hcju/isabelleps huggingface dataset:
inconsistent_fm: fixes S :: "('c, 'd) fm set"   and p :: "('c, 'd) fm" assumes "consistent S"   and "\<not> consistent ({p} \<union> S)" obtains S' :: "('c, 'd) fm list" where "set S' \<subseteq> S" and "p # S' \<turnstile> \<^bold>\<bottom>"
poly_mult_r_distr': fixes R :: "('c, 'd) ring_scheme"   and p1 :: "'c list"   and p2 :: "'c list"   and p3 :: "'c list" assumes "domain R"   and "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R"   and "set p3 \<subseteq> carrier R" shows "ring.poly_mult R p1 (ring.poly_add R p2 p3) = ring.poly_add R (ring.poly_mult R p1 p2) (ring.poly_mult R p1 p3)"
subset_interior_of_eq: fixes S :: "'c set"   and X :: "'c topology" shows "(S \<subseteq> X interior_of S) = openin X S"
indicator_sums: fixes A :: "nat \<Rightarrow> 'b set"   and x :: "'b" assumes "\<And>i j. i \<noteq> j \<Longrightarrow> A i \<inter> A j = {}" shows "(\<lambda>i. indicat_real (A i) x) sums indicat_real (\<Union> (range A)) x"
subint1I: fixes G :: "prog"   and I :: "qtname"   and i :: "iface"   and J :: "qtname" assumes "iface G I = Some i"   and "J \<in> set (isuperIfs i)" shows "(I, J) \<in> subint1 G"
m_comm: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "Group.comm_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"
infinite_countable_subset': fixes X :: "'c set" assumes "infinite X" shows "\<exists>C\<subseteq>X. countable C \<and> infinite C"
of_int_ceiling_le_add_one: fixes r :: "'b" shows "of_int \<lceil>r\<rceil> \<le> r + (1::'b)"
poly_subst_times: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f (p * q) = poly_subst f p * poly_subst f q"
Suc: fixes r :: "mregex"   and s :: "mregex" shows "RPDi i (MTimes r s) \<subseteq> {MTimes r s} \<union> MTimesL r (\<Union>j\<le>i. RPDi j s) \<union> (\<Union>j\<le>i. RPDi j r)"
symrun_interp_formula_stuttering: fixes \<gamma> :: "'a constr"   and \<Gamma> :: "'a constr list" assumes "\<gamma> \<in> set \<Gamma>" shows "\<lbrakk>\<lbrakk> \<gamma> # \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
expand_or_propag: fixes \<mu> :: "'b ltlr"   and \<eta> :: "'b ltlr"   and n_ns :: "'b node \<times> 'b node set" assumes "\<mu> or\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<inter> (old (fst n_ns) \<union> new (fst n_ns)) \<noteq> {}"   and "\<forall>nd\<in>snd n_ns. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {}" shows "expand n_ns \<le> SPEC (\<lambda>nm_nds. \<forall>nd\<in>snd nm_nds. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {})"
S_inductive_Sup_in_ChainTr: fixes X :: "'a set"   and c :: "'a" assumes "S_inductive_set D"   and "Chain D X"   and "c \<in> carrier (Iod D (insert (Algebra2.Sup D X) X))"   and "Algebra2.Sup D X \<notin> X"   and "\<forall>y\<in>carrier (Iod D (insert (Algebra2.Sup D X) X)). c \<prec>\<^bsub>Iod D (insert (Algebra2.Sup D X) X)\<^esub> y \<longrightarrow> \<not> y \<prec>\<^bsub>Iod D (insert (Algebra2.Sup D X) X)\<^esub> Algebra2.Sup D X" shows "c \<in> upper_bounds D X"
idom_tOp_nonzeros: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e" assumes "Idomain R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>"   and "y \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<noteq> \<zero>\<^bsub>R\<^esub>"
dickson_less_pD1: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "dickson_less_p d m p q" shows "p \<in> dgrad_p_set d m"
