hcju/isabelleps huggingface dataset:
measurable_restrict_mono: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and A :: "'b set"   and N :: "'c measure"   and B :: "'b set" assumes "f \<in> restrict_space M A \<rightarrow>\<^sub>M N"   and "B \<subseteq> A" shows "f \<in> restrict_space M B \<rightarrow>\<^sub>M N"
confc: shows "conf_clinit P sh ((stk, loc, C, M, pc, ics) # frs)"
match_iface_case_wildcard_length: fixes i :: "char list"   and p_i :: "char list" assumes "iface_name_is_wildcard i"   and "match_iface (Iface i) p_i" shows "length i - 1 \<le> length p_i"
Arg2pi_eq: fixes z :: "complex" shows "z = complex_of_real (cmod z) * exp (\<i> * complex_of_real (Arg2pi z))"
assms: shows "c \<noteq> (0::'a) \<or> p \<noteq> 0"
B: shows "xs @ y # ipurge_tr I D (D y) zs \<in> traces P"
assms_7: fixes x :: "'a" assumes "P x"   and "x \<le> lfp f" shows "\<alpha> (f x) = g (\<alpha> x)"
real_sqrt_sum_squares_mult_squared_eq: fixes x :: "real"   and y :: "real"   and xa :: "real"   and ya :: "real" shows "(sqrt ((x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)))\<^sup>2 = (x\<^sup>2 + y\<^sup>2) * (xa\<^sup>2 + ya\<^sup>2)"
kE_int: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "I.wtE (kE \<xi>)"   and "wt T" shows "I.int (kE \<xi>) T = fromFull (tpOf T) (F.int \<xi> T)"
abs_0_eq: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "(zero = abs a) = (a = zero)"
Pool_2: fixes w1 :: "nat \<Rightarrow> real"   and w2 :: "nat \<Rightarrow> real" assumes "\<And>i. i < count_weights s m2 \<Longrightarrow> w1 i = w2 i" shows "insert_weights s m2 w1 = insert_weights s m2 w2"
induct: fixes P :: "fm \<Rightarrow> bool"   and fm :: "fm" assumes "P T"   and "P F"   and "\<And>x. P (Le x)"   and "\<And>x. P (Lt x)"   and "\<And>x. P (Eq x)"   and "\<And>x. P (NEq x)"   and "\<And>x. P x \<Longrightarrow> P (NOT x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (And x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Or x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Imp x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Iff x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (E x)"   and "\<And>x. P x \<Longrightarrow> P (A x)" shows "P fm"
indRelRPO_is_preorder: shows "preorder indRelRPO"
suc_any: fixes Xa :: "'a set"   and Ya :: "'a set"   and xa :: "'a"   and ya :: "'a"   and Z :: "'a set"   and z :: "'a" assumes "derivation Xa"   and "derivation Ya"   and "xa \<in> Xa"   and "ya \<in> Ya"   and "Z = {z \<in> Xa. z \<sqsubset> xa}"   and "extreme Z (\<sqsubseteq>) z"   and "xa = f z"   and "z \<sqsubset> ya \<and> z \<in> Ya \<or> z = ya \<or> ya \<sqsubset> z \<and> ya \<in> Xa" shows "ya \<sqsubset> xa \<and> ya \<in> Xa \<or> xa \<sqsubseteq> ya"
approx_6: fixes i :: "nat" shows "ereal (u i) < b"
length_gt_zero_last_concat: fixes t :: "'b list"   and s :: "'b list" assumes "0 < length t" shows "last (s @ t) = last t"
append_eq_conv_conj: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
gd_term_id: fixes cmp_term :: "('b \<times> 'c) nat_term_order" shows "gd_term (\<lambda>x. x) (\<lambda>x. x) (\<lambda>s t. le_of_nat_term_order cmp_term (s, the_min) (t, the_min)) (\<lambda>s t. lt_of_nat_term_order cmp_term (s, the_min) (t, the_min)) (le_of_nat_term_order cmp_term) (lt_of_nat_term_order cmp_term)"
