hcju/isabelleps huggingface dataset:
seq_app_first: fixes s :: "'b"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(s ## \<sigma>) 0 = s"
info_2_unique: fixes x :: "real_alg_2"   and y :: "real_alg_2" assumes "invariant_2 x"   and "invariant_2 y"   and "real_of_2 x = real_of_2 y" shows "info_2 x = info_2 y"
set_mset_subset_singletonD: fixes A :: "'b multiset"   and x :: "'b" assumes "set_mset A \<subseteq> {x}" shows "A = replicate_mset (size A) x"
PossWorldX: shows "[PossibleWorld (x\<^sup>P) in v]"
rep_eq: fixes x :: "'a fls"   and xa :: "'a fls" shows "fls_nth (x + xa) = (\<lambda>n. fls_nth x n + fls_nth xa n)"
ipset_from_cidr_large_pfxlen: fixes n :: "nat"   and ip :: "'a word" assumes "LENGTH('a) \<le> n" shows "ipset_from_cidr ip n = {ip}"
topological_basisE: fixes "open" :: "'b set \<Rightarrow> bool"   and B :: "'b set set"   and O' :: "'b set"   and x :: "'b" assumes "class.topological_space open"   and "topological_space.topological_basis open B"   and "open O'"   and "x \<in> O'" obtains B' :: "'b set" where "B' \<in> B" and "x \<in> B'" and "B' \<subseteq> O'"
dim_sums_Int: fixes S :: "'a set"   and T :: "'a set" assumes "subspace S"   and "subspace T" shows "dim {x + y |x y. x \<in> S \<and> y \<in> T} + dim (S \<inter> T) = dim S + dim T"
dvd_by_all2: fixes M :: "nat" shows "\<exists>N>0. \<forall>m. 0 < m \<and> m \<le> M \<longrightarrow> m dvd N"
\<Phi>_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "\<Phi> (g \<cdot> f) = \<Phi> g \<cdot>\<acute> \<Phi> f"
chop_add1: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<longrightarrow> |i| = |j| + |k|"
successive_step: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" assumes "successive P (x # xs)"   and "\<not> P x" shows "successive P xs"
gram_schmidt_result_3: fixes ws :: "'a vec list"   and n :: "nat"   and us :: "'a vec list" assumes "set ws \<subseteq> carrier_vec n"   and "distinct ws"   and "\<not> module.lin_dep class_ring (module_vec TYPE('a) n) (set ws)"   and "us = gram_schmidt n ws" shows "set us \<subseteq> carrier_vec n"
eq_OO: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(=) OO R = R"
Upow: fixes u :: "ZF" assumes "Elem u U" shows "Elem (Power u) U"
reflclp_tranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "r\<^sup>+\<^sup>+\<^sup>=\<^sup>= = r\<^sup>*\<^sup>*"
bot_entails_all: fixes Bot :: "'b set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and B :: "'b"   and N1 :: "'b set" assumes "consequence_relation Bot entails"   and "B \<in> Bot" shows "entails {B} N1"
absolutely_integrable_abs_iff: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "(f absolutely_integrable_on S) = (f integrable_on S \<and> (\<lambda>x. \<Sum>i\<in>Basis. \<bar>f x \<bullet> i\<bar> *\<^sub>R i) integrable_on S)"
arr_to_hfun_comp: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" shows "arr_to_hfun (local.comp g f) = hcomp (arr_to_hfun g) (arr_to_hfun f)"
mul_ext_cong: fixes xs1 :: "'b list"   and ys1 :: "'b list"   and xs2 :: "'b list"   and ys2 :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "mset xs1 = mset ys1"   and "mset xs2 = mset ys2"   and "\<And>x x'. \<lbrakk>x \<in> set ys1; x' \<in> set ys2\<rbrakk> \<Longrightarrow> f x x' = g x x'" shows "mul_ext f xs1 xs2 = mul_ext g ys1 ys2"
smult_altdef: fixes c :: "'a"   and p :: "'a poly" shows "smult c p = [:c:] * p"
IN_Edge: fixes x :: "'v"   and y :: "'v" shows "\<^bold>I\<^bold>N\<^bsub>\<Delta>''\<^esub> (Edge x y) = (if edge \<Delta> x y then {Vertex x} else {})"
