hcju/isabelleps huggingface dataset:
typings_induct: fixes x4 :: "binding list"   and x5 :: "(char list \<times> trm) list"   and x6 :: "(char list \<times> POPLmarkRecord.type) list"   and P2 :: "binding list \<Rightarrow> (char list \<times> trm) list \<Rightarrow> (char list \<times> POPLmarkRecord.type) list \<Rightarrow> bool" assumes "x4 \<turnstile> x5 [:] x6"   and "\<And>\<Gamma>. \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<Longrightarrow> P2 \<Gamma> [] []"   and "\<And>\<Gamma> t T fs fTs l. \<lbrakk>\<Gamma> \<turnstile> t : T; \<Gamma> \<turnstile> fs [:] fTs; P2 \<Gamma> fs fTs; fs\<langle>l\<rangle>\<^sub>? = \<bottom>\<rbrakk> \<Longrightarrow> P2 \<Gamma> ((l, t) \<Colon> fs) ((l, T) \<Colon> fTs)" shows "P2 x4 x5 x6"
hyp_ind: shows "\<exists>\<sigma>. ground_on A \<sigma>"
l'_def: shows "l' \<equiv> relabel_effect f l u"
normalize: fixes A :: "'a set" shows "normalize (Gcd\<^sub>f\<^sub>i\<^sub>n A) = Gcd\<^sub>f\<^sub>i\<^sub>n A"
cases: fixes m :: "'m"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option" assumes "m\<mu>t m ts ts'" obtains t :: "'t" and x :: "'x" and x' :: "'x" and ln' :: "'l \<Rightarrow>f nat" and ln :: "'l \<Rightarrow>f nat" where "finite (dom ts)" and "ts t = \<lfloor>(x, ln)\<rfloor>" and "ts' t = \<lfloor>(x', ln')\<rfloor>" and "\<mu> (x, m) (x', m)" and "\<And>t'. t' \<noteq> t \<Longrightarrow> ts t' = ts' t'"
inj_p_fun: fixes x :: "'a" shows "is_inj x = is_p_fun (x\<^sup>\<smile>)"
octo_in_Reals_if_Re_con: fixes q :: "real" assumes "Ree (octo_of_real q) = q" shows "q \<in> \<real>"
index_of_c_snd_main: shows "nat_to_pr index_of_c_snd = c_snd"
reduction_signs_are_distinct_R: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list" assumes "satisfy_equation_R p qs subsets signs"   and "invertible_mat (M_mat_R signs subsets)"   and "distinct signs" shows "distinct (reduction_signs_R p qs signs subsets (M_mat_R signs subsets))"
differentiableI: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "(f has_derivative f') net" shows "f differentiable net"
axioms_2: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "sigma_algebra \<Omega> M" shows "sigma_algebra_axioms M"
GET: fixes y :: "Var"   and f :: "Field"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var" assumes "CONTEXT y = low"   and "GAMMA f = low" shows "G \<rhd> Get x y f : Sec (\<lambda>(s, t, \<beta>). \<exists>r l C Flds v. fst r y = RVal (Loc l) \<and> lookup (snd r) l = Some (C, Flds) \<and> lookup Flds f = Some v \<and> r \<equiv>\<^sub>\<beta> t \<and> s = (update (fst r) x v, snd r))"
residual_rep_abs: fixes r :: "'b \<times> 'c" shows "alpha_residual (rep_residual (abs_residual r)) r"
arr_char: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.arr f = (f \<noteq> Maps.Null \<and> Maps.Dom f \<in> Collect obj \<and> Maps.Cod f \<in> Collect obj \<and> Maps.Map f \<in> Maps.Hom (Maps.Dom f) (Maps.Cod f))"
butlast_splice: fixes xs :: "'b list"   and ys :: "'b list" shows "butlast (splice xs ys) = (if length ys < length xs then splice (butlast xs) ys else splice xs (butlast ys))"
Morse_Gromov_theorem: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and deltaG :: "'b itself \<Rightarrow> real"   and lambda :: "real"   and C :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and G :: "'b set" assumes "class.Gromov_hyperbolic_space_geodesic dist uniformity open deltaG"   and "lambda C -quasi_isometry_on {a..b} f"   and "geodesic_segment_between G (f a) (f b)" shows "hausdorff_distance (f ` {a..b}) G \<le> 92 * lambda\<^sup>2 * (C + deltaG TYPE('b))"
