hcju/isabelleps huggingface dataset:
iptables_bigstep_induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<And>t. P [] t t"   and "\<And>m a. \<lbrakk>Semantics.matches \<gamma> m p; a = action.Accept\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalAllow)"   and "\<And>m a. \<lbrakk>Semantics.matches \<gamma> m p; a = action.Drop \<or> a = Reject\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalDeny)"   and "\<And>m a. \<lbrakk>Semantics.matches \<gamma> m p; a = Log \<or> a = Empty\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a. \<not> Semantics.matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs rs\<^sub>1 rs\<^sub>2 t t'. \<lbrakk>rs = rs\<^sub>1 @ rs\<^sub>2; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'\<rbrakk> \<Longrightarrow> P rs Undecided t'"   and "\<And>m a chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>Semantics.matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); Semantics.matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; P rs\<^sub>1 Undecided Undecided\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a chain rs t. \<lbrakk>Semantics.matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided t" shows "P rs s t"
inverts: fixes int :: "int"   and int' :: "int" shows "(MkI\<cdot>int \<sqsubseteq> MkI\<cdot>int') = (int \<sqsubseteq> int')"
tMap_is_tInit: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace"   and s :: "'b" shows "(tMap f t = tInit s) = (\<exists>s'. t = tInit s' \<and> f s' = s)"
simps_2: shows "eval_ternary_Not TernaryFalse = TernaryTrue"
gcdI2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a gcdof b c"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<sim> somegcd G b c"
mult2_ns_eq_mult2_ns_alt: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "ns O s \<subseteq> s"   and "refl ns"   and "trans s" shows "mult2_alt_ns ns s = mult2_ns ns s"
goodE: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" assumes "good P f" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
pred_True: shows "pred_literal (\<lambda>_. True) = (\<lambda>_. True)"
test_comp: fixes x :: "'a"   and y :: "'a" assumes "is_test x"   and "is_test y" shows "is_test (x ; y)"
divide_by_norm_row_def: fixes A :: "(('c, 'd) vec, 'e) vec"   and a :: "'e" shows "divide_by_norm_row A a = (\<chi>b. ((1 / norm (column b A)) *\<^sub>R column b A) $ a)"
pred_less_imp_le: fixes m :: "nat"   and n :: "nat" assumes "m - Suc 0 < n" shows "m \<le> n"
esinvQ: shows "ts_inv P I (thr s) (shr s)"
t_id_t_proper_2: fixes v :: "i"   and \<tau> :: "\<kappa>"   and \<tau>' :: "\<kappa>" shows "[\<tau> \<^bold>= \<tau>' \<^bold>\<rightarrow> (\<^bold>\<exists>\<beta>. \<beta>\<^sup>P \<^bold>= \<tau>') in v]"
minus_mod: fixes b :: "nat"   and a :: "nat" assumes "b < a" shows "[a - b mod q = a - b] (mod q)"
hom_mset_factors: fixes hom :: "'a \<Rightarrow> 'b"   and F :: "'a multiset"   and p :: "'a" assumes "irreducibility_hom hom"   and "mset_factors F p" shows "mset_factors (image_mset hom F) (hom p)"
gamma2_smooth: shows "gamma2 piecewise_C1_differentiable_on {0..1}"
2: fixes q :: "'a poly"   and r :: "'a poly" assumes "0 < degree q"   and "degree q < degree p"   and "0 < degree r"   and "degree r < degree p"   and "p = q * r" shows "False"
period_set_def: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a" shows "MC_syntax.period_set K x = {i. 0 < i \<and> 0 < MC_syntax.p K x x i}"