after_futures: fixes xs :: "'e list"   and P :: "'e process"   and ys :: "'e list" assumes "xs \<in> traces P" shows "futures (P \<setminus> xs) ys = futures P (xs @ ys)"
ht_hash_in_dom_in_dom_bounded_hashcode_nat: fixes l :: "('a \<times> 'b) list list"   and k :: "'a" assumes "ht_hash l"   and "k \<in> dom (map_of (concat l))" shows "k \<in> dom (map_of (l ! bounded_hashcode_nat (length l) k))"
has_bochner_integral_zero: fixes M :: "'c measure" shows "has_bochner_integral M (\<lambda>x. 0::'d) (0::'d)"
assms: shows "\<forall>x\<in>set (x # I). case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"
orbit_altdef_self_in: fixes s :: "'e"   and f :: "'e \<Rightarrow> 'e" assumes "s \<in> orbit f s" shows "orbit f s = {(f ^^ n) s |n. True}"
order_0_multiple_of_one: fixes A :: "'b tensor" assumes "order A = 0" obtains a :: "'b" where "A = a \<cdot> 1"
ss_def: shows "ss \<equiv> split 0 X (lpp ` G)"
pr_gr_1: fixes x :: "nat" shows "c_is_sub_fun (pr_gr x) univ_for_pr"
isSucc_succ: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "isSucc r (succ r i)"
validity_def: fixes i :: "('p, 'v, 's) configuration"   and c :: "('p, 'v, 's) configuration" shows "validity i c \<equiv> initial i \<and> reachable i c \<longrightarrow> (\<forall>v. <\<bottom>, outM v> \<in># msgs c \<longrightarrow> (\<exists>p. <p, inM v> \<in># msgs i))"
y1_prop: shows "r < y1 \<and> (\<forall>(d, e, f)\<in>set b. \<forall>x\<in>{r<..y1}. d * x\<^sup>2 + e * x + f < 0)"
pmf_integral_code_unfold: fixes p :: "'e pmf" shows "pmf_integral p = measure_pmf.expectation p"
divisor_add_diff_mod_if: fixes m :: "nat"   and b :: "nat"   and a :: "nat" shows "(m + b mod m - a mod m) mod m = (if a mod m \<le> b mod m then b mod m - a mod m else m + b mod m - a mod m)"
exp_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "exp_aform_err p X = (let ivl = ivl_of_aform_err p X in min_range_mono p floatarith.Exp floatarith.Exp (lower ivl) (upper ivl) X)"
assms_2: shows "seq \<pp>\<^sub>1[b, c] f"
eval_oracle_def: fixes c_o :: "'c_o"   and c_a :: "'c_a" shows "eval_oracle c_o c_a = (\<lambda>(b, s') guess. map_spmf (\<lambda>b'. ((), b \<or> b', s')) (eval c_o c_a s' guess))"
lookup_correct: fixes m :: "('b, 'c) RBT.rbt"   and k :: "'b" assumes "rm.invar m" shows "rm.lookup k m = rm.\<alpha> m k"
advantage_def: fixes key_gen :: "'a spmf"   and "prf" :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and rand :: "'c spmf"   and \<A> :: "(bool, 'b, 'c) gpv" shows "Pseudo_Random_Function.prf.advantage key_gen prf rand \<A> = \<bar>spmf (Pseudo_Random_Function.prf.game_0 key_gen prf \<A>) True - spmf (Pseudo_Random_Function.prf.game_1 rand \<A>) True\<bar>"
dropWhile_eq_Nil_conv: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(dropWhile P xs = []) = (\<forall>x\<in>set xs. P x)"
map_id: fixes t :: "('d, 'e) optionT" shows "map_optionT' id id t = t"
nat_pow_pow: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier G" shows "(x [^] n) [^] m = x [^] (n * m)"
mkarr_rep_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f" shows "free_monoidal_category.mkarr C (free_monoidal_category.rep C f) = f"
d_def: fixes dims :: "nat list" shows "state_sig.d dims = prod_list dims"
simps: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and xs :: "'varSort"   and x :: "'var"   and qX :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm" shows "qInitAbs MOD (qAbs xs x qX) = igAbs MOD xs x (qInit MOD qX)"
reachableNodes_def: fixes u :: "nat" shows "reachableNodes u \<equiv> {v. connected u v}"