frac_less2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less_eq x y"   and "less zero w"   and "less w z" shows "less (divide x z) (divide y w)"
smult_distrib_right_diff: fixes r :: "'r"   and s :: "'r"   and m :: "'m" assumes "r \<in> R"   and "s \<in> R"   and "m \<in> M" shows "(r - s) \<cdot> m = r \<cdot> m - s \<cdot> m"
simps_1: shows "Nat_Bijection.list_encode [] = 0"
int_bit_induct: fixes P :: "int \<Rightarrow> bool"   and k :: "int" assumes "P 0"   and "P (- 1)"   and "\<And>k. \<lbrakk>P k; k \<noteq> 0\<rbrakk> \<Longrightarrow> P (k * 2)"   and "\<And>k. \<lbrakk>P k; k \<noteq> - 1\<rbrakk> \<Longrightarrow> P (1 + k * 2)" shows "P k"
listall_def: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "listall P xs \<equiv> \<forall>i<length xs. P (xs ! i)"
merge_size: fixes l :: "'b tree"   and r :: "'b tree" shows "size (merge l r) = size l + size r"
ad1: shows "length (blockmat w) = rat_poly.row_length (kauff_mat ws)"
valid_plan_from_def: fixes M :: "object atom formula set"   and \<pi>s :: "plan_action list" shows "valid_plan_from M \<pi>s = (\<exists>M'. plan_action_path M \<pi>s M' \<and> M' \<^sup>c\<TTurnstile>\<^sub>= goal P)"
simps: fixes a :: "'a trm" shows "NF a = (a = Unit \<or> (\<exists>x. a = Var x) \<or> (\<exists>A x T A' B. a = App A B \<and> A \<noteq> Fn x T A' \<and> NF A \<and> NF B) \<or> (\<exists>A x T. a = Fn x T A \<and> NF A) \<or> (\<exists>A B. a = SimplyTyped.Pair A B \<and> NF A \<and> NF B) \<or> (\<exists>P A B. a = Fst P \<and> P \<noteq> SimplyTyped.Pair A B \<and> NF P) \<or> (\<exists>P A B. a = Snd P \<and> P \<noteq> SimplyTyped.Pair A B \<and> NF P))"
rfr_disc_equity_market_axioms: shows "rfr_disc_equity_market M G Mkt r risk_free_asset"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "continuous_on U f"   and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b"   and "\<And>x. x \<in> S \<Longrightarrow> f x = a"   and "\<And>x. x \<in> T \<Longrightarrow> f x = b" shows "thesis"
bij_betw_apply: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and a :: "'b" assumes "bij_betw f A B"   and "a \<in> A" shows "f a \<in> B"
inverse_arrows_can: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "free_monoidal_category.can C f" shows "category.inverse_arrows (free_monoidal_category.comp C) f (free_monoidal_category.mkarr C (monoidal_language.Inv C (monoidal_language.red C (monoidal_language.Dom C (free_monoidal_category.rep C f))) \<^bold>\<cdot> monoidal_language.Diagonalize C (free_monoidal_category.rep C f) \<^bold>\<cdot> monoidal_language.red C (monoidal_language.Cod C (free_monoidal_category.rep C f))))"
assms: fixes i :: "nat" assumes "i < length xs" shows "xs ! i \<in> S"
connectedin_continuous_map_image: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "continuous_map X Y f"   and "connectedin X S" shows "connectedin Y (f ` S)"
safetyI: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>\<sigma>. \<not> P \<sigma> \<Longrightarrow> \<exists>i. \<forall>\<beta>. \<not> P (stake i \<sigma> @- \<beta>)" shows "safety P"
gbinomial_pochhammer': fixes a :: "'a"   and k :: "nat" shows "a gchoose k = pochhammer (a - of_nat k + (1::'a)) k / fact k"
kat_while_inv_def: fixes p :: "'a"   and i :: "'a"   and x :: "'a" shows "while p inv i do x = while p do x"
eSuc_minus_1: fixes n :: "enat" shows "eSuc n - 1 = n"
