hcju/isabelleps huggingface dataset:
nth_take_lemma: fixes k :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "k \<le> length xs"   and "k \<le> length ys"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
AimpB_p: fixes \<tau> :: "\<zero> \<Rightarrow> i \<Rightarrow> bool" shows "\<lfloor>(\<lambda>w. (\<forall>x. (\<^bold>\<box>x (\<lambda>x. \<lparr>\<tau> x w\<rparr>) \<^bold>\<rightarrow> \<^bold>\<box>x \<^bold>\<down> \<tau>) w) \<longrightarrow> (\<forall>x. (\<^bold>\<box>x \<^bold>\<down> \<tau> \<^bold>\<rightarrow> \<^bold>\<box>x (\<lambda>x. \<lparr>\<tau> x w\<rparr>)) w))\<rfloor>"
larger: shows "wp a Q (\<phi> s) < wp b (Q \<circ> \<phi>) s"
partitioned_offending: shows "\<forall>F\<in>set_offending_flows \<lparr>nodes = V, edges = E \<union> X\<rparr> nP. F \<subseteq> X"
modus_tollens_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> in v]"   and "[\<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
remdups_removeAll: fixes x :: "'b"   and xs :: "'b list" shows "remdups (removeAll x xs) = remove1 x (remdups xs)"
preserves_lunit_1: fixes f :: "'c" assumes "C.ide f" shows "F \<l>\<^sub>C[f] = \<l>\<^sub>D[F f] \<cdot>\<^sub>D (D.inv (unit (trg\<^sub>C f)) \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv (\<Phi> (trg\<^sub>C f, f))"
source_in: fixes x :: "'v" shows "\<not> edge (residual_network f) x (source (residual_network f))"
divide_strict_right_mono_neg: 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 b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less b a"   and "less c zero" shows "less (divide a c) (divide b c)"
sup_commut_basic_type: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" shows "\<tau> \<squnion> \<sigma> = \<sigma> \<squnion> \<tau>"
eq_id_iff: fixes f :: "'d \<Rightarrow> 'd" shows "(\<forall>x. f x = x) = (f = id)"
index_def: shows "index = (\<lambda>t. SOME i. t = real (fst i) / 2 ^ snd i * dist x0 y0 \<and> fst i \<le> 2 ^ snd i)"
assms_3: shows "f w \<bullet> rot (x - y) < 0"
Nil_prefix: fixes xs :: "'b list" shows "\<epsilon> \<le>p xs"
Rf_limit_Sup: fixes Ns :: "'a literal multiset set llist" assumes "chain sr_ext.derive Ns" shows "sr.Rf (Liminf_llist Ns) = sr.Rf (Sup_llist Ns)"
dist_eq_0_iff: fixes dist :: "'g \<Rightarrow> 'g \<Rightarrow> real"   and uniformity :: "('g \<times> 'g) filter"   and "open" :: "'g set \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.metric_space dist uniformity open" shows "(dist x y = 0) = (x = y)"
neutral: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x\<in>A. g x = (1::'b)" shows "prod g A = (1::'b)"
wt_s_t_le_\<delta>_t: shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) + eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef s 0)) * eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t)) \<le> zhmset_of \<delta>\<^sub>h + eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t))"
cond_skip: fixes b :: "'b hrel" assumes "out\<alpha> \<sharp> b" shows "(b \<and> II) = (II \<and> b\<^sup>-)"
absorb_iff2: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd" shows "less_eq_prod a b = (sup_prod a b = b)"
absorb2: fixes b :: "'c word"   and a :: "'c word" assumes "b \<le>s a" shows "signed.min a b = b"
left_translations_of_monoid_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "left_translations_of_monoid M composition unit \<equiv> monoid M composition unit"
interval_integral_substitution_finite: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})"   and "continuous_on (g ` {a..b}) f"   and "continuous_on {a..b} g'" shows "LBINT x=ereal a..ereal b. g' x *\<^sub>R f (g x) = interval_lebesgue_integral lborel (ereal (g a)) (ereal (g b)) f"
assms_4: shows "f \<in> A \<rightarrow> UNIV"
trace_decomp_head: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration" assumes "trace c (ev # t) c'" shows "\<exists>c''. c \<turnstile> ev \<mapsto> c'' \<and> trace c'' t c'"