is_acc_def: fixes r :: "nat \<Rightarrow> 'Q" shows "is_acc r \<equiv> \<forall>A\<in>F. \<exists>\<^sub>\<infinity>i. r i \<in> A"
axiom_subst_typ': fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and A :: "Core.term"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "A \<in> axioms \<Theta>"   and "inst_ok \<Theta> insts" shows "\<Theta>,\<Gamma> \<turnstile> subst_typ' insts A"
g: shows "g \<in> M \<rightarrow>\<^sub>M N"
S_bij_betw_chamber0: fixes S :: "'b set" assumes "CoxeterComplex S" shows "bij_betw (\<lambda>s. \<langle>S - {s}\<rangle>) S (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) 0)"
filtercomap_bot: fixes f :: "'a \<Rightarrow> 'b" shows "filtercomap f bot = bot"
m1_def: shows "m1 \<equiv> x$ = Suc<$x> \<and> y$ = $y"
Liminf_upto_llist_infinity: fixes Xs :: "'a set llist" shows "Liminf_upto_llist Xs \<infinity> = Liminf_llist Xs"
countableI_bij1: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B"   and "countable A" shows "countable B"
inverse_arrows_def: fixes f :: "'a"   and g :: "'a" shows "A.inverse_arrows f g \<equiv> A.ide (g \<cdot>\<^sub>A f) \<and> A.ide (f \<cdot>\<^sub>A g)"
comm_neutral: fixes a :: "'d" shows "a + (0::'d) = a"
iso_runit: fixes f :: "'a arrow_of_spans_data" assumes "ide f" shows "local.iso \<r>[f]"
hyps: shows "{\<lambda>nl. nl = 0 \<up> n @ x # lm} [Dec n 2, Goto 0] {\<lambda>nl. nl = 0 # 0 \<up> n @ lm}"
word_bl_Rep': fixes x :: "'b word" shows "length (to_bl x) = LENGTH('b)"
dual_eqI: fixes x :: "'a dual"   and y :: "'a dual" assumes "undual x = undual y" shows "x = y"
zip_fun_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "f \<parallel> g \<equiv> \<lambda>x. (f x, g x)"
well_com_mat_O: shows "well_com (Utrans_P vars1 mat_O)"
\<psi>_def: shows "\<psi> = Matrix.vec N (\<lambda>x. complex_of_real (1 / sqrt (real N)))"
square_free_factorization_def: fixes p :: "'a poly"   and cbs :: "'a \<times> ('a poly \<times> nat) list" shows "square_free_factorization p cbs \<equiv> case cbs of (c, bs) \<Rightarrow> p = Polynomial.smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i) \<and> (p = 0 \<longrightarrow> c = (0::'a) \<and> bs = []) \<and> (\<forall>a i. (a, i) \<in> set bs \<longrightarrow> square_free a \<and> 0 < degree a) \<and> (\<forall>a i b j. (a, i) \<in> set bs \<longrightarrow> (b, j) \<in> set bs \<longrightarrow> (a, i) \<noteq> (b, j) \<longrightarrow> algebraic_semidom_class.coprime a b) \<and> distinct bs"
gen_check_filter: fixes as :: "nat list" shows "gen_check as = filter (suffs C as) (alls B as)"
n1: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> G" shows "nprv F \<psi>"
SC_completeness: fixes ps :: "(nat, nat) form list"   and p :: "(nat, nat) form" assumes "\<forall>e f g. list_all (eval e f g) ps \<longrightarrow> eval e f g p" shows "\<turnstile> p # map compl ps"
assms_3: shows "\<forall>\<^sub>F i in I. \<forall>x\<in>X. f i x \<le> h x"
surj_hom_induced_retraction_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and p :: "int" assumes "retraction_map X Y f" shows "carrier (homology_group p Y) = hom_induced p X {} Y {} f ` carrier (homology_group p X)"
stable_rank_bounded: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and x :: "nat"   and j :: "nat"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "semi_mojmir.stable_rank \<Sigma> \<delta> q\<^sub>0 w x j"   and "infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)"   and "q\<^sub>0 \<notin> F" shows "j < i"
