hcju/isabelleps huggingface dataset:
usubst_upd_comm: fixes x :: "'e \<Longrightarrow> 'f"   and y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'h \<Rightarrow> 'f"   and u :: "('e, 'h) uexpr"   and v :: "('g, 'h) uexpr" assumes "x \<bowtie> y" shows "\<sigma>(x \<mapsto>\<^sub>s u, y \<mapsto>\<^sub>s v) = \<sigma>(y \<mapsto>\<^sub>s v, x \<mapsto>\<^sub>s u)"
list_all_filter_eq: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "list_all P xs" shows "filter P xs = xs"
to_bl_numeral: fixes bin :: "num" shows "to_bl (numeral bin) = bin_to_bl LENGTH('b) (numeral bin)"
fsmult_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and v :: "'c"   and a :: "'b" assumes "FGModule G smult V"   and "v \<in> V" shows "aezfun_scalar_mult.fsmult smult a v \<in> V"
BetaNegOr: fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list"   and B :: "('a, 'b) form" assumes "\<turnstile> Neg A # G"   and "\<turnstile> Neg B # G" shows "\<turnstile> Neg (Or A B) # G"
cut_less_image: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and I :: "'a set"   and n :: "'a" assumes "SetInterval2.strict_mono_on f A"   and "I \<subseteq> A"   and "n \<in> A" shows "(f ` I) \<down>< f n = f ` I \<down>< n"
eqvtI_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and perm :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'" shows "perm \<bullet> P \<Longrightarrow>\<^sub>lperm \<bullet> u in perm \<bullet> P''\<rightarrow>perm \<bullet> a<perm \<bullet> x> \<prec> perm \<bullet> P'"
Gamma_fact: fixes n :: "nat" shows "Gamma ((1::'a) + of_nat n) = fact n"
CL_imp_ex_isLub: fixes S :: "'a set" assumes "S \<subseteq> A" shows "\<exists>L. isLub S cl L"
is_Inner_def: fixes vertex :: "'b vertex" shows "is_Inner vertex = (\<exists>x3. vertex = Inner x3)"
FAddSA_NoCycles: fixes SA :: "('d, 'e, 'f) seqauto"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and S :: "'d"   and F :: "('d, 'e, 'f) seqauto set" assumes "States SA \<inter> dom G = {}"   and "S \<in> dom G"   and "dom G = \<Union> (States ` F)"   and "NoCycles F G" shows "NoCycles (insert SA F) (G [f+] (S, SA))"
hyps_2: shows "w =\<^sub>F w\<^sub>1 @ w\<^sub>2"
get_get: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> ('k \<Rightarrow> 'a option) \<Rightarrow> 'm" shows "get (\<lambda>s. get (f s)) = get (\<lambda>s. f s s)"
disc_6: fixes x2 :: "nat" shows "\<not> is_HOLD (WAIT x2)"
measurable_Diff_null_set: fixes B :: "'a set"   and M :: "'a measure"   and A :: "'a set" assumes "B \<in> null_sets M" shows "(A - B \<in> fmeasurable M \<and> A \<in> sets M) = (A \<in> fmeasurable M)"
disc_2: fixes x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm" shows "is_App (App x21 x22)"
obj_def': fixes a :: "'a" shows "B.obj a = (B.arr a \<and> trg a = a)"
rp_bad_simps: fixes bad :: "bool"   and \<sigma> :: "'a \<Rightarrow> 'a option"   and x :: "'a" shows "rp_bad (bad, \<sigma>) x = (case \<sigma> x of None \<Rightarrow> spmf_of_set A \<bind> (\<lambda>y. if y \<in> ran \<sigma> then map_spmf (\<lambda>y'. (y', True, \<sigma>(x \<mapsto> y'))) (spmf_of_set (A - ran \<sigma>)) else return_spmf (y, bad, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, bad, \<sigma>))"
group_hom_axioms_def: fixes G :: "('f, 'g) monoid_scheme"   and H :: "('h, 'i) monoid_scheme"   and h :: "'f \<Rightarrow> 'h" shows "group_hom_axioms G H h \<equiv> h \<in> hom G H"
min_key_val_def: fixes ko :: "'o"   and xs :: "'x" shows "min_key_val ko xs = min_key_val_raw ko (list_of_oalist xs)"
T1_1: fixes w :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" shows "[\<lparr>F,x\<rparr> in w] = (\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.ex1 r w)"