Bag_def: fixes xs :: "('b, nat) alist" shows "Bag xs = Abs_multiset (count_of (alist.impl_of xs))"
partition_filter1: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "fst (partition P xs) = filter P xs"
zip_replicate1: fixes n :: "nat"   and x :: "'g"   and ys :: "'h list" shows "zip (replicate n x) ys = map (Pair x) (take n ys)"
th2: fixes i :: "nat"   and z :: "nat"   and n :: "nat" assumes "i \<le> z div n" shows "i * n \<le> z"
someI2: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c"   and Q :: "'c \<Rightarrow> bool" assumes "P a"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (SOME x. P x)"
wf_tuple_map_Some: fixes xs :: "'a list"   and n :: "nat"   and A :: "nat set" assumes "length xs = n"   and "{0..<n} \<subseteq> A" shows "wf_tuple n A (map Some xs)"
simps_90: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x91 :: "'c formula"   and x92 :: "\<I>"   and x93 :: "'c formula" shows "(case Until x91 x92 x93 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f9 x91 x92 x93"
bidel_arc_biconnected_diff_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" shows "bidel_arc_biconnected_diff G M a \<equiv> bidel_arc_biconnected_non_triv G M a \<and> bidel_arc_diff_face G M a"
poly_exp_expand: shows "p = (\<Prod>(a, i)\<in>as. a ^ i) * (\<Prod>(a, i)\<in>as. a)"
is_path_loop: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "is_path \<pi>"   and "0 < i"   and "\<pi> i = \<pi> 0" shows "is_path (\<lambda>n. \<pi> (n mod i))"
casts_conf: 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 T :: "ty"   and v :: "val"   and v' :: "val"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and T' :: "ty" assumes "wwf_prog P"   and "P \<turnstile> T casts v to v' "   and "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor>"   and "P \<turnstile> T' \<le> T" shows "P,h \<turnstile> v' :\<le> T"
det_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A $h i $h p i))"
typing_alt_simps_1: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Literal NullLiteral : \<tau>) = (\<tau> = OclVoid[?])"
to_tree_order_impl: shows "to_tree_order = local.a_to_tree_order"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s1)))))))) \<and> t2 = snd (fst (set_annul True (snd (fst (write_cpu (cpu_reg_val PC s2 + sign_ext24 (ucast (get_operand_w22 (snd instr ! Suc 0)) << 2)) nPC (snd (fst (write_cpu (cpu_reg_val nPC s2) PC s2))))))))"
keys_diff: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and b :: "'b \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.keys (a - b) \<subseteq> Poly_Mapping.keys a \<union> Poly_Mapping.keys b"
no_loose_bvar1_subst_bv2_decr: fixes t :: "Core.term"   and i :: "nat"   and x :: "Core.term" assumes "\<not> loose_bvar1 t i" shows "subst_bv2 t i x = decr i t"
s1': shows "s1' = (ls1', (ts1', m1'), ws1', is1')"
content_dvd_coeff: fixes p :: "'a poly"   and n :: "nat" shows "content p dvd coeff p n"
comp_by_index_def: shows "comp_by_index = (\<lambda>n1 n2. c_pair 4 (c_pair n1 n2))"
Singl: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and trn :: "('a, 'b, 'c) trans" assumes "IO_Automaton.validTrans step trn" shows "IO_Automaton.valid step [trn]"
QI_w: fixes i :: "int"   and i' :: "int"   and j :: "int"   and j' :: "int" assumes "i \<le> i'"   and "i' \<le> j"   and "j \<le> j'" shows "w i j + w i' j' \<le> w i' j + w i j'"
SUP_upper: fixes i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd \<times> 'e" assumes "s2p A i" shows "less_eq_prod (f i) (Sup_prod (\<P> f A))"
product_simps_1: fixes AA :: "('c, 'd) dca list" shows "dgca.alphabet (union_list' AA) = \<Inter> (dca.alphabet ` set AA)"
simps: fixes xs :: "'c list" shows "msort xs = (let n = length xs in if n \<le> 1 then xs else merge (msort (take (n div 2) xs)) (msort (drop (n div 2) xs)))"