vars_partitioned_def: fixes Cs :: "('c, 'd) Term.term literal multiset list" shows "vars_partitioned Cs = (\<forall>i<length Cs. \<forall>j<length Cs. i \<noteq> j \<longrightarrow> vars_clause (Cs ! i) \<inter> vars_clause (Cs ! j) = {})"
B: shows "foldl (+) 0 ns + n \<le> p"
bd_cts_trD: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and M :: "nat \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "bd_cts_tr T"   and "\<And>i. le_trans (M i) (M (Suc i))"   and "\<And>i. feasible (M i)" shows "equiv_trans (T (Sup_trans (range M))) (Sup_trans (range (T \<circ> M)))"
simps_11: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Sqrt a) vs = sqrt (interpret_floatarith a vs)"
annotated_no_stop: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and upd :: "'Var ModeUpd" assumes "\<not> has_annotated_stop (c@[upd])" shows "\<not> has_annotated_stop c"
union_le_diff_plus: fixes P :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "P \<subseteq># M"   and "N < P" shows "M - P + N < M"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "F.wtE \<xi> \<equiv> \<forall>x. intT (tpOfV x) (\<xi> x)"
dvd_all_coeffs_imp_dvd: fixes p :: "'a poly"   and c :: "'a" assumes "\<forall>a\<in>set (coeffs p). c dvd a" shows "[:c:] dvd p"
T_star: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = length qs - 2"
predict_stoch_proc_def: fixes F :: "nat \<Rightarrow> 'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> 'e"   and N :: "'e measure" shows "predict_stoch_proc F X N = (X 0 \<in> F 0 \<rightarrow>\<^sub>M N \<and> (\<forall>n. X (Suc n) \<in> F n \<rightarrow>\<^sub>M N))"
runit_ide_simp: fixes f :: "'a" assumes "ide f" shows "Cop.runit f = \<l>[f]"
normal_series_extend: fixes H :: "'a set"   and \<HH> :: "'a set list" assumes "normal_series (G\<lparr>carrier := H__, carrier := H\<rparr>) \<HH>"   and "H \<lhd> G\<lparr>carrier := H__\<rparr>" shows "normal_series (G\<lparr>carrier := H__\<rparr>) (\<HH> @ [carrier (G\<lparr>carrier := H__\<rparr>)])"
Application: fixes y :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<Theta> :: "(Vars.var \<times> exp) list"   and z :: "exp"   and e' :: "exp" assumes "atom y \<sharp> (\<Gamma>, e, x, L, \<Delta>, \<Theta>, z)"   and "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'"   and "\<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z" shows "\<Gamma> : App e x \<Down>\<^bsub>L\<^esub> \<Theta> : z"
elims_1: fixes x :: "('a, 'b) lit"   and y :: "bool" assumes "M.wtL x = y" obtains a :: "('a, 'b) atm" where "x = Pos a__" and "y = M.wtA a__"   | a :: "('a, 'b) atm" where "x = Neg a" and "y = M.wtA a"
coprime_eq_empty_prime_inter: fixes n :: "nat"   and m :: "nat" assumes "n \<noteq> 0"   and "m \<noteq> 0" shows "coprime n m = (prime_factors n \<inter> prime_factors m = {})"
qVar_alpha_iff: fixes xs :: "'f"   and x :: "'g"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qVar xs x #= X') = (X' = qVar xs x)"
execute_parallel_operator_cons: fixes s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and ops :: "'a strips_operator list" shows "execute_parallel_operator s (op # ops) = execute_parallel_operator (s ++ map_of (effect_to_assignments op)) ops"
nodes_of_phis'_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and next' :: "'val"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "uninst_code.nodes_of_phis' g next next' nodes_of_phis = (let old_phis = Set.remove next (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> id x) in Let (Mapping.delete (snd next) nodes_of_phis) (Mapping.map_default next' {} (\<lambda>ns. Set.remove next ns \<union> old_phis)))"
