalize_non_0_smult: fixes p :: "'a poly" shows "\<exists>a. a \<noteq> (0::'a) \<and> Polynomial.smult a (primitive_part p) = p"
ksimplex: shows "kuhn_simplex p n b_s u_s s"
that: assumes "prefix xs ys"   and "xs \<noteq> ys" shows "thesis"
listIntersectionWithSet: fixes l :: "'b list"   and X :: "'b set" shows "set (takeAll (\<lambda>x. x \<in> X) l) \<subseteq> X \<inter> set l"
ncols_notzero: fixes A :: "'b matrix"   and m :: "nat"   and n :: "nat" assumes "Rep_matrix A m n \<noteq> (0::'b)" shows "n < ncols A"
of_nat_less_top: fixes i :: "nat" shows "of_nat i < top"
design_defs_4: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and H :: "('c \<times> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<times> 'c \<Rightarrow> bool" shows "P is H healthy \<equiv> P = H P"
simps_3: fixes v :: "'b val" shows "flat (val.Left v) = flat v"
path_connected_punctured_ball: fixes a :: "'a"   and r :: "real" assumes "2 \<le> DIM('a)" shows "path_connected (ball a r - {a})"
disjoint: shows "{0..<Suc k} \<inter> {Suc k..<n} = {}"
bot_unique: fixes a :: "'a" shows "(a \<le> 0) = (a = 0)"
inject_2: fixes x21 :: "tf"   and x22 :: "host"   and y21 :: "tf"   and y22 :: "host" shows "(Func_host x21 x22 = Func_host y21 y22) = (x21 = y21 \<and> x22 = y22)"
induct: fixes x1 :: "nat \<Rightarrow> 'n"   and x2 :: "nat"   and P :: "(nat \<Rightarrow> 'n) \<Rightarrow> nat \<Rightarrow> bool" assumes "(x1, x2) \<in> scp"   and "\<And>h \<sigma> n. \<lbrakk>h \<in> local.hvars; h \<in> local.reads (path \<sigma> n); \<forall>k<n. h \<notin> local.writes (path \<sigma> k)\<rbrakk> \<Longrightarrow> P (path \<sigma>) n"   and "\<And>\<pi> m n. \<lbrakk>(\<pi>, m) \<in> scp; P \<pi> m; n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m\<rbrakk> \<Longrightarrow> P \<pi> n"   and "\<And>\<pi> m n v. \<lbrakk>(\<pi>, m) \<in> scp; P \<pi> m; n dd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m\<rbrakk> \<Longrightarrow> P \<pi> n"   and "\<And>\<pi> m \<pi>' m' n v. \<lbrakk>(\<pi>, m) \<in> scp; P \<pi> m; (\<pi>', m') \<in> scp; P \<pi>' m'; n dcd\<^bsup>\<pi>,v\<^esup>\<rightarrow> m via \<pi>' m'\<rbrakk> \<Longrightarrow> P \<pi> n" shows "P x1 x2"
simps_3: shows "ternary_to_bool TernaryUnknown = None"
fresh_heap_expr: fixes a :: "atom"   and \<Gamma> :: "('a \<times> 'b) list"   and x :: "'a"   and e :: "'b" assumes "a \<sharp> \<Gamma>"   and "(x, e) \<in> set \<Gamma>" shows "a \<sharp> e"
pd_node1: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and y :: "'a"   and x :: "'a" assumes "IFC prob"   and "IFC_def.is_pd prob y x" shows "y \<in> IFC_def.nodes prob"
filterlim_def: fixes f :: "'b \<Rightarrow> 'c"   and F2 :: "'c filter"   and F1 :: "'b filter" shows "filterlim f F2 F1 = (filtermap f F1 \<le> F2)"
not_comp_hd_imp_Var: fixes \<xi> :: "('s, 'v) hd"   and \<zeta> :: "('s, 'v) hd" assumes "\<not> \<xi> \<le>\<ge>\<^sub>h\<^sub>d \<zeta>" shows "is_Var \<zeta> \<or> is_Var \<xi>"
residuated_conjr1: fixes y :: "'a" shows "residuated ((\<rhd>) y)"
orderE: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" obtains "a = a \<sqinter> b"
span_is_submodule: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and S :: "'i set" assumes "module R M"   and "S \<subseteq> carrier M" shows "LinearCombinations.submodule R (LinearCombinations.module.span R M S) M"
ht_distinct_ls_insls: fixes l :: "('a \<times> 'b) list list"   and ls :: "('a \<times> 'b) list" assumes "ht_distinct l" shows "ht_distinct (ls_insls ls l)"
prems: shows "\<not> is_final (steps0 (1, tp) A (Suc stp))"
prems_2: shows "wellorder_class.enumerate S (Suc i) < x"
is_merge_status_1: fixes a :: "status"   and st' :: "status" shows "is_failed (merge_status a st') = (is_failed a \<or> is_failed st')"
set_bset_to_set_bset: fixes A :: "'c set" assumes "|A| <o natLeq +c |UNIV|" shows "set_bset (the_inv set_bset A) = A"
bn_comp_snd_less: fixes r :: "nat"   and q :: "nat"   and n :: "nat"   and p :: "nat" assumes "r < q" shows "snd (bn_comp n p q r) < q"
simps_5: fixes t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "insert\<langle>t,s\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = insert\<langle>t \<cdot> \<theta>,s \<cdot> \<theta>\<rangle>"
bij_betw_implies_inj_on_and_card_eq: fixes B :: "'c set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "finite B"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "bij_betw f A B = (inj_on f A \<and> card A = card B)"
bi_unique_intersect_r: fixes T :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e set"   and a' :: "'f set"   and b :: "'e set"   and b' :: "'f set"   and xr :: "'f set" assumes "bi_unique T"   and "rel_set T a a'"   and "rel_set T b b'"   and "rel_set T (a \<inter> b) xr" shows "a' \<inter> b' = xr"
ACI_norm_lang: fixes n :: "nat"   and r :: "'b rexp" shows "lang n \<guillemotleft>r\<guillemotright> = lang n r"
star_inductr': fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
Red_F_def: fixes N :: "'a literal multiset set" shows "G.Red_F N = {C. \<exists>DD\<subseteq>N. DD \<TTurnstile>e {C} \<and> (\<forall>D\<in>DD. D < C)}"
1: shows "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k \<and> \<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
image_Collect_mem: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set" shows "{f x |x. x \<in> S} = f ` S"
foldr_option_def: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a option"   and xs :: "'b list"   and e :: "'a" shows "foldr_option f xs e = foldr (\<lambda>a b. Option.bind b (f a)) xs (Some e)"
pfxm_length_def: fixes prefix_match :: "'b prefix_match" shows "pfxm_length prefix_match = (case prefix_match of PrefixMatch x1 x2 \<Rightarrow> x2)"
flowsto_step: fixes A :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and S :: "real set"   and B :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and C :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and D :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and T :: "real set"   and E :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set"   and F :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "flowsto A S B C"   and "flowsto D T E F" shows "flowsto A (S \<union> {s + t |s t. s \<in> S \<and> t \<in> T}) (B \<union> E \<union> C \<inter> D) (C - D \<union> F)"
m_def: shows "m = (LEAST m. 0 < m \<and> fds_nth f m \<noteq> (0::'a))"