hcju/isabelleps huggingface dataset:
axioms_3: fixes \<eta> :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and G' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'" shows "Group_Theory.group G' composition' unit'"
hyps: shows "a ^ n = (\<Sum>k = 0..n. of_nat (Stirling n k) * ffact k a)"
assms_1: shows "ipassmt_sanity_nowildcards ipassmt"
add_diff_assoc2: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subseteq># b" shows "b - a + c = b + c - a"
characterize_consistent_signs_at_roots_def: fixes p :: "real poly"   and qs :: "real poly list" shows "characterize_consistent_signs_at_roots p qs = remdups (map (signs_at qs) (characterize_root_list_p p))"
nonzero_memI: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "a \<in> nonzero R"
extremed_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "extremed A r \<equiv> \<exists>e. extreme A r e"
B: shows "\<forall>\<^sub>F n in sequentially. Sigma_Algebra.measure (distr P borel (\<lambda>x. (g x + birkhoff_sum f n x) / B n)) {x. (L - 1) * a \<le> \<bar>x\<bar>} < alpha"
rconj_relator_freeword: fixes S :: "'b set"   and R :: "'b list set"   and rs :: "'b list"   and xs :: "('b \<times> bool) list" assumes "GroupWithGeneratorsRelators S R"   and "rs \<in> R"   and "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "rconjby (Abs_freeword xs) (Abs_freeword (map (charpair S) rs)) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
simps_1: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat option"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and i :: "(String.literal \<times> value list) stream" shows "shd (make_full_observation e s d p i) = \<lparr>statename = s, datastate = d, action = shd i, output = p\<rparr>"
pts_symbolic_c_preserves_well_analyzed: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "well_analyzed \<A>" shows "well_analyzed \<A>'"
pwd_spy_id_prik_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "let (s, S, A, B, C, D) = pwd_spy_id_prik n; SK = (Some S, {A, B}, {C, D}) in pwd_spy_id_prik_state n (S, A, B, C, D) \<subseteq> s \<and> {SesKey SK} \<in> key_sets (Pwd n) (crypts (Log -` spied s))"
prism_sumr_def: shows "Inr\<^sub>\<triangle> = \<lparr>prism_match = \<lambda>v. case v of Inl x \<Rightarrow> Map.empty x | Inr x \<Rightarrow> Some x, prism_build = Inr\<rparr>"
set_delete_axioms: shows "set_delete rs.\<alpha> rs.invar rs.delete"
proj2_intersection_incident_2: fixes l :: "proj2_line"   and m :: "proj2_line" shows "proj2_incident (proj2_intersection l m) m"
reflp_eq: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "reflp R = ((=) \<le> R)"
preserves_reflects_arr: fixes f :: "'a" shows "hom.arr (\<phi>.map f) = hom.arr f"
H_cons: fixes p :: "'a"   and p' :: "'a"   and q' :: "'a"   and q :: "'a"   and x :: "'a" assumes "tp \<le> tp'"   and "tq' \<le> tq"   and "H p' x q'" shows "H p x q"
prems: shows "(c, (a # w1) @ w2, c') \<in> trcl t"
T: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "e2ennreal (esssup M (\<lambda>x. ereal \<bar>(f + g) x\<bar>)) \<le> e2ennreal (esssup M (\<lambda>x. ereal \<bar>f x\<bar>)) + e2ennreal (esssup M (\<lambda>x. ereal \<bar>g x\<bar>))"
keys_impl_of: fixes t :: "('b, 'c) RBT.rbt" shows "RBT_Impl.keys (rbt.impl_of t) = RBT.keys t"
assms_2: shows "plossless_converter \<I>' \<I> conv"
Or_propos': fixes x :: "'a ltl" assumes "x \<in> set xs" shows "propos x \<subseteq> propos (Or xs)"
less_eq_cset_def: shows "csubseteq \<equiv> map_fun rcset (map_fun rcset id) (\<subseteq>)"
right_idem: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<inter># b \<inter># b = a \<inter># b"
neutral: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>A. g x = (0::'d)" shows "sum g A = (0::'d)"