var_not_in_subst_dom: fixes v :: "'a"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "v \<notin> subst_domain s" shows "s v = Var v"
cut_greater_Max_empty: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I \<le> t" shows "I \<down>> t = {}"
fsubset_finsertI: fixes B :: "'x fset"   and a :: "'x" shows "B |\<subseteq>| finsert a B"
Bernstein_changes_01_def': fixes p :: "nat"   and P :: "real poly" shows "Bernstein_changes_01 p P = int (nat (changes (map (\<lambda>j. inverse (real (p choose j)) * coeff (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]) (p - j)) [0..<p + 1])))"
normal_exp: fixes x :: "ordinal" assumes "oSuc 0 < x" shows "normal ((**) x)"
max_mult_left: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x" shows "x * max y z = max (x * y) (x * z)"
that: fixes s :: "'a llist" assumes "s \<in> A\<^sup>\<infinity>"   and "t = r @@ s" shows "thesis"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> analz H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> analz H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> analz H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> analz H; P (Crypt K X); Key (invKey K) \<in> analz H; P (Key (invKey K))\<rbrakk> \<Longrightarrow> P X" shows "P x"
assms_2: shows "\<forall>f. s@@staticLoc f = t@@staticLoc f"
productive_in_N: fixes C :: "'a literal multiset" assumes "production C \<noteq> {}" shows "C \<in> N"
sel_3: fixes x21 :: "'f"   and x22 :: "'d" shows "continuation (IO x21 x22) = x22"
dma_correct: fixes mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var" shows "dma mem x = (if \<forall>e\<in>dma_type x. ev\<^sub>B mem e then Low else High)"
DisjI: fixes A :: "fm"   and B :: "fm" assumes "ss_fm A"   and "ss_fm B" shows "ss_fm (A OR B)"
usolves_odeD_3: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and t0 :: "real"   and T :: "real set"   and X :: "'b set" assumes "(y usolves_ode f from t0) T X" shows "is_interval T"
complete_Lin: fixes a :: "alt"   and b :: "alt" assumes "a \<noteq> b" shows "\<exists>L\<in>Lin. a <\<^bsub>L\<^esub> b"
not_interval_UNIV2_1: fixes a :: "'c"   and b :: "'c" shows "UNIV \<noteq> cbox a b"
nth_bounded: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" assumes "x !! n"   and "x < 2 ^ m"   and "m \<le> LENGTH('b)" shows "n < m"
part_UNlift_UNpart: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "properL dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n)" shows "part {..<brnL dl (length dl)} (UNlift cl dl P F ` UNpart cl P)"
elapsed_ge_pos: fixes u :: "'c \<Rightarrow> real"   and u' :: "'c \<Rightarrow> real"   and c :: "'c" assumes "eq_elapsed u u'"   and "c \<in> \<X>"   and "u \<in> V"   and "u' \<in> V" shows "elapsed u u' \<le> u' c"
effect__strips_iii_c: fixes s' :: "'a \<Rightarrow> bool option"   and s :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator"   and v :: "'a" assumes "s' = s \<then> op"   and "v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "s' v = s v"
preserves_runit_1: fixes f :: "'a" assumes "C.ide f" shows "F \<r>\<^sub>C[f] = \<r>\<^sub>D[F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (unit (src\<^sub>C f))) \<cdot>\<^sub>D D.inv (\<Phi> (f, src\<^sub>C f))"
Red1Seq: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and v :: "'addr val"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Val v;; e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>e,s\<rangle>"
trace_callee_conv_trace_callee_aux: fixes callee :: "'b \<Rightarrow> 'c \<Rightarrow> ('a \<times> 'b) spmf"   and p :: "'b spmf"   and xs :: "('c \<times> 'a) list"   and a :: "'c" shows "trace_callee callee p xs a = trace_callee_aux callee p xs \<bind> (\<lambda>s. map_spmf fst (callee s a))"