paths_with_new_start_in_v0: fixes xs :: "'a list" assumes "xs \<in> paths_with_new" shows "hd xs = v0"
Lcm_upto_smallo: fixes c :: "real" assumes "exp 1 < c" shows "(\<lambda>n. real (Lcm {1..n})) \<in> o((^) c)"
SUP_const_minus_ennreal: fixes I :: "'a set"   and c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "(\<Squnion>x\<in>I. c - f x) = c - \<Sqinter> (f ` I)"
upds_map_Dummy: fixes l :: "Vars.var list" shows "upds (map Dummy l) = {}"
in_JF1rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d JF1"   and y :: "'e JF1" shows "JF1rel R x y = (\<exists>z. z \<in> JF1in {(x, y). R x y} \<and> JF1map fst z = x \<and> JF1map snd z = y)"
xs: shows "xs = <drop n lm> @ Bk \<up> rn"
nilCases': fixes Rs :: "residual"   and P :: "bool" assumes "\<zero> \<longmapsto> Rs" shows "P"
xa_sub_x_ya: shows "set_mset (X + Ya + (Y - Ya) - Y) \<subseteq> set_mset (X + Ya)"
sorted_merge: fixes xs :: "'c list"   and ys :: "'c list" shows "sorted (merge xs ys) = (sorted xs \<and> sorted ys)"
p: fixes k :: "nat" assumes "k < n" shows "positive (adjoint (M k) * (1\<^sub>m d - P k) * M k)"
ring_n_prod_primesTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)" shows "\<forall>j\<le>n. (\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (mL K P I) (Kb\<^bsub>K n P\<^esub>) n) = (\<nu>\<^bsub>K P j\<^esub>) (mIg\<^bsub> K n P I\<^esub>)"
diff_is_0_eq: fixes m :: "nat"   and n :: "nat" shows "(m - n = 0) = order.greater_eq n m"
upper_bound: shows "\<forall>A\<in>sets M. emeasure N A \<le> emeasure (density M f) A"
simps_1: fixes uu :: "'b extC" shows "(extC.Bot \<le> uu) = True"
finite_nat_bounded: fixes S :: "nat set" assumes "finite S" shows "\<exists>k. S \<subseteq> {..<k}"
perm_cols_transpose: fixes M :: "('c, 'd) sq_matrix"   and p :: "'d \<Rightarrow> 'd" shows "perm_cols (Square_Matrix.transpose M) p = Square_Matrix.transpose (perm_rows M p)"
mem_equal_annul: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32"   and new_state_var :: "sparc_state_var"   and new_cpu_reg2 :: "CPU_register \<Rightarrow> word32"   and new_state_var2 :: "sparc_state_var" assumes "mem_equal s1 s2 pa" shows "mem_equal (s1\<lparr>state_var := new_state_var, cpu_reg := new_cpu_reg\<rparr>) (s2\<lparr>state_var := new_state_var2, cpu_reg := new_cpu_reg2\<rparr>) pa"
i_th_strict1: fixes i :: "nat discr Box" shows "\<bottom> !! i = \<bottom>"
Abs_stone_phi_pair_inverse: fixes y :: "'c regular \<times> 'c dense Filters.filter" assumes "y \<in> stone_phi.pairs" shows "Rep_stone_phi_pair (Abs_stone_phi_pair y) = y"
Picard_step_def: fixes X0 :: "'c set"   and t0 :: "'d"   and h :: "'d"   and X :: "'c set" shows "Picard_step X0 t0 h X = SPEC (\<lambda>R. case R of None \<Rightarrow> True | Some R \<Rightarrow> nonempty R \<and> compact R \<and> R \<subseteq> Csafe \<and> (\<forall>x0\<in>X0. \<forall>h'\<in>{t0..t0 + h}. \<forall>phi\<in>Abstract_Rigorous_Numerics.cfuncset t0 h' X. x0 + integral {t0..h'} (\<lambda>t. ode (phi t)) \<in> R))"
last_quotient: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" assumes "normal_series G \<GG>"   and "1 < length \<GG>" shows "last (normal_series.quotients G \<GG>) = G Mod \<GG> ! (length \<GG> - 1 - 1)"
max_input_list_cons: fixes a :: "vname gexp"   and G :: "vname gexp list" shows "max_input_list (a # G) = max (GExp.max_input a) (max_input_list G)"
bind_eq_return_pmf_None: fixes p :: "'i spmf"   and f :: "'i \<Rightarrow> 'h spmf" shows "(p \<bind> f = return_pmf None) = (\<forall>x\<in>set_spmf p. f x = return_pmf None)"
