hcju/isabelleps huggingface dataset:
read_only_unowned: shows "read_only_unowned (snd (snd c\<^sub>0)) (fst c\<^sub>0)"
Quotient_alt_def2: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and Abs :: "'g \<Rightarrow> 'h"   and Rep :: "'h \<Rightarrow> 'g"   and T :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (T x (Abs y) \<and> T y (Abs x))))"
simps_21: fixes x4 :: "val"   and x1 :: "char list" shows "Val x4 \<noteq> new x1"
Cons_1: fixes us :: "'a vec list"   and us' :: "'a vec list" assumes "gram_schmidt_sub0 n us ws = us'"   and "set ws \<subseteq> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us"   and "lin_indpt (set us)"   and "corthogonal us" shows "set us' \<subseteq> carrier_vec n \<and> distinct us' \<and> corthogonal us' \<and> local.span (set (us @ ws)) = local.span (set us')"
FP: fixes n :: "nat" assumes "n < length dl" shows "{} \<notin> F n ` P n"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and e\<^sub>0 :: "'b"   and d\<^sub>0 :: "'b"   and \<eta>\<^sub>0 :: "'b"   and \<epsilon>\<^sub>0 :: "'b"   and e\<^sub>1 :: "'b"   and d\<^sub>1 :: "'b"   and \<eta>\<^sub>1 :: "'b"   and \<epsilon>\<^sub>1 :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "equivalence_in_bicategory V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0"   and "equivalence_in_bicategory V H \<a> \<i> src trg e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1" shows "two_equivalences_in_bicategory V H \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0 e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1"
wellformed_and_finite_2: fixes p :: "real"   and E :: "nat set set"   and n :: "nat" assumes "edge_space p"   and "E \<in> Pow (edge_space.S_edges n)" shows "uwellformed (edge_space.edge_ugraph n E)"
et: shows "\<forall>v \<rho>. v \<in> E e \<rho> \<longrightarrow> Tenv \<rho> \<turnstile> e : T v"
axioms_2: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
fds_nth_scaleR: fixes c :: "real"   and f :: "'b fds" shows "fds_nth (c *\<^sub>R f) = (\<lambda>n. c *\<^sub>R fds_nth f n)"
assms_2: shows "r\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> = x"
OKI_checkI: fixes einp :: "'d \<Rightarrow> 'c withERR option" assumes "liftAll (\<lambda>X. X \<noteq> ERR) einp" shows "OKI (checkI einp) = einp"
of_int_monom: fixes k :: "int" shows "of_int k = monom (of_int k) 0"
X_3: shows "J \<noteq> {} \<or> I = {}"
simps_9: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm" assumes "countable_bipartite_web \<Gamma>" shows "(case bi.SINK of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = f2"
hyps: fixes n :: "nat" shows "isOK (forallM_index_aux P n xs) = (\<forall>i<length xs. isOK (P (xs ! i) (i + n)))"
moebius_mu_smallo: assumes "prime_number_theorem" shows "sum_upto moebius_mu \<in> o(\<lambda>x. x)"
singleton_lepoll: fixes x :: "'b"   and y :: "'c"   and A :: "'c set" shows "{x} \<lesssim> insert y A"
deadlockD2: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and l :: "'l" assumes "deadlock s"   and "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "0 < ln $ l"   and "\<not> waiting (wset s t)" obtains l' :: "'l" and t' :: "'t" where "0 < ln $ l'" and "t \<noteq> t'" and "thr s t' \<noteq> None" and "has_lock (locks s $ l') t'"
firstPosEqual: fixes a :: "'b"   and l :: "'b list"   and b :: "'b" assumes "a \<in> set l"   and "b \<in> set l" shows "(firstPos a l = firstPos b l) = (a = b)"
ex1_implies_ex: fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "\<exists>x. P x"
head_in_verts: fixes e :: "'b" assumes "e \<in> arcs G" shows "head G e \<in> verts G"
wf_inv_image: fixes r :: "('r \<times> 'r) set"   and f :: "'s \<Rightarrow> 'r" assumes "wf r" shows "wf (inv_image r f)"
height_split_min: fixes t :: "'c tree234"   and x :: "'c"   and t' :: "'c up\<^sub>d" assumes "split_min t = (x, t')"   and "0 < height t"   and "bal t" shows "height t' = height t"