det_code_fst_sel_fun: fixes A :: "'a mat" shows "det A = det_code fst_sel_fun trivial_mute_fun A"
inject_1: fixes x8 :: "'a"   and y8 :: "'a" shows "(\<langle>x8\<rangle>\<^sub>\<T> = \<langle>y8\<rangle>\<^sub>\<T>) = (x8 = y8)"
ModuleHom_axioms: shows "ModuleHom (\<cdot>) V (\<cdot>) T"
assms_2: shows "\<forall>s\<in>S. T (Abs_freeletter s) = f s"
assms_3: fixes x :: "'a" assumes "x \<in> I" shows "(0::'b) \<le> f x"
rep_eq: fixes x :: "uint32" shows "lsb x = lsb (Rep_uint32 x)"
case_3: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm"   and x3 :: "'n" assumes "countable_bipartite_web \<Gamma>" shows "(case bi.Inner x3 of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = f3 x3"
dnso3: fixes x :: "'a" shows "ad (ad x) + (1::'a) = (1::'a)"
OrdInd2H: fixes H :: "fm set"   and A :: "fm"   and i :: "name" assumes "H \<turnstile> A(i::=Zero)"   and "H \<turnstile> SyntaxN.All i (OrdP (Var i) IMP A IMP A(i::=SUCC (Var i)))" shows "insert (OrdP (Var i)) H \<turnstile> A"
roots_of_real_main: fixes p :: "real poly" assumes "p \<noteq> 0"   and "degree p \<le> 2 \<or> set (coeffs p) \<subseteq> \<rat>" shows "set (roots_of_real_main p) = {x. poly p x = 0}"
assms_1: shows "\<forall>n. f n \<noteq> l"
lsplit_def: shows "lsplit = (\<lambda>f l. f (hd l) (tl l))"
a2: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>' m. m \<le> n' \<and> ( \<Phi>' + \<Gamma>' \<Rightarrow>* \<Psi>' + \<Delta>', m) \<in> derivable  R* \<and> p = ( \<Phi>' \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Psi>')"
sqrt_int_floor_pos: fixes x :: "int" assumes "0 \<le> x" shows "sqrt_int_floor_pos x = \<lfloor>sqrt (real_of_int x)\<rfloor>"
rtrancl_beta_Abs: fixes s :: "dB"   and s' :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sup>* s'" shows "dB.Abs s \<rightarrow>\<^sub>\<beta>\<^sup>* dB.Abs s'"
Guarantee: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> {s. s \<in> g \<longrightarrow> s \<in> R}"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Guard f g c Q,A"
case_nat_comb_seq: fixes s' :: "'a"   and n :: "nat"   and \<omega> :: "nat \<Rightarrow> 'a"   and \<omega>' :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "(case i + n of 0 \<Rightarrow> s' | Suc x \<Rightarrow> comb_seq n \<omega> \<omega>' x) = (case i of 0 \<Rightarrow> case n of 0 \<Rightarrow> s' | Suc x \<Rightarrow> \<omega> x | Suc x \<Rightarrow> \<omega>' x)"
closeWS: fixes P :: "complex mat" assumes "is_quantum_predicate P" shows "is_quantum_predicate (wlp S P)"
simps: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list" shows "rhsq q \<rightarrow> l qs = qs"
permute_Tree_tConj: fixes p :: "perm"   and tset :: "('b, 'c, 'd) Tree set['b]" shows "p \<bullet> tConj tset = tConj (p \<bullet> tset)"
natural_transformation_axioms: shows "natural_transformation A B H F \<tau>'\<sigma>'.map"
E17: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>\<box>F \<longrightarrow> \<box>\<diamond>F"
simps_2: fixes P :: "'c \<Rightarrow> 'c list"   and f :: "'c"   and T :: "('c, 'd) Term.term list" shows "term_variants P (Fun f T) = (let S = product_lists (map (term_variants P) T) in map (Fun f) S @ concat (map (\<lambda>g. map (Fun g) S) (P f)))"
spr_sim_range: shows "sim_range SPR.MC spr_simMC spr_sim"
Poincare_recurrence_thm_7: fixes A :: "'a set" assumes "A \<in> sets M" shows "AE x\<in>A in M. x \<in> recurrent_subset_infty A"
lex_ext_unbounded_stri_imp_nstri: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and xs :: "'a list"   and ys :: "'a list" assumes "fst (lex_ext_unbounded f xs ys)" shows "snd (lex_ext_unbounded f xs ys)"
ZFact_defs_1: fixes k :: "int" shows "ZFact k = \<Z> Quot Idl\<^bsub>\<Z>\<^esub> {k}"