e: shows "echelon_form (echelon_form_of A bezout)"
ObjCf: shows "cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
simps_2: fixes i :: "'b memref"   and "is" :: "'b memref list" shows "all_acquired (i # is) = (case i of Write\<^sub>s\<^sub>b volatile x xa xb A L R W \<Rightarrow> if volatile then A \<union> all_acquired is else all_acquired is | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> A \<union> all_acquired is | _ \<Rightarrow> all_acquired is)"
fds_const_zero: shows "fds_const (0::'b) = 0"
hurwitz_zeta_integral_Gamma_def: fixes z :: "complex"   and a :: "real" assumes "1 < Re z"   and "0 < a" shows "hurwitz_zeta a z = rGamma z * (CLINT x:{0<..}|lebesgue. complex_of_real x powr (z - 1) * complex_of_real (exp (- a * x) / (1 - exp (- x))))"
subtype_reflexivity_semiautomated: fixes \<Gamma> :: "binding list"   and T :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "T closed_in \<Gamma>" shows "\<Gamma>\<turnstile>T<:T"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg s\<^sub>1 r\<^sub>0"
pref_iffD1: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "|~ F = G" shows "|~ F \<longrightarrow> G"
vangle_def: fixes u :: "'b"   and v :: "'b" shows "vangle u v = (if u = (0::'b) \<or> v = (0::'b) then pi / 2 else arccos (u \<bullet> v / (norm u * norm v)))"
norm_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and u :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "norm u = 1"
run_V0: fixes G :: "('b, 'c) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "graph_defs.is_run G r" shows "r 0 \<in> g_V0 G"
cut_apply: fixes x :: "'a"   and a :: "'a"   and R :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" assumes "(x, a) \<in> R" shows "cut f R a x = f x"
prems_1: shows "c \<subseteq>\<^sub>g Catch c1' c2'"
inorder_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree" shows "Tree2.inorder (join l r) = Tree2.inorder l @ Tree2.inorder r"
orthogonal_clauses_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "real_inner.orthogonal inner a zero"
step: shows "(m, sb, \<O>) \<rightarrow>\<^sub>f (m', sb', \<O>')"
Deduct_axioms: shows "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"
simps_2: fixes n :: "nat"   and us :: "'a vec list"   and w :: "'a vec"   and ws :: "'a vec list" shows "gram_schmidt_sub0 n us (w # ws) = (let w' = adjuster n w us + w in if vec_is_zero n w' then gram_schmidt_sub0 n us ws else gram_schmidt_sub0 n (w' # us) ws)"
psi'_def: shows "psi' = \<psi> and\<^sub>p OR {G\<^sub>p chi val |val. val \<subseteq> A} or\<^sub>p OR {chi val and\<^sub>p chi val U\<^sub>p (\<psi> and\<^sub>p chi val') |val val'. val \<subseteq> A \<and> val' \<subseteq> A \<and> val' \<noteq> val}"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_uminus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vrat_uminus" shows "b = c"
simps_11: fixes f1 :: "int \<Rightarrow> int list \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> 'a"   and x21 :: "int"   and x22 :: "int"   and x23 :: "int list" shows "(case Dvd x21 x22 x23 of Le x xa \<Rightarrow> f1 x xa | Dvd x xa xb \<Rightarrow> f2 x xa xb | NDvd x xa xb \<Rightarrow> f3 x xa xb) = f2 x21 x22 x23"
Diff_transfer: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (rel_set A)) (-) (-)"
