hcju/isabelleps huggingface dataset:
single_eq_add_mset: fixes a :: "'c"   and b :: "'c"   and M :: "'c multiset" shows "({#a#} = add_mset b M) = (b = a \<and> M = {#})"
wlog_linorder_le: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "\<And>a b. a \<le> b \<Longrightarrow> P a b"   and "P b a \<Longrightarrow> P a b" shows "P a b"
gc_approx_def: fixes prog :: "'a program_scheme" shows "gc_approx prog \<equiv> {\<lparr>global_config.processes = lcs, state = \<lparr>global_state.variables = v\<rparr>\<rparr> |lcs v. size lcs \<le> length (program.processes prog) \<and> set_mset lcs \<subseteq> lc_approx prog \<and> dom v \<subseteq> global_var_approx prog}"
binop_pdevs_raw_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'c"   and y :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "binop_pdevs_raw f x y i = (if x i = (0::'c) \<and> y i = (0::'d) then 0::'b else f (x i) (y i))"
1: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "\<gamma> - 2 / 2 ^ m + 1 / 2 ^ n \<le> d (F m n)"
ld: shows "LD' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 \<tau>1 \<tau>2 \<tau>3"
psi_def: fixes n :: "nat" shows "psi n = sum mangoldt {1..n}"
Exists: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula" assumes "wf_mformula \<sigma> j (Suc n) (lift_envs R) \<phi> \<phi>'" shows "wf_mformula \<sigma> j n R (MExists \<phi>) (formula.Exists \<phi>')"
qmpt_factorE_1: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "qmpt_factor proj M2 T2" shows "proj \<in> quasi_measure_preserving M M2"
Sim: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> Rel" shows "P \<leadsto><Rel'> Q"
SC_def: fixes X :: "'p set"   and Y :: "'p set" shows "SC X Y \<equiv> \<exists>w x y z. X = {w, x} \<and> Y = {y, z} \<and> w x \<equiv> y z"
J: shows "triangular_to_jnf_vector A = concat (map (jnf_vector \<circ> step_3) Cs)"
dom_in_domains: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "domains f \<noteq> {}" shows "local.dom f \<in> domains f"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" shows "wtE \<xi> \<equiv> \<forall>x. intT (tpOfV x) (\<xi> x)"
nextl: fixes u :: "'a list" shows "MN.nextl (init DFA) u = h (R `` {u})"
inf_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B = Collect (inf (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
incrIndexList_help4: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "|ls| = m"
intros_5: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "i_set_indp [ r, mod m, c ]"
terminates_merge_guards_to_terminates_Normal: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>merge_guards c \<down> Normal s" shows "\<Gamma>\<turnstile>c \<down> Normal s"
t_twice_compile_def: shows "t_twice_compile = tm_of abc_twice @ shift (mopup 1) (length (tm_of abc_twice) div 2)"
nonzero_inverse_minus_eq: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "inverse (- a) = - inverse a"
relatesI: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a stream" assumes "P (shd s) (shd (stl s))" shows "relates P s"
FrameStatImpRefl: fixes F :: "'b frame" shows "F \<hookrightarrow>\<^sub>F F"
Neg: shows "\<forall>v\<in>vars_lit (Neg x). \<sigma> v = \<tau> v"
T_slow_median_le: fixes xs :: "'a list" shows "T_slow_median xs \<le> (length xs)\<^sup>2 + 3 * length xs + 4"
finite_iff_cardOf_nat: fixes A :: "'a set" shows "finite A = (|A| <o |UNIV|)"
that: fixes s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1' \<approx> s2'" shows "thesis"
trim: fixes s :: "char list" shows "\<exists>w. s = w @ trim s"
card_of_equiv_class: fixes B :: "'c set"   and F :: "('d \<Rightarrow> 'c) set"   and A :: "'d set" assumes "finite B"   and "F \<in> {f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // range_permutation A B" shows "card F = fact (card B)"
nth_root_nat_aux_gt: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m" shows "n < (nth_root_nat_aux m k (k ^ m) n + 1) ^ m"
respection: shows "rel_weakly_respects_barbs (indRelLSTPO SRel TRel) (STCalWB SWB TWB)"
