hcju/isabelleps huggingface dataset:
fbox_star_unfoldr: fixes z :: "'a"   and x :: "'a" shows " |(1::'a)] z \<cdot>  |x\<^sup>\<star>]  |x] z =  |x\<^sup>\<star>] z"
in_diff_count: fixes a :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" shows "(a \<in># M - N) = (count N a < count M a)"
mult_minus_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = uminus (times a b)"
none_MT_rulessubset: fixes a :: "((int \<times> int) set set, int) Combinators list"   and b :: "((int \<times> int) set set, int) Combinators list" assumes "none_MT_rules C a"   and "set b \<subseteq> set a" shows "none_MT_rules C b"
tiny_digraphI: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<CC>"   and "vsv (\<CC>\<lparr>Dom\<rparr>)"   and "vsv (\<CC>\<lparr>Cod\<rparr>)"   and "vcard \<CC> = 4\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "\<CC>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<CC>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "tiny_digraph \<alpha> \<CC>"
assms: shows "Preliminaries.abs x + Preliminaries.abs y = 0"
simps_1: fixes p :: "perm"   and tset :: "('e, 'f, 'g) Tree set['e]" shows "p \<bullet> tConj tset = tConj (map_bset (permute p) tset)"
r_def: shows "r \<equiv> poly_of_vec (vec_last (vec n c) d)"
cases: fixes a :: "'a set \<Rightarrow> 'a set" assumes "IK_nf a" obtains "a = id"   | "a = I"   | "a = K"   | "a = I \<circ> K"   | "a = K \<circ> I"   | "a = I \<circ> K \<circ> I"   | "a = K \<circ> I \<circ> K"
bind_game_le_inv_q: fixes q :: "nat"   and \<A> :: "nat \<times> nat \<Rightarrow> (nat \<times> nat \<times> (nat \<times> nat) \<times> nat \<times> nat \<times> nat) spmf" assumes "rivest q" shows "spmf (abstract_commitment.bind_game (rivest.key_gen q) (rivest.verify q) (rivest.valid_msg q) \<A>) True \<le> 1 / real q"
equivalence_in_bicategory_axioms: shows "D.internal_equivalence (F f) g' \<phi> \<psi>"
assms: shows "finite_SMP_representation arity Ana \<Gamma> M"
rcs_nonempty: fixes H :: "'a set"   and X :: "'a set" assumes "G \<guillemotright> H "   and "X \<in> set_rcs G H" shows "X \<noteq> {}"
le_less_trans: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't"   and z :: "'loc \<times> 't" assumes "x\<le>\<^sub>py"   and "y<\<^sub>pz" shows "x<\<^sub>pz"
simps_1: fixes x :: "'b"   and y :: "'b" shows "(extT.Val x \<le> extT.Val y) = (x \<le> y)"
\<Delta>\<Phi>_link: fixes h1 :: "'b hp"   and h2 :: "'b hp" shows "\<Phi>_hp (link h1 h2) - \<Phi>_hp h1 - \<Phi>_hp h2 \<le> 2 * log 2 (real (Pairing_Heap_List2_Analysis.size_hp h1 + Pairing_Heap_List2_Analysis.size_hp h2))"
len_lexext_wf: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP (\<lambda>x y. gt y x)" shows "wfP (\<lambda>xs ys. len_lexext gt ys xs)"
distinct_279: fixes x131 :: "floatarith"   and x132 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Powr x131 x132 \<noteq> x151 ^\<^sub>e x152"
Iic_subset_Iio_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.linorder less_eq less" shows "(ord.atMost less_eq a \<subseteq> ord.lessThan less b) = less a b"
i_join_finite_nth_ge_card_eq_nth_Max: fixes I :: "nat set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "finite I"   and "I \<noteq> {}"   and "card I \<le> Suc n" shows "(f \<Join>\<^sub> I) n = f (Max I)"
succ_edge: fixes G :: "nat graph_rec"   and n :: "nat" assumes "cfg_doms G" shows "\<forall>v\<in>cfg_doms.succs G n. (n, v) \<in> g_E G"
assms: shows "crdt_ops (xs @ [(oid, oper)] @ ys) deps"