le_supE: fixes a :: "'c multiset"   and b :: "'c multiset"   and x :: "'c multiset" assumes "a \<union># b \<subseteq># x" obtains "a \<subseteq># x" and "b \<subseteq># x"
fringe_of_empty_red1: fixes prb :: "('d, 'e, 'f, 'g) pre_RedBlack_scheme" assumes "edges (red prb) = {}"   and "subs prb = {}"   and "marked prb = (\<lambda>rv. False)"   and "out_edges (black prb) (fst (root (red prb))) \<noteq> {}" shows "fringe prb = {root (red prb)}"
daMM: shows "adjoint M * M \<in> carrier_mat n n"
distinct_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list" assumes "f permutes {..<length xs}" shows "distinct (permute_list f xs) = distinct xs"
FIN_succ: fixes u :: "nat" shows "finite (E `` {u})"
positive_rat: fixes a :: "int"   and b :: "int" shows "Rat.positive (Fract a b) = (0 < a * b)"
frange_fconverse: fixes r :: "V" shows "\<R>\<^sub>\<bullet> (r\<inverse>\<^sub>\<bullet>) = \<D>\<^sub>\<bullet> r"
a: shows "[\<lparr>F,x\<^sup>P\<rparr> \<^bold>\<equiv> \<lbrace>x\<^sup>P,H\<rbrace> in v]"
Cons1_2: shows "separateAtoms (At # L) = (a, b, c, d)"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "VVV.arr f = (arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f)))"
simps: fixes a :: "'b list"   and A :: "'b set" shows "(a \<in> lists A) = (a = \<epsilon> \<or> (\<exists>aa l. a = aa # l \<and> aa \<in> A \<and> l \<in> lists A))"
axioms_1: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "RModule R smult M"
ccExp_Lam: fixes y :: "Vars.var"   and e :: "exp"   and n :: "Arity" shows "\<G>\<^bsub>pred\<cdot>n\<^esub> e G|` fv (Lam [y]. e) \<sqsubseteq> \<G>\<^bsub>n\<^esub> (Lam [y]. e)"
exec: shows "exec_meth ci P (ins @ ins') (xt @ shift (length ins) xt') t h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')"
expectation_gpv_parametric': fixes C :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'i \<Rightarrow> 'l \<Rightarrow> bool" shows "((=) ===> rel_\<I> C R ===> (A ===> (=)) ===> rel_gpv'' A C R ===> (=)) expectation_gpv expectation_gpv"
intros_2: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.transT tval aval (c1, s) s'" shows "PL.transC tval aval (c1 ;; c2, s) (c2, s')"
strict_assoc: fixes a0 :: "'a"   and a1 :: "'a"   and a2 :: "'a" assumes "ide a0"   and "ide a1"   and "ide a2" shows "ide \<a>[a0, a1, a2]"
n_o_narrow: fixes X :: "char list set"   and S1 :: "ivl Abs_State.st option"   and S2 :: "ivl Abs_State.st option" assumes "finite X"   and "domo S1 \<subseteq> X"   and "domo S2 \<subseteq> X"   and "S2 \<sqsubseteq> S1"   and "\<not> S1 \<sqsubseteq> S1 \<triangle> S2" shows "n_o (n_st n_ivl X) (S1 \<triangle> S2) < n_o (n_st n_ivl X) S1"
sets_lebesgue_outer_open: fixes S :: "'b set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'b set" where "open T" and "S \<subseteq> T" and "T - S \<in> lmeasurable" and "emeasure lebesgue (T - S) < ennreal e"
inner_prod_with_itself_Re: fixes u :: "complex Matrix.vec" shows "0 \<le> Re \<langle>u|u\<rangle>"
increasing_measurable_info: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and f :: "'b \<Rightarrow> 'd"   and s :: "'c"   and N :: "'d measure"   and t :: "'c" assumes "filtrated_prob_space M F"   and "f \<in> F s \<rightarrow>\<^sub>M N"   and "s \<le> t" shows "f \<in> F t \<rightarrow>\<^sub>M N"
dither_loops_aux: fixes m :: "nat"   and stp :: "nat" shows "steps0 (1, Bk \<up> m, [Oc]) dither stp = (1, Bk \<up> m, [Oc]) \<or> steps0 (1, Bk \<up> m, [Oc]) dither stp = (2, Oc # Bk \<up> m, [])"
le_sup_iff: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "(x \<oplus> y \<preceq> z) = (x \<preceq> z \<and> y \<preceq> z)"
