hcju/isabelleps huggingface dataset:
finite_imageI: fixes F :: "'b set"   and h :: "'b \<Rightarrow> 'c" assumes "finite F" shows "finite (h ` F)"
zero_le_pprt: 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero (lattice_ab_group_add.pprt zero sup a)"
exponential_order_uminus_iff: fixes M :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" shows "exponential_order M a (\<lambda>x. - f x) = exponential_order M a f"
ord_pow: fixes x :: "'a"   and k :: "nat" assumes "x \<in> carrier G"   and "k dvd ord x"   and "k \<noteq> 0" shows "ord (x [^] k) = ord x div k"
b2_le_c2: shows "b\<^sup>2 \<le> c\<^sup>2"
residual_network_sel_3: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "source (RES.residual_network f) = source (residual_network f)"
prems_6: shows "Marker \<in> set (msgs (S t j) cid)"
S_comm: fixes n :: "nat" shows "S x n * y = y * S x n"
fimage_finsert: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and B :: "'b fset" shows "f |`| finsert a B = finsert (f a) (f |`| B)"
triangle_F: shows "Adj.\<epsilon>FoF\<eta>.map = F"
case_1: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x1 :: "'d" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
small_imp_big: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> l F g" shows "f \<in> L F g"
leof_trans: fixes a :: "'a nres"   and X :: "'a set"   and c :: "'a nres" assumes "a \<le>\<^sub>n RES X"   and "RES X \<le>\<^sub>n c" shows "a \<le>\<^sub>n c"
assms_4: shows "x ; 1 \<cdot> y\<^sup>T ; 1 = 0"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "sum g {m..n} = g m + sum g {Suc m..n}"
fresh_ssubst_dbfm: fixes i :: "'a"   and V :: "name set"   and A :: "dbfm" assumes "atom i \<sharp> p \<bullet> V"   and "V \<subseteq> Vs" shows "atom i \<sharp> ssubst (vquot_dbfm V A) V F"
homeomorphic_simply_connected: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "simply_connected S" shows "simply_connected T"
preserves_dom: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.dom (V.map f) = V.map (V.dom f)"
exists_algorithm_is_SNF_imp_admits_diagonal_reduction_all: fixes m :: "nat"   and n :: "nat" assumes "\<exists>algorithm. \<forall>A\<in>carrier_mat m n. is_SNF A (algorithm A)" shows "\<forall>A\<in>carrier_mat m n. admits_diagonal_reduction A"
R_subset_Field: fixes R :: "('a \<times> 'a) set" shows "R \<subseteq> Field R \<times> Field R"
real_binomial_eq_mult_binomial_Suc: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "real (n choose k) = (real n + 1 - real k) / (real n + 1) * real (Suc n choose k)"
Zn_iso_DirProds_prime_powers': fixes n :: "nat" assumes "n \<noteq> 0" shows "Z n \<cong> DirProds Z ((\<lambda>p. p ^ multiplicity p n) ` prime_factors n)"
f10: fixes A :: "'a set" shows "A - c1 - c1 = A - c1"
fds_nth_zeta_pos: fixes n :: "nat" assumes "0 < n" shows "fds_zeta $ n = (1::'b)"
ln_minus_ln_floor_bigo: shows "(\<lambda>x. ln x - ln (real (nat \<lfloor>x\<rfloor>))) \<in> O(\<lambda>_. 1)"
assms: fixes e :: "'a \<times> 'a" shows "f e \<le> g e"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> ('b, 'c) antecedent \<Rightarrow> 'a"   and out_port :: "('b, 'c) out_port" shows "P (case out_port of Reg x \<Rightarrow> f1 x | Hyp x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. out_port = Reg x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. out_port = Hyp x21 x22 \<longrightarrow> P (f2 x21 x22)))"
of_nat_mult: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m * n) = times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