that: fixes c :: "nat"   and n :: "nat" assumes "0 < c"   and "\<And>m. n < m \<Longrightarrow> f m \<le> c * g m"   and "\<And>c n. 0 < c \<Longrightarrow> \<exists>m>n. c * f m < g m" shows "thesis"
s_Pair1: fixes \<pi> :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e\<^sub>1' :: "Syntax.term"   and e\<^sub>2 :: "Syntax.term" assumes "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater>" shows "\<lless>\<pi>, Syntax.Pair e\<^sub>1 e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', Syntax.Pair e\<^sub>1' e\<^sub>2\<ggreater>"
Conj_def: fixes xset :: "('a, 'b, 'c, 'd) formula set['a]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
derives_def: fixes u :: "symbol list"   and v :: "symbol list" shows "derives u v = ((u, v) \<in> derivations)"
WHILE_PARALLEL_tp: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and q :: "L"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr" assumes "X \<rhd> c : Sec (Q p G) (EQ G q) (EQ G q) \<Phi>"   and "(G, b, p) \<in> HS_B"   and "LEQ p q" shows "X \<rhd> While b c : Sec (Q p G) (EQ G q) (EQ G q) (PhiWhileP (EQ G q) b \<Phi>)"
count_image_mset_le_count_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a multiset"   and y :: "'b" assumes "inj_on f (set_mset M)" shows "count (image_mset f M) y \<le> count M (inv_into (set_mset M) f y)"
trace_minus_linear: fixes A :: "'c mat"   and n :: "nat"   and B :: "'c mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "trace (A - B) = trace A - trace B"
sum_distrib_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and r :: "'b" assumes "class.semiring_0 plus zero times" shows "times (comm_monoid_add.sum plus zero f A) r = comm_monoid_add.sum plus zero (\<lambda>n. times (f n) r) A"
simps_2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "removen n (x # xs) = (if n = 0 then xs else x # removen (n - 1) xs)"
per_compare_def: fixes R :: "('c \<times> 'd) set"   and a :: "'c"   and b :: "'d" shows "per_compare R a b \<equiv> (a, b) \<in> R"
sumpar: shows "\<forall>n. partial_density_operator (matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n)"
root_mk: fixes f :: "'b"   and ts :: "'a list" assumes "(f, length ts) \<in> F" shows "root (mk f ts) = (f, length ts)"
bisimSubstTransitive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim>\<^sub>s Q"   and "\<Psi> \<rhd> Q \<sim>\<^sub>s R" shows "\<Psi> \<rhd> P \<sim>\<^sub>s R"
carrier_vec_induct: fixes v :: "'a vec"   and n :: "nat"   and P :: "nat \<Rightarrow> 'a vec \<Rightarrow> bool" assumes "v \<in> carrier_vec n"   and "P 0 vNil"   and "\<And>n a v. \<lbrakk>v \<in> carrier_vec n; P n v\<rbrakk> \<Longrightarrow> P (Suc n) (vCons a v)" shows "P n v"
B: shows "AE x in S \<Otimes>\<^sub>M T. Py (snd x) = 0 \<longrightarrow> Pxy x = 0"
Graph_Invariant_Strong_axioms: shows "Graph_Invariant_Strong E P"
assms: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
less_eq_top_Top: fixes x :: "'b top" shows "x \<le> top.Top"
sym: fixes S :: "('d, 'e) eq_object_scheme"   and x :: "'d"   and y :: "'d" assumes "equivalence S"   and "x .=\<^bsub>S\<^esub> y"   and "x \<in> carrier S"   and "y \<in> carrier S" shows "y .=\<^bsub>S\<^esub> x"
associatedD1: fixes a :: "'b poly"   and b :: "'b poly" assumes "smult (inverse (lead_coeff a)) a = smult (inverse (lead_coeff b)) b" shows "a dvd b"
wf_interp_for_formula_any_Inr: fixes w :: "'a list"   and P :: "nat set"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "wf_interp_for_formula (w, Inr P # I) \<phi>" shows "\<forall>P\<subseteq>{0..length w - 1}. wf_interp_for_formula (w, Inr P # I) \<phi>"