bconf_LAss: fixes P :: "'a prog"   and sh :: "sheap"   and V :: "'b"   and e :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (V:=e,b) \<surd>) = (P,sh \<turnstile>\<^sub>b (e,b) \<surd>)"
isometry_on_continuous: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f" shows "continuous_on X f"
reduc_saturated_def: fixes N :: "'f set" shows "reduc_calc.reduc_saturated N = (Inf_from (N - Red_F N) \<subseteq> Red_Red_I N)"
assms_1: shows "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"
bounded_path_image: fixes g :: "real \<Rightarrow> 'b" assumes "path g" shows "bounded (path_image g)"
pfp: shows "step (local.\<gamma>\<^sub>o \<top>) (local.\<gamma>\<^sub>c C) \<le> local.\<gamma>\<^sub>c C"
UN_iterate_less: fixes x :: "'a"   and S :: "'a set"   and f :: "'a set \<rightarrow> 'a set" assumes "x \<in> S"   and "\<And>y. y \<subseteq> S \<Longrightarrow> f\<cdot>y \<subseteq> S" shows "(\<Union>i. f\<^bsup>i\<^esup>\<cdot>{x}) \<subseteq> S"
sort_ex_code: fixes rel :: "('a \<times> 'a) list"   and S :: "'a set" shows "sort_ex (set rel) S = exesort_ex rel S"
put_surjectivity: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "wb_lens x" shows "\<exists>\<rho> v. put\<^bsub>x\<^esub> \<rho> v = \<sigma>"
subgroup_subgroup_generated: fixes S :: "'c set" shows "subgroup (carrier (subgroup_generated H S)) H"
distinguishing_weak_formula_distinguishes: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and P :: "'c"   and Q :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition"   and "\<not> indexed_weak_nominal_ts.weakly_logically_equivalent TYPE('a) \<tau> satisfies transition P Q" shows "indexed_weak_nominal_ts.is_distinguishing_formula satisfies transition (indexed_weak_nominal_ts.distinguishing_weak_formula \<tau> satisfies transition P Q) P Q"
insert_ident: fixes x :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
open_countable_basis_ex: fixes p :: "'b set \<Rightarrow> bool"   and B :: "'b set set"   and X :: "'b set" assumes "countable_basis p B"   and "p X" shows "\<exists>B'\<subseteq>B. X = \<Union> B'"
map_add_upd: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b" shows "f ++ g(x \<mapsto> y) = (f ++ g)(x \<mapsto> y)"
inf_def: fixes a :: "'a G"   and b :: "'a G" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
MakeNT_cod: fixes f :: "('j, 'g, 'k, 'h, 'l, 'i) NatTrans" shows "NTCatCod f = NTCatCod (MakeNT f)"
many_strongly_connected_implies_13_eq: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>T ; x ; x\<^sup>\<star> = x\<^sup>+"
clinear_def: fixes f :: "'c \<Rightarrow> 'd" shows "clinear f \<equiv> Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) f"
bind_resumption_cong: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and y :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption"   and g :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "x = y"   and "\<And>z. z \<in> results y \<Longrightarrow> f z = g z" shows "x \<bind> f = y \<bind> g"
can_sync_def: fixes t :: "'t"   and x :: "'x"   and m :: "'m"   and LT :: "('l + 't + 't) set" shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong> \<equiv> \<exists>taa x' m'. t \<turnstile> \<langle>x, m\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle> \<and> LT = collect_waits taa"
fold_set_insert_dj: shows "insert = op_set_insert_dj"
simps: fixes a1 :: "('a, 'b) Term.term"   and a2 :: "('a, 'c) Term.term" shows "emv a1 a2 = ((\<exists>x y. a1 = Var x \<and> a2 = Var y) \<or> (\<exists>f g ss ts. a1 = Fun f ss \<and> a2 = Fun g ts \<and> f = g \<and> length ss = length ts \<and> (\<forall>i<length ts. emv (ss ! i) (ts ! i))))"
Im_Ln_pos_lt: fixes z :: "complex" assumes "z \<noteq> 0" shows "(0 < Im (Ln z) \<and> Im (Ln z) < pi) = (0 < Im z)"