gamma1_def: shows "gamma1 = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
eucl_induct: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize"   and "\<And>b. P b zero"   and "\<And>a b. \<lbrakk>b \<noteq> zero; P b (modulo a b)\<rbrakk> \<Longrightarrow> P a b" shows "P a b"
Entry_reachesE: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)"   and "invar g" obtains ns :: "'node list" where "g \<turnstile> Entry g-ns\<rightarrow>n" and "EntryPath g ns"
cases: fixes x :: "ereal" obtains r :: "real" where "x = ereal r"   | "x = - \<infinity>"   | "x = \<infinity>"
zero_less_one: fixes one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
igSubstIGVar2_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
arr_mset_split: fixes l :: "int"   and m :: "int"   and r :: "int"   and arr :: "int \<Rightarrow> 'a" assumes "l \<le> m"   and "m \<le> r" shows "arr_mset arr l r = arr_mset arr l m + arr_mset arr m r"
rev_iffD2: fixes Q :: "bool"   and P :: "bool" assumes "Q"   and "P = Q" shows "P"
lim_stream_eq_coinduct: fixes R :: "'a \<Rightarrow> 'a stream measure \<Rightarrow> bool"   and x :: "'a"   and B :: "'a stream measure" assumes "R x B"   and "x \<in> space M"   and "\<And>x B. R x B \<Longrightarrow> \<exists>B'\<in>M \<rightarrow>\<^sub>M prob_algebra (stream_space M). (AE y in K x. R y (B' y) \<or> lim_stream y = B' y) \<and> B = K x \<bind> (\<lambda>y. B' y \<bind> (\<lambda>\<omega>. return (stream_space M) (y ## \<omega>)))" shows "lim_stream x = B"
assms_1: shows "(\<langle>A\<^sub>F, \<Psi>\<rangle>) \<turnstile>\<^sub>F \<phi>"
list_update_nonempty: fixes xs :: "'g list"   and k :: "nat"   and x :: "'g" shows "(xs[k := x] = []) = (xs = [])"
assms_4: shows "b \<succeq>[\<R>] w \<and> (w, b) \<notin> \<R>"
divide_nonpos_neg: fixes x :: "'a"   and y :: "'a" assumes "x \<le> (0::'a)"   and "y < (0::'a)" shows "(0::'a) \<le> x / y"
rel_inject: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a list"   and x2 :: "nat"   and y1 :: "'b list"   and y2 :: "nat" shows "rel_fundef R (Fundef x1 x2) (Fundef y1 y2) = (list_all2 R x1 y1 \<and> x2 = y2)"
inducts: fixes x :: "ty"   and P :: "ty \<Rightarrow> bool" assumes "is_refT x"   and "P NT"   and "\<And>C. P (Class C)"   and "\<And>A. P (A\<lfloor>\<rceil>)" shows "P x"
cnj_cancel_iff: fixes x :: "quat"   and y :: "quat" shows "(Quaternions.cnj x = Quaternions.cnj y) = (x = y)"
subspace_scale: fixes S :: "'b set"   and x :: "'b"   and c :: "'a" assumes "local.subspace S"   and "x \<in> S" shows "c *s x \<in> S"
bag_bound_V: fixes t :: "'c" assumes "t \<in> V\<^bsub>T\<^esub>" shows "card (bag t) \<le> card V"
iso_dghm_ArrMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
WHILET_def: shows "aWHILET \<equiv> aWHILEIT (\<lambda>_. True)"
not_INFM: fixes P :: "'b \<Rightarrow> bool" shows "(\<not> (\<exists>\<^sub>\<infinity>x. P x)) = (\<forall>\<^sub>\<infinity>x. \<not> P x)"
distinct_filter: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "distinct xs" shows "distinct (filter P xs)"
hnr_comp1_aux: fixes P :: "'a \<Rightarrow> bool"   and R1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and c :: "'b \<Rightarrow> 'c Heap"   and R1p :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and b :: "'a \<Rightarrow> 'd nres"   and Q :: "'e \<Rightarrow> bool"   and R1' :: "('a \<times> 'e) set"   and a :: "'e \<Rightarrow> 'f nres"   and R' :: "('d \<times> 'f) set"   and a1 :: "'e"   and c1 :: "'b" assumes "\<And>b1 c1. P b1 \<Longrightarrow> hn_refine (hn_ctxt R1 b1 c1) (c c1) (hn_ctxt R1p b1 c1) R (b $ b1)"   and "\<And>a1 b1. \<lbrakk>Q a1; (b1, a1) \<in> R1'\<rbrakk> \<Longrightarrow> (b $ b1, a $ a1) \<in> \<langle>R'\<rangle>nres_rel"   and "\<And>a1 b1. \<lbrakk>Q a1; (b1, a1) \<in> R1'\<rbrakk> \<Longrightarrow> P b1"   and "Q a1" shows "hn_refine (hr_comp R1 R1' a1 c1) (c c1) (hr_comp R1p R1' a1 c1) (hr_comp R R') (a a1)"