refl_onD1: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "refl_on A r"   and "(x, y) \<in> r" shows "x \<in> A"
bprv_\<phi>G_eqv_not_exi_PPf: shows "bprv (eqv \<phi>G (neg (exi yy (PPf (Var yy) \<langle>\<phi>G\<rangle>))))"
sas_plus_equivalent_to_strips_i: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) \<pi>" shows "\<Psi>\<^sub>G\<^sub>+ \<subseteq>\<^sub>m execute_parallel_plan_sas_plus (\<Psi>\<^sub>I\<^sub>+) (\<phi>\<^sub>P\<inverse> \<Psi> \<pi>)"
monoD: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> y" shows "F x \<le> F y"
not_inf: shows "\<not> \<Gamma>\<turnstile> (c, s) \<rightarrow> \<dots>(\<infinity>)"
monoD: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<sqsubseteq> f y"
rename_E_aimpl_alt: shows "rename_E f E = E_of_succ (\<lambda>v. case fi_map v of None \<Rightarrow> {} | Some u \<Rightarrow> f ` succ_of_E E u)"
monom_is_polynomial: fixes K :: "'a set"   and a :: "'a"   and n :: "nat" assumes "subring K R"   and "a \<in> K - {\<zero>}" shows "polynomial K (local.monom a n)"
h3: shows "(\<Sum>i = 0..<n + 1. f i * (B ^ n / B ^ i)) = (\<Sum>i = 0..<n + 1. f i * B ^ (n - i))"
units_of_def: fixes G :: "('a, 'b) monoid_scheme" shows "units_of G = \<lparr>carrier = Units G, mult = (\<otimes>\<^bsub>G\<^esub>), one = \<one>\<^bsub>G\<^esub>\<rparr>"
1: shows "((ereal \<circ> (\<lambda>x. log q (1 - x)) \<circ> real_of_ereal) \<longlongrightarrow> 0) (at_right 0)"
jkbpDet: shows "\<forall>a. \<forall>t\<in>SPR.jkbpC. length (jAction SPR.MC t a) \<le> 1"
hermitean_mk_circline': fixes A :: "complex"   and D :: "complex"   and B :: "complex" assumes "is_real A"   and "is_real D" shows "hermitean (A, B, cnj B, D)"
in_prime_factors_imp_prime: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "normalization_semidom.prime zero times one normalize p"
emeasure_interval_measure_Icc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x y. x \<le> y \<Longrightarrow> F x \<le> F y"   and "continuous_on UNIV F" shows "emeasure (interval_measure F) {a..b} = ennreal (F b - F a)"
simps: fixes V :: "nat set"   and s :: "nat set"   and t :: "'b option list set" shows "New_max.projectTable V (s, t) = (s \<inter> V, restrict V ` t)"
lift_Suc_antimono_le: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "\<And>n. f (Suc n) \<le> f n"   and "n \<le> n'" shows "f n' \<le> f n"
sign_def: fixes p :: "'a \<Rightarrow> 'a" shows "??.Permutations.sign p = (if evenperm p then 1 else - 1)"
neg_cycle_empty: fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and xs :: "nat list"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "set xs \<subseteq> {0..n}"   and "i \<le> n"   and "len M i i xs < \<one>" shows "[M]\<^bsub>v,n\<^esub> = {}"
monom_numeral: fixes n :: "num" shows "monom 0 (numeral n) = numeral n"
fv_map_fv_set: fixes L :: "('b, 'a) Term.term list" shows "\<Union> (set (map fv L)) = fv\<^sub>s\<^sub>e\<^sub>t (set L)"
Cauchy_Schwarz_ineq: fixes x :: "'b"   and y :: "'b" shows "real_of (cnj_class.cnj (inner x y) * inner x y) \<le> real_of (inner x x) * real_of (inner y y)"
Pi_UNIV: fixes A :: "'b set" shows "A \<rightarrow> UNIV = UNIV"
leading_coeff_0_iff: fixes p :: "'a poly" shows "(Polynomial.lead_coeff p = (0::'a)) = (p = 0)"
ConsThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es :: "expr list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>throw e' # es,s\<^sub>1\<rangle>"
Cd_range': fixes x :: "'a"   and d :: "'b"   and X :: "'a set" assumes "x \<in> Bossy'.Cd d X" shows "x \<in> X \<inter> Field (BPd' d)"
