hcju/isabelleps huggingface dataset:
nds_invars_def: fixes nds :: "('a, 'b) node_scheme set" shows "nds_invars nds \<equiv> inj_on name nds \<and> finite nds \<and> expand_init \<notin> name ` nds \<and> (\<forall>nd\<in>nds. finite (old nd) \<and> incoming nd \<subseteq> insert expand_init (name ` nds))"
map_poly_idI: fixes p :: "'b poly"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>x. x \<in> set (coeffs p) \<Longrightarrow> f x = x" shows "map_poly f p = p"
stc_inverse: fixes x :: "complex star" assumes "x \<in> HFinite"   and "stc x \<noteq> 0" shows "stc (inverse x) = inverse (stc x)"
wf_merge: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and n :: "nat"   and I :: "nat set"   and t1 :: "'b option list"   and J :: "nat set"   and t2 :: "'b option list"   and V :: "nat set"   and t :: "'b option list" assumes "getIJ getIJ"   and "wf_tuple n I t1"   and "wf_tuple n J t2"   and "V = I \<union> J"   and "t = New_max.merge t1 t2" shows "wf_tuple n V t"
test_bit_split': fixes c :: "'d word"   and a :: "'b word"   and b :: "'c word" shows "word_split c = (a, b) \<longrightarrow> (\<forall>n m. b !! n = (n < size b \<and> c !! n) \<and> a !! m = (m < size a \<and> c !! (m + size b)))"
map_id: fixes t :: "('d, 'e, 'f) generat" shows "map_generat id id id t = t"
left_neutral: fixes a :: "'q" shows "(0::'q) + a = a"
openin_map_topology: fixes p :: "'d \<Rightarrow> 'c"   and X :: "'d topology" shows "openin (map_topology p X) = (\<lambda>U. U \<subseteq> p ` topspace X \<and> openin X (p -` U \<inter> topspace X))"
max_input_Times: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_input (Times a1 a2) = max (max_input a1) (max_input a2)"
id1: shows "(V = [arith_ops_record.one ff_ops]) = (v = 1)"
qPsubstAll_preserves_alphaAll_qSwapAll: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodEnv rho" shows "(qGood X \<longrightarrow> (\<forall>Y. X #= Y \<longrightarrow> X #[[rho]] #= Y #[[rho]]) \<and> (\<forall>xs z1 z2. qFreshEnv xs z1 rho \<and> qFreshEnv xs z2 rho \<longrightarrow> X #[[z1 \<and> z2]]_xs #[[rho]] #= X #[[rho]] #[[z1 \<and> z2]]_xs)) \<and> (qGoodAbs A \<longrightarrow> (\<forall>B. A $= B \<longrightarrow> A $[[rho]] $= B $[[rho]]) \<and> (\<forall>xs z1 z2. qFreshEnv xs z1 rho \<and> qFreshEnv xs z2 rho \<longrightarrow> A $[[z1 \<and> z2]]_xs $[[rho]] $= A $[[rho]] $[[z1 \<and> z2]]_xs))"
t: shows "T\<^sub>p_on_rand' BIT (type0 init x y) qs = 0"
ax5: fixes v :: "'c \<Rightarrow> 'd" shows "\<turnstile> \<box>[v$ \<noteq> $v]_v"
fresh_psubst: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho" shows "fresh zs z X #[rho] = (\<forall>ys y. fresh ys y X \<or> freshImEnvAt zs z rho ys y)"
substitute_substitute'': fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and t :: "'a ttree" shows "(xs \<in> paths (substitute f T t)) = (\<exists>xs'\<in>paths t. substitute'' f T xs' xs)"
log_of_power_less: fixes m :: "nat"   and b :: "real"   and n :: "nat" assumes "real m < b ^ n"   and "1 < b"   and "0 < m" shows "log b (real m) < real n"
set_iterator_def: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set" shows "set_iterator it S0 \<equiv> set_iterator_genord it S0 (\<lambda>_ _. True)"
glbound_in_ofD_lbound: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "ordering.glbound_in_of less_eq P x y b" shows "ordering.lbound_of less_eq x y b"
sup_left_commute: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
invertible_proportionality_def: shows "invertible_proportionality \<equiv> Restr proportionality (Collect invertible)"
f_mono: shows "mono (\<lambda>S. W \<union> S \<inter> V \<union> directly_attracted p (S \<inter> V))"
right_uniqueI: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x y; A x z\<rbrakk> \<Longrightarrow> y = z" shows "right_unique A"
write_cpu_pc_privilege: fixes s' :: "'b sparc_state"   and w :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_cpu w PC s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