simps_9: fixes f :: "'c \<Rightarrow> 'a"   and x21 :: "'c"   and x22 :: "'c list" shows "map f (x21 # x22) = f x21 # map f x22"
intro: fixes x :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma> v u. put\<^bsub>x\<^esub> (put\<^bsub>x\<^esub> \<sigma> v) u = put\<^bsub>x\<^esub> \<sigma> u" shows "mwb_lens_axioms x"
starP_star_of: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "(*p* P) (star_of x) = P x"
Commutativity: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatEq SImp (SCompose \<Psi> \<Psi>') (SCompose \<Psi>' \<Psi>)"
distinct_17: fixes x101 :: "'b ltln"   and x102 :: "'b ltln" shows "true\<^sub>n \<noteq> x101 W\<^sub>n x102"
rep_eq: fixes x :: "'a reg_lan"   and xa :: "'a reg_lan" shows "(x < xa) = (Rep_reg_lan x \<subset> Rep_reg_lan xa)"
S5n_common_knowledge_fixed_point_simpler: fixes M :: "('b, 'c, 'd) KripkeStructure"   and w :: "'d"   and a :: "'b"   and as :: "'b list"   and \<phi> :: "('b, 'c) Kform" assumes "S5n M"   and "w \<in> worlds M"   and "a \<in> set as" shows "M, w \<Turnstile> \<^bold>C\<^bsub>as\<^esub> \<phi> = M, w \<Turnstile> \<^bold>K\<^sub>a \<^bold>C\<^bsub>as\<^esub> \<phi>"
card_UNIV: fixes finite_UNIV :: "('a, bool) phantom"   and card_UNIV :: "('a, nat) phantom" assumes "class.card_UNIV finite_UNIV card_UNIV" shows "card_UNIV = Phantom('a) CARD('a)"
take_0: fixes xs :: "'k list" shows "take 0 xs = []"
hyp_foldl: shows "foldl Gauss_Jordan_column_k (0, A) [0..<Suc k] = (if \<forall>m. is_zero_row_upt_k m (Suc k) (snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])) then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n (Suc k) (snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]))) + 1, snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]))"
assms_2: assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)" shows "P"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "enat \<Rightarrow> 'c"   and enat :: "enat" shows "P (case enat of 0 \<Rightarrow> f1 | eSuc x \<Rightarrow> f2 x) = (\<not> (enat = 0 \<and> \<not> P f1 \<or> (\<exists>x2. enat = eSuc x2 \<and> \<not> P (f2 x2))))"
vempty_Z_def: shows "[]\<^sub>\<circ> = ZFC_in_HOL.set {x. x \<noteq> x}"
HoVH_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "H.HoVH \<mu> \<equiv> if VVV.arr \<mu> then fst \<mu> \<otimes> fst (snd \<mu>) \<otimes> snd (snd \<mu>) else null"
r_result1_converg_phi: fixes i :: "nat"   and x :: "nat"   and v :: "nat" assumes "\<phi> i x \<down>= v" shows "\<exists>t. (\<forall>t'\<ge>t. eval r_result1 [t', i, x] \<down>= Suc v) \<and> (\<forall>t'<t. eval r_result1 [t', i, x] \<down>= 0)"
simps_1: fixes s :: "g_state" shows "g_step s Any = (case s of Even \<Rightarrow> Odd | Odd \<Rightarrow> Even)"
assms_4: shows "finite X \<longrightarrow> n < card X"
multiple_t_design: fixes n :: "nat" assumes "0 < n" shows "t_design \<V> (multiple_blocks n) \<k> 2 (\<Lambda> * int n)"
IH1: shows "quicksort_post R h lo (p - Suc 0) xs' xs''"
\<omega>_space: shows "\<omega> \<in> space (Pi\<^sub>M UNIV M)"
infsetsum_mono: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and g :: "'c \<Rightarrow> real" assumes "f abs_summable_on A"   and "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x" shows "infsetsum f A \<le> infsetsum g A"
Par_perm: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "A B Par C D" shows "A B Par C D \<and> B A Par C D \<and> A B Par D C \<and> B A Par D C \<and> C D Par A B \<and> C D Par B A \<and> D C Par A B \<and> D C Par B A"