simps: fixes x :: "const"   and y :: "const" shows "x \<squnion> y = (if x = y then x else Any)"
WF_always: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<box>WF(A)_v = WF(A)_v"
less_imp_le: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x <\<cdot> y" shows "x \<le>\<cdot> y"
jviewInit: shows "\<forall>a s. jviewInit a (envObs a s) = jview a (tInit s)"
t3: shows "term_ok \<Theta> (Abs \<tau>' t $ Fv x \<tau>')"
IH_3: assumes "prod f X dvd (1::'b)"   and "x \<in> X" shows "f x dvd (1::'b)"
map_disc_iff: fixes f :: "'f \<Rightarrow> 'e"   and a :: "'f list" shows "(map f a = []) = (a = [])"
fold_mset_empty: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e" shows "fold_mset f s {#} = s"
augment_edge_rev_other: fixes u :: "nat"   and v :: "nat"   and e' :: "nat \<times> nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "(u, v) \<notin> E"   and "e' \<noteq> (v, u)" shows "augment_edge f (u, v) \<Delta> e' = f e'"
comp_inv_arr': fixes f :: "'b" assumes "D.iso f" shows "D.inv f \<cdot>\<^sub>D f = D.dom f"
initial_segment_Nil: fixes ys :: "'a list" shows "initial_segment [] ys"
IH: assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A))" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t (to_st A))"
WT_resource_of_oracle: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and s :: "'a" assumes "callee_invariant_on callee I \<I>"   and "I s" shows "\<I> \<turnstile>res RES callee s \<surd>"
distinct_1: shows "ccode.Gt \<noteq> ccode.Lt"
simps_1: fixes P :: "'b prog"   and vs :: "'a \<times> addr_loc \<Rightarrow> 'a val set"   and ad :: "'a"   and al :: "addr_loc"   and v :: "'a val" shows "w_value P vs (NormalAction (WriteMem ad al v)) = vs((ad, al) := insert v (vs (ad, al)))"
hd_pow: fixes n :: "nat"   and u :: "'b list" assumes "n \<noteq> 0" shows "hd (u \<^sup>@ n) = hd u"
quotient_map_fst: fixes X :: "'d topology"   and Y :: "'e topology" shows "quotient_map (prod_topology X Y) X fst = (topspace Y = {} \<longrightarrow> topspace X = {})"
put\<^sub>N\<^sub>o\<^sub>d\<^sub>e_def: fixes node_ptr :: "(_) node_ptr"   and node :: "'c RNode_scheme" shows "put node_ptr node = put (cast node_ptr) (cast node)"
cut_le_Int_left: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<le> t = A \<down>\<le> t \<inter> B"
path_verts: fixes G :: "nat graph_rec"   and p :: "nat list"   and n :: "nat" assumes "cfg_doms G"   and "path_entry (g_E G) p n" shows "n \<in> set (g_V G)"
assms_2: shows "Fun c [] \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t subst_range \<delta>"
finished_item_def: fixes \<alpha> :: "symbol list" shows "finished_item \<alpha> = Item (\<SS>, \<alpha>) (length \<alpha>) 0 (length Doc)"
res_opp_fold_A_A_im_opp_fold_A_A_im: shows "restrict1 opp_fold_A (\<Union> A) \<turnstile> A = opp_fold_A \<turnstile> A"
len_takeWhile_comp: fixes n1 :: "'a"   and xs :: "'a list"   and n2 :: "'a" assumes "n1 \<in> set xs"   and "n2 \<in> set xs"   and "n1 \<noteq> n2" shows "length (takeWhile ((\<noteq>) n1) xs) \<noteq> length (takeWhile ((\<noteq>) n2) xs)"
safe_form_impl_def: shows "safe_form_impl = (\<lambda>(ode_ops, uu_). safe_form ode_ops)"
divides_aux_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and q :: "'a"   and r :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "unique_euclidean_semiring_numeral.divides_aux zero (q, r) = (r = zero)"