hyps_2: fixes x :: "name"   and M :: "trm" assumes "x:M \<in> \<parallel>(A)\<parallel>" shows "SNa M"
finite_cross_intros_3: fixes g1 :: "real \<Rightarrow> complex"   and z :: "complex"   and g2 :: "real \<Rightarrow> complex" assumes "finite_axes_cross g1 z"   and "finite_axes_cross g2 z" shows "finite_axes_cross (g1 +++ g2) z"
set_insert_tries: fixes key :: "'a \<Rightarrow> 'b list"   and xs :: "'a list"   and t :: "('b, 'a list) trie" shows "set_tries (inserts_tries key xs t) = set xs \<union> set_tries t"
inj_onf: shows "inj_on f (cball a r \<inter> T)"
oalist_inv_alt: fixes xs :: "('a \<times> 'e) list"   and ko :: "'o" shows "oalist_inv (xs, ko) = oalist_inv_raw ko xs"
alphaAbs_qAbs_iff_alphaAbs_ex_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_ex_equal_or_qFresh xs x X xs' x' X'"
finite_bck_img_single: shows "\<forall>x\<in>{0..1}. finite (\<phi> -` {x})"
flush_cache_all_privilege: fixes s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "get_S (cpu_reg_val PSR s) = 0"   and "s' = flush_cache_all s" shows "get_S (cpu_reg_val PSR s') = 0"
normalI: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G"   and "\<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H" shows "H \<lhd> G"
union_iff: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "(a \<in># A + B) = (a \<in># A \<or> a \<in># B)"
C1: shows "C.commutative_square \<nu>'.cod.leg0 \<mu>'.cod.leg1 (\<nu>'.chine \<cdot> \<p>\<^sub>1[\<nu>'.dom.leg0, \<mu>'.dom.leg1]) (\<mu>'.chine \<cdot> \<p>\<^sub>0[\<nu>'.dom.leg0, \<mu>'.dom.leg1])"
winding_number_zero_in_outside: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "z \<in> outside (path_image \<gamma>)" shows "winding_number \<gamma> z = 0"
ladder_L_n_pairwise_le: fixes u :: "nat"   and v :: "nat" assumes "u \<le> v"   and "v < length L" shows "ladder_n L u \<le> ladder_n L v"
Restrp_antisymmetric: shows "antisymmetric UNIV ((\<sqsubseteq>) \<restriction> A)"
inj_f: shows "inj_on (\<lambda>v. vec n (\<lambda>i. coeff (v mod m i) 0)) (carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u}))"
carrier_stalk_def: shows "stx.carrier_stalk \<equiv> dlim \<FF> \<rho> (neighborhoods x)"
concat_map_nth0: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b list" assumes "xs \<noteq> []"   and "f (xs ! 0) \<noteq> []" shows "concat (map f xs) ! 0 = f (xs ! 0) ! 0"
simps_20: fixes P :: "'a prog"   and h :: "'heap"   and e :: "('b, 'c, 'addr) exp"   and e' :: "('b, 'c, 'addr) exp"   and e'' :: "('b, 'c, 'addr) exp" shows "\<tau>move1 P h (if (e) e' else e'') = (\<tau>move1 P h e \<or> final e)"
T_composition_borel_1: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. f (T x)) \<in> borel_measurable M"
I_def: shows "I = sample_primes \<bind> (\<lambda>P. sample_set_excl prime_set {P} \<bind> (\<lambda>Q. let N = P * Q; N' = (P - 1) * (Q - 1) in sample_coprime N' \<bind> (\<lambda>e. let d = nat (fst (bezw e N') mod int N') in return_spmf ((N, e), d))))"
wqo_wf_and_no_antichain_conv: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "qo_on P A" shows "wqo_on P A = (wfp_on (strict P) A \<and> (\<nexists>f. antichain_on P f A))"