X_def: shows "X \<equiv> Matrix.mat 2 2 (\<lambda>(i, j). if i = j then 0 else 1)"
mset_map_split_orig: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b multiset"   and M1 :: "'a multiset"   and M2 :: "'a multiset" assumes "f `# P = M1 + M2" obtains P1 :: "'b multiset" and P2 :: "'b multiset" where "P = P1 + P2" and "f `# P1 = M1" and "f `# P2 = M2"
select_edge_def: fixes PDPE :: "'a set list \<times> 'a set \<times> ('a \<times> 'a) set" shows "select_edge PDPE \<equiv> let (p, D, pE) = PDPE in SELECT (\<lambda>e. e \<in> pE \<inter> last p \<times> UNIV) \<bind> (\<lambda>e. case e of None \<Rightarrow> RETURN (None, p, D, pE) | Some (u, v) \<Rightarrow> RETURN (Some v, p, D, pE - {(u, v)}))"
dvdmE': fixes m :: "int"   and p :: "int poly"   and q :: "int poly" assumes "poly_mod.dvdm m p q" obtains r :: "int poly" where "poly_mod.Mp m q = poly_mod.Mp m (p * poly_mod.Mp m r)"
assms_3: shows "e = the (eval r_enumdom [i, x])"
Sup_3: fixes x :: "'a" assumes "x \<in> M" shows "x \<in> iterates_above a"
rep_eq: fixes x :: "('e, 'c) lgenerator'"   and xa :: "('f, 'd) lgenerator'" shows "lgenerator (lzip_trans' x xa) = zip_raw (lgenerator x) (lgenerator xa)"
PK: fixes r :: "'a pfm"   and i :: "'a"   and p :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! K\<^sub>! i p \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i ([r]\<^sub>! p))"
absorb1: fixes b :: "'a::semilattice_sup"   and a :: "'a::semilattice_sup" assumes "(b::'a::semilattice_sup) \<le> (a::'a::semilattice_sup)" shows "(a::'a::semilattice_sup) \<squnion> (b::'a::semilattice_sup) = a"
SimpleDef: assumes "v \<in> defs g n"   and "v' = (v, n, SimpleDef)" shows "thesis"
ss_proper_implies_sub: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "proper_sub_set_system \<U> \<A> \<V> \<B>" shows "sub_set_system \<U> \<A> \<V> \<B>"
RBT_inverse: fixes y :: "('a, 'b) RBT_Impl.rbt" assumes "y \<in> {t. is_rbt t}" shows "impl_of (RBT y) = y"
prod_simps_2: fixes f0 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f1 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f0"   and "arr f1" shows "local.dom (HF'.prod f0 f1) = HF'.prod (local.dom f0) (local.dom f1)"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case EV_CONSUME_ONE of EV_CONSUME_ALL \<Rightarrow> f1 | EV_CONSUME_ONE \<Rightarrow> f2) = f2"
cl_inv_trans: fixes g :: "'a gState_scheme"   and g' :: "'b gState_scheme"   and p :: "pState" assumes "length (gState.channels g) \<le> length (gState.channels g')"   and "cl_inv (g, p)" shows "cl_inv (g', p)"
simps_3: fixes uv :: "'a extT" shows "extT.Top + uv = extT.Top"
select_convs_8: fixes counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and more :: "'d" shows "cross_edges \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, \<dots> = more\<rparr> = cross_edges"
cblinfun_blinfun_transfer_uminus: shows "rel_fun blinfun_cblinfun_eq blinfun_cblinfun_eq uminus uminus"
compare_self_below_EQ: fixes x :: "'a" shows "compare\<cdot>x\<cdot>x \<sqsubseteq> EQ"
assms_3: shows "recfn (Suc n) (Pr n f g)"
nextl_state: fixes q :: "hf"   and xs :: "'a list" assumes "q \<in> states Accessible_dfa" shows "Accessible.nextl q xs \<in> states Accessible_dfa"
cosh_square_eq: shows "(cosh x)\<^sup>2 = (sinh x)\<^sup>2 + (1::'a)"
inject_4: fixes x4 :: "real \<times> real \<times> real"   and y4 :: "real \<times> real \<times> real" shows "(NeqUni x4 = NeqUni y4) = (x4 = y4)"