cases_scheme: fixes r :: "'a StoreImpl_scheme" obtains newOID :: "CTypeId \<Rightarrow> nat" and newAID :: "Arraytype \<Rightarrow> nat" and vals :: "Location \<Rightarrow> Value" and more :: "'a" where "r = \<lparr>newOID = newOID, newAID = newAID, vals = vals, \<dots> = more\<rparr>"
BRBangCases': fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, !Q) \<in> bangRel Rel" obtains P :: "pi" and Q :: "pi" where "Pb__ = !P" and "!Qb__ = !Q" and "(P, Q) \<in> Rel"
rep_eq: fixes x :: "'a sq_mtx" shows "(*\<^sub>V) x = (*v) (to_vec x)"
assms_1: shows "val_process Mkt pf n y \<noteq> 0"
multiplicity_geI: fixes x :: "'a"   and p :: "'a"   and n :: "nat" assumes "x \<noteq> (0::'a)"   and "\<not> is_unit p"   and "p ^ n dvd x" shows "n \<le> multiplicity p x"
chine_composites_2: shows "Chn \<lparr>Chn = \<mu>_\<nu>\<pi>_\<rho>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<mu> \<star> (\<nu> \<star> \<pi>) \<star> \<rho>)\<rparr> = \<mu>_\<nu>\<pi>_\<rho>.chine_assoc"
set_inorder: fixes t :: "'a tree" shows "set (inorder t) = set_tree t"
zero_nc: fixes x :: "'a" shows "x \<cdot> (0::'a) \<sqinter> nc = (0::'a)"
two_comparisons_into_case_order_10: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and Q :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if x = y then P else if lt_of_comp comp x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
inj_def: fixes f :: "'b \<Rightarrow> 'c" shows "inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
simps: fixes C :: "'b literal set"   and X :: "'b trm set" shows "trms_ecl (Ecl C X) = X"
left: fixes \<sigma>' :: "'state"   and coll' :: "'coll" assumes "\<exists>\<sigma>1 coll1 coll2. (\<sigma>1, coll1) \<in> csmall P \<sigma> \<and> (\<sigma>', coll2) \<in> csmall_nstep P \<sigma>1 (Suc n) \<and> combine coll1 coll2 = coll'" shows "(\<sigma>', coll') \<in> csmall_nstep P \<sigma> (Suc (Suc n))"
rw1: shows "fst (foldl echelon_form_of_column_k_det_iarrays (1::'a, matrix_to_iarray A, 0, bezout) [0..<Suc k]) = fst (foldl (echelon_form_of_column_k_det bezout) (1::'a, A, 0) [0..<Suc k])"
f: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<and> (\<forall>x. \<guillemotleft>x : unity \<rightarrow> a\<guillemotright> \<longrightarrow> f \<cdot> x = F x)"
continuous_map_compose: fixes X :: "'d topology"   and X' :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and X'' :: "'f topology"   and g :: "'e \<Rightarrow> 'f" assumes "continuous_map X X' f"   and "continuous_map X' X'' g" shows "continuous_map X X'' (g \<circ> f)"
hfun_to_arr_def: fixes B :: "hf"   and C :: "hf"   and F :: "hf" shows "hfun_to_arr B C F = mkArr (UP ` hfset B) (UP ` hfset C) (\<lambda>x. UP (happ F (DOWN x)))"
elims: fixes x :: "('a \<times> 'c) list"   and xa :: "'c \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "evalHeap x xa = y" obtains uu_ :: "'c \<Rightarrow> 'b" where "xa__ = []" and "xa = uu_" and "y = \<bottom>"   | x :: "'a" and e :: "'c" and h :: "('a \<times> 'c) list" and eval :: "'c \<Rightarrow> 'b" where "xa__ = (x, e) # h" and "xa = eval" and "y = (evalHeap h eval)(x := eval e)"
\<delta>_def: shows "\<delta> \<equiv> e / B / real DIM('b)"
MPLS_charles_def: fixes PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set" shows "MPLS_charles PROB \<equiv> {Inf (PLS_charles (fst p) (snd p) PROB) |p. fst p \<in> valid_states PROB \<and> snd p \<in> valid_plans PROB}"