mpoly_induct: fixes P :: "'a mpoly \<Rightarrow> bool"   and p :: "'a mpoly" assumes "\<And>m a. P (MPoly_Type.monom m a)"   and "\<And>p1 p2 m a. \<lbrakk>P p1; P p2; p2 = MPoly_Type.monom m a; m \<notin> monomials p1; a \<noteq> (0::'a)\<rbrakk> \<Longrightarrow> P (p1 + p2)" shows "P p"
distinct_53: fixes x5 :: "'a OCL_Types.type"   and x6 :: "'a OCL_Types.type" shows "Set x5 \<noteq> OrderedSet x6"
assms_1: shows "\<not> list_all (\<lambda>x. x = dir.Left) p"
iLastStrong_iEx_conv: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLastStrong t0 I P = (\<diamond> t {iprev t0 I} \<down>< t0. P t)"
assms_3: shows "a = (0::'a) \<or> b \<in> local.span S"
sel_3: fixes x21 :: "'c"   and x22 :: "'c llist" shows "ltl (LCons x21 x22) = x22"
up_z: shows "hom_boundary (int n) (nsphere n) (equator n) up = z"
StableD: fixes F :: "'c program"   and A :: "'c set" assumes "F \<in> Stable A" shows "F \<in> A Co A"
nth_map: fixes n :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "n \<sqsubset> length xs" shows "map f xs ! n = f (xs ! n)"
onorm_scaleR: fixes f :: "'b \<Rightarrow> 'c"   and r :: "real" assumes "bounded_linear f" shows "onorm (\<lambda>x. r *\<^sub>R f x) = \<bar>r\<bar> * onorm f"
frontier_minimal_separating_closed: fixes S :: "'a set"   and C :: "'a set" assumes "closed S"   and "\<not> connected (- S)"   and "C \<in> components (- S)"   and "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected (- T)" shows "frontier C = S"
ring_hom_a_inv: fixes S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c"   and b :: "'a" assumes "ring S"   and "h \<in> ring_hom R S"   and "b \<in> carrier R" shows "h (\<ominus> b) = \<ominus>\<^bsub>S\<^esub> h b"
leibniz_rule'_higher_interval: fixes S :: "'b set"   and a :: "'c"   and b :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and k :: "nat" assumes "convex S"   and "open S"   and "higher_differentiable_on (S \<times> {a..b}) (\<lambda>(x, t). f x t) k" shows "higher_differentiable_on S (\<lambda>x. integral {a..b} (f x)) k"
bot: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f \<in> o[bot](g)"
natural_transformation_\<T>e: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b" assumes "yoneda_lemma_fixed_e C S \<phi> F a e" shows "natural_transformation (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F (transformation_by_components.map (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) (yoneda_lemma.\<T>o C S \<phi> F a e))"
tC_satC: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('fsym, 'psym) lit list" shows "TE.satC \<xi> (tC c) = Ik.satC \<xi> c"
bound: shows "uint w \<in> {2147483649..4294967295}"
elims: fixes G1 :: "'b poly"   and G2 :: "'b poly"   and x :: "nat"   and y :: "'b fract" assumes "h G1 G2 x = y" obtains i :: "nat" where "x = i" and "y = (if i \<le> 1 then 1 else if i = 2 then lead_coeff (F G1 G2 2) ^ (degree (F G1 G2 1) - degree (F G1 G2 2)) else if F G1 G2 i = 0 then 1 else lead_coeff (F G1 G2 i) ^ (degree (F G1 G2 (i - 1)) - degree (F G1 G2 i)) / h G1 G2 (i - 1) ^ (degree (F G1 G2 (i - 1)) - degree (F G1 G2 i) - 1))"
simps_2: fixes a :: "'d"   and s :: "'e" shows "fix_step a (Skip s) = Skip (a, s)"
f_def: shows "f = (\<lambda>x. \<lfloor>(x - a) / d\<rfloor>)"