l1_step4_refines_a0i_commit_skip_r: fixes A :: "agent"   and B :: "agent"   and gnx :: "msg"   and Rb :: "fid_t" shows "{R01iar \<inter> UNIV \<times> l1_inv6} (a0i_commit A B (Exp gnx (NonceF (Rb $ ny))))\<^sup>=, l1_step4 Rb A B gnx {> R01iar}"
a_idem: fixes x :: "'a" shows "ad x \<cdot> ad x = ad x"
S0'_OK: shows "S0' = \<alpha> ` S0"
ccompatible1: fixes c :: "real"   and k :: "'d \<Rightarrow> nat"   and x :: "'d"   and X :: "'d set" assumes "c \<le> real (k x)"   and "c \<in> \<nat>"   and "x \<in> X" shows "ccompatible {region X I r |I r. valid_region X k I r} (EQ x c)"
inclusion_preserves_inverse: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b"   and g :: "'b" assumes "subcategory C Arr"   and "category.inverse_arrows (subcategory.comp C Arr) f g" shows "category.inverse_arrows C f g"
resolve_def: fixes C :: "'a literal list"   and D :: "'a literal list" shows "resolve C D = concat (map (\<lambda>L. case L of Pos A \<Rightarrow> [] | Neg A \<Rightarrow> if maximal_wrt A (mset D) then resolve_on A (remove1 L D) C else []) D)"
fds_const_abs_converges: fixes c :: "'a"   and s :: "'a" shows "fds_abs_converges (fds_const c) s"
sys_xcpts_defs_2: shows "ClassCast \<equiv> ''ClassCast''"
conc_pow_comm: fixes A :: "'b list set"   and n :: "nat" shows "A @@ A ^^ n = A ^^ n @@ A"
getPos_NegPos_map_simp2: fixes X :: "'e \<Rightarrow> 'd"   and src :: "'e negation_type list" shows "getPos (NegPos_map X src) = map X (getPos src)"
strict_trans: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subset># b"   and "b \<subset># c" shows "a \<subset># c"
simps_2: fixes l :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "rbt_in_traverse_pairs (Node l c k v r) = rbt_in_traverse_pairs l @ (k, v) # rbt_in_traverse_pairs r"
infsetsum_Sigma: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "countable A"   and "\<And>i. countable (B i)"   and "f abs_summable_on Sigma A B" shows "infsetsum f (Sigma A B) = (\<Sum>\<^sub>ax\<in>A. \<Sum>\<^sub>ay\<in>B x. f (x, y))"
Split_def: fixes c :: "('b, 'c) node set \<Rightarrow> ('b, 'c) node set \<Rightarrow> 'a"   and M :: "('b, 'c) node set" shows "Split c M \<equiv> THE u. \<exists>x y. M = Scons x y \<and> u = c x y"
simps_4: fixes c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" shows "redex (Seq c\<^sub>1 c\<^sub>2) = redex c\<^sub>1"
normal_invE_1: fixes N :: "'a set" assumes "N \<lhd> G\<lparr>carrier := H\<rparr>" shows "subgroup N (G\<lparr>carrier := H\<rparr>)"
OtherMessagesOnlyGrowing: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration"   and m' :: "('a, 'c) message" assumes "asynchronousSystem.steps trans sends cfg1 m cfg2"   and "m \<noteq> m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
base_1: shows "a = (\<Sum>i\<in>S. x i * Measure.measure M (A i))"
of_nat_Suc: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc m) = plus one (semiring_1.of_nat one plus zero m)"
diff_face_not_in_2: shows "a' \<notin> G.face_cycle_set a"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "\<lbrace>t \<^bold>\<otimes> u\<rbrace> = D.tensor \<lbrace>t\<rbrace> \<lbrace>u\<rbrace>"
prod_prime_factors: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring_multiplicative divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "comm_monoid_mult.prod times one (\<lambda>p. power.power one times p (factorial_semiring.multiplicity times one p x)) (set_mset (factorial_semiring.prime_factorization zero times one normalize x)) = normalize x"
phi_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and phis :: "'c \<Rightarrow> 'd \<times> 'b \<Rightarrow> 'b list option"   and g :: "'c"   and v :: "'b" shows "CFG_SSA_wf_base.phi \<alpha>n defs phis g v = phis g (CFG_SSA_wf_base.defNode \<alpha>n defs phis g v, v)"
is_circle_mk_circle: fixes a :: "complex"   and r :: "real" shows "is_circle (mk_circle a r)"