chain_def: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" shows "Complete_Partial_Order.chain ord S = (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
assms: shows "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}"
AnfangTauschA: fixes \<DD> :: "'b formula"   and \<Delta> :: "'b formula list"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<DD> # \<Delta> @ \<Gamma> \<Rightarrow> \<Theta>" shows "\<Delta> @ \<DD> # \<Gamma> \<Rightarrow> \<Theta>"
conjugate_mat_def: fixes m :: "'a mat" shows "conjugate m = mat (dim_row m) (dim_col m) (\<lambda>(i, j). conjugate (m $$ (i, j)))"
mtf2_forward_effect3: fixes q :: "'a"   and xs :: "'a list"   and n :: "nat"   and i :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "index xs q - n \<le> i \<and> i < index xs q" shows "index (mtf2 n q xs) (xs ! i) = Suc (index xs (xs ! i)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q"
lcm_proj1_if_dvd: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b a" shows "lcm a b = normalize a"
nth_prefixes_is_prefix: fixes n :: "nat"   and p :: "'b list" assumes "n < length p" shows "prefix (prefixes p ! n) (prefixes p ! Suc n)"
main_def: shows "main \<equiv> println STR ''Hello World! What is your name?'' \<bind> (\<lambda>_. getLine \<bind> (\<lambda>name. println (STR ''Hello, '' + name + STR ''!'')))"
iprange_interval_uncurry: fixes ipcidr :: "'b word \<times> 'b word" shows "iprange_interval ipcidr = uncurry WordInterval ipcidr"
imgTr10: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and H :: "'e set" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "F \<guillemotright> H " shows "\<one>\<^bsub>G\<^esub> \<in> f ` H"
N: shows "n1 = (replicate (length (fst M)) False)[q := True]"
gcd_neg2: fixes a :: "'a"   and b :: "'a" shows "gcd a (- b) = gcd a b"
prems_5: shows "\<forall>t\<in>fst ` set (tx # txs). y \<notin> FvarsT t"
unch_defs: fixes v :: "'b \<Rightarrow> 'c"   and w :: "nat \<Rightarrow> 'b" shows "(Unchanged v) w = (v (second w) = v (first w))"
path_component_path_component: fixes S :: "'b set"   and x :: "'b" shows "path_component_set (path_component_set S x) x = path_component_set S x"
as2: fixes s :: "'a event list"   and b :: "'a event set" assumes "front_tickFree s"   and "\<exists>aa\<in>UNIV. ev aa \<notin> b" shows "(s, b) \<in> F (DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)"
prod_ex: shows "\<forall>A\<in>set As. \<exists>D. production N D = {A}"
ncols_not_0: fixes A :: "(('c, 'd) vec, 'e) vec" shows "0 \<noteq> ncols A"
case_prod_beta': fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
cong_modulus_minus_iff: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and uminus :: "'b \<Rightarrow> 'b"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo b c (uminus a) = unique_euclidean_semiring.cong modulo b c a"
rel_G_conversep: fixes L1 :: "'w \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'x \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'y \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'z \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'aa \<Rightarrow> 't \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'u \<Rightarrow> bool" shows "rel_G L1\<inverse>\<inverse> L2\<inverse>\<inverse> Co1\<inverse>\<inverse> Co2\<inverse>\<inverse> Contra1\<inverse>\<inverse> Contra2\<inverse>\<inverse> = (rel_G L1 L2 Co1 Co2 Contra1 Contra2)\<inverse>\<inverse>"
Trg_Src: fixes t :: "'a E.term" assumes "E.Arr t" shows "E.Trg (E.Src t) = E.Src t"
remove_child_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"
S_def: shows "S = (\<lambda>x. \<Sum>p\<in>Pr x. ln (1 - 1 / real p))"
hmem_hadd_E: fixes l :: "hf"   and x :: "hf"   and y :: "hf" assumes "l \<^bold>\<in> x @+ y" obtains "l \<^bold>\<in> x"   | z :: "hf" where "z \<^bold>\<in> y" and "l = x @+ z"
eint_iless: fixes n :: "eint"   and m :: "int" assumes "n < eint m" shows "\<exists>k. n = eint k"