qItem_simps_13: fixes xsa :: "'e"   and xa :: "'f"   and X :: "('c, 'd, 'e, 'f, 'g) qTerm"   and xs' :: "'e"   and x' :: "'f"   and X' :: "('c, 'd, 'e, 'f, 'g) qTerm" shows "(qAbs xsa xa X $= qAbs xs' x' X') = (xsa = xs' \<and> (\<exists>y. y \<notin> {xa, x'} \<and> qAFresh xsa y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> xa]]_xsa #= X' #[[y \<and> x']]_xs'))"
cong_mult_self_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times b a) zero a"
card_eq_0_iff: fixes A :: "'a set" shows "(card A = 0) = (A = {} \<or> infinite A)"
diff_fun_times: fixes a :: "'a \<Rightarrow> 'c"   and b :: "'a \<Rightarrow> 'c" assumes "diff_fun k charts a"   and "diff_fun k charts b" shows "diff_fun k charts (a * b)"
tendsto_mult_ereal_pos: fixes f :: "'b \<Rightarrow> ereal"   and l :: "ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal"   and m :: "ereal" assumes "(f \<longlongrightarrow> l) F"   and "(g \<longlongrightarrow> m) F"   and "0 < l"   and "0 < m" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
reds_to_red_n: fixes c :: "com"   and s :: "char list \<Rightarrow> val option"   and c' :: "com"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>c,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>" shows "\<exists>n. \<langle>c,s\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>c',s'\<rangle>"
Maclaurin_exp_le: fixes x :: "real"   and n :: "nat" shows "\<exists>t. \<bar>t\<bar> \<le> \<bar>x\<bar> \<and> exp x = (\<Sum>m<n. x ^ m / fact m) + exp t / fact n * x ^ n"
has_derivative_compose: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and g :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x))" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
span_is_subspace: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n" shows "subspace class_ring (local.span S) V"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" shows "(\<Psi>', R \<parallel> S, S \<parallel> R) \<in> Rel"
rtrancl_eta_subst': fixes s :: "dB"   and t :: "dB"   and u :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "s[u/i] \<rightarrow>\<^sub>\<eta>\<^sup>* t[u/i]"
list_vec: fixes xs :: "'b list" shows "list_of_vec (vec_of_list xs) = xs"
asymp_equiv_harm: shows "harm \<sim>[sequentially] (\<lambda>n. ln (real n))"
Loop_sound_lemma: fixes c :: "stmt"   and n :: "nat"   and P :: "assn"   and x :: "vname"   and s :: "state"   and c0 :: "stmt"   and n0 :: "nat"   and t :: "state" assumes "\<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<and> s<x> \<noteq> Null \<longrightarrow> P t"   and "s -c0-n0\<rightarrow> t"   and "P s"   and "c0 = While (x) c"   and "n0 = n" shows "P t \<and> t<x> = Null"
lex_scaleR_nonneg: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and r :: "real" assumes "Counterclockwise_2D_Arbitrary.lex a b"   and "0 \<le> r" shows "Counterclockwise_2D_Arbitrary.lex a (a + r *\<^sub>R (b - a))"
subgroup_Inter: fixes A :: "'f set set"   and G :: "('f, 'g) monoid_scheme" assumes "\<And>H. H \<in> A \<Longrightarrow> subgroup H G"   and "A \<noteq> {}" shows "subgroup (\<Inter> A) G"
has_fields_fun: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
intro: fixes r :: "('d \<times> 'd) set" assumes "Well_order r" shows "wo_rel r"
d_divergences: fixes P :: "'a process" assumes "deterministic P" shows "divergences P = {}"
set_sel_1: fixes a :: "'a list" assumes "a \<noteq> []" shows "hd a \<in> list.set a"
fin_not_empty: shows "finite {i. F i \<noteq> {}}"
MPair_imp_eq_left: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" assumes "MPair X Y = MPair X' Y'" shows "X = X'"
Expr2ZFDepthSuc: fixes e :: "(ZF, ZF) Expression"   and n :: "nat" assumes "ZFDepth (Expr2ZF e) = Suc n" shows "ZFType (Expr2ZF e) = 1"
lift_Suc_mono_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less (f n) (f (Suc n))"   and "n < n'" shows "less (f n) (f n')"