linear_comb_eq: fixes H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "g \<in> {j. j \<le> n} \<rightarrow> H"   and "\<forall>j\<in>{j. j \<le> n}. f j = g j" shows "l_comb R M n s f = l_comb R M n s g"
src_def: fixes \<mu> :: "'a" shows "S.src \<mu> = (if S.arr \<mu> then src\<^sub>B \<mu> else S.null)"
nonconsequence_remainder: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn" shows "(Tarskian_logic.remainder Cn A \<phi> = {A}) = (\<phi> \<notin> Cn A)"
inangle1123: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B" shows "A InAngle A B C"
inf_seq_distrib: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "((c\<^sub>0::'a::refinement_lattice) \<sqinter> (c\<^sub>1::'a::refinement_lattice)) ; (d::'a::refinement_lattice) = c\<^sub>0 ; d \<sqinter> c\<^sub>1 ; d"
FactGroup_onto: assumes "f ` carrier G = carrier H" shows "(\<lambda>X. the_elem (f ` X)) ` carrier (G Mod kernel G H f) = carrier H"
ball_empty: fixes P :: "'g \<Rightarrow> bool" shows "Ball {} P = True"
4: fixes y :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd"   and x :: "('t \<Rightarrow>\<^sub>0 'c) \<times> nat \<times> 'd" assumes "(y, x) \<in>\<^sub>p snd ` set ps" shows "x \<in> set (gs @ bs @ hs)"
S: fixes u :: "'a" assumes "(x, u) \<in> E" shows "P x u"
preserves_dom: fixes f :: "'a" assumes "arr f" shows "local.dom (R f) = R (local.dom f)"
finite_set_of_vector: fixes X :: "('a, 'b) vec" shows "finite (set_of_vector X)"
roots_inclI: fixes p :: "'a list list"   and q :: "'a list list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "q \<in> carrier (poly_ring (poly_ring R))"   and "q \<noteq> []"   and "\<And>a. \<lbrakk>a \<in> carrier (poly_ring R); p \<noteq> []\<rbrakk> \<Longrightarrow> [\<one>\<^bsub>poly_ring R\<^esub>, \<ominus>\<^bsub>poly_ring R\<^esub> a] [^]\<^bsub>poly_ring (poly_ring R)\<^esub> UP.alg_mult p a pdivides\<^bsub>poly_ring R\<^esub> q" shows "UP.roots p \<subseteq># UP.roots q"
hdTriv: shows "hd (\<HH> @ [carrier G]) = {\<one>}"
foundation10': fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(\<tau> \<Turnstile> A and B) = (\<tau> \<Turnstile> A \<and> \<tau> \<Turnstile> B)"
get_root_node_si_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_root_node_si ptr" shows "ptr |\<in>| object_ptr_kinds h"
lookup_add: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and g :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "poly_mapping.lookup (f + g) k = poly_mapping.lookup f k + poly_mapping.lookup g k"
annot_def: fixes t :: "('c, 'b) FingerTreeStruc" shows "annot t = gmft t"
separating_cut: fixes \<Delta> :: "('a, 'b) network_scheme"   and S :: "('a \<times> 'a) set" assumes "separating (web_of_network \<Delta>) S"   and "source \<Delta> \<noteq> sink \<Delta>"   and "\<And>x. \<not> edge \<Delta> (sink \<Delta>) x" shows "cut \<Delta> (cut_of_sep \<Delta> S)"
GF_advice_prop_congruent_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "\<phi>[X]\<^sub>\<nu> \<sim>\<^sub>P \<psi>[X]\<^sub>\<nu>"
less: shows "\<forall>(q, s')\<in>set (step pc (\<tau>s ! pc)). s' \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! q"
subst_subst_domain: fixes x :: "'a"   and t :: "('b, 'a) Term.term" shows "subst_domain (subst x t) = (if t = Var x then {} else {x})"
distinct_sorted: fixes xs :: "'a list" assumes "Sorted_Less.sorted xs" shows "distinct xs"
poly2_poly_y_x: fixes p :: "'c poly poly"   and x :: "'c"   and y :: "'c" shows "poly2 (poly_y_x p) x y = poly2 p y x"
add_match_add_missing_ret_unfoldings_rot: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (add_missing_ret_unfoldings rs1 rs2), s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_missing_ret_unfoldings (Rule (MatchNot m) Return # rs1) rs2, s\<rangle> \<Rightarrow> t"
pseudominimal_inD1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ordering less_eq less"   and "ordering.pseudominimal_in less_eq less P x" shows "x \<in> P"