strong_induct: fixes x1 :: "(Vars.var \<times> exp) list"   and x2 :: "exp"   and x3 :: "Vars.var list"   and x4 :: "(Vars.var \<times> exp) list"   and x5 :: "exp"   and P :: "'b \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Vars.var list \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> bool"   and c :: "'b" assumes "x1 : x2 \<Down>\<^bsub>x3\<^esub> x4 : x5"   and "\<And>\<Gamma> x e L c. P c \<Gamma> (Lam [x]. e) L \<Gamma> (Lam [x]. e)"   and "\<And>y \<Gamma> e x L \<Delta> \<Theta> z e' c. \<lbrakk>{atom y} \<sharp>* c; atom y \<sharp> (\<Gamma>, e, x, L, \<Delta>, \<Theta>, z); \<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'; \<And>c. P c \<Gamma> e L \<Delta> (Lam [y]. e'); \<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c \<Delta> e'[y::=x] L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (App e x) L \<Theta> z"   and "\<And>\<Gamma> x e L \<Delta> z c. \<lbrakk>map_of \<Gamma> x = Some e; delete x \<Gamma> : e \<Down>\<^bsub>x # L\<^esub> \<Delta> : z; \<And>c. P c (delete x \<Gamma>) e (x # L) \<Delta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) L ((x, z) # \<Delta>) z"   and "\<And>\<Delta> \<Gamma> L body \<Theta> z c. \<lbrakk>atom ` domA \<Delta> \<sharp>* (\<Gamma>, L); \<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c (\<Delta> @ \<Gamma>) body L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (Let \<Delta> body) L \<Theta> z"   and "\<And>\<Gamma> b L c. P c \<Gamma> (Bool b) L \<Gamma> (Bool b)"   and "\<And>\<Gamma> scrut L \<Delta> b e\<^sub>1 e\<^sub>2 \<Theta> z c. \<lbrakk>\<Gamma> : scrut \<Down>\<^bsub>L\<^esub> \<Delta> : Bool b; \<And>c. P c \<Gamma> scrut L \<Delta> (Bool b); \<Delta> : (if b then e\<^sub>1 else e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z; \<And>c. P c \<Delta> (if b then e\<^sub>1 else e\<^sub>2) L \<Theta> z\<rbrakk> \<Longrightarrow> P c \<Gamma> (scrut ? e\<^sub>1 : e\<^sub>2) L \<Theta> z" shows "P c x1 x2 x3 x4 x5"
rev_H_uv_2: shows "rev_H' (rev_I uv) = rev_I uv"
subprob_space_distr: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and M' :: "'b measure" assumes "subprob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'"   and "space M' \<noteq> {}" shows "subprob_space (distr M M' f)"
emeasure_Collect_T: fixes P :: "'s stream \<Rightarrow> bool"   and s :: "'s" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (T s) {x \<in> space (T s). P x} = \<integral>\<^sup>+ t. emeasure (T t) {x \<in> space (T t). P (t ## x)} \<partial>measure_pmf (K s)"
Par1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto> a<\<nu>x> \<prec> (P' \<parallel> Q)"
elims: fixes x :: "'a poly"   and xa :: "'a poly"   and y :: "'a poly" assumes "gcd_poly_code_aux x xa = y" obtains p :: "'a poly" and q :: "'a poly" where "x = p" and "xa = q" and "y = (if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
Abs_cong: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and x' :: "'var" assumes "good X"   and "good X'"   and "fresh xs y X"   and "fresh xs y X'"   and "X #[y \<and> x]_xs = X' #[y \<and> x']_xs" shows "Abs xs x X = Abs xs x' X'"
congruentP_def: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "f respectsP \<phi> \<equiv> \<forall>x y. \<phi> x y \<longrightarrow> f x = f y"
has_integral_monotone_convergence_increasing: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and x :: "nat \<Rightarrow> real"   and s :: "'c set"   and g :: "'c \<Rightarrow> real"   and x' :: "real" assumes "\<And>k. (f k has_integral x k) s"   and "\<And>k x. x \<in> s \<Longrightarrow> f k x \<le> f (Suc k) x"   and "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"   and "x \<longlonglongrightarrow> x'" shows "(g has_integral x') s"
mult_less_cancel2_hmset: fixes m :: "hmultiset"   and k :: "hmultiset"   and n :: "hmultiset" shows "(m * k < n * k) = (0 < k \<and> m < n)"