Image_Int_subset: fixes R :: "('d \<times> 'c) set"   and A :: "'d set"   and B :: "'d set" shows "R `` (A \<inter> B) \<subseteq> R `` A \<inter> R `` B"
mult_strict_left_mono_neg: fixes 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 abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less b a"   and "less c zero" shows "less (times c a) (times c b)"
polyfun_matrix_deep_model: fixes i :: "nat"   and j :: "nat" assumes "i < last rs ^ N_half"   and "j < last rs ^ N_half" shows "polyfun {..<weight_space_dim} (\<lambda>f. A' f $$ (i, j))"
pt_same: shows "\<forall>pt. pt \<notin> set_of lt \<longrightarrow> low pt = lowa pt \<and> high pt = higha pt"
update_convs_3: fixes lim' :: "nat \<Rightarrow> nat"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'a" shows "lim_update lim' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays, refs = refs, lim = lim' lim, \<dots> = more\<rparr>"
yun_monic_factorization: fixes f :: "'a poly" assumes "monic f" shows "yun_gcd.yun_monic_factorization gcd (map_poly hom f) = map (\<lambda>(f, i). (map_poly hom f, i)) (yun_gcd.yun_monic_factorization gcd f)"
H1: shows "f = {(tr, ref). tr = [] \<and> ref \<inter> ev ` A = {}} \<union> {(tr, ref). tr \<noteq> [] \<and> hd tr \<in> ev ` A \<and> (\<exists>a. ev a = hd tr \<and> (tl tr, ref) \<in> F (P a))}"
ev_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.ev \<phi> \<equiv> Shallow.until Shallow.tr \<phi>"
less_le_not_le: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
rep_eq: fixes x :: "'a"   and xa :: "'a antichain" shows "(x \<in>\<^sub>A xa) = (x \<in> set_antichain xa)"
devExt_same: fixes n' :: "nat"   and n'' :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n\<^sub>s :: "nat"   and n''' :: "nat" assumes "\<forall>n'''>n'. n''' \<le> n'' \<longrightarrow> Option.is_none (devBC t n nid n''')"   and "n\<^sub>s \<le> n'"   and "n''' \<le> n''"   and "n' \<le> n'''" shows "devExt t n nid n\<^sub>s (n''' - n\<^sub>s) = devExt t n nid n\<^sub>s (n' - n\<^sub>s)"
vseq_implies_hpar_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.seq \<nu> \<mu>" shows "src\<^sub>C \<nu> = src\<^sub>C \<mu>"
ih_16: shows "getSATFlag state' = UNDEF \<longrightarrow> InvariantEquivalentZL (getF state') (getM state') F0'"
simps: fixes i :: "nat"   and l :: "nat"   and gsa :: "int vec iarray"   and dmusa :: "int iarray iarray"   and acc :: "int vec" shows "gso_array.while i l gsa dmusa acc = (if i \<le> l then acc else let v = dmusa !! l !! l \<cdot>\<^sub>v acc - dmusa !! i !! l \<cdot>\<^sub>v gsa !! l in Let (map_vec (\<lambda>k. k div dmusa !! (l - 1) !! (l - 1)) v) (gso_array.while i (l + 1) gsa dmusa))"
simps_2: fixes x :: "'b equation" shows "negative_literal (Neg x) = True"
mem: shows "smult (inverse (x - x')) [:- x', 1::'a:] \<in> set (map (\<lambda>xi. smult (inverse (x - xi)) [:- xi, 1::'a:]) (filter (\<lambda>c. c \<noteq> x) (map fst xs_ys)))"
ctr: fixes xs :: "'d list" shows "cycle xs = hd xs ## cycle (tl xs @ [hd xs])"
shunt2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> - y \<le> z) = (x \<le> y \<squnion> z)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "total_quasi_ordered_set A less_eq"   and "antisymmetric A less_eq" shows "total_ordered_set A less_eq"