Oadmit_Var: fixes \<sigma> :: "('a, 'b, 'c) subst"   and c :: "'c"   and U :: "('c + 'c) set" shows "Oadmit \<sigma> (OVar c) U"
map_norm_in_poly_ring_carrier: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "map UP.poly_of_const p \<in> carrier (poly_ring (K [X]\<^bsub>K [X]\<^esub>))"
Fun_\<Phi>: fixes f :: "'s"   and x :: "'s" assumes "S.arr f"   and "x \<in> S.set (S.dom f)" shows "S'.Fun (\<Phi> f) (\<phi> x) = \<Phi>a f (\<phi> x)"
from_nat_to_nat_id: fixes a :: "'a" shows "from_nat (to_nat a) = a"
leftderives_implies_LeftDerivation: fixes a :: "symbol list"   and b :: "symbol list" assumes "leftderives a b" shows "\<exists>D. LeftDerivation a D b"
chop_single: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<and> |i| = 1 \<longrightarrow> |j| = 0 \<or> |k| = 0"
triangle_left: shows "(\<tau> \<star> h) \<cdot> (h \<star> \<sigma>) = h"
balanced_build: fixes ps :: "(real, 'a) vec list"   and ks :: "'a list" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)" shows "balanced (build ks ps)"
pos: fixes x :: "real" assumes "x \<in> C" shows "0 \<le> f'' x"
tminus_sound: fixes P :: "'b \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "sound (\<lambda>s. P s \<ominus> c)"
ex_retract: fixes x :: "'c \<Longrightarrow> 'd \<times> 'd" assumes "vwb_lens x"   and "Idempotent id"   and "uex x \<circ> id = id \<circ> uex x" shows "retract ((uex x \<circ> id) \<Leftarrow>\<langle>uex x,id\<rangle>\<Rightarrow> id)"
hyps: shows "monom P \<one> (n_ + m) = monom P \<one> n_ \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
is_monomial_def: fixes p :: "'c \<Rightarrow>\<^sub>0 'd" shows "is_monomial p = (card (keys p) = 1)"
sign_compose: fixes p :: "'a \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a" assumes "permutation p"   and "permutation q" shows "sign (p \<circ> q) = sign p * sign q"
nv2L_nvL: fixes l :: "('c, 'psym) lit" shows "nv2L (gL l) = nvL l"
DisjAssoc: fixes A :: "fm"   and B :: "fm"   and C :: "fm" shows "A OR B OR C IMP (A OR B) OR C \<in> boolean_axioms"
supp_prod: fixes x :: "'b"   and y :: "'c" shows "supp (x, y) = supp x \<union> supp y"
sccs_verts_conv: shows "H.sccs_verts = verts ` H.sccs"
bridge_branch_nominals: fixes xs :: "(nat \<times> nat) set"   and k :: "'b"   and j :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list" shows "branch_nominals (mapi_branch (bridge k j xs) branch) \<union> {k, j} = branch_nominals branch \<union> {k, j}"
Rep_singleton_matrix: fixes j :: "nat"   and i :: "nat"   and e :: "'b"   and m :: "nat"   and n :: "nat" shows "Rep_matrix (singleton_matrix j i e) m n = (if j = m \<and> i = n then e else (0::'b))"
q_def: shows "q = Poly (map (\<lambda>k. cs ! (N - k)) [0..<Suc N])"
take_Suc_Cons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "take (Suc n) (x # xs) = x # take n xs"
in_fset: fixes S :: "'b fset"   and x :: "'b" shows "(x |\<in>| S) = (x \<in> fset S)"
map_to_set_inverse: fixes m :: "'a \<Rightarrow> 'b option" shows "set_to_map (map_to_set m) = m"
assms_2: shows "SubRel.subpath g v1 [] v2 subs'"
Pi_def: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'e set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
COD_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "COD (mkarr t) = Cod t"
finite_gpvI: fixes gpv :: "('b, 'out, 'in) gpv" assumes "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.finite_gpv (c input)" shows "local.finite_gpv gpv"