intros_1: fixes a :: "'b"   and b :: "'b"   and xs :: "'b list" shows "a # b # xs \<approx>2 b # a # xs"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'a"   and xs :: "'a list" shows "ipurge_tr_rev_aux I D U (x # xs) = (if \<exists>v\<in>sources_aux I D U xs. (D x, v) \<in> I then x # ipurge_tr_rev_aux I D U xs else ipurge_tr_rev_aux I D U xs)"
euclid_ext_poly_i_def: fixes ops :: "'b arith_ops_record" shows "euclid_ext_poly_i ops = arith_ops.euclid_ext_aux_i (poly_ops ops) (arith_ops_record.one (poly_ops ops)) (arith_ops_record.zero (poly_ops ops)) (arith_ops_record.zero (poly_ops ops)) (arith_ops_record.one (poly_ops ops))"
assms_1: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> t"
ab_semigroup_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "ab_semigroup_add_hom hom \<equiv> semigroup_add_hom hom"
vars1: shows "free_vars e1 \<subseteq> V \<union> V'"
simps_1: fixes M :: "('b, 'c) model"   and uu :: "'d \<Rightarrow> 'b"   and w :: "'b"   and x :: "'c" shows "(M, uu, w \<Turnstile> Pro x) = V M w x"
UN_constant_eq: fixes a :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd set"   and c :: "'d set" assumes "a \<in> A"   and "\<forall>y\<in>A. f y = c" shows "\<Union> (f ` A) = c"
coeff_const: fixes a :: "'c"   and i :: "nat" shows "coeff [:a:] i = (if i = 0 then a else (0::'c))"
E: shows "Suc k + (length (drop (Suc k) cfs) - 1) = length cfs - 1"
finite_insert: fixes a :: "'k"   and A :: "'k set" shows "finite (insert a A) = finite A"
subs: shows "\<Union> (A ` F) - A x \<subseteq> \<Union> (A ` F)"
vcomp_ide_dom: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B F \<tau> = \<tau>"
P_def: shows "P \<equiv> \<lambda>x. if x \<in> (\<lambda>x. (X x, Y x)) ` space M then Px x else 0"
LeadsTo_Basis: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A Ensures B" shows "F \<in> A \<longmapsto>w B"
fsubseteq_dom_subset: fixes f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" assumes "f \<subseteq>\<^sub>f g" shows "fdom f \<subseteq> fdom g"
fmdom'I: fixes m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f"   and y :: "'e" assumes "fmlookup m x = Some y" shows "x \<in> fmdom' m"
pre_order_pair_def: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" shows "pre_order_pair S NS \<equiv> refl NS \<and> trans S \<and> trans NS"
image_monic_is_auto_reduced: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_auto_reduced B" shows "is_auto_reduced (monic ` B)"
space_measure_of_conv: fixes \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "space (measure_of \<Omega> A \<mu>) = \<Omega>"
assms_3: shows "\<guillemotleft>\<nu> : v \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright>"
add_eq_conv_diff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and N :: "'b multiset" shows "(add_mset a M = add_mset b N) = (M = N \<and> a = b \<or> M = add_mset b (N - {#a#}) \<and> N = add_mset a (M - {#b#}))"
not_less_Least: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
path_from_to_def: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and xs :: "'c list"   and w :: "'c" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w \<equiv> (Digraph.walk G xs \<and> distinct xs) \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