a_rcos_equation: fixes ha :: "'a"   and a :: "'a"   and h :: "'a"   and b :: "'a"   and hb :: "'a" assumes "ha \<oplus> a = h \<oplus> b"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "h \<in> I"   and "ha \<in> I"   and "hb \<in> I" shows "hb \<oplus> a \<in> (\<Union>h\<in>I. {h \<oplus> b})"
wca_on_rwp_on_rspR_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "wca_on A f"   and "f_range_on A f"   and "decisive_on A f" shows "rwp_on A f = rspR_on A f"
ag_r_inv1: fixes x :: "'a" assumes "x \<in> carrier A" shows "x \<plusminus> -\<^sub>a x = \<zero>"
bv_chopI: fixes w :: "bit list"   and w1 :: "bit list"   and w2 :: "bit list"   and i :: "nat" assumes "w = w1 @ w2"   and "i = length w2" shows "bv_chop w i = (w1, w2)"
assms_3: shows "passive_subset (Liminf_llist (lmap snd TNs)) = {}"
abs_eq_iff: fixes x :: "('a, 'b, 'c) Formula.Tree"   and y :: "('a, 'b, 'c) Formula.Tree" shows "(Formula.abs_Tree\<^sub>\<alpha> x = Formula.abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
height_balance_list: fixes xs :: "'a list" shows "height (balance_list xs) = nat \<lceil>log 2 (real (length xs + 1))\<rceil>"
kcomp_klift: fixes f :: "'d \<Rightarrow> 'f set"   and g :: "'f \<Rightarrow> 'e set" shows "f \<circ>\<^sub>K g = bd\<^sub>\<F> g \<circ> f"
exec_wfp_always_in_states: fixes p :: "prog"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "wfp p"   and "\<sigma> \<in> states p" shows "\<exists>\<sigma>'. body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<and> \<sigma>' \<in> states p"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "Error x1 \<noteq> Ok x2"
exhaust: fixes y :: "('d, 'e, 'a, 'b, 'f, 'c) paramS" obtains x1 :: "'a \<Rightarrow> 'b list" and x2 :: "'c \<Rightarrow> ('d, 'e, 'a, 'b, 'f) qTerm set list" and x3 :: "'a \<times> 'c \<Rightarrow> ('d, 'e, 'a, 'b, 'f) qAbs set list" and x4 :: "('a \<Rightarrow> 'b \<Rightarrow> ('d, 'e, 'a, 'b, 'f) qTerm set option) list" where "y = ParS x1 x2 x3 x4"
assms_1: fixes R :: "('a \<times> 'a) set" assumes "R \<in> Rs" shows "sorter = sorter' R"
r_def: shows "r \<equiv> \<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2)"
select_convs_2: fixes igWls :: "'b \<Rightarrow> 'h \<Rightarrow> bool"   and igWlsAbs :: "'a \<times> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and igVar :: "'a \<Rightarrow> 'g \<Rightarrow> 'h"   and igAbs :: "'a \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'c"   and igOp :: "'f \<Rightarrow> ('d \<Rightarrow> 'h option) \<Rightarrow> ('e \<Rightarrow> 'c option) \<Rightarrow> 'h"   and igFresh :: "'a \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and igFreshAbs :: "'a \<Rightarrow> 'g \<Rightarrow> 'c \<Rightarrow> bool"   and igSwap :: "'a \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'h"   and igSwapAbs :: "'a \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'c \<Rightarrow> 'c"   and igSubst :: "'a \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'h"   and igSubstAbs :: "'a \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'i" shows "igWlsAbs \<lparr>igWls = igWls, igWlsAbs = igWlsAbs, igVar = igVar, igAbs = igAbs, igOp = igOp, igFresh = igFresh, igFreshAbs = igFreshAbs, igSwap = igSwap, igSwapAbs = igSwapAbs, igSubst = igSubst, igSubstAbs = igSubstAbs, \<dots> = more\<rparr> = igWlsAbs"
finite_UNIV_sum_def: shows "finite_UNIV = Phantom('a + 'b) (of_phantom finite_UNIV \<and> of_phantom finite_UNIV)"
assms_2: fixes u :: "'a llist" assumes "u \<le> v"   and "lfinite u" shows "P u"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated G A) = A"
hyps_7: shows "\<forall>j'\<in>{LEAST i'. k' < i' \<and> (\<exists>i. cs\<^bsup>path \<sigma>\<^esup> i = cs\<^bsup>path \<sigma>'\<^esup> i')..<n'}. v \<notin> local.writes (path \<sigma>' j')"
dgrad_setI: fixes d :: "'d \<Rightarrow> nat"   and t :: "'d"   and m :: "nat" assumes "d t \<le> m" shows "t \<in> dgrad_set d m"
commutative_square_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "category C" shows "category.commutative_square C f g h k \<equiv> (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g) \<and> (partial_magma.arr C h \<and> partial_magma.arr C k \<and> partial_magma.dom C h = partial_magma.dom C k) \<and> partial_magma.dom C f = partial_magma.cod C h \<and> C f h = C g k"