emb_induct_reverse: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and P :: "('c, 'd) tm \<Rightarrow> ('c, 'd) tm \<Rightarrow> bool" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "\<And>t. P t t"   and "\<And>t u s. \<lbrakk>t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b u; u \<unrhd>\<^sub>e\<^sub>m\<^sub>b s; P u s\<rbrakk> \<Longrightarrow> P t s" shows "P t s"
polyn_addTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "pol_coeff S (n, f)" shows "\<forall>g. pol_coeff S (n + m, g) \<longrightarrow> polyn_expr R X n (n, f) \<plusminus> polyn_expr R X (n + m) (n + m, g) = polyn_expr R X (n + m) (add_cf S (n, f) (n + m, g))"
one_less_of_natD: fixes n :: "nat" assumes "(1::'a) < of_nat n" shows "1 < n"
iso_can: fixes f :: "'c local.term set" assumes "\<F>C.can f" shows "\<F>C.iso f"
mod_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo zero a = zero"
hb: shows "\<forall>v. insertion (nth_default 0 (L[var := v])) b = B"
simple_integral_cong: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal"   and g :: "'b \<Rightarrow> ennreal" assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" shows "integral\<^sup>S M f = integral\<^sup>S M g"
of_nat_gbinomial: fixes n :: "nat"   and k :: "nat" shows "of_nat (n gchoose k) = of_nat n gchoose k"
upd_cols_transpose: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d set"   and p :: "'d \<Rightarrow> ('c, 'd) vec" shows "upd_cols (Square_Matrix.transpose M) i p = Square_Matrix.transpose (upd_rows M i p)"
n\<phi>_2: shows "\<phi> \<in> fmla"
prod_ord_pair_ko_ntm_def: fixes ko :: "'a nat_term_order" shows "prod_ord_pair_ko_ntm ko \<equiv> comparator.prod_ord_pair (nat_term_compare_inv ko)"
intro: fixes T :: "'a \<Rightarrow> 'a"   and V :: "'a set" assumes "T ` V \<subseteq> V" shows "FGModuleEnd_axioms V T"
OrdP_Mem_imp_Subset: fixes H :: "fm set"   and k :: "tm"   and l :: "tm" assumes "H \<turnstile> k IN l"   and "H \<turnstile> OrdP l" shows "H \<turnstile> k SUBS l"
zero_if_Limit_eq_Limit_plus_vnat: fixes \<alpha> :: "V"   and \<beta> :: "V"   and n :: "V" assumes "Limit \<alpha>"   and "Limit \<beta>"   and "\<alpha> = \<beta> + n"   and "n \<in>\<^sub>\<circ> \<omega>" shows "n = 0"
rep_eq: fixes x :: "'k word" shows "the_int x = take_bit LENGTH('k) (Word.rep x)"
is_singletonI: fixes x :: "'a" shows "is_singleton {x}"
ce_rel_lm_19: fixes x :: "nat"   and r :: "(nat \<times> nat) set" assumes "(c_fst x, c_snd x) \<in> r" shows "x \<in> ce_rel_to_set r"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "P TBase"   and "P TUnit"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1\<rightarrow>ty2)" shows "P ty"
reachable_in_oreachable: fixes s :: "'a \<times> 'b"   and A :: "('a \<times> 'b, 'c) automaton"   and I :: "'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "s \<in> reachable A I" shows "s \<in> oreachable A (act I) U"
exec_preserves_disj: fixes c :: "com"   and ps :: "char list \<Rightarrow> int option"   and t :: "nat"   and ps' :: "char list \<Rightarrow> int option"   and ps'' :: "char list \<Rightarrow> int option" assumes "(c, ps) \<Rightarrow>\<^sub>A t \<Down> ps'"   and "ps'' ## ps" shows "ps'' ## ps'"
cp0: fixes X :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and Y :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and \<tau> :: "'d state \<times> 'd state" shows "(X \<doteq> Y) \<tau> = ((\<lambda>_. X \<tau>) \<doteq> (\<lambda>_. Y \<tau>)) \<tau>"
embed_bool_cancel: fixes G :: "'a \<Rightarrow> bool"   and s :: "'a" shows "\<guillemotleft> G \<guillemotright> s * \<guillemotleft> \<N> G \<guillemotright> s = 0"