iso_fixespointwise_chamber_in_int_apartments: fixes A :: "'a set set"   and A' :: "'a set set"   and C :: "'a set"   and f :: "'a \<Rightarrow> 'a" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A'"   and "ChamberComplexIsomorphism A A' f"   and "fixespointwise f C" shows "fixespointwise f (\<Union> (A \<inter> A'))"
f_good: fixes xs :: "'a list" assumes "xs \<in> paths" shows "f xs \<in> S \<and> f xs \<in> set xs"
BuildSeq2_combine: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s1 :: "hf"   and k1 :: "hf"   and y1 :: "hf"   and y1' :: "hf"   and s2 :: "hf"   and k2 :: "hf"   and y2 :: "hf"   and y2' :: "hf"   and y :: "hf"   and y' :: "hf" assumes "BuildSeq2 B C s1 k1 y1 y1'"   and "BuildSeq2 B C s2 k2 y2 y2'"   and "C y y' y1 y1' y2 y2'" shows "BuildSeq2 B C (insf (seq_append (succ k1) s1 s2) (succ (succ (k1 @+ k2))) \<langle>y, y'\<rangle>) (succ (succ (k1 @+ k2))) y y'"
rep_eq: fixes x :: "'c"   and xa :: "'d"   and xb :: "('c, 'd) mapping" shows "mapping.rep (Mapping.update x xa xb) = mapping.rep xb(x \<mapsto> xa)"
composite_simps_1: shows "src (f \<star> g \<star> h) = src h"
fmmap_fset_of_fmap: fixes f :: "'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fset_of_fmap (fmmap f m) = (\<lambda>(k, v). (k, f v)) |`| fset_of_fmap m"
next_change_multiplicity'_unique: fixes n :: "int"   and c :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t" assumes "n \<noteq> 0"   and "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc) \<and> t' \<le> t" shows "\<exists>!c'. next_change_multiplicity' c c' loc t n"
eventually_at_left: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eventually P (at_left x) = (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
zero_closed: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_monoid G" shows "\<zero>\<^bsub>G\<^esub> \<in> carrier G"
d_1: shows "fst x = fst (normalize_quot x) * d"
mem_restr_UNIV: fixes x :: "'a option list" shows "mem_restr UNIV x"
simps_12: fixes x1 :: "'b"   and x31 :: "'b local.term"   and x32 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> x31 \<^bold>\<otimes> x32"
partition_split_def: fixes r :: "'a \<times> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "partition_split r x xs \<equiv> foldr (partition_split_body r x) xs ([], [])"
rep_eq: shows "Rep_lifted_pair bot = (\<lambda>f. (bot, Filters.filter.Abs_filter UNIV))"
eventually_open_cball: fixes X :: "'b set"   and x :: "'b" assumes "open X"   and "x \<in> X" shows "\<forall>\<^sub>F e in at_right 0. cball x e \<subseteq> X"
B3a: shows "\<^bold>\<forall>x. x \<cdot> (dom x) \<cong> x"
simple_functionD_1: fixes M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "simple_function M g" shows "finite (g ` space M)"
PhiWhileOp_Monotone: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "Monotone (PhiWhileOp b \<Phi>)"
iter_tensor_of_H_rep_is_correct: fixes n :: "nat" assumes "1 \<le> n" shows "H \<otimes>\<^bsup>n\<^esup> = Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n))"
ord_Zp_mult: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp" shows "ord_Zp (x \<otimes> y) = ord_Zp x + ord_Zp y"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"
p1_def: shows "p1 = except p {u. deg_pm u < poly_deg p}"
simps_5: fixes M :: "nat \<Rightarrow> complex mat"   and S :: "com"   and P :: "complex mat" shows "wlp (While M S) P = wlp_while (M 0) (M 1) (wlp S) P"
Suc_n_not_le_n: fixes n :: "nat" shows "\<not> Suc n \<le> n"