mu_repr_def: fixes n :: "nat"   and m :: "nat"   and mu :: "int iarray iarray"   and fs :: "int vec list" shows "LLL.mu_repr n m mu fs = (mu = IArray.of_fun (\<lambda>i. IArray.of_fun (LLL.d\<mu> n fs i) i) m)"
quot_of_ratfps_quot_to_ratfps: fixes x :: "'b poly \<times> 'b poly" assumes "constant_term (snd x) \<noteq> (0::'b)"   and "x \<in> normalized_fracts" shows "quot_of_ratfps (quot_to_ratfps x) = x"
avg_aux_correct: fixes s :: "'s" assumes "invar s" shows "avg_aux s = (card (\<alpha> s), \<Sum> (\<alpha> s))"
le3456_lt__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B Lt C D"   and "C D Le E F" shows "A B Lt E F"
x_txs: fixes ta :: "'trm"   and xa :: "'var" assumes "(ta, xa) \<in> set txs" shows "x \<noteq> xa"
AAcc_\<tau>red1t_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and i :: "'addr expr1"   and xs :: "'addr val list"   and i' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val" assumes "\<tau>red1gt uf P t h (i, xs) (i', xs')" shows "\<tau>red1gt uf P t h (Val a\<lfloor>i\<rceil>, xs) (Val a\<lfloor>i'\<rceil>, xs')"
reads_subset: fixes S :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and f :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a"   and S' :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set" assumes "reads S f h h'"   and "\<forall>P\<in>S' - S. reflp P \<and> transp P"   and "S \<subseteq> S'" shows "reads S' f h h'"
fsubseteq_ffun_upd1: fixes f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun"   and x :: "'a"   and v :: "'b" assumes "f \<subseteq>\<^sub>f g"   and "x \<notin> fdom g" shows "f \<subseteq>\<^sub>f g(x \<mapsto> v)\<^sub>f"
hyps: fixes \<S> :: "nat \<Rightarrow> bool option" assumes "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only \<S> \<union> all_shared sb"
c_def: fixes i :: "int"   and j :: "int" assumes "i < j" shows "c i j = Min (c_k i j ` {i + 1..j})"
assoc_add_vec: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>3 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 + v\<^sub>2 + v\<^sub>3 = v\<^sub>1 + (v\<^sub>2 + v\<^sub>3)"
h5: shows "invertible_mat (get_matrix_R (reduce_system_R p (qs, matrix, subsets, signs)))"
antisym: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "ko.le ko x y"   and "ko.le ko y x" shows "x = y"
in_prime_factors_imp_prime: fixes p :: "'a"   and x :: "'a" assumes "p \<in> prime_factors x" shows "prime p"
Red_I_without_red_F: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "Red_I (N - Red_F N) = Red_I N"
that: assumes "verts H \<subseteq> verts G"   and "arcs H \<subseteq> arcs G"   and "compatible G H"   and "wf_digraph H"   and "wf_digraph G" shows "thesis"
poly_limit_aux: fixes p :: "real poly" shows "((\<lambda>x. poly p x / x ^ degree p) \<longlongrightarrow> lead_coeff p) at_infinity"
inv_eq_self: fixes x :: "int" assumes "x \<in> Units R"   and "x = inv x" shows "x = \<one> \<or> x = \<ominus> \<one>"
differentiable_on_openD: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and x :: "'b" assumes "f differentiable_on X"   and "open X"   and "x \<in> X" shows "f differentiable at x"
A2: shows "\<lfloor>(\<lambda>w. universal (\<lambda>x xa. (\<P> x \<^bold>\<and> \<^bold>\<box>(\<lambda>v. \<forall>xb. existsAt xb v \<longrightarrow> x xb \<longrightarrow> xa xb) \<^bold>\<rightarrow> \<P> xa) w))\<rfloor>"
density_RN_deriv: fixes N :: "'a measure" assumes "absolutely_continuous N N"   and "sets N = N.events" shows "density N (RN_deriv N N) = N"
Copied: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and v :: "nat"   and qs :: "('b, 'c) fm list"   and a :: "'c"   and v' :: "nat"   and p :: "('b, 'c) fm"   and k :: "'c"   and i :: "'c"   and xs :: "(nat \<times> nat) set"   and w :: "nat"   and w' :: "nat"   and rs :: "('b, 'c) fm list"   and b :: "'c" assumes "branch !. v = Some (qs, a)"   and "qs !. v' = Some p"   and "descendants k i branch xs"   and "(w, w') \<in> xs"   and "branch !. w = Some (rs, b)"   and "rs !. w' = Some p"   and "Nom a at b in' set branch" shows "descendants k i branch ({(v, v')} \<union> xs)"