proj_set_def: fixes x :: "'c"   and A :: "'c set" shows "proj_set x A = {y \<in> A. dist x y = infdist x A}"
BuildSeq_exI: fixes B :: "hf \<Rightarrow> bool"   and t :: "hf"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "B t" shows "\<exists>s k. BuildSeq B C s k t"
span_superset: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale" shows "S \<subseteq> module.span scale S"
intensional_simps_16: fixes y :: "'a \<Rightarrow> bool" shows "y \<and> #True \<equiv> y"
card_cartesian_product: fixes A :: "'b set"   and B :: "'c set" shows "card (A \<times> B) = card A * card B"
AA4: fixes A :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'f"   and w :: "'d \<Rightarrow> 'e" shows "\<turnstile> \<diamond>\<langle>\<langle>A\<rangle>_v\<rangle>_w \<longrightarrow> \<diamond>\<langle>\<langle>A\<rangle>_w\<rangle>_v"
alllstsE: fixes x :: "'b llist"   and A :: "'b set" assumes "x \<in> A\<^sup>\<infinity>" obtains "x \<in> A\<^sup>\<star>"   | "x \<in> A\<^sup>\<omega>"
Min_singleton: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.linorder less_eq less" shows "linorder.Min less_eq {x} = x"
standard_uniqueness_isomorphs: fixes X :: "'b set set"   and W :: "'c set set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and C :: "'c set" assumes "ThinChamberComplex X"   and "ChamberComplexIsomorphism W X f"   and "ChamberComplexIsomorphism W X g"   and "SimplicialComplex.maxsimp W C"   and "fun_eq_on f g C" shows "fun_eq_on f g (\<Union> W)"
L_delta: fixes a :: "'b"   and s :: "(bool \<times> 'b) rexp \<times> bool" shows "L_b (delta_b a s) = Deriv a (L_b s)"
opartial_net_preserves_subnets: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and t :: "'e net_state"   and a :: "'f node_action"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and st' :: "'e net_state"   and np :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree" assumes "((\<sigma>, SubnetS s t), a, \<sigma>', st') \<in> opnet_sos (automaton.trans (opnet np p1)) (automaton.trans (opnet np p2))" shows "\<exists>s' t'. st' = SubnetS s' t'"
L_def: shows "L \<equiv> {x. \<exists>w. (x, w) \<in> Rel}"
ipt_iprange_to_set_uncurry_IpAddrNetmask: fixes a :: "'a word \<times> nat" shows "ipt_iprange_to_set (uncurry IpAddrNetmask a) = uncurry ipset_from_cidr a"
check_OR_def: fixes X :: "'pub0 \<times> 'pub1"   and A :: "'msg0 \<times> 'msg1"   and s :: "'bool"   and Z :: "('bool \<times> 'response0) \<times> 'bool \<times> 'response1" shows "check_OR X A s Z = (s = fst (fst Z) \<oplus> fst (snd Z) \<and> fst (fst Z) \<in> challenge_space \<and> fst (snd Z) \<in> challenge_space \<and> check0 (fst X) (fst A) (fst (fst Z)) (snd (fst Z)) \<and> check1 (snd X) (snd A) (fst (snd Z)) (snd (snd Z)))"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>1 f = n \<and> f \<cdot>\<^sub>1 n = n"
refl_indis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set" assumes "PL_Indis cval indis" shows "refl indis"
vec_inv_dim: fixes v :: "'b vec" shows "dim_vec (vec_inv v) = dim_vec v"
insert_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
xsys: shows "Missing_List.min_list xs \<le> Missing_List.min_list ys"
abst_rept_id: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<mu> \<in> \<K> (abss s)" shows "abst (rept s \<mu>) = \<mu>"
empty_subsetI: fixes A :: "'e set" shows "{} \<subseteq> A"
Fi: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "map of_int_hom.vec_hom fs ! i \<in> Rn"
fermat_liar_witness_of_composition_1: fixes a :: "nat"   and n :: "nat" shows "fermat_liar a n \<noteq> fermat_witness a n"