mr_first: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and b :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and c :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and s :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp b)"   and "\<forall>r. r \<in> set c \<longrightarrow> x \<notin> dom (Cp r)"   and "s = b # c" shows "applied_rule_rev Cp x s = \<lfloor>b\<rfloor>"
foldr_conv_fold: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list" shows "foldr f xs = fold f (rev xs)"
i_less: fixes a :: "real"   and b :: "real pdevs"   and i :: "nat" assumes "(a, b) \<in> set XS"   and "i < degree b" shows "i < min d d'"
tagged_division_of_finite: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "s tagged_division_of i" shows "finite s"
path_measure_eq_repcs''_new: fixes s :: "'s \<times> ('c \<Rightarrow> real)"   and cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and \<psi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool" assumes "abss s = state cfg"   and "cfg \<in> R_G.valid_cfg"   and "repcs s cfg \<in> MDP.valid_cfg"   and "\<And>x. pred_stream (\<lambda>s. s \<in> S) x \<Longrightarrow> pred_stream (\<lambda>s. \<phi> (reps (abss s)) = \<phi> s) (state (repcs s cfg) ## x)"   and "\<And>x. pred_stream (\<lambda>s. s \<in> S) x \<Longrightarrow> pred_stream (\<lambda>s. \<psi> (reps (abss s)) = \<psi> s) (state (repcs s cfg) ## x)" shows "emeasure (R_G.T cfg) {x \<in> space R_G.St. (holds (absp \<phi>) suntil holds (absp \<psi>)) (state cfg ## x)} = emeasure (MDP.T (repcs s cfg)) {x \<in> space MDP.St. (holds \<phi> suntil holds \<psi>) (state (repcs s cfg) ## x)}"
min_enat_simps_5: fixes q :: "enat" shows "min \<infinity> q = q"
diff_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and d :: "'c"   and c :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "less_eq a b"   and "less_eq d c" shows "less_eq (minus a c) (minus b d)"
cont_on_right: shows "continuous_on (path_image (\<lambda>y. twoC (1, y))) F"
map_1: fixes f :: "'d \<Rightarrow> 'b" shows "map_option f None = None"
X: shows "X \<in> sets (N \<Otimes>\<^sub>M M)"
induct: fixes x1 :: "Val list"   and x2 :: "Var list"   and x3 :: "(Var \<times> Val) list"   and x4 :: "Val list"   and P :: "Val list \<Rightarrow> Var list \<Rightarrow> (Var \<times> Val) list \<Rightarrow> Val list \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> Frame"   and "\<And>oo ops. oo = ops \<Longrightarrow> P ops [] emp oo"   and "\<And>oo par S ops R x v. \<lbrakk>(oo, par, S, ops) \<in> Frame; P oo par S ops; R = S[x\<mapsto>v]\<rbrakk> \<Longrightarrow> P (v # oo) (x # par) R ops" shows "P x1 x2 x3 x4"
\<alpha>_\<beta>: shows "LeftDerivation \<alpha> ((i, r) # D) \<beta>"
isotopic_amicable_pair: fixes m :: "nat"   and n :: "nat"   and g :: "nat"   and M :: "nat"   and N :: "nat"   and h :: "nat" assumes "m Amic n"   and "1 \<le> m"   and "1 \<le> n"   and "m = g * M"   and "n = g * N"   and "real (Esigma h) = real h / real g * real (Esigma g)"   and "h \<noteq> g"   and "1 < h"   and "1 < g"   and "gcd g M = 1"   and "gcd g N = 1"   and "gcd h M = 1"   and "gcd h N = 1" shows "(h * M) Amic (h * N)"
ds_faithful: fixes x :: "'a"   and y :: "'a" assumes "ds x \<le> ds y" shows "x \<le> y"
dbl_simps_3: shows "neg_numeral_class.dbl (1::'a) = (2::'a)"
and_mask2: fixes w :: "'a word"   and n :: "nat" shows "w << n >> n = w && mask (size w - n)"