locate_def: fixes f :: "'b \<Rightarrow> nat"   and al :: "'b list"   and i :: "nat" shows "locate f al i \<equiv> SOME n_j. locate_pred f al i n_j"
measurable_bounded_lemma: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> real"   and a :: "'c"   and b :: "'c" assumes "f \<in> borel_measurable lebesgue"   and "g integrable_on cbox a b"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x" shows "f integrable_on cbox a b"
simps: fixes p :: "'b list"   and q :: "'b list"   and r :: "'b list"   and s :: "'b list" shows "is_awalk_cyc_decomp p (q, r, s) = (p = q @ r @ s \<and> (\<exists>u v w. awalk u q v \<and> awalk v r v \<and> awalk v s w) \<and> 0 < length r \<and> (\<exists>u. distinct (awalk_verts u q)))"
no_better_than_def: fixes x :: "'b"   and B :: "'b set"   and P :: "('b \<times> 'b) set" shows "no_better_than x B P = {y \<in> B. x \<succeq>[P] y}"
mat_of_cols_list_def: fixes nr :: "nat"   and cs :: "complex list list" shows "Tensor.mat_of_cols_list nr cs = Matrix.mat nr (length cs) (\<lambda>(i, j). cs ! j ! i)"
divisors_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors plus zero times"   and "times a b = zero" shows "a = zero \<or> b = zero"
wf_bs_polyneg: fixes bs :: "'b list"   and p :: "poly" assumes "wf_bs bs p" shows "wf_bs bs (~\<^sub>p p)"
db\<^sub>s\<^sub>s\<^sub>t_no_upd_append: fixes B :: "('e, 'f) stateful_strand_step list"   and A :: "('e, 'f) stateful_strand_step list" assumes "\<forall>b\<in>set B. \<not> is_Insert b \<and> \<not> is_Delete b" shows "db'\<^sub>s\<^sub>s\<^sub>t A = db'\<^sub>s\<^sub>s\<^sub>t (A @ B)"
leg_simps_3: shows "C.arr HH\<mu>\<nu>H\<pi>\<rho>.dom.leg1"
no_self_loop_in_tree: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c" assumes "DFS_invar G param s" shows "(v, v) \<notin> tree_edges s"
MergeGuards_iff: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P merge_guards c Q,A = \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
incseqD: fixes f :: "nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat" assumes "incseq f"   and "i \<le> j" shows "f i \<le> f j"
assms_1: shows "M1 \<preceq>\<lbrakk>T\<rbrakk> M2"
Confidentiality_Auth_A: fixes Aa :: "agent"   and authKa :: "nat"   and Peer :: "msg"   and Taa :: "msg"   and evsa :: "event list"   and A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Crypt (shrK Aa) \<lbrace>Key authKa, Peer, Taa\<rbrace> \<in> parts (knows Spy evsa)"   and "Aa \<notin> bad"   and "evsa \<in> kerbV"   and "\<And>x. Says Kas Aa \<lbrace>Crypt (shrK Aa) \<lbrace>Key authKa, Peer, Taa\<rbrace>, x\<rbrace> \<in> set evsa \<Longrightarrow> Says Kas A \<lbrace>Crypt Ka \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, authTicket\<rbrace> \<in> set evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "evs \<in> kerbV" shows "Key authK \<notin> analz (knows Spy evs)"
PO_m1_refines_ri_a0i: shows "refines (R_a0im1_ri \<inter> a0i_inv1_iagree \<times> m1_inv0_fin) med_a0im1_ri a0i m1"
incseq_convergent: fixes X :: "nat \<Rightarrow> real"   and B :: "real" assumes "incseq X"   and "\<forall>i. X i \<le> B" obtains L :: "real" where "X \<longlonglongrightarrow> L" and "\<forall>i. X i \<le> L"
released_def: fixes t :: "'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g \<times> 'h \<times> 'b" shows "released t = (let (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = t in \<R>)"
contains_def2: fixes f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" shows "contains f iA n = (iA \<in> set (nforms (f n)))"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "prefixBeforeLastMarked (h # t) = (if marked h \<and> markedElements t = [] then [] else h # prefixBeforeLastMarked t)"