intros_1: fixes x :: "'b set"   and B :: "'b set set" assumes "x \<in> B" shows "x \<in> topo B"
Meth: fixes A :: "triple set"   and C :: "cname"   and P :: "assn"   and m :: "mname"   and Q :: "assn" assumes "\<forall>D. A \<turnstile> {\<lambda>s'. \<exists>s a. s<This> = Addr a \<and> D = obj_class s a \<and> D \<preceq>C C \<and> P s \<and> s' = init_locs D m s} Impl (D, m) {Q}" shows "A \<turnstile> {P} Meth (C, m) {Q}"
prems_2: fixes k :: "nat" assumes "k < Suc n" shows "g k \<in> carrier_mat d d"
state_set_card_i: fixes a :: "'a"   and X :: "'a list" shows "[a] \<notin> (#) a ` state_set X"
continuous_from_closed_graph: fixes T :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "compact T"   and "f ` S \<subseteq> T"   and "closed ((\<lambda>x. (x, f x)) ` S)" shows "continuous_on S f"
U_3: shows "U \<subseteq> - T \<union> - cball 0 r"
index_hma_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and i :: "'d"   and j :: "'c" shows "index_hma A i j \<equiv> A $h i $h j"
inf_continuous_gfp: fixes F :: "'a \<Rightarrow> 'a" assumes "inf_continuous F" shows "gfp F = (\<Sqinter>i. (F ^^ i) \<top>)"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c Predicate.pred \<Rightarrow> 'b"   and f3 :: "'c Predicate.pred \<Rightarrow> 'c Predicate.seq \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c Predicate.pred" shows "(case seq.Insert x21 x22 of seq.Empty \<Rightarrow> f1 | seq.Insert x xa \<Rightarrow> f2 x xa | seq.Join x xa \<Rightarrow> f3 x xa) = f2 x21 x22"
epath_subset_E: fixes E :: "'b uprod set"   and u :: "'b"   and p :: "'b uprod list"   and v :: "'b" assumes "epath E u p v" shows "set p \<subseteq> E"
redundant_clause_def: fixes C :: "'a eclause"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" shows "redundant_clause C S \<sigma> C' = (\<exists>S'\<subseteq>instances S. set_entails_clause (clset_instances S') (cl_ecl C) \<and> (\<forall>x\<in>S'. subterms_inclusion (subst_set (trms_ecl (fst x)) (snd x)) (trms_ecl C)) \<and> (\<forall>x\<in>S'. (mset_ecl (fst x, snd x), mset_cl (C', \<sigma>)) \<in> mult (mult trm_ord) \<or> mset_ecl (fst x, snd x) = mset_cl (C', \<sigma>)))"
pfp_def: fixes f :: "'a \<Rightarrow> 'a" shows "pfp f = while_option (\<lambda>x. \<not> f x \<le> x) f"
SUP_cong: fixes A :: "'f set"   and B :: "'f set"   and C :: "'f \<Rightarrow> 'g"   and D :: "'f \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
vintersection_vunion_right: fixes A :: "V"   and B :: "V"   and C :: "V" shows "A \<inter>\<^sub>\<circ> (B \<union>\<^sub>\<circ> C) = A \<inter>\<^sub>\<circ> B \<union>\<^sub>\<circ> A \<inter>\<^sub>\<circ> C"
closure_closed: fixes T :: "'b set set"   and a :: "'b set" assumes "topology T" shows "closure\<^bsub>T\<^esub> a closed\<^bsub>T\<^esub>"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "(a \<cong> a') = (\<exists>f. \<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> local.iso f)"
pareto_loser_in_alts: fixes x :: "'alt pmf" assumes "x \<in> pareto_losers (SD \<circ> R)" shows "x \<in> lotteries"
LeastI2_order: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y"   and "\<And>x. \<lbrakk>P x; \<forall>y. P y \<longrightarrow> less_eq x y\<rbrakk> \<Longrightarrow> Q x" shows "Q (ord.Least less_eq P)"
le_funI: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
vimage_vintersection: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<circ> (A \<inter>\<^sub>\<circ> B) \<subseteq>\<^sub>\<circ> r `\<^sub>\<circ> A \<inter>\<^sub>\<circ> r `\<^sub>\<circ> B"
assms_3: shows "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
All2_cong: fixes H :: "fm set"   and t :: "tm"   and t' :: "tm"   and A :: "fm"   and A' :: "fm"   and i :: "name" assumes "H \<turnstile> t EQ t'"   and "H \<turnstile> A IFF A'"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> All2 i t A IFF All2 i t' A'"