filter_distinct_at_special: fixes xs :: "'a list"   and as :: "'a list"   and u :: "'a"   and bs :: "'a list"   and us :: "'a list"   and zs :: "'a list"   and hd_us :: "'a"   and tl_us :: "'a list" assumes "distinct xs"   and "xs = as @ u # bs"   and "filter (\<lambda>v. v = u \<or> v \<in> set us) xs = u # us"   and "set zs \<inter> set xs \<subseteq> {u} \<union> set as"   and "us = hd_us # tl_us" shows "filter (\<lambda>v. v \<in> set us) (zs @ bs) = us \<and> hd_us \<in> set bs"
cp_def: shows "cp \<equiv> if 0 \<le> n then reversepath (n_circ (nat n)) else n_circ (nat (- n))"
assms: shows "\<forall>x. x \<in> s \<longrightarrow> x < k"
hyps_2: fixes s :: "'a regex" assumes "local.safe_regex Past Lax r2"   and "s \<in> rpd test i r2" shows "local.safe_regex Past Lax s"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "infinite A" shows "sum g A = (0::'d)"
eNorm_zero: fixes N :: "'c quasinorm" shows "eNorm N (0::'c) = 0"
sams_lea2_suma2__lea456: fixes A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p"   and G' :: "'p"   and H' :: "'p"   and I' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p" assumes "A' B' C' LeA A B C"   and "G H I LeA G' H' I'"   and "SAMS A B C D E F"   and "A B C D E F SumA G H I"   and "A' B' C' D' E' F' SumA G' H' I'" shows "D E F LeA D' E' F'"
denseI: fixes A :: "'a set" assumes "carrier \<subseteq> closure A" shows "A dense"
finprod_closed: fixes f :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "f \<in> A \<rightarrow> carrier M" shows "finsum M f A \<in> carrier M"
lessThan_def: fixes less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and u :: "'e" shows "ord.lessThan less u \<equiv> {x. less x u}"
atLeastAtMost_iff: fixes i :: "'c word"   and l :: "'c word"   and u :: "'c word" shows "(i \<in> signed.atLeastAtMost l u) = (l \<le>s i \<and> i \<le>s u)"
thread_oks_ts_change: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ts' :: "'f \<Rightarrow> ('i \<times> 'j \<Rightarrow>f nat) option"   and tas :: "('f, 'k, 'l) new_thread_action list" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "thread_oks ts tas = thread_oks ts' tas"
is_parallel_solution_for_problem_def: fixes \<Pi> :: "('c, 'd) strips_problem_scheme"   and \<pi> :: "'c strips_operator list list" shows "STRIPS_Semantics.is_parallel_solution_for_problem \<Pi> \<pi> \<equiv> \<Pi>\<^sub>G \<subseteq>\<^sub>m execute_parallel_plan (\<Pi>\<^sub>I) \<pi> \<and> list_all (list_all (\<lambda>op. ListMem op (\<Pi>\<^sub>\<O>))) \<pi>"
fixp_def: fixes Sup :: "'g set \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'g" assumes "class.ccpo Sup less_eq less" shows "ccpo.fixp Sup less_eq f = Sup (ccpo.iterates Sup less_eq f)"
Ide_implies_Arr: fixes t :: "'a E.term" assumes "E.Ide t" shows "E.Arr t"
has_domain_iff_arr: fixes f :: "'b" shows "(\<phi>.B.domains f \<noteq> {}) = \<phi>.B.arr f"
mono_transI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P Q. \<lbrakk>sound P; sound Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "mono_trans t"
Gamma_series'_nonpos_Ints_LIMSEQ: fixes z :: "'a" assumes "z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "Gamma_series' z \<longlonglongrightarrow> (0::'a)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'c \<Rightarrow> 'a"   and option :: "'c option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (\<forall>x2. option = Some x2 \<longrightarrow> P (f2 x2)))"
hyps: fixes v :: "'v" assumes "finite E"   and "finite V"   and "is_trail v xs v'"   and "even (degree v' G)" shows "num_of_odd_nodes (rem_unPath xs G) = num_of_odd_nodes G + (if even (degree v G) \<and> v \<noteq> v' then 2 else 0)"