list_augment_idem: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "list_augment xs i (xs ! i) = xs"
split_block_4: fixes A :: "'a mat"   and sr1 :: "nat"   and sc1 :: "nat"   and A1 :: "'a mat"   and A2 :: "'a mat"   and A3 :: "'a mat"   and A4 :: "'a mat"   and sr2 :: "nat"   and sc2 :: "nat" assumes "split_block A sr1 sc1 = (A1, A2, A3, A4)"   and "dim_row A = sr1 + sr2"   and "dim_col A = sc1 + sc2" shows "A4 \<in> carrier_mat sr2 sc2"
prob_space_axioms: shows "prob_space (distr M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)))"
hpsafe_ODE_simps: fixes ODE :: "('d, 'f) ODE"   and p :: "('d, 'e, 'f) formula" shows "hpsafe (EvolveODE ODE p) = (osafe ODE \<and> fsafe p)"
jw_App: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty"   and e' :: "Syntax.term" assumes "\<Gamma> \<turnstile>\<^sub>W e : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "\<Gamma> \<turnstile>\<^sub>W e' : \<tau>\<^sub>1" shows "\<Gamma> \<turnstile>\<^sub>W App e e' : \<tau>\<^sub>2"
nat_le_linear: fixes m :: "nat"   and n :: "nat" shows "order.greater_eq n m \<or> order.greater_eq m n"
Z_def: shows "Z = {t \<in> {0..min (dist x y) (dist x z)}. g t = h t}"
z': shows "\<forall>s. (Q \<and>* (\<lambda>s. True)) (fst s, snd s div k) \<longrightarrow> (Q \<and>* (\<lambda>s. True)) s"
oghoare_atom_com_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "'e set"   and a :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a c Q, A"   and "atom_com c" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c Q, A"
Cast\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "val list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"
sup_set_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
space_Sup_measure'2: fixes M :: "'b measure set" shows "space (Sup_measure' M) = \<Union> (space ` M)"
inj_on_Span_iff_trivial_ker: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "inj_on h (R.Span K Us) = (a_kernel (R\<lparr>carrier := R.Span K Us\<rparr>) S h = {\<zero>})"
intersect_normal: fixes K :: "'a set" assumes "finite (carrier G)"   and "K \<lhd> G" shows "composition_series (G\<lparr>carrier := K\<rparr>) (remdups_adj (map ((\<inter>) K) \<HH>))"
cindex_pathE_linepath: fixes z :: "complex"   and a :: "complex"   and b :: "complex" assumes "z \<notin> closed_segment a b" shows "cindex_pathE (linepath a b) z = (let c1 = Re a - Re z; c2 = Re b - Re z; c3 = Im a * Re b + Re z * Im b + Im z * Re a - Im z * Re b - Im b * Re a - Re z * Im a; d1 = Im a - Im z; d2 = Im b - Im z in if 0 < c1 \<and> c2 < 0 \<or> c1 < 0 \<and> 0 < c2 then if 0 < c3 then 1 else - 1 else if (c1 = 0) = (c2 \<noteq> 0) \<and> (c1 = 0 \<longrightarrow> d1 \<noteq> 0) \<and> (c2 = 0 \<longrightarrow> d2 \<noteq> 0) then if c1 = 0 \<and> (0 < c2) = (0 < d1) \<or> c2 = 0 \<and> (0 < c1) = (d2 < 0) then 1 / 2 else - 1 / 2 else 0)"
map_of_aset_delete: fixes S :: "('b \<times> 'c) set"   and k :: "'b"   and v :: "'c" assumes "unique_keys_set S"   and "(k, v) \<in> S" shows "map_of_aset (S - {(k, v)}) = delete_map k (map_of_aset S)"
subset_\<J>\<I>k: fixes k :: "nat"   and u :: "nat" shows "\<J> k u \<subseteq> \<I> k"
least_simps_3: fixes x' :: "'c"   and y' :: "'c" shows "least (Some x') (Some y') = (if x' \<le> y' then Some x' else Some y')"
assms_1: shows "\<forall>x\<in>X. finite x"
dec: shows "decseq (\<lambda>i. PF.emb UNIV (J i) (X' i))"
wlt_def: fixes F :: "'c program"   and B :: "'c set" shows "wlt F B \<equiv> \<Union> {A. F \<in> A \<longmapsto> B}"
spr_sim_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agent :: "'b"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'f list"   and envTrans :: "'f \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g" assumes "FiniteSingleAgentEnvironment jkbp agent" shows "FiniteSingleAgentEnvironment.spr_sim jkbp envInit envAction envTrans envVal envObs agent \<equiv> \<lambda>t. (FiniteSingleAgentEnvironment.spr_abs jkbp envInit envAction envTrans envVal envObs agent t, tLast t)"
res_prime_units_eq: shows "Units R = {1..int (p - 1)}"
prod_pos_nat_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat" assumes "finite A" shows "(0 < prod f A) = (\<forall>a\<in>A. 0 < f a)"
AE_2_4: shows "AE x in M. 0 \<le> - subcocycle_lim u x"
type4_def: fixes init :: "'a list"   and x :: "'a"   and y :: "'a" shows "type4 init x y = bernoulli_pmf (5 / 10) \<bind> (\<lambda>a. bernoulli_pmf (5 / 10) \<bind> (\<lambda>b. return_pmf (if \<not> [a, b] ! index init y then ([x, y], [a, b], init) else ([y, x], [a, b], init))))"
continuous_uminus_ereal: fixes A :: "ereal set" shows "continuous_on A uminus"
f1: fixes e :: "('a \<times> ('a, 'b) operation) event" assumes "e \<in> set pre" shows "e \<in> set (history k)"
star_subdist_var_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.left_pre_kleene_algebra plus times one less_eq less star" shows "less_eq x (star (plus x y))"
Basis_list_nth_nonzero: fixes i :: "nat" assumes "i < length Basis_list" shows "Basis_list ! i \<noteq> (0::'a)"
norm_fstzero_zerosnd_1: shows "norm no_real_inner.fstzero = 1"
f_2: shows "cols A = permute_list f (map ((!) (cols A)) (filter (\<lambda>i. i \<notin> set ids) [0..<length (cols A)]) @ ss)"
star_inductl_var: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z + x \<odot> y \<le> y" shows "x\<^sup>\<star> \<odot> z \<le> y"
nat_to_set_is_finite: fixes u :: "nat" shows "finite (nat_to_set u)"
null_setsD1: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> null_sets M" shows "emeasure M A = 0"
dim_row_take_cols: fixes A :: "'b mat"   and ls :: "nat list" shows "dim_row (take_cols A ls) = dim_row A"
mono_def: fixes Retr :: "'c \<Rightarrow> 'd" shows "Bisim.mono Retr \<equiv> \<forall>theta theta'. theta \<le> theta' \<longrightarrow> Retr theta \<le> Retr theta'"
\<pi>_upper_bound_aux: fixes c :: "real"   and x :: "real" assumes "0 < c"   and "0 < x" shows "x powr - c * ln x \<le> 1 / (c * exp 1)"
dense_pp: fixes x :: "'a" shows "dense x = (- - x = top)"
cos_x_y_le_one: fixes x :: "real"   and y :: "real" shows "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
align\<^sub>d_N: shows "local.aligned\<^sub>d e N"
assms: shows "(x, y) \<in> shuffle_to_top Y d"
poincare_distance_satisfies_pred: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance_pred u v (poincare_distance u v)"
profile_mono\<^sub>d_def: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "profile_mono\<^sub>d f g \<equiv> profile_mono_scheme_defined f g \<and> profile_mono\<^sub>d_axioms g"
cases: fixes a1 :: "('d \<times> 'e) set"   and a2 :: "('d \<times> 'e) list"   and a3 :: "('d \<times> 'e, 'f) ta_rule set"   and a4 :: "('d \<times> 'e) set"   and a5 :: "('d \<times> 'e) list"   and a6 :: "('d \<times> 'e, 'f) ta_rule set"   and \<delta>1 :: "('d, 'f) ta_rule set"   and \<delta>2 :: "('e, 'f) ta_rule set" assumes "((a1, a2, a3), a4, a5, a6) \<in> frp_step \<delta>1 \<delta>2" obtains W :: "('d \<times> 'e) list" and q1 :: "'d" and q2 :: "'e" and Wtl :: "('d \<times> 'e) list" and Wn :: "('d \<times> 'e) list" and Q :: "('d \<times> 'e) set" and W' :: "('d \<times> 'e) list" and Q' :: "('d \<times> 'e) set" and \<delta>d' :: "('d \<times> 'e, 'f) ta_rule set" and \<delta>d :: "('d \<times> 'e, 'f) ta_rule set" where "a1 = Q" and "a2 = W" and "a3 = \<delta>d" and "a4 = Q'" and "a5 = W'" and "a6 = \<delta>d'" and "W = (q1, q2) # Wtl" and "distinct Wn" and "set Wn = f_succ (\<delta>_prod \<delta>1 \<delta>2) `` {(q1, q2)} - Q" and "W' = Wn @ Wtl" and "Q' = Q \<union> f_succ (\<delta>_prod \<delta>1 \<delta>2) `` {(q1, q2)}" and "\<delta>d' = \<delta>d \<union> {r \<in> \<delta>_prod \<delta>1 \<delta>2. lhs r = (q1, q2)}"
convex_rel_interior_inter: fixes I :: "'a set set" assumes "\<forall>S\<in>I. convex S"   and "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}" shows "rel_interior (\<Inter> I) \<subseteq> \<Inter> {rel_interior S |S. S \<in> I}"
diff_union_swap: fixes a :: "'b"   and b :: "'b"   and M :: "'b multiset" assumes "a \<noteq> b" shows "add_mset b (M - {#a#}) = add_mset b M - {#a#}"
br_chain: fixes \<beta> :: "'d \<Rightarrow> 'f"   and J :: "'d \<Rightarrow> bool"   and \<alpha> :: "'f \<Rightarrow> 'e"   and I :: "'f \<Rightarrow> bool" shows "br \<beta> J O br \<alpha> I = br (\<alpha> \<circ> \<beta>) (\<lambda>s. J s \<and> I (\<beta> s))"
inPlus_alt: fixes r :: "'c rexp"   and s :: "'c rexp" shows "inPlus r s = (let X = toplevel_summands (Plus r s) - {Zero} in PLUS (sorted_list_of_set (if Full \<in> X then {Full} else X)))"
assms: shows "(\<sigma>, s) \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U"
branchI: fixes f :: "nat \<Rightarrow> 'b"   and Gamma :: "'b"   and subs :: "'b \<Rightarrow> 'b set" assumes "f 0 = Gamma"   and "\<forall>n. terminal subs (f n) \<longrightarrow> f (Suc n) = f n"   and "\<forall>n. \<not> terminal subs (f n) \<longrightarrow> f (Suc n) \<in> subs (f n)" shows "branch subs Gamma f"
all_subset_def: fixes I :: "'a interval list"   and J :: "'a interval list" shows "(I all_subset J) = (length I = length J \<and> (\<forall>i<length I. set_of (I ! i) \<subseteq> set_of (J ! i)))"
no_inter': shows "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) xs \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb = {}"
pr_is_total_rec: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "total_recursive f"
\<Sigma>_comp_of_int: fixes f :: "'b \<Rightarrow> 'a" shows "\<Sigma> (f \<circ> of_int) = \<Sigma> f"
Hancl3: fixes A :: "real"   and d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and s :: "nat" assumes "1 < A"   and "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "\<forall>n. 0 < b n"   and "0 < s"   and "(\<lambda>n. real_of_int (a n) powr (1 / real_of_int (2 ^ n))) \<longlonglongrightarrow> A"   and "\<forall>n\<ge>s. (\<Prod>j. d (n + j)) < A / real_of_int (a n) powr (1 / real_of_int (2 ^ n))"   and "LIM n sequentially. d n ^ 2 ^ n / real_of_int (b n) :> at_top"   and "convergent_prod d" shows "(\<Sum>n. real_of_int (b n) / real_of_int (a n)) \<notin> \<rat>"
wt_var: fixes \<Gamma> :: "(nat \<times> ty) list"   and x :: "nat"   and T :: "ty" assumes "lookup \<Gamma> x = Some T" shows "\<Gamma> \<turnstile> EVar x : T"
all_nth_imp_all_set: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and x :: "'g" assumes "\<forall>i<length xs. P (xs ! i)"   and "x \<in> set xs" shows "P x"
hasw_def: fixes xs :: "'c list"   and S :: "'c set" shows "hasw xs S \<equiv> set xs \<subseteq> S \<and> distinct xs"
comm_monoidI: fixes G :: "('e, 'f) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
iso_iff_section_and_retraction: fixes f :: "'a \<times> 'a" shows "VV.iso f = (VV.section f \<and> VV.retraction f)"
norm_eq_iff_face_cong: fixes xs :: "nat list"   and ys :: "nat list" assumes "distinct xs"   and "xs \<noteq> []"   and "ys \<noteq> []" shows "(rotate_min xs = rotate_min ys) = xs \<cong> ys"
lb3_lb: shows "0 \<le> real_of_float (lapprox_rat prec 1 3)"