single_subset_iff: fixes a :: "'a"   and M :: "'a multiset" shows "(\<LM> a  \<RM> \<subseteq># M) = (a \<in># M)"
length_replicate: fixes n :: "nat"   and x :: "'a" shows "length (x \<up> n) = n"
powhr_less_cancel_iff: fixes x :: "real star"   and a :: "real star"   and b :: "real star" assumes "1 < x" shows "(x powhr a < x powhr b) = (a < b)"
cases: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c + 'e"   and a2 :: "'d + 'f" assumes "rel_sum R1 R2 a1 a2" obtains a :: "'c" and c :: "'d" where "a1 = Inl a" and "a2 = Inl c" and "R1 a c"   | b :: "'e" and d :: "'f" where "a1 = Inr b" and "a2 = Inr d" and "R2 b d"
skel_def: fixes X :: "('e, 'f, 'g, 'h, 'i) qTerm set" shows "skel X \<equiv> qSkel (pick X)"
lasso_accepted: fixes ma :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and m :: "'a"   and L :: "('Q \<times> nat) lasso" assumes "gb_graph.is_lasso (b_graph.to_gbg_ext (degeneralize_ext ma) m) L" shows "gb_graph.is_acc_run (b_graph.to_gbg_ext (degeneralize_ext ma) m) (run_of_lasso L)"
orthonormal_basis_subspace: fixes S :: "'c set" assumes "subspace S" obtains B :: "'c set" where "B \<subseteq> S" and "pairwise orthogonal B" and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1" and "independent B" and "card B = dim S" and "span B = S"
S'S: shows "gs.span (fs.gs.gso ` {0..<i - 1}) = gs.span ((!) (map of_int_hom.vec_hom fs) ` {0..<i - 1})"
type_copy_Rep_o_Abs: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a" assumes "type_definition Rep Abs UNIV" shows "Rep \<circ> Abs = id"
Busemann_function_le_Gromov_product: fixes xi :: "'b Gromov_completion"   and y :: "'b"   and x :: "'b" shows "ereal (- Busemann_function_at xi y x / 2) \<le> extended_Gromov_product_at x xi (to_Gromov_completion y)"
lossless: fixes \<eta> :: "nat" shows "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) (\<A> \<eta>)"
sel_rews_5: fixes slist :: "[:'a:]"   and a :: "'a" assumes "slist \<noteq> \<bottom>" shows "shead\<cdot>(a :# slist) = a"
assms_4: shows "it \<subseteq> {r \<in> \<delta>. q \<in> set (rhsq r)}"
elims: fixes x :: "('a, 'b) rbt"   and y :: "('a \<times> 'b) list" assumes "rbt_in_traverse_pairs x = y" obtains "x = Leaf" and "y = []"   | l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "y = rbt_in_traverse_pairs l @ (k, v) # rbt_in_traverse_pairs r"
real_interval_plus: fixes a :: "float interval"   and b :: "float interval" shows "real_interval (a + b) = real_interval a + real_interval b"
restore_new_paras: fixes gs :: "recf list"   and ffp :: "nat"   and xs :: "nat list"   and anything :: "nat list" assumes "length gs \<le> ffp" shows "{\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything} mv_boxes (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) 0 (length gs) {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ 0 # xs @ anything}"
perm_compose': fixes pi2 :: "(name \<times> name) list"   and pi1 :: "(name \<times> name) list"   and x :: "'a" shows "(pi2 \<bullet> pi1) \<bullet> x = pi2 \<bullet> pi1 \<bullet> rev pi2 \<bullet> x"
coboundedI: fixes A :: "'a word set"   and a :: "'a word" assumes "finite A"   and "a \<in> A" shows "signed.Min A \<le>s a"
Abs_Measurement_System_ext_inverse: fixes y :: "'a \<times> 'b" assumes "y \<in> UNIV" shows "Rep_Measurement_System_ext (Abs_Measurement_System_ext y) = y"
beukers_nn_integral3_def: shows "local.beukers_nn_integral3 = \<integral>\<^sup>+x\<in>D'. ennreal (case x of (w, xa, y) \<Rightarrow> (xa * (1 - xa) * y * (1 - y) * w * (1 - w)) ^ n / (1 - (1 - xa * y) * w) ^ (n + 1))\<partial>lborel"