adjoint_preds: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real"   and p :: "char" shows "Preds (USubst.adjoint \<sigma> I \<omega>) p = (case SPreds \<sigma> p of None \<Rightarrow> Preds I p | Some r \<Rightarrow> \<lambda>d. \<omega> \<in> fml_sem (repc I dotid d) r)"
v1_in_type: shows "(left (ext v), left (ext v) + x) \<in> {r. fst r \<le> snd r}"
Q\<^sub>L_def: fixes \<Sigma> :: "'b list"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" shows "Q\<^sub>L \<Sigma> \<delta> q\<^sub>0 = (if \<Sigma> \<noteq> [] then gen_dfs (\<lambda>q. map (\<delta> q) \<Sigma>) insert (\<in>) {} [q\<^sub>0] else {})"
finite: fixes A :: "'d set" assumes "class.finite TYPE('d)" shows "finite A"
integrable_measure_pmf_finite: fixes M :: "'b pmf"   and f :: "'b \<Rightarrow> 'c" assumes "finite (set_pmf M)" shows "integrable (measure_pmf M) f"
compose_chart_apply_chart: fixes p :: "'c \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'c"   and c :: "('d, 'c) chart" assumes "homeomorphism UNIV UNIV p p'" shows "apply_chart (compose_chart p p' c) = p \<circ> apply_chart c"
star_slide_var1: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x \<le> x \<cdot> x\<^sup>\<star>"
flift2_bottom_iff: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c lift" shows "(flift2 f\<cdot>x = \<bottom>) = (x = \<bottom>)"
sum_interval_distl: fixes m :: "nat"   and n :: "nat"   and x :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "m \<le> n" shows "x \<cdot> sum f {m..n} = (\<Sum>i = m..n. x \<cdot> f i)"
resCap_cf_def: fixes cf :: "'c \<Rightarrow> 'b"   and p :: "'c list" shows "resCap_cf cf p \<equiv> Min {cf e |e. e \<in> set p}"
valid_indexE: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" obtains "is = []" and "dsa__ = []"   | isa :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "is = i # isa" and "dsa__ = d # ds" and "isa \<lhd> ds" and "i < d"
sd_power_prop1: fixes act :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d"   and i :: "nat" assumes "unital_quantale_module act" shows "power.power sd_unit (unital_quantale_module.sd_prod act) (x, y) (Suc i) = (x ^ Suc i, \<Squnion> {act (x ^ k) y |k. k \<le> i})"
source_out: fixes y :: "'v" shows "edge \<Delta> (source \<Delta>) y = (y = x)"
subst_eq: fixes k :: "nat"   and u :: "db" shows "DBVar k<u/k> = u"
inv_safe_def: fixes c :: "('loc, 't) configuration" shows "inv_safe c = (\<forall>loc1 loc2 t s. 0 < zcount (c_pts c loc1) t \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<and> worklists_vacant_to c (results_in t s) \<longrightarrow> (\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c loc2)))"
simps: fixes prec :: "nat"   and x :: "float" shows "ub_ln prec x = (if x \<le> 0 then None else if x < 1 then Some (- the (lb_ln prec (float_divl (max prec 1) 1 x))) else let horner = \<lambda>x. float_round_up prec (x * ub_ln_horner prec (get_odd prec) 1 x) in if x \<le> Float 3 (- 1) then Some (horner (x - 1)) else if x < Float 1 1 then Some (float_round_up prec (horner (Float 1 (- 1)) + horner (x * rapprox_rat prec 2 3 - 1))) else let l = bitlen (mantissa x) - 1 in Some (float_plus_up prec (float_round_up prec (ub_ln2 prec * Float (exponent x + l) 0)) (horner (Float (mantissa x) (- l) - 1))))"
eq: fixes U :: "'a set" assumes "closedin X U" shows "{x \<in> topspace Y. g x \<in> U} = f ` U"
incseq_Suc_iff: fixes f :: "nat \<Rightarrow> 'b" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