ask_writer_def: fixes f :: "'r \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.ask_writer f = WriterT (ask (\<lambda>r. run_writer (f r)))"
extremum_unique: fixes a :: "'e multiset" shows "(a \<subseteq># {#}) = (a = {#})"
cases: fixes a :: "expr" assumes "lc a" obtains xT :: "char list \<times> Unary_PCF.type" where "a = \<langle>xT\<rangle>"   | b :: "bool" where "a = b"   | e1 :: "expr" and e2 :: "expr" where "a = e1__ ? e2__" and "lc e1__" and "lc e2__"   | e1 :: "expr" and e2 :: "expr" where "a = e1 \<cdot> e2" and "lc e1" and "lc e2"   | T :: "Unary_PCF.type" and X :: "(char list \<times> Unary_PCF.type) fset" and e :: "expr" where "a = \<Lambda>\<langle>T\<rangle> e" and "\<forall>x. (x, T) |\<notin>| X \<longrightarrow> lc (open0_Var (x, T) e)"
mem_cone_hull: fixes x :: "'c"   and S :: "'c set"   and c :: "real" assumes "x \<in> S"   and "0 \<le> c" shows "c *\<^sub>R x \<in> cone hull S"
ex1_class_image: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c"   and X :: "'b set" assumes "refl_on A r"   and "f respects r"   and "X \<in> A // r" shows "\<exists>!b. b \<in> f ` X"
has_domain_iff_arr: fixes f :: "'s" shows "(domains f \<noteq> {}) = arr f"
assms_1: fixes n :: "nat" shows "U n \<in> events"
process_surj_pair: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "(FAILURES P, DIVERGENCES P) = P"
rev_subsetD: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
qFreshEnv_def: fixes zs :: "'f"   and z :: "'g"   and rho :: "'f \<Rightarrow> 'g \<Rightarrow> ('h, 'i, 'f, 'g, 'j) qTerm option" shows "qFreshEnv zs z rho \<equiv> rho zs z = None \<and> (\<forall>xs. liftAll (qFresh zs z) (rho xs))"
two_cong_0_iff: fixes p :: "nat" shows "[2 = 0] (mod p) = (p = 1 \<or> p = 2)"
add_left_mono: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> b" shows "dbm_add c a \<preceq> dbm_add c b"
run_eval_match_def: shows "run_eval_match \<equiv> SOME f. \<forall>env v pes err_v s. evaluate_match True env s v pes err_v (f env v pes err_v s)"
bin_join_table: assumes "table n A X"   and "table n B Y" shows "bin_join n A X b B Y = join X b Y"
gcd_idem_nat: fixes x :: "nat" shows "gcd x x = x"
complete_lattice_partial_function_definitions_dual: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "partial_function_definitions (\<lambda>x y. less_eq y x) Inf"
1: shows "\<forall>a\<in>{a \<in> set as. is_dvd a}. hd_coeff a = 1"
right_neutral: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and a :: "'c" assumes "monoid f z" shows "f a z = a"
complex_of_real_mult_Complex: fixes r :: "real"   and x :: "real"   and y :: "real" shows "cor r * Complex x y = Complex (r * x) (r * y)"
fv_subterms_set: fixes M :: "('d, 'c) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms\<^sub>s\<^sub>e\<^sub>t M) = fv\<^sub>s\<^sub>e\<^sub>t M"
decs_clock_monotone: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s d res"   and "ck = True" shows "clock (fst res) \<le> clock s"
rbt_less_prop: fixes k :: "'a"   and t :: "('a, 'c) rbt" shows "(t |\<guillemotleft> k) = (\<forall>x\<in>set (keys t). x < k)"
purgeIdle_simps_1: shows "GM_sec_model.purgeIdle [] = []"
size_less_than_median_of_medians: fixes d :: "nat"   and med :: "'b list \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" assumes "\<And>xs. \<lbrakk>xs \<noteq> []; length xs \<le> Suc (2 * d)\<rbrakk> \<Longrightarrow> is_median (med xs) xs"   and "is_median x (map med (chop (Suc (2 * d)) xs))" shows "size {#y \<in># mset xs. y < x#} \<le> nat \<lceil>(3 * real d + 1) / real (2 * (2 * d + 1)) * real (length xs)\<rceil> + 5 * d div 2 + 1"
s1: shows "set (labels \<sigma>3'') \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
cases: fixes a1 :: "'a list"   and a2 :: "sequent" assumes "(a1, a2) \<in> Ax" obtains i :: "nat" and xs :: "NominalSequents.var list" where "a1 = []" and "a2 = ( \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>)"   | "a1 = []" and "a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>)"