h_simp: fixes E :: "'a set" shows "rupture_surj A P ` poly_of_const ` E = (rupture_surj A P \<circ> poly_of_const) ` E"
image_set: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "f ` set xs = set (map f xs)"
map_eq_imp_length_eq: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and g :: "'d \<Rightarrow> 'b"   and ys :: "'d list" assumes "map f xs = map g ys" shows "length xs = length ys"
F_pos: fixes n :: "nat"   and x :: "'a" shows "0 \<le> - u n x / real n"
inj_comm_monoid_add_hom_axioms: shows "inj_comm_monoid_add_hom (map_poly (\<lambda>a. [:a:]))"
neq_Nil_conv: fixes xs :: "'g list" shows "(xs \<noteq> []) = (\<exists>y ys. xs = y # ys)"
env_delete_below_arg: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e" shows "env_delete x m \<sqsubseteq> m"
A_def: shows "A = {x \<in> space M. f x = g x}"
\<W>_elem_in_TokensAt: fixes P :: "(symbol \<times> character list) list set"   and u :: "symbol \<times> character list"   and k :: "nat" assumes "P \<subseteq> \<PP>"   and "u \<in> \<W> P k" shows "u \<in> TokensAt k (Gen P)"
map_cong: fixes x :: "'a pmf"   and ya :: "'a pmf"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> set_pmf ya \<Longrightarrow> f z = g z" shows "map_pmf f x = map_pmf g ya"
inf0: fixes a :: "'a" shows "a \<equiv> a \<sqinter> \<top>"
replicate_mset_eq_empty_iff: fixes n :: "nat"   and a :: "'a" shows "(replicate_mset n a = {#}) = (n = 0)"
less_div_imp_mult_add_divisor_le: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x < n div m" shows "x * m + m \<le> n"
sfis_mono: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> sfis f M"   and "b \<in> sfis g M"   and "f \<le> g" shows "a \<le> b"
bin_add_not: fixes x :: "int" shows "x + NOT x = - 1"
freshAbs_psubstAbs_E2: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "goodAbs A"   and "goodEnv rho"   and "rho ys y = Some Y"   and "freshAbs zs z A $[rho]" shows "freshAbs ys y A \<or> fresh zs z Y"
has_contour_integral_bound_linepath: fixes f :: "complex \<Rightarrow> complex"   and i :: "complex"   and a :: "complex"   and b :: "complex"   and B :: "real" assumes "(f has_contour_integral i) (linepath a b)"   and "0 \<le> B"   and "\<And>x. x \<in> closed_segment a b \<Longrightarrow> cmod (f x) \<le> B" shows "cmod i \<le> B * cmod (b - a)"
vfsequence_vlrestriction_vcard: shows "T \<restriction>\<^sup>l\<^sub>\<circ> vcard T = T"
mod_mult_self2: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times b c)) b = modulo a b"
exi: shows "\<exists>!Q. Q \<in> subgroups_of_size q"
eq: shows "enn2real (1 / U' y y) = - 1 / - r"
self_in_orbit_step: fixes s :: "'c"   and f :: "'c \<Rightarrow> 'c" assumes "s \<in> orbit f s" shows "orbit f (f s) = orbit f s"
count_roots_above: fixes a :: "real" assumes "p' \<noteq> 0" shows "sign_changes (sturm_squarefree p) a - sign_changes_inf (sturm_squarefree p) = card {x. a < x \<and> poly p' x = 0}"
not_def: fixes P :: "'b \<Rightarrow> bool" shows "\<not> P \<equiv> \<lambda>A. \<not> P A"
conductor_ge_iff: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat" assumes "dcharacter n \<chi>" shows "(a \<le> dcharacter.conductor n \<chi>) = (\<forall>d. dcharacter.induced_modulus n \<chi> d \<longrightarrow> a \<le> d)"
countably_subadditive_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> 'c" shows "countably_subadditive M f = (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M \<longrightarrow> f (\<Union> (range A)) \<le> (\<Sum>i. f (A i)))"