feasible_boundedD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and b :: "real"   and P :: "'b \<Rightarrow> real" assumes "feasible t"   and "bounded_by b P"   and "nneg P" shows "bounded_by b (t P)"
polychain_of_ccw: fixes e :: "real \<times> real \<Rightarrow> real"   and qs :: "(real \<times> real) list"   and Pc :: "real \<times> real" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "ccw'.sortedP 0 qs"   and "length qs \<noteq> 1" shows "list_all (\<lambda>(xi, xj). ccw' xi xj (Pc + (\<Sum>P\<in>set qs. e P *\<^sub>R P))) (polychain_of Pc qs)"
aform_val_def: fixes e :: "nat \<Rightarrow> real"   and X :: "'b \<times> 'b pdevs" shows "aform_val e X = fst X + pdevs_val e (snd X)"
int_simps_28: fixes P :: "'b \<Rightarrow> bool" shows "\<turnstile> (P \<or> \<not> P) = #True"
divide_le_eq_1_pos: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a" shows "(b / a \<le> (1::'a)) = (b \<le> a)"
inj_on_iff_eq_card: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A" shows "inj_on f A = (card (f ` A) = card A)"
inv_iso: shows "inv_into (carrier G) \<phi> \<in> Group.iso H G"
prefix_less_le_not_le: fixes xs :: "'a list"   and zs :: "'a list" shows "(xs < zs) = (xs \<le> zs \<and> \<not> zs \<le> xs)"
less_lengthI: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set"   and p :: "nat" assumes "xs \<in> list n A"   and "p < n" shows "p < length xs"
derangements_fixpoint: fixes d :: "nat \<Rightarrow> nat"   and A :: "nat set"   and x :: "nat" assumes "d \<in> derangements A"   and "x \<notin> A" shows "d x = x"
cong: fixes g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "\<And>a. g a = h a" shows "Sum_any g = Sum_any h"
simps_3: fixes b1 :: "bool"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (Bc b1) (Gt v va) = True"
inj_map_strong: fixes x :: "'a fset"   and xa :: "'a fset"   and f :: "'a \<Rightarrow> 'b"   and fa :: "'a \<Rightarrow> 'b" assumes "\<And>z za. \<lbrakk>z \<in> fset x; za \<in> fset xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "f |`| x = fa |`| xa" shows "x = xa"
caratheodory: fixes p :: "'b set" shows "convex hull p = {x. \<exists>S. finite S \<and> S \<subseteq> p \<and> card S \<le> DIM('b) + 1 \<and> x \<in> convex hull S}"
mult_left_commute: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> V" shows "a \<cdot> b \<cdot> x = b \<cdot> a \<cdot> x"
obt_6: shows "(\<Sum>v\<in>S. u v *\<^sub>R v) = y"
face_cycle_succ_cyclic: fixes a :: "'b" shows "cyclic_on G.face_cycle_succ (G.face_cycle_set a)"
predicate2I: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y. P x y \<Longrightarrow> Q x y" shows "P \<le> Q"
get_root_node_si_def: shows "get_root_node_si \<equiv> l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_si Shadow_DOM.get_parent get_host"
dist_not_less_zero: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.metric_space dist uniformity open" shows "\<not> dist x y < 0"
deriv_add: fixes f :: "'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at z"   and "g field_differentiable at z" shows "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
web_of_network_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "web_of_network \<Delta> = \<lparr>edge = \<lambda>(x, y) (y', z). y' = y \<and> edge \<Delta> x y \<and> edge \<Delta> y z, weight = capacity \<Delta>, A = {(source \<Delta>, x) |x. edge \<Delta> (source \<Delta>) x}, B = {(x, sink \<Delta>) |x. edge \<Delta> x (sink \<Delta>)}, \<dots> = network.more \<Delta>\<rparr>"