same_orbit_char: fixes m :: "'a set"   and n :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)"   and "n \<in> subgroups_of_size (p ^ a)" shows "((m, n) \<in> conj.same_orbit_rel) = (\<exists>g\<in>carrier G. conjugation_action (p ^ a) g n = m)"
projectTable_idle: fixes n :: "nat"   and A :: "nat set"   and X :: "'a option list set"   and I :: "nat set" assumes "table n A X"   and "A \<subseteq> I" shows "projectTable I (A, X) = (A, X)"
inject_4: fixes x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree"   and y41 :: "'g"   and y42 :: "('e, 'f, 'g) Tree" shows "(tAct x41 x42 = tAct y41 y42) = (x41 = y41 \<and> x42 = y42)"
iface_name_is_wildcard_fst: fixes i :: "char"   and "is" :: "char list" assumes "iface_name_is_wildcard (i # is)"   and "is \<noteq> []" shows "iface_name_is_wildcard is"
row_formula: fixes M1 :: "'a list list"   and M2 :: "'a list list" shows "\<forall>i. i < row_length M1 * row_length M2 \<and> mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2 \<longrightarrow> row (M1 \<otimes> M2) i = vec_vec_Tensor (row M1 (i div row_length M2)) (row M2 (i mod row_length M2))"
IH: assumes "S_ \<inter> B = {}" shows "swapidseq_ext (S_ \<union> B) (n_ + m) (p_ \<circ> q)"
before_in_setD2: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" assumes "x < y in xs" shows "y \<in> set xs"
showsp_real_alg_def: fixes p :: "nat"   and x :: "real_alg"   and y :: "char list" shows "showsp_real_alg p x y = show_real_alg x @ y"
add_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a \<squnion> c \<le> b \<squnion> c"
card_of_Plus_ordLeq_infinite: fixes C :: "'c set"   and A :: "'d set"   and B :: "'e set" assumes "infinite C"   and "|A| \<le>o |C|"   and "|B| \<le>o |C|" shows "|A <+> B| \<le>o |C|"
differentiable_bound_segment: fixes x0 :: "'a"   and a :: "'a"   and G :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and B :: "real" assumes "\<And>t. t \<in> {0..1} \<Longrightarrow> x0 + t *\<^sub>R a \<in> G"   and "\<And>x. x \<in> G \<Longrightarrow> (f has_derivative f' x) (at x within G)"   and "\<And>x. x \<in> {0..1} \<Longrightarrow> onorm (f' (x0 + x *\<^sub>R a)) \<le> B" shows "norm (f (x0 + a) - f x0) \<le> norm a * B"
restrict_chart_in_atlas: fixes c :: "('a, 'b) chart"   and S :: "'a set" assumes "c \<in> f.src.atlas" shows "restrict_chart S c \<in> f.src.atlas"
chain_lSup_lprefix: fixes A :: "'f llist set"   and zs :: "'f llist" assumes "Complete_Partial_Order.chain lprefix A"   and "\<And>xs. xs \<in> A \<Longrightarrow> lprefix xs zs" shows "lprefix (lSup A) zs"
exhaust: fixes y :: "chan" obtains x1 :: "tag" and x2 :: "agent" and x3 :: "agent" and x4 :: "msg" where "y = Chan x1 x2 x3 x4"
succ_diff: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "succ r i \<noteq> i"
lossless_R2: fixes m1 :: "'msg1"   and m2 :: "'msg2" shows "lossless_spmf (R2 m1 m2)"
annos: shows "P,E \<turnstile> es [\<leadsto>] es'"
int: fixes n :: "nat" shows "integrable M (\<lambda>x. \<bar>f n x - g x\<bar> powr p)"
constant_function_def: fixes S :: "'d set"   and a :: "'e" shows "constant_function S a = (\<lambda>x\<in>S. a)"
All2_Eats_I: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and H :: "fm set"   and A :: "fm" assumes "atom i \<sharp> (t, u)"   and "H \<turnstile> All2 i t A"   and "H \<turnstile> A(i::=u)" shows "H \<turnstile> All2 i (Eats t u) A"
nres_rel_comp: fixes A :: "('g \<times> 'i) set"   and B :: "('i \<times> 'h) set" shows "\<langle>A\<rangle>nres_rel O \<langle>B\<rangle>nres_rel = \<langle>A O B\<rangle>nres_rel"
obtain_elem_frontier: fixes M :: "'b zmultiset"   and t :: "'b" assumes "0 < zcount M t" obtains s :: "'b" where "s \<le> t \<and> s \<in>\<^sub>A frontier M"