right_idem: fixes a :: "'c"   and b :: "'c" shows "inf (inf a b) b = inf a b"
t_def: shows "t = Syntax.Let (Syntax.Let (Auth (Inj1 Unit)) y (Unauth (Var y))) x (Syntax.Let (Syntax.Let (Auth Unit) z (Unauth (Var z))) y (Var x))"
defined: shows "f ` f.src.carrier \<subseteq> f.dest.carrier"
diff_fun_space_def: shows "dest.diff_fun_space = {f. diff_fun k charts2 f \<and> extensional0 dest.carrier f}"
d_in_ab: shows "d \<in> ideal_generated {a, b}"
linear_strict_order_without_irreflexive: fixes x :: "'a" shows "((x \<le> - top \<and> x \<sqinter> x \<le> x) \<and> x \<squnion> id x = - top) = (x \<sqinter> x \<le> x \<and> x \<squnion> id x = - top)"
1: shows "(\<Sum>m | m \<in> {1..k} \<and> coprime m k. \<Phi> m * unity_root d (int m)) = sum (sum (\<lambda>m. \<Phi> m * unity_root d (int m))) (rcosets\<^bsub>G\<^esub> kernel G H f)"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> G"   and "v \<in> G" shows "sub.inverse u = v"
hfref_cons: fixes f :: "'a \<Rightarrow> 'b Heap"   and g :: "'c \<Rightarrow> 'd nres"   and P :: "'c \<Rightarrow> bool"   and R :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S :: "'d \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'c \<Rightarrow> bool"   and R' :: "('c \<Rightarrow> 'a \<Rightarrow> assn) \<times> ('c \<Rightarrow> 'a \<Rightarrow> assn)"   and S' :: "'d \<Rightarrow> 'b \<Rightarrow> assn" assumes "(f, g) \<in> [P]\<^sub>a R \<rightarrow> S"   and "\<And>x. P' x \<Longrightarrow> P x"   and "\<And>x y. fst R' x y \<Longrightarrow>\<^sub>t fst R x y"   and "\<And>x y. snd R x y \<Longrightarrow>\<^sub>t snd R' x y"   and "\<And>x y. S x y \<Longrightarrow>\<^sub>t S' x y" shows "(f, g) \<in> [P']\<^sub>a R' \<rightarrow> S'"
Bseq_mult: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "Bseq f"   and "Bseq g" shows "Bseq (\<lambda>x. f x * g x)"
assms: shows "filter is_mw_Phase (sys_mem_store_buffers p s) = []"
lNact_notActive: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and k :: "nat" assumes "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> k"   and "k < n" shows "\<not> \<parallel>c\<parallel>\<^bsub>t k\<^esub>"
simps_5: fixes \<phi> :: "'a MFOTL.formula" shows "MFOTL.future_reach (formula.Exists \<phi>) = MFOTL.future_reach \<phi>"
ample_independent: fixes q :: "'state"   and A :: "'action set"   and w :: "'action list" assumes "ample_set q A"   and "A \<subset> {a. en a q}"   and "path w q"   and "A \<inter> set w = {}" shows "Ind A (set w)"
unitary_simps_2: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "unitary A" shows "A * adjoint A = 1\<^sub>m n"
ereal_mult_minus_left: fixes a :: "ereal"   and b :: "ereal" shows "- a * b = - (a * b)"
poly_div_def: fixes p :: "'a poly"   and q :: "'a poly" shows "poly_div p q = resultant (poly_x_mult_y p) (poly_lift q)"
fdomain_vunion: fixes A :: "V"   and B :: "V" shows "\<D>\<^sub>\<bullet> (A \<union>\<^sub>\<circ> B) = \<D>\<^sub>\<bullet> A \<union>\<^sub>\<circ> \<D>\<^sub>\<bullet> B"
Sum_any_product: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite {a. f a \<noteq> (0::'c)}"   and "finite {b. g b \<noteq> (0::'c)}" shows "Sum_any f * Sum_any g = (\<Sum>a. \<Sum>b. f a * g b)"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'ma" shows "bind (return u) f = f u"
WT_resource_of_oracle: fixes s :: "'s_rest'" assumes "IR2 s" shows "\<I>RA \<oplus>\<^sub>\<I> \<I>RU \<turnstile>res RES (callee_of_rest rest2) s \<surd>"
PO_l3_inv8: shows "reach l3 \<subseteq> l3_inv8"