ads_d_def: fixes x :: "('a \<times> 'a) set" shows "rdom x = rel_antidomain_kleene_algebra.t x"
all_set_conv_all_nth: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>set xs. P x) = (\<forall>i<length xs. P (xs ! i))"
FW_neg_cycle_detect: 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 n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<not> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> less_eq zero (linordered_ab_monoid_add.len plus m i i xs))" shows "\<exists>i\<le>n. less (linordered_ab_monoid_add.fw plus less_eq m n n i i) zero"
complete_if_size_height: fixes kdt :: "'b kdt" assumes "KD_Tree.size_kdt kdt = 2 ^ height kdt" shows "KD_Tree.complete kdt"
typ_of_typ_def: fixes T :: "typ" shows "typ_of T = T"
Sigma_mset_empty_iff: fixes I :: "'b multiset"   and X :: "'b \<Rightarrow> 'c multiset" shows "(Sigma_mset I X = {#}) = (\<forall>i\<in>#I. X i = {#})"
base: fixes ds :: "'d set"   and a :: "'x set" shows "COP.iterates_abovep ds a a"
Cnext_solve_timedelayed: fixes K\<^sub>1 :: "clock"   and n :: "nat"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and \<delta>\<tau> :: "'c tag_const"   and K\<^sub>2 :: "clock"   and K\<^sub>3 :: "clock"   and \<Phi> :: "'c TESL_atomic list" shows "{K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>, K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi>"
quat_of_real_times_commute: fixes r :: "real"   and q :: "quat" shows "quat_of_real r * q = q * quat_of_real r"
identity: shows "map_fun (\<lambda>x. x) (\<lambda>x. x) = id"
maybe_or_invalid: fixes x :: "trilean"   and y :: "trilean" shows "(x \<or>? y = invalid) = (x = invalid \<or> y = invalid)"
is_Sup_def: fixes A :: "'b set"   and sup :: "'b" shows "is_Sup A sup = ((\<forall>x\<in>A. x \<sqsubseteq> sup) \<and> (\<forall>z. (\<forall>x\<in>A. x \<sqsubseteq> z) \<longrightarrow> sup \<sqsubseteq> z))"
sorted_sorted_list_of_set: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
ZObis_prefix: shows "ZObis \<subseteq> ZOretr ZObis"
sum_mult_is_ideal_prod: fixes A :: "'a set"   and B :: "'a set" assumes "ideal R A"   and "ideal R B" shows "sum_mult R A B = A \<diamondsuit>\<^sub>r B"
subst_def: fixes f :: "'c \<Rightarrow> 'd option"   and mem :: "'c \<Rightarrow> 'd" shows "subst f mem = (\<lambda>x. case f x of None \<Rightarrow> mem x | Some v \<Rightarrow> v)"
independent_sets_mono: fixes vs :: "nat set"   and G :: "nat set \<times> nat set set"   and us :: "nat set" assumes "vs \<in> independent_sets G"   and "us \<subseteq> vs" shows "us \<in> independent_sets G"
zmult_neg_pos: fixes i :: "int"   and j :: "int" assumes "i < 0"   and "0 < j" shows "i * j < 0"
dim_col_transfer_rule: fixes A :: "'e mat"   and A' :: "(('e, 'f) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_col A = CARD('f)"
cblinfun_of_matrix_minus: fixes x :: "'b \<Rightarrow> 'a \<Rightarrow> complex"   and y :: "'b \<Rightarrow> 'a \<Rightarrow> complex" shows "cblinfun_of_matrix x - cblinfun_of_matrix y = cblinfun_of_matrix (x - y)"
Seq_bury: fixes bc\<^sub>1 :: "com"   and bc\<^sub>2 :: "com"   and c :: "com"   and X :: "char list set" shows "(bc\<^sub>1;; bc\<^sub>2 = bury c X) = (\<exists>c\<^sub>1 c\<^sub>2. c = c\<^sub>1;; c\<^sub>2 \<and> bc\<^sub>2 = bury c\<^sub>2 X \<and> bc\<^sub>1 = bury c\<^sub>1 (L c\<^sub>2 X))"
simps: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set" shows "threshold \<phi> w \<G> = Preliminaries2.index (\<PP> \<phi> \<G> w)"