simps_3: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and signalsInit2 :: "signal \<Rightarrow> nat"   and signalsResp2 :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "l2_state.more \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, chan = chan, bad = bad, \<dots> = more\<rparr> = more"
nths_count: fixes xs :: "'c list"   and A :: "nat set"   and x :: "'c" shows "count (mset (nths xs A)) x = count (mset xs) x - card {i. i < length xs \<and> i \<notin> A \<and> xs ! i = x}"
swapInp_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" shows "inp %[x \<and> y]_xs \<equiv> lift (swap xs x y) inp"
test_absorb_4: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot" shows "inf (sup x y) x = x"
len_lexext_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> len_lexext gt xs xs"
natural_transformation_axioms: shows "natural_transformation J.COMP (\<cdot>) mkCone_\<chi>.A.map local.map (mkCone (\<chi> (J.MkIde False)) (\<chi> (J.MkIde True)))"
flist_def: fixes xs :: "'a fset" shows "flist xs = sorted_list_of_set \<langle>xs\<rangle>\<^sub>f"
tabulate2_def: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "tabulate2 m n f = IArray.of_fun (\<lambda>i. IArray.of_fun (f i) n) m"
poly_deg_def: fixes p :: "('b \<Rightarrow>\<^sub>0 'a) \<Rightarrow>\<^sub>0 'c" shows "poly_deg p = (if keys p = {} then 0::'a else Max (deg_pm ` keys p))"
marker_must_be_delivered_2_trace: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace c t c'"   and "Marker \<in> set (msgs c i)"   and "Marker \<notin> set (msgs c' i)"   and "channel i = Some (p, q)" shows "\<exists>ev\<in>set t. \<exists>p q. ev = RecvMarker i p q"
rule_sub_nec: fixes \<phi> :: "\<o> \<Rightarrow> \<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and \<Theta> :: "bool \<Rightarrow> bool"   and v :: "i" assumes "Substable Substable_Cond \<phi>"   and "\<And>v. [\<psi> \<^bold>\<equiv> \<chi> in v]"   and "\<Theta> [\<phi> \<psi> in v]" shows "\<Theta> [\<phi> \<chi> in v]"
cgenideal_self: fixes R :: "('e, 'f) ring_scheme"   and i :: "'e" assumes "ring R"   and "i \<in> carrier R" shows "i \<in> PIdl\<^bsub>R\<^esub> i"
K_bisim_unique: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and \<mu> :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and x :: "'s \<times> ('c \<Rightarrow> real)"   and x' :: "'s \<times> ('c \<Rightarrow> real)" assumes "s \<in> S"   and "\<mu> \<in> K s"   and "x \<in> set_pmf \<mu>"   and "x' \<in> set_pmf \<mu>"   and "x ~ x'" shows "x = x'"
Adj_def: shows "Adj = (\<lambda>p i. if 0 < i then cfn p (\<theta> p i) - PC p (te p i) else 0)"
DynT_mheadsE: fixes G :: "prog"   and mode :: "inv_mode"   and invC :: "qtname"   and statT :: "ref_ty"   and C :: "qtname"   and L :: "lenv"   and e :: "expr"   and statDeclT :: "ref_ty"   and sm :: "mhead"   and sig :: "sig"   and s :: "st"   and a' :: "val"   and declC :: "qtname" assumes "G\<turnstile>mode\<rightarrow>invC\<preceq>statT"   and "wf_prog G"   and "\<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>e\<Colon>-RefT statT"   and "(statDeclT, sm) \<in> mheads G C statT sig"   and "mode = invmode sm e"   and "invC = invocation_class mode s a' statT"   and "declC = invocation_declclass G mode s a' statT sig" obtains dm :: "qtname \<times> methd" where "methd G declC sig = Some dm" and "dynlookup G statT invC sig = Some dm" and "G\<turnstile>resTy (mthd dm)\<preceq>resTy sm" and "wf_mdecl G declC (sig, mthd dm)" and "declC = declclass dm" and "is_static dm = is_static sm" and "is_class G invC" and "is_class G declC" and "G\<turnstile>invC\<preceq>\<^sub>C declC" and "if invmode sm e = IntVir then \<forall>statC. statT = ClassT statC \<longrightarrow> G\<turnstile>invC\<preceq>\<^sub>C statC else ((\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC) \<or> (\<forall>statC. statT \<noteq> ClassT statC \<and> declC = Object)) \<and> statDeclT = ClassT (declclass dm)"