r_prefixes_def: shows "r_prefixes \<equiv> Pr 1 r_singleton_encode (Cn 3 r_next [recf.Id 3 1])"
intuitionisticI: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>h h'. \<lbrakk>P h; h \<preceq> h'\<rbrakk> \<Longrightarrow> P h'" shows "intuitionistic P"
xs_i_ge_a: shows "\<forall>i<length xs. a \<le> xs ! i"
E: shows "offs_set_prev ns (x # xs) index key mi ma i = {}"
assms_1: shows "isLastAssertedLiteral literal clause (elements M)"
monomial_decomp_imp_hom_decomp: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "monomial_decomp ps" shows "hom_decomp ps"
square_test_correct: fixes n :: "nat" shows "square_test n = is_square n"
FVv: fixes V :: "char list" shows "fv (Var V) = {V}"
WhileCondThrow: fixes 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 e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and c :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
preserves_modes_mem_RelS_Eg1Eg2: shows "preserves_modes_mem RelS_Eg1Eg2"
5: shows "(w \<sqinter> - prim_EP w v e)\<^sup>+ * prim_P w v e\<^sup>T\<^sup>\<star> = (w \<sqinter> - prim_EP w v e)\<^sup>+"
env_restr_HSem: fixes \<Gamma> :: "('b \<times> 'a) list"   and S :: "'b set"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "domA \<Gamma> \<inter> S = {}" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` S = \<rho> f|` S"
insert_subset_eq_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(add_mset a A \<subseteq># B) = (a \<in># B \<and> A \<subseteq># remove1_mset a B)"
invertible_mat: shows "invertible_mat (M_mat [[1], [- 1]] [[], [0]])"
map_cong: fixes x :: "'b \<times> 'c"   and ya :: "'b \<times> 'c"   and f1 :: "'b \<Rightarrow> 'd"   and g1 :: "'b \<Rightarrow> 'd"   and f2 :: "'c \<Rightarrow> 'e"   and g2 :: "'c \<Rightarrow> 'e" assumes "x = ya"   and "\<And>z1. z1 \<in> Basic_BNFs.fsts ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> Basic_BNFs.snds ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_prod f1 f2 x = map_prod g1 g2 ya"
inv: shows "Q \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"
reds: shows "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
circline_type_circline_point_0: shows "circline_type circline_point_0 = 0"
uniform_limit_uminus: fixes g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and X :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter" assumes "uniform_limit X g l F" shows "uniform_limit X (\<lambda>a b. - g a b) (\<lambda>a. - l a) F"
scaleR: fixes r :: "real"   and b :: "real" shows "f' (r *\<^sub>R b) = r *\<^sub>R f' b"
lset_llist_of_tllist: fixes xs :: "('a, 'b) tllist" shows "lset (llist_of_tllist xs) = tset xs"
eq_abs_iff': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "(a = abs b) = (less_eq zero a \<and> (b = a \<or> b = uminus a))"
f_def: shows "f \<equiv> prime_sum_upto (\<lambda>p. 1 / real p)"
bres_def: fixes x :: "'a \<times> 'b"   and z :: "'a \<times> 'b" shows "prod.bres x z = Sup_prod {y. less_eq_prod (times_prod' x y) z}"
simps: fixes a :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" shows "(a \<in> R*) = (\<exists>r seq. a = extendRule seq r \<and> r \<in> R)"
ddh_0_def: fixes \<G> :: "'b cyclic_group"   and \<A> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool spmf" shows "ddh.ddh_0 \<G> \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> x) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> (x * y))))"
f1: shows "\<forall>p pa. (\<not> p (VV pa p)) = pa (VV pa p) \<or> Collect p = Collect pa"
res_to_fun_monom: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and k :: "nat"   and n :: "nat" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp"   and "a k = b k" shows "(monom (UP Zp) c n \<bullet> a) k = (monom (UP Zp) c n \<bullet> b) k"
associated_polynomials_imp_same_length: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p \<sim>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> q" shows "length p = length q"