remove_singleton_from_division_of: fixes A :: "real set set"   and a :: "real"   and b :: "real"   and x :: "real" assumes "A division_of {a..b}"   and "a < b"   and "x \<in> {a..b}" shows "\<exists>c d. c < d \<and> {c..d} \<in> A \<and> x \<in> {c..d}"
pdevs_val_perm: fixes xs :: "'a list"   and ys :: "'a list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "xs <~~> ys"   and "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
matrix_to_iarray_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "matrix_to_iarray A = IArray (map (vec_to_iarray \<circ> ($) A \<circ> mod_type_class.from_nat) [0..<CARD('f)])"
monom_mult_uminus_left: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.monom_mult (- c) t p = - local.punit.monom_mult c t p"
excess_non_negative': shows "\<forall>v\<in>V - {s}. (0::'capacity) \<le> excess f v"
out_arcs_int_arcs: fixes G :: "('f, 'e) pre_digraph"   and u :: "'f" shows "out_arcs G u \<inter> arcs G = out_arcs G u"
power_mult: fixes p :: "int poly fract"   and m :: "nat"   and n :: "nat" shows "(p ^ m) ^ n = p ^ (m *n n)"
less_eq_multiset_plus_right: fixes M :: "'a multiset"   and N :: "'a multiset" shows "M \<le> M + N"
axioms_2: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "ideal I R" shows "ring R"
inner_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"   and v :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis"   and "v \<in> Basis" shows "inner u v = (if u = v then 1 else 0)"
int_shift_equiv: fixes v :: "'d \<Rightarrow> real"   and R :: "('d \<Rightarrow> real) set"   and v' :: "'d \<Rightarrow> real"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and t :: "int" assumes "v \<in> R"   and "v' \<in> R"   and "R \<in> {region X I r |I r. valid_region X k I r}"   and "0 \<le> t" shows "v' \<oplus> real_of_int t \<in> [v \<oplus> real_of_int t]\<^sub>{region X I r |I r. valid_region X k I r}"
that: fixes r :: "V"   and A :: "V" assumes "x = \<langle>\<langle>r, A\<rangle>, r \<restriction>\<^sup>l\<^sub>\<circ> A\<rangle>"   and "r \<in>\<^sub>\<circ> R"   and "A \<in>\<^sub>\<circ> X" shows "thesis"
arg_min_if_finite_2: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "S \<noteq> {}" shows "\<not> (\<exists>x\<in>S. f x < f (arg_min_on f S))"
real_linorder_cases: fixes a :: "'a"   and b :: "'a" assumes "is_real a"   and "is_real b" obtains "a < b"   | "a = b"   | "b < a"
no_await_trancl: fixes ctx :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and ctx' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "ctx \<leadsto>\<^sup>+ ctx'"   and "no_await (fst (fst ctx))" shows "no_await (fst (fst ctx'))"
map\<^sub>0_def: fixes a :: "'b" shows "\<tau>.F.map\<^sub>0 a \<equiv> src\<^sub>C (F a)"
ennreal_one_neq_top: shows "1 \<noteq> \<top>"
funcset_Un_left: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'e set"   and C :: "'f set" shows "(f \<in> A \<union> B \<rightarrow> C) = (f \<in> A \<rightarrow> C \<and> f \<in> B \<rightarrow> C)"
set_iterator_image_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and S' :: "'f set" assumes "set_iterator it S"   and "inj_on g S"   and "S' = g ` S" shows "set_iterator (set_iterator_image g it) S'"
card_S: shows "card {f \<in> {f. \<exists>s\<in>simplices. face f s}. face f s \<and> rl ` f = {..n}} = card {a \<in> s. rl ` (s - {a}) = {..n}}"
refl_assert: fixes P :: "cell list \<times> cell list \<Rightarrow> bool" shows "P \<mapsto> P"