map_2: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b msllist" shows "mslmap f (MSLCons x21 x22) = MSLCons (f x21) (mslmap f x22)"
simps_4: fixes uu :: "nat"   and v :: "name" shows "wellformed' uu (Const v) = True"
legs_form_cospan_1: shows "C.cospan cods.\<nu>.leg0 cods.\<pi>.leg1"
res_def: fixes S :: "'b literal set set" shows "res S = (\<Union>C1\<in>S. \<Union>C2\<in>S. \<Union>L1\<in>C1. \<Union>L2\<in>C2. case (L1, L2) of (i\<^sup>+, a\<^sup>+) \<Rightarrow> {} | (i\<^sup>+, j\<inverse>) \<Rightarrow> if i = j then {C1 - {i\<^sup>+} \<union> (C2 - {j\<inverse>})} else {} | (aa\<inverse>, b) \<Rightarrow> {})"
pdivides_iff_shell: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "p pdivides q = p divides\<^bsub>K [X]\<^esub> q"
\<gamma>_def: shows "\<gamma> = Infer {#(C + {#L#}) \<cdot> \<mu>#} ((D + {#L'#}) \<cdot> \<sigma> \<cdot> \<mu>) (C \<cdot> \<mu>)"
A: fixes v :: "'a" shows "{i. i < Suc (length xs) \<and> i \<notin> A \<and> (v # xs) ! i = x} = {i. i < Suc (length xs) \<and> i \<notin> A \<and> (v # xs) ! i = x} \<inter> {0} \<union> {i. i < Suc (length xs) \<and> i \<notin> A \<and> (v # xs) ! i = x} \<inter> {i. \<exists>j. i = Suc j}"
simps_1: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "inst \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = inst"
analz_crypt_out: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invK K) \<notin> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
mkIde_as_mkArr: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>" shows "mkArr A A (\<lambda>x. x) = mkIde A"
assms_4: shows "ns1 O ns1 \<subseteq> ns1"
frag_cmul_cmul: fixes c :: "int"   and d :: "int"   and x :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul c (frag_cmul d x) = frag_cmul (c * d) x"
nth_zip: fixes i :: "nat"   and xs :: "'d list"   and ys :: "'e list" assumes "i < length xs"   and "i < length ys" shows "zip xs ys ! i = (xs ! i, ys ! i)"
restrict_self_UNIV: fixes m :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "m |` (dom m - S) = m |` (UNIV - S)"
Rep_vector: fixes x :: "'c vector" shows "Rep_vector x \<in> vectors"
diff_add_mset_swap: fixes b :: "'d"   and A :: "'d multiset"   and M :: "'d multiset" assumes "b \<notin># A" shows "add_mset b M - A = add_mset b (M - A)"
nl32: fixes l' :: "real"   and r' :: "real"   and lr :: "real"   and r :: "real" assumes "1 \<le> l'"   and "1 \<le> r'"   and "1 \<le> lr"   and "1 \<le> r" shows "4 * (l' + r') * (lr + r) * (lr + r' + r) \<le> (l' + lr + r' + r) ^ 3"
listset_doubletonE: fixes ys :: "'b list"   and A :: "'b set"   and B :: "'b set" assumes "ys \<in> listset [A, B]" obtains a :: "'b" and b :: "'b" where "a \<in> A" and "b \<in> B" and "ys = [a, b]"
par_distrib_axioms: shows "par_distrib (\<parallel>) skip"
IH2: assumes "Vagree \<nu> \<nu>' (FVDiff t2)" shows "frechet I t2 (fst \<nu>) (snd \<nu>) = frechet I t2 (fst \<nu>') (snd \<nu>')"
rExt: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi> \<otimes> \<Psi>', P, Q) \<in> X \<or> \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q"