called_by_chain_def: fixes \<Gamma> :: "char list \<Rightarrow> 'b rule list option" shows "called_by_chain \<Gamma> = _Collect (callee, caller) (case \<Gamma> caller of None \<Rightarrow> False | Some rs \<Rightarrow> \<exists>m. Rule m (Call callee) \<in> set rs)"
S_O_rtrancl_NS_1: shows "S O NS\<^sup>* = S"
strategies_continue: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and v :: "'c"   and w :: "'c"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "WellOrderedStrategies G S p good r"   and "v \<in> S"   and "v \<rightarrow>\<^bsub>G\<^esub> w"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>) \<Longrightarrow> \<sigma> v = w"   and "\<sigma> \<in> good v" shows "\<sigma> \<in> good w"
cases: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.discr0 tval aval indis a" obtains c :: "('b, 'c) com" where "a = c" and "\<And>s c' s'. \<lbrakk>PL.mustT tval aval c s; PL.transC tval aval (c, s) (c', s')\<rbrakk> \<Longrightarrow> (s, s') \<in> indis \<and> PL_Indis.discr0 tval aval indis c'" and "\<And>s s'. \<lbrakk>PL.mustT tval aval c s; PL.transT tval aval (c, s) s'\<rbrakk> \<Longrightarrow> (s, s') \<in> indis"
bilinear_radd: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "bilinear h" shows "h x (y + z) = h x y + h x z"
monadic_filter_rev_def: shows "monadic_filter_rev = monadic_filter_rev_aux []"
streams_sset: fixes s :: "'a stream"   and A :: "'a set" assumes "s \<in> streams A" shows "i.R s \<subseteq> A"
open_Un: fixes S :: "'a set"   and T :: "'a set" assumes "open S"   and "open T" shows "open (S \<union> T)"
mult_isol_var: fixes u :: "'a"   and x :: "'a"   and v :: "'a"   and y :: "'a" assumes "u \<le> x"   and "v \<le> y" shows "u ; v \<le> x ; y"
refl_onD2: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "refl_on A r"   and "x \<succeq>[r] y" shows "y \<in> A"
brn_ParT_simp: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brn (ParT cl) = brnL cl (length cl)"
of_int_less_1_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) one = (z < 1)"
ax_Methd1: fixes G :: "prog"   and A :: "'b triple set"   and P :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and Q :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and ms :: "(qtname \<times> sig) set"   and C :: "qtname"   and sig :: "sig" assumes "G,A \<union> {{P} Methd-\<succ> {Q} | ms}|\<turnstile>{{P} body G-\<succ> {Q} | ms}"   and "(C, sig) \<in> ms" shows "G,A\<turnstile>{Normal (P C sig)} Methd C sig-\<succ> {Q C sig}"
curryI: fixes f :: "'c \<times> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "f (a, b)" shows "curry f a b"
nres_order_simps_20: fixes x :: "'b"   and y :: "'b" shows "(RETURN x \<le> RETURN y) = (x = y)"
restrict_map_empty: fixes D :: "'c set" shows "Map.empty |` D = Map.empty"
word_and_le': fixes b :: "'b word"   and c :: "'b word"   and a :: "'b word" assumes "b \<le> c" shows "a AND b \<le> c"
wf_cdecl_all: fixes p :: "prog" assumes "p = tprg" shows "Ball (set Classes) (wf_cdecl p)"
red_rtrancl_repE: fixes d :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow> nat"   and G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and m :: "nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and r :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "G \<subseteq> i.punit.dgrad_p_set d m"   and "finite G"   and "p \<in> i.punit.dgrad_p_set d m"   and "(i.punit.red G)\<^sup>*\<^sup>* p r" obtains q :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "p = r + (\<Sum>g\<in>G. punit.mult_scalar (q g) g)" and "\<And>g. q g \<in> i.punit.dgrad_p_set d m" and "\<And>g. lex_pm (i.lpp (punit.mult_scalar (q g) g)) (i.lpp p)"
I: fixes n :: "nat"   and s0 :: "state"   and L :: "lenv"   and accC :: "qtname"   and C :: "assigned"   and s1 :: "state"   and Y :: "vals"   and Z :: "'a" assumes "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"   and "s0\<Colon>\<preceq>(G, L)"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c\<Colon>\<surd>"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>In1r c\<guillemotright> C"   and "G\<turnstile>s0 \<midarrow>c\<midarrow>n\<rightarrow> s1"   and "P Y s0 Z" shows "Q \<diamondsuit> s1 Z \<and> s1\<Colon>\<preceq>(G, L)"