simps_1: fixes f\<^sub>1 :: "'c val \<Rightarrow> 'c val"   and r\<^sub>2 :: "'b rexp"   and f\<^sub>2 :: "'c val \<Rightarrow> 'c val" shows "simp_Plus (Zero, f\<^sub>1) (r\<^sub>2, f\<^sub>2) = (r\<^sub>2, F_RIGHT f\<^sub>2)"
mult_isol: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.pre_dioid plus times less_eq less"   and "less_eq x y" shows "less_eq (times z x) (times z y)"
sum: fixes g :: "'d \<Rightarrow> 'b"   and S :: "'d set" shows "construct B f (sum g S) = (\<Sum>a\<in>S. construct B f (g a))"
asymmetric_conv_closed: fixes x :: "'a" assumes "asymmetric x" shows "asymmetric (x\<^sup>T)"
Suc: fixes x :: "real" shows "bernpoly n (x + 1) - bernpoly n x = real n * x ^ (n - 1)"
no_spoofing_algorithm_sound_generalized: fixes rs1 :: "'b common_primitive rule list"   and rs2 :: "'b common_primitive rule list"   and iface :: "iface"   and allowed :: "'b word set"   and denied1 :: "'b word set"   and ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option" assumes "simple_ruleset rs1"   and "simple_ruleset rs2"   and "\<forall>r\<in>set rs2. normalized_nnf_match (get_match r)"   and "setbydecision TYPE('c) iface rs1 FinalAllow \<subseteq> allowed"   and "denied1 \<subseteq> setbydecision_all TYPE('c) iface rs1 FinalDeny"   and "no_spoofing_algorithm iface ipassmt rs2 allowed denied1" shows "nospoof TYPE('c) iface ipassmt (rs1 @ rs2)"
find_remove_length: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and res :: "'a option"   and xs' :: "'a list" assumes "find_remove P xs = (res, xs')" shows "length xs' \<le> length xs"
set_empty: fixes xs :: "'a list" shows "(set xs = {}) = (xs = [])"
Ln_of_nat_over_of_nat: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "0 < n" shows "Ln (of_nat m / of_nat n) = complex_of_real (ln (real m) - ln (real n))"
delete_no_failure: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "'a"   and n :: "'a"   and j :: "nat"   and s :: "('a \<times> 'b \<times> bool) list" assumes "es @ [Deliver (i, Delete n)] prefix of j"   and "apply_operations es = Some s" shows "\<exists>ys. delete s n = Some ys"
vequalityCE: fixes A :: "V"   and B :: "V"   and c :: "V" assumes "A = B" obtains "c \<in>\<^sub>\<circ> A" and "c \<in>\<^sub>\<circ> B"   | "c \<notin>\<^sub>\<circ> A" and "c \<notin>\<^sub>\<circ> B"
union: fixes M :: "'a set set" assumes "\<And>t. t \<in> M \<Longrightarrow> t open" shows "\<Union> M open"
appendPath_append_last: fixes Ds :: "char list list"   and Cs :: "char list list" assumes "Ds \<noteq> []" shows "(Cs @\<^sub>p Ds) @\<^sub>p [last Ds] = Cs @\<^sub>p Ds"
is_processT3: fixes P :: "'a process" shows "\<forall>s t. (s @ t, {}) \<in> F P \<longrightarrow> (s, {}) \<in> F P"
kon_graph_def: shows "kon_graph \<equiv> \<lparr>nodes = {a, b, c, d}, edges = {(a, ab1, b), (b, ab1, a), (a, ab2, b), (b, ab2, a), (a, ac1, c), (c, ac1, a), (a, ac2, c), (c, ac2, a), (a, ad1, d), (d, ad1, a), (b, bd1, d), (d, bd1, b), (c, cd1, d), (d, cd1, c)}\<rparr>"
rI: shows "rel_set A (X \<inter> Collect (Domainp A)) Y"
invertible_right_inverse2: fixes u :: "'a"   and v :: "'a" assumes "sub.invertible u"   and "u \<in> H"   and "v \<in> H" shows "u \<cdot> (sub.inverse u \<cdot> v) = v"
unique_idx_ab: fixes ab :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h"   and hs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data' :: "nat \<times> 'h"   and aux :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h" assumes "ab_spec ab"   and "unique_idx (gs @ bs) data"   and "(hs, data') = add_indices aux data" shows "unique_idx (gs @ ab gs bs hs data') data'"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a"   and llist :: "'b llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (llist = LCons (lhd llist) (ltl llist) \<longrightarrow> P (f2 (lhd llist) (ltl llist))))"