that: fixes i :: "nat"   and j :: "nat" assumes "n = i + j"   and "xs = replicate i x"   and "ys = replicate j x" shows "thesis"
frechet_derivative_sin: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "f differentiable at x" shows "frechet_derivative (\<lambda>x. sin (f x)) (at x) = (\<lambda>xa. frechet_derivative f (at x) xa * cos (f x))"
fully_faithful_functor_axioms: shows "fully_faithful_functor A B F"
to_fun_is_Fun: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)" shows "(\<bullet>) x \<in> carrier (function_ring (carrier Zp) Zp)"
simps_1: fixes e' :: "'a exp" shows "throw_of (throw e') = \<lfloor>e'\<rfloor>"
\<Z>_subset_\<X>: fixes k :: "nat"   and n :: "nat" shows "\<Z> k n \<subseteq> \<X> k"
dres_unit_simps: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'a dres" shows "dRETURN u \<bind> f = f ()"
random_dictatorship_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "RD R = pmf_of_set agents \<bind> (\<lambda>i. pmf_of_set (favorites R i))"
less_imp_le: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x \<subset># y" shows "x \<subseteq># y"
ocondition_NF_wp: fixes L :: "'a \<Rightarrow> bool"   and l :: "'a \<Rightarrow> 'b option"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'b option"   and C :: "'a \<Rightarrow> bool" assumes "ovalidNF L l Q"   and "ovalidNF R r Q" shows "ovalidNF (\<lambda>s. if C s then L s else R s) (ocondition C l r) Q"
finite_GIrrRep_repset: shows "finite GIrrRep_repset"
if_SE_split: fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and B\<^sub>1 :: "'d \<Rightarrow> (bool \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> (bool \<times> 'd) option" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2) = ((P \<sigma> \<longrightarrow> (\<sigma> \<Turnstile> B\<^sub>1)) \<and> (\<not> P \<sigma> \<longrightarrow> (\<sigma> \<Turnstile> B\<^sub>2)))"
t3: fixes X :: "real set" assumes "bdd_above X" shows "bdd_above (sqrt ` X)"
intros_1: fixes A :: "'a set" shows "[] \<in> lists A"
surjective_sum: fixes f :: "'a + 'b \<Rightarrow> 'c" shows "case_sum (\<lambda>x. f (Inl x)) (\<lambda>y. f (Inr y)) = f"
attract: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubset> x"   and "x \<sqsubset> y"   and "z \<sqsubset> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubset> x"
yk_1: shows "y = nat (fst res) + k1 * p"
vimage_Diff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" shows "f -` (A - B) = f -` A - f -` B"
char_approx2: fixes n :: "nat"   and t :: "real" assumes "\<And>k. k \<le> n \<Longrightarrow> integrable M (\<lambda>x. x ^ k)" shows "cmod (char M t - (\<Sum>k\<le>n. (\<i> * complex_of_real t) ^ k / fact k * complex_of_real (expectation (\<lambda>x. x ^ k)))) \<le> \<bar>t\<bar> ^ n / fact (Suc n) * expectation (\<lambda>x. min (2 * \<bar>x\<bar> ^ n * real (Suc n)) (\<bar>t\<bar> * \<bar>x\<bar> ^ Suc n))"
affine_binop'_def: fixes p :: "nat"   and X :: "real \<times> real pdevs"   and Y :: "real \<times> real pdevs"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real"   and k :: "nat" shows "affine_binop' p X Y a b c d k = (let (r, e1) = trunc_bound_eucl p (a * fst X + b * fst Y + c); (Z, e2) = trunc_bound_pdevs p (add_pdevs (scaleR_pdevs a (snd X)) (scaleR_pdevs b (snd Y))) in (r, pdev_upd Z k (sum_list' p [e1, e2, d])))"
healthy_monoD2: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "healthy t"   and "sound P"   and "sound Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
sources_sinks_aux: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and xs :: "'b list" shows "sources_aux I D U xs = sinks_aux (I\<inverse>) D U (rev xs)"
induct: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a list" assumes "\<And>x. P x []"   and "\<And>x y ys. P x ys \<Longrightarrow> P x (y # ys)" shows "P a0 a1"
Abs_bit0'_code: fixes x :: "int" shows "Rep_bit0 (Abs_bit0' x) = x mod int CARD('b bