hcju/isabelleps huggingface dataset:
pre_adj_if_postR: fixes lv :: "('b \<times> nat) tree"   and l :: "'b"   and a :: "nat"   and r :: "('b \<times> nat) tree"   and r' :: "('b \<times> nat) tree" assumes "invar \<langle>lv, (l, a), r\<rangle>"   and "post_del r r'" shows "pre_adjust \<langle>lv, (l, a), r'\<rangle>"
Ci_stays: fixes t :: "nat" assumes "enat (t_C + t_Cj + t) < llength Sts" shows "(C, j) \<in># wP_of_wstate (lnth Sts (t_C + t_Cj + t))"
rel_inject: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b stream"   and y1 :: "'c"   and y2 :: "'c stream" shows "stream_all2 R (x1 ## x2) (y1 ## y2) = (R x1 y1 \<and> stream_all2 R x2 y2)"
funcset_comp: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and C :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "g \<circ> f \<in> A \<rightarrow> C"
simps: fixes a1 :: "'a list"   and a2 :: "sequent" shows "((a1, a2) \<in> nprovRules) = (\<exists>c F x A ps. a1 = ps \<and> a2 = c \<and> NominalSequents.mset c = \<LM> F \<nabla> [x].A  \<RM> \<and> ps \<noteq> [])"
ia: shows "ia < IArray.length (matrix_to_iarray (Gram_Schmidt_column_k_efficient A k) !! i)"
0: shows "0 \<in> interior (convex hull {a - z, b - z, c - z})"
wary_AppE: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "wary (App s t)" obtains "wary s" and "wary t" and "enat (num_args s) < arity_hd (head s)"
d: shows "CARD('a) = CARD('b) * CARD('d)"
nf_inres_def: fixes m :: "'c nres"   and x :: "'c" shows "nf_inres m x \<equiv> nofail m \<and> inres m x"
poly_square_subst_coeff: fixes f :: "'a poly"   and i :: "nat" shows "poly.coeff (poly_square_subst f) i = poly.coeff f (2 * i)"
fst_splitE: fixes s' :: "'a \<times> 'b"   and x' :: "'a" assumes "fst s' = x'" obtains x :: "'a" and s :: "'b" where "s' = (x, s)" and "x = x'"
liar_imp_coprime: fixes a :: "nat"   and n :: "nat" assumes "a \<in> {1..<n}"   and "fermat_liar a n" shows "coprime a n"
cases: fixes x :: "('b, 'c) arr" obtains uu_ :: "'b" and uv_ :: "'b" and F :: "'c" where "x = MkArr uu_ uv_ F"   | "x = Null"
ring_is_ag: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "aGroup R"
assms: shows "lead_coeff p * lead_coeff q \<noteq> (0::'a)"
pure_fresh: fixes a :: "atom"   and x :: "'a" shows "a \<sharp> x"
subst0_I: fixes p :: "fm"   and bbs :: "bool list"   and b :: "int"   and bs :: "int list"   and a :: "Cooper.num" assumes "qfree p" shows "Ifm bbs (b # bs) (subst0 a p) = Ifm bbs (Inum (b # bs) a # bs) p"
rel_prod_sel: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'b \<times> 'd"   and q :: "'c \<times> 'e" shows "rel_prod R1 R2 p q = (R1 (fst p) (fst q) \<and> R2 (snd p) (snd q))"
vfsequence_vrange_vconsI: fixes X :: "V"   and x :: "V" assumes "\<R>\<^sub>\<circ> \<SS>\<^sub>\<int> \<subseteq>\<^sub>\<circ> X"   and "x \<in>\<^sub>\<circ> X" shows "\<R>\<^sub>\<circ> (\<SS>\<^sub>\<int> #\<^sub>\<circ> x) \<subseteq>\<^sub>\<circ> X"
tree_has_edge: fixes G :: "('a, 'b) Graph_scheme" assumes "Tree G"   and "1 < card V\<^bsub>G\<^esub>" shows "\<exists>v w. v \<rightarrow>\<^bsub>G\<^esub> w"
absorb_iff1: fixes b :: "'d"   and a :: "'d" shows "(b \<le> a) = (sup a b = a)"
integral_fst: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f x y)" shows "LINT x|M2. integral\<^sup>L M1 (f x) = LINT (x, y)|(M2 \<Otimes>\<^sub>M M1). f x y"
ordRels_def_2: shows "ordLess = {(r, r'). Well_order r \<and> Well_order r' \<and> (\<exists>f. embedS r r' f)}"
cyclic_on_def: fixes f :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "cyclic_on f S = (\<exists>s\<in>S. S = orbit f s)"
continuous_map_eq_topcontinuous_at: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = (\<forall>x\<in>topspace X. topcontinuous_at X Y f x)"
inj_on_image_set_diff: fixes f :: "'a \<Rightarrow> 'b"   and C :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "inj_on f C"   and "A - B \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A - B) = f ` A - f ` B"
of_bl_eq: fixes bs :: "bool list" shows "of_bl bs = horner_sum of_bool 2 (rev bs)"
x_less_2_0_1': fixes x :: "'a word" assumes "LENGTH('a) \<noteq> 1"   and "x < 2" shows "x = 0 \<or> x = 1"
AboveS_Field: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "AboveS r A \<subseteq> Field r"
fps_conv_radius_sin: fixes c :: "'a" shows "fps_conv_radius (fps_sin c) = \<infinity>"
B: shows "\<forall>n<Suc (length ws). P ((w # ws) ! n) (drop n ws @ xs)"
insert_DiffM2: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "M - \<LM> x  \<RM> + \<LM> x  \<RM> = M"
assms_1: shows "minimal_sequence_to_failure_extending V M1 M2 vs xs"
monotone_fst: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "monotone (rel_prod orda ordb) orda fst"
design_def: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool"   and Q :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "(P \<turnstile> Q) \<equiv> \<lambda>(A, A'). ok A \<and> P (A, A') \<longrightarrow> ok A' \<and> Q (A, A')"
inf_absorb2: fixes y :: "'a multiset"   and x :: "'a multiset" assumes "y \<subseteq># x" shows "x \<inter># y = y"
dist_triangle_le: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and z :: "'a"   and y :: "'a"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z \<le> e" shows "dist x y \<le> e"
that: fixes y' :: "nat itrm" assumes "itrm_brackets vs x = Some y'"   and "itrm_bracket v y' = Some x'" shows "thesis"
NF: shows "nofail (m0 \<bind> WHILE\<^sub>T cond step)"
assms_2: assumes "0 \<le> x"   and "y \<le> 0" shows "P (gcd x (- y))"
pred_def: fixes P :: "'BExp set" shows "pred P \<equiv> \<lambda>mem. \<forall>p\<in>P. ev\<^sub>B mem p"
int_float_disjoint: fixes t :: "t" shows "is_int_t t = - is_float_t t"
monom_mult_uminus_left: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult (- c) t p = - MPoly_Type_Class.punit.monom_mult c t p"
Eq\<^sub>1S: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" shows "[F \<^bold>=\<^sub>1 G in v] = (F = G)"
g2_ord_X: shows "[g2[b .. a\<^sub>n ..  a\<^sub>1]insert b Y]"
is_Proj_I: fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "P o\<^sub>C\<^sub>L P = P"   and "P* = P" shows "is_Proj P"
ipurge_tr_rev_t_out_def: shows "ipurge_tr_rev_t_out \<equiv> Out"
extensional_merge: fixes I :: "'b set"   and J :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" shows "merge I J (x, y) \<in> extensional (I \<union> J)"
add_neg_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less a zero"   and "less b zero" shows "less (plus a b) zero"
span_zero: fixes S :: "('a, 'b) vec set" shows "0 \<in> vec.span S"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) local.map \<Omega>.map \<tau>"
l7_2: fixes M :: "'p"   and A :: "'p"   and B :: "'p" assumes "M Midpoint A B" shows "M Midpoint B A"
pair_pseudo_graph_axioms: shows "pair_pseudo_graph (subdivide G e w)"
cexp_cong: fixes p1 :: "('c \<times> 'c) set"   and r1 :: "('d \<times> 'd) set"   and p2 :: "('e \<times> 'e) set"   and r2 :: "('f \<times> 'f) set" assumes "p1 =o r1"   and "p2 =o r2"   and "Card_order r2"   and "Card_order p2" shows "p1 ^c p2 =o r1 ^c r2"
expectation_return_pmf: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e" shows "measure_pmf.expectation (return_pmf x) f = f x"
cases: fixes a1 :: "'a scheduler"   and a2 :: "'a scheduler" assumes "eq_scheduler a1 a2" obtains sc1 :: "'a scheduler" and sc2 :: "'a scheduler" and D :: "'a pmf" where "a1 = sc1" and "a2 = sc2" and "action_sch sc1 = D" and "action_sch sc2 = D" and "\<forall>s\<in>set_pmf D. eq_scheduler (cont_sch sc1 s) (cont_sch sc2 s)"
DSourcesA72_L1: shows "DSources level1 sA72 = {sA6}"
maxsimpchain_in_subcomplex: fixes Y :: "'a set set"   and ys :: "'a set list" assumes "order.greater_eq X Y \<and> SimplicialComplex Y"   and "order.greater_eq Y (set ys)"   and "folding.gallery ys" shows "SimplicialComplex.maxsimpchain Y ys"
fin_dim_def: shows "gs.fin_dim = (\<exists>A. finite A \<and> A \<subseteq> Rn \<and> gs.span A = Rn)"
dom_release_rels_eq: fixes \<R>' :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and \<S> :: "nat set" assumes "dom \<R>' = dom \<R>" shows "dom (release sb \<S> \<R>') = dom (release sb \<S> \<R>)"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>list z. P z (TVar list)"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1 \<rightarrow> ty2)" shows "P z ty"
VarNonOccForm_imp_fresh: fixes v :: "hf"   and x :: "hf"   and e :: "(name, hf) finfun" assumes "VarNonOccForm v x" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e" and "atom (decode_Var v) \<sharp> A"
genby_genset_subset: fixes S :: "'b set" shows "order.greater_eq \<langle>S\<rangle> S"
in_Chains_conv_chain: fixes M :: "'b set"   and r :: "('b \<times> 'b) set" shows "(M \<in> Chains r) = Complete_Partial_Order.chain (\<lambda>x y. (x, y) \<in> r) M"
is_le_rel_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.is_le_rel less_eq less rel = (rel = (=) \<or> rel = less_eq \<or> rel = less)"
n_as: shows "n_as = as @ (m, lam) # bs"
AnalBinds_Cons: fixes exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "ExpAnalysis.AnalBinds exp ((x, e) # \<Gamma>)\<cdot>ae = (ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae)(x := fup\<cdot>(exp e)\<cdot>(ae x))"
widen_antisym: fixes G :: "prog"   and S :: "ty"   and T :: "ty" assumes "G\<turnstile>S\<preceq>T"   and "G\<turnstile>T\<preceq>S"   and "ws_prog G" shows "S = T"
finfun_upd_apply_other: fixes a :: "'a"   and a' :: "'a"   and f :: "('a, 'b) finfun"   and b :: "'b" assumes "a \<noteq> a'" shows "finfun_apply (finfun_update f a b) a' = finfun_apply f a'"
luckyFind'_eval: fixes xs :: "real list"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "length xs = var" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (luckyFind' var L F) (xs @ x # \<Gamma>))"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "List.find P (x # xs) = (if P x then Some x else List.find P xs)"
\<phi>_in_hom_2: shows "\<guillemotleft>r\<^sub>0s\<^sub>1.\<phi> : r\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
tendsto_Im2: fixes g :: "'a \<Rightarrow> quat"   and a :: "quat"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im2 (g x)) \<longlongrightarrow> Im2 a) F"
order_trans_rules_28: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a = b"   and "b < c" shows "a < c"
a: shows "\<forall>i\<in>S. a i \<in> sfis (f i) M"
edges_less_eq_linorder: shows "is_linorder_rel edges_less_eq"
finprod_empty: fixes f :: "'c \<Rightarrow> 'b" shows "sum f {} = (0::'b)"
diamond_inf_word_step: fixes a :: "'action"   and w :: "'action stream"   and p :: "'state" assumes "Ind {a} (sset w)"   and "en a p"   and "run w p" shows "run w (ex a p)"
elims_2: fixes x :: "('d, 'e, 'f) FSM" assumes "observable x" obtains M :: "('d, 'e, 'f) FSM" where "x = M" and "\<forall>t s1. succ M t s1 = {} \<or> (\<exists>s2. succ M t s1 = {s2})"
zero_im_sheaf_def: shows "zero_im_sheaf \<equiv> \<lambda>V. \<zero>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
claim1: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list"   and \<upsilon>' :: "'a list" assumes "trans r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<sigma> @ \<tau>'| + r|\<upsilon>'| -s r \<down>l (\<sigma> @ \<tau>') \<inter>s r \<down>l \<tau>, r|\<sigma>| + r|\<tau>|) \<in> mul_eq r"
word_shiftr_lt: fixes w :: "'b word"   and n :: "nat" shows "unat (w >> n) < 2 ^ (LENGTH('b) - n)"
WT: shows "WT_core \<I>_adv \<I>_usr I core"
comp_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>D f : a \<Rightarrow>\<^sub>D c\<guillemotright>"
cases: fixes a1 :: "'a"   and a2 :: "'b \<Rightarrow> nat"   and a3 :: "'a"   and a4 :: "'b \<Rightarrow> nat"   and M :: "('a \<times> ('a, 'b) Op) set" assumes "((a1, a2), a3, a4) \<in> step M" obtains s :: "'a" and v :: "'b" and s' :: "'a" and vs :: "'b \<Rightarrow> nat" where "a1 = sa__" and "a2 = vs__" and "a3 = s'" and "a4 = (\<lambda>x. if x = v__ then Suc (vs__ v__) else vs__ x)" and "(sa__, Inc v__ s') \<in> M"   | s :: "'a" and v :: "'b" and sn :: "'a" and sz :: "'a" and vs :: "'b \<Rightarrow> nat" and n :: "nat" where "a1 = sb__" and "a2 = vsa__" and "a3 = sn__" and "a4 = (\<lambda>x. if x = va__ then n else vsa__ x)" and "(sb__, Dec va__ sn__ sz__) \<in> M" and "vsa__ va__ = Suc n"   | s :: "'a" and v :: "'b" and sn :: "'a" and sz :: "'a" and vs :: "'b \<Rightarrow> nat" where "a1 = s" and "a2 = vs" and "a3 = sz" and "a4 = vs" and "(s, Dec v sn sz) \<in> M" and "vs v = 0"
nz: fixes i :: "'n" assumes "i \<in> T" shows "i \<noteq> z"
matching_dsts_pull_out_accu: fixes s :: "'a word"   and rs :: "'a simple_rule list"   and a1 :: "'a wordinterval"   and a2 :: "'a wordinterval" shows "wordinterval_to_set (matching_dsts s rs (wordinterval_union a1 a2)) = wordinterval_to_set (matching_dsts s rs a2) - wordinterval_to_set a1"
lift_state_oracle_def: fixes F :: "('e \<Rightarrow> 'c \<Rightarrow> (('d \<times> 'a) \<times> 'e) spmf) \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> (('d \<times> 'a) \<times> 'b) spmf"   and "oracle" :: "'a \<times> 'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a \<times> 'e) spmf" shows "lift_state_oracle F oracle = (\<lambda>(t, s') a. map_spmf rprodl (F (map_fun (Pair t) (map_fun id (map_spmf lprodr)) oracle) s' a))"
inj_on_one_iff: shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>)"
i: fixes s :: "(char list \<Rightarrow> int option) \<times> nat" shows "(case s of (s, n) \<Rightarrow> I (s, n) \<and> lmaps_to_axpr b True s) \<longrightarrow> pre C (I \<and>* $ 1) s"
update: fixes nts' :: "nat list"   and aux' :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and zs :: "event_data option list set list"   and aux'' :: "(nat \<times> event_data option list set list \<times> event_data option list set) list"   and buf' :: "event_data option list set list list" assumes "eval_matchF I mr (case nts' of [] \<Rightarrow> \<tau> \<sigma> j | nt # x \<Rightarrow> nt) aux' = (zs, aux'')"   and "rel_mapping (\<le>) P P'"   and "mbufnt_take (update_matchF n I mr mrs) aux (mbufn_add (map (fst \<circ> meval n (\<tau> \<sigma> j) db) \<phi>s) buf) (nts @ [\<tau> \<sigma> j]) = (aux', buf', nts')" shows "wf_matchF_aux \<sigma> V n R I r aux'' (Monitor.progress \<sigma> P' (formula.MatchF I r) (Suc j)) 0 \<and> Monitor.progress \<sigma> P (formula.MatchF I r) j + length zs = Monitor.progress \<sigma> P' (formula.MatchF I r) (Suc j) \<and> Monitor.progress \<sigma> P (formula.MatchF I r) j + length zs + length aux'' = progress_regex \<sigma> P' r (Suc j) \<and> list_all2 (\<lambda>i. qtable n (fv_regex r) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i (formula.MatchF I r))) [Monitor.progress \<sigma> P (formula.MatchF I r) j..<Monitor.progress \<sigma> P (formula.MatchF I r) j + length zs] zs"
left_secure: shows "component_secure_prot n (Pi n) Sec attack"
perfect_sec_P2_def: fixes funct_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "OT_14_sim.perfect_sec_P2 funct_14_OT m1 m2 \<equiv> R2_14_OT m1 m2 = funct_14_OT m1 m2 \<bind> (\<lambda>(s1, s2). S2_14_OT m2 s2)"
asymp_equiv_sandwich_real: fixes l :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and u :: "'b \<Rightarrow> real"   and f :: "'b \<Rightarrow> real" assumes "l \<sim>[F] g"   and "u \<sim>[F] g"   and "\<forall>\<^sub>F x in F. f x \<in> {l x..u x}" shows "f \<sim>[F] g"
summableI_nonneg_bounded: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "\<And>n. (0::'b) \<le> f n"   and "\<And>n. sum f {..<n} \<le> x" shows "summable f"
product_indicator: fixes A :: "'b set"   and x :: "'b"   and B :: "'b set" shows "rind A x * rind B x = rind (A \<inter> B) x"
assms_2: shows "g' \<in> rcoset_rel H `` {g}"
simps_1: fixes uu :: "nat \<Rightarrow> val"   and v :: "val" shows "expr_sem_rf uu (Val v) = v"
C_of_def: fixes A :: "'a" shows "C_of A = {#L \<in># CA_of A. L \<noteq> Pos A#}"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "\<ll>.map f = null"
sorted_list_of_multiset_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq {#} = []"
divide_nonneg_nonpos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less_eq y zero" shows "less_eq (divide x y) zero"
nth_prefix_length: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. \<not> P x" shows "\<not> P (xs ! prefix_length P xs)"
max_tstp_d_d: fixes d :: "nat + nat" shows "max_tstp d d = d"
simps_5: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (Lt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
finite_imp_FoldSet: fixes A :: "'b set"   and h :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "finite A" shows "\<exists>x. FoldSet h z A x"
maxsimp_singleton: fixes w :: "'w" assumes "w \<in> W" shows "chamber (dual_order.poset_simplex_map \<P> {w})"
leg0_uniquely_isomorphic_1: shows "s\<^sub>0 \<star> w \<cong> r\<^sub>0"
power_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq a b"   and "less_eq zero a" shows "less_eq (power.power one times a n) (power.power one times b n)"
lea_trans: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A1 :: "'a"   and B1 :: "'a"   and C1 :: "'a"   and A2 :: "'a"   and B2 :: "'a"   and C2 :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C A1 B1 C1"   and "Tarski_neutral_dimensionless.LeA Bet Cong A1 B1 C1 A2 B2 C2" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C A2 B2 C2"
is_basis: fixes p :: "'d set \<Rightarrow> bool"   and B :: "'d set set" assumes "countable_basis p B" shows "topological_space.topological_basis p B"
a_kernel_def: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"
search: fixes P :: "nat \<Rightarrow> bool"   and y :: "nat" assumes "\<And>x. P x \<or> \<not> P x" shows "(\<exists>x<y. P x) \<or> \<not> (\<exists>x<y. P x)"
local_help1: fixes vs :: "'c list"   and c :: "'c list"   and r :: "'c"   and d :: "'c list"   and a :: "'c list"   and b :: "'c list" assumes "vs = c @ r # d"   and "vs = a @ r # b"   and "r \<notin> set a"   and "r \<notin> set b" shows "a = c"
poly_isCont: fixes x :: "'a"   and p :: "'a poly" shows "isCont (poly p) x"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "A.arr f"   and "A.dom f = a"   and "A.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>"
norm_def: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real" shows "Normed_Space.norm V norm \<equiv> seminorm V norm \<and> norm_axioms V norm"
simps_2: fixes l :: "'c tree"   and a :: "'c"   and r :: "'c tree" shows "height \<langle>l, a, r\<rangle> = max (height l) (height r) + 1"
Gromov_completion_complete: assumes "complete UNIV" shows "complete UNIV"
continuous_at: fixes x :: "'a"   and T :: "'a set" assumes "x \<in> src.carrier" shows "continuous (at x within T) f"
sel_3: fixes x1 :: "nat"   and x2 :: "(nat \<times> linear_poly) list"   and x3 :: "('a \<times> QDelta atom) list"   and x4 :: "linear_poly \<Rightarrow> nat option"   and x5 :: "'a list" shows "Atoms (IState x1 x2 x3 x4 x5) = x3"
graphIntersection: fixes X :: "'c set"   and Y :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "graph (X \<inter> Y) f \<subseteq> graph X f || Y"
prems_2: shows "\<forall>op'\<in>set (op' # ops'). op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"
invar_def: fixes q :: "('e, 'f) BinomialTree list" shows "invar q \<equiv> queue_invar q \<and> rank_invar q"
vs_conf_insert_iff: fixes P :: "'b prog"   and h :: "'heap"   and vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set"   and ad :: "'addr"   and al :: "addr_loc"   and v :: "'addr val" shows "h.vs_conf P h (vs((ad, al) := insert v (vs (ad, al)))) = (h.vs_conf P h vs \<and> (\<exists>T. h.addr_loc_type P h ad al T \<and> h.conf P h v T))"
less_imp_not_eq2: fixes x :: "'a list"   and y :: "'a list" assumes "x <lex y" shows "(y = x) = False"
distinct_3: fixes x1 :: "'d"   and x3 :: "'e" shows "Normal x1 \<noteq> Fault x3"
closedin_trans: fixes T :: "'a set"   and S :: "'a set"   and U :: "'a set" assumes "closedin (top_of_set T) S"   and "closedin (top_of_set U) T" shows "closedin (top_of_set U) S"
is_set_to_sorted_list_def: fixes ordR :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rk :: "('e \<times> 'd) set"   and Rs :: "('e \<times> 'd) set \<Rightarrow> ('f \<times> 'd set) set"   and tsl :: "'f \<Rightarrow> 'e list" shows "is_set_to_sorted_list ordR Rk Rs tsl \<equiv> \<forall>s s'. (s, s') \<in> \<langle>Rk\<rangle>Rs \<longrightarrow> (\<exists>l'. (tsl s, l') \<in> \<langle>Rk\<rangle>list_rel \<and> RETURN l' \<le> it_to_sorted_list ordR s')"
prune_successors: fixes v :: "nat \<times> 'a"   and V :: "(nat \<times> 'a) set"   and u :: "nat \<times> 'a"   and w :: "'b stream"   and A :: "('b, 'a) nba" assumes "v \<in> V"   and "u \<in> gusuccessors A w v"   and "u \<in> prune A w V" shows "v \<in> prune A w V"