perm_restrict_simps_1: fixes x :: "'c"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "x \<in> S" shows "perm_restrict f S x = f x"
im_jointfunTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and g :: "nat \<Rightarrow> 'c" shows "jointfun n f m g ` {i. i \<le> n} = f ` {i. i \<le> n}"
circ_reflexive: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_idempotent_semiring.reflexive (x\<^sup>\<odot>)"
ev_A_norm_v: fixes j :: "'n" shows "map_vector cmod (cA *v v) $h j \<le> (A *v map_vector cmod v) $h j"
increasing: fixes F :: "ordinal \<Rightarrow> ordinal"   and x :: "ordinal" assumes "normal F" shows "x \<le> F x"
cases: fixes a1 :: "cmd"   and a2 :: "cmd" assumes "subcmd a1 a2" obtains c :: "cmd" where "a1 = ca__" and "a2 = Skip"   | c :: "cmd" where "a1 = cb__" and "a2 = cb__"   | c1 :: "cmd" and c :: "cmd" and c2 :: "cmd" where "a1 = c1__;; c2__" and "a2 = cc__" and "subcmd c1__ cc__"   | c2 :: "cmd" and c :: "cmd" and c1 :: "cmd" where "a1 = c1a__;; c2a__" and "a2 = cd__" and "subcmd c2a__ cd__"   | c1 :: "cmd" and c :: "cmd" and b :: "expr" and c2 :: "cmd" where "a1 = if (b__) c1b__ else c2b__" and "a2 = ce__" and "subcmd c1b__ ce__"   | c2 :: "cmd" and c :: "cmd" and b :: "expr" and c1 :: "cmd" where "a1 = if (ba__) c1 else c2" and "a2 = cf__" and "subcmd c2 cf__"   | c' :: "cmd" and c :: "cmd" and b :: "expr" where "a1 = while (b) c'" and "a2 = c" and "subcmd c' c"
iMODb_pred_insert_conv: fixes m :: "nat"   and r :: "nat"   and c :: "nat" assumes "m \<le> r" shows "insert (r - m) [ r, mod m, c ] = [ r - m, mod m, Suc c ]"
is_operator_applicable_in_def: fixes s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator" shows "is_operator_applicable_in s op \<equiv> map_of (precondition_of op) \<subseteq>\<^sub>m s"
count_drop: fixes n :: "nat"   and cs :: "'a list"   and x :: "'a" shows "count_list (drop n cs) x \<le> count_list cs x"
inject_1: fixes x1 :: "bool"   and y1 :: "bool" shows "(Det x1 = Det y1) = (x1 = y1)"
WTrtCond: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and e\<^sub>1 :: "expr"   and T :: "ty"   and e\<^sub>2 :: "expr" assumes "P,E,h \<turnstile> e : Boolean"   and "P,E,h \<turnstile> e\<^sub>1 : T"   and "P,E,h \<turnstile> e\<^sub>2 : T" shows "P,E,h \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 : T"
tllist_ord_TNil_TNil: fixes b :: "'f"   and b1 :: "'f"   and b2 :: "'f" shows "tllist_ord b (TNil b1) (TNil b2) = flat_ord b b1 b2"
\<pi>f: shows "\<forall>i. \<exists>k\<le>n. \<pi> k = f i"
is_extensional: fixes f :: "'j" assumes "\<not> J.arr f" shows "\<gamma>.map f = C.null"
halt_state_keep_steps_add: fixes m :: "nat"   and lm :: "'b"   and stpa :: "nat"   and n :: "nat" assumes "nonstop m (bl2wc (<lm>)) stpa = 0" shows "conf m (bl2wc (<lm>)) stpa = conf m (bl2wc (<lm>)) (stpa + n)"
is_natural_2: fixes f :: "'a" assumes "A.arr f" shows "\<tau> (A.cod f) \<cdot>\<^sub>B G f = \<tau> f"
normalize_gives_polynomial: fixes p :: "'a list"   and K :: "'a set" assumes "set p \<subseteq> K" shows "polynomial K (local.normalize p)"
show_lawE: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and x :: "'a"   and p :: "nat"   and y :: "char list"   and z :: "char list" assumes "show_law s x" obtains "s p x (y @ z) = s p x y @ z"