countable_acc: fixes X :: "'s set" assumes "countable X" shows "countable (acc `` X)"
finprod_cong: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier G) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finprod G f A = finprod G g B"
prems_3: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state')"
s_ce: shows "ce_rel_to_set s \<in> ce_sets"
det_swaprows: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and A :: "'a mat" assumes "k < n"   and "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (swaprows k l A) = - det A"
is_zero_row_utp_0': fixes A :: "(('b, 'c) vec, 'a) vec" shows "\<forall>m. is_zero_row_upt_k m 0 A"
non_zero: assumes "n \<noteq> 0" shows "coeff P p n \<noteq> \<zero>"
is_inf_uniq: fixes x :: "'a"   and y :: "'a"   and i :: "'a"   and i' :: "'a" assumes "is_inf x y i"   and "is_inf x y i'" shows "i = i'"
abs_eq: fixes xa :: "'b"   and x :: "'b" shows "(to_dual_ord xa < to_dual_ord x) = (x < xa)"
subst_apply_pairs_pset_subst: fixes F :: "(('a, 'c) Term.term \<times> ('a, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "set (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>) = set F \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta>"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and s :: "('a, 'b) state" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>s\<^sub>e s = (v \<Turnstile>\<^sub>t \<T> s \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>b\<^sub>e \<B>\<I> s)"
refl: fixes n :: "'a LDCFG_node"   and S :: "'a LDCFG_node set" assumes "CFGExit_wf.valid_node n"   and "n \<in> S" shows "n \<in> CFGExit_wf.wod_backward_slice S"
map_scale_single: fixes k :: "'b"   and x :: "'a"   and l :: "'b" shows "k \<cdot> monomial l x = monomial (k * l) x"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (\<tau> f) = \<tau> (A.cod f)"
mcont_cont: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "mcont luba orda lubb ordb f" shows "cont luba orda lubb ordb f"
lex_append_leftD: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "\<forall>x. (x, x) \<notin> r"   and "(xs @ ys, xs @ zs) \<in> lex r" shows "(ys, zs) \<in> lex r"
set_comp_def: fixes g :: "'b set_arrow"   and f :: "'b set_arrow" shows "g \<odot> f = \<lparr>set_dom = set_dom f, set_func = compose (set_dom f) (set_func g) (set_func f), set_cod = set_cod g\<rparr>"
eval_fds_const: fixes c :: "'b" shows "eval_fds (fds_const c) = (\<lambda>_. c)"
vreal_mult_def: shows "vreal_mult = (\<lambda>x\<in>\<^sub>\<circ>\<real>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (real_of_vreal (x\<lparr>0\<^sub>\<nat>\<rparr>) * real_of_vreal (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<real>)"
insert_absorb: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "a \<triangleright> A = A"
FL_valid_Not: fixes P :: "'state"   and x :: "('a, 'pred, 'act, 'effect) FL_Formula.formula" shows "P \<Turnstile> FL_Formula.Not x = (\<not> P \<Turnstile> x)"
components_are_iso: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "natural_isomorphism A B F G \<tau>"   and "partial_magma.ide A a" shows "category.iso B (\<tau> a)"
det3_nonneg_segments_of_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "length (half_segments_of_aform X) \<noteq> 1" shows "list_all (\<lambda>seg. 0 \<le> det3 (fst seg) (snd seg) (aform_val e X)) (segments_of_aform X)"