inner_zero_left: fixes x :: "real" shows "0 * x = 0"
genideal_one: shows "Idl\<^bsub>\<Z>\<^esub> {1} = UNIV"
prems_3: shows "\<forall>p1 p2. p1 \<in> insert p P \<and> p2 \<in> insert p P \<and> p1 \<noteq> p2 \<longrightarrow> comm_monoid_mult_class.coprime p1 p2"
igFreshIGAbs2STR_imp_igFreshIGAbs2: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igFreshIGAbs2STR MOD" shows "igFreshIGAbs2 MOD"
intros_1: fixes \<delta> :: "real \<Rightarrow> real"   and a :: "real" shows "fine \<delta> (a, a) []"
s_Unroll_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Unroll e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Unroll e'a" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "e = Roll e'" and "\<pi>' = \<pi>" and "value e'"
evaluate_match_total: fixes env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v" shows "\<exists>s' r. evaluate_match True env s v pes v' (s', r)"
eq_TT_dest: fixes x :: "'b"   and y :: "'b" assumes "eq\<cdot>x\<cdot>y = TT" shows "x = y"
IntVal_in_type_universe: fixes v :: "int" shows "IntVal v \<in> type_universe INTEG"
is_ddi_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and v :: "'b"   and m :: "nat" shows "IFC_def.is_ddi prob n \<pi> v m = (IFC_def.is_path prob \<pi> \<and> m < n \<and> v \<in> IFC_def.reads prob (\<pi> n) \<inter> IFC_def.writes prob (\<pi> m) \<and> (\<forall>l\<in>{m<..<n}. v \<notin> IFC_def.writes prob (\<pi> l)))"
rGamma_1: shows "rGamma (1::'a) = (1::'a)"
flopped_half_chamber_systems_fg: shows "folding_g.\<C> - f \<turnstile> folding_g.\<C> = g \<turnstile> folding_g.\<C>"
prime_factors_gcd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize (gcd a b)) = set_mset (factorial_semiring.prime_factorization zero times one normalize a) \<inter> set_mset (factorial_semiring.prime_factorization zero times one normalize b)"
one_eq_numeral_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "Num.num" assumes "class.semiring_char_0 one times plus zero" shows "(one = numeral.numeral one plus n) = (num.One = n)"
remove1_filter_not: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "\<not> P x" shows "remove1 x (filter P xs) = filter P xs"
times: fixes ops :: "'c arith_ops_record"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R ===> R) (arith_ops_record.times ops) (*)"
preserves_hseq: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.arr (G \<mu>) \<and> D.arr (G \<nu>) \<and> src\<^sub>D (G \<mu>) = trg\<^sub>D (G \<nu>)"
rt_fun: fixes t :: "'a"   and t' :: "'a"   and u :: "'a" assumes "R\<^sup>*\<^sup>* t t'" shows "R\<^sup>*\<^sup>* (app t u) (app t' u)"
siso_Par: fixes cl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "sisoL cl" shows "siso (Par cl)"
k_def: shows "k = (LEAST k. 0 < k \<and> [a ^ k = 1] (mod n))"
irreducible\<^sub>d_lifting: fixes n :: "nat"   and f :: "int poly" assumes "n \<noteq> 0"   and "degree (poly_mod.Mp (p ^ n) f) = degree_m f"   and "irreducible\<^sub>d_m f" shows "poly_mod.irreducible\<^sub>d_m (p ^ n) f"
simps: fixes s1 :: "'e"   and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" shows "P14 (s1, s2) = (initialized s2 \<longrightarrow> initVals s2 \<noteq> {})"
at_simp: fixes a :: "'a"   and j :: "'j"   and \<tau> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A.arr a"   and "J.arr j"   and "A_B.arr (\<tau> j)" shows "at a \<tau> j = A_B.Map (\<tau> j) a"
a1: shows "s' = snd (fst (logical_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0"
open_Collect_const: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "bool" assumes "class.topological_space open" shows "open {x. P}"
aezfun_decomp_aezdeltafun: fixes c :: "('a, 'b) aezfun" shows "\<exists>ras. set (map snd ras) = supp (aezfun c) \<and> c = (\<Sum>(r, a)\<leftarrow>ras. r \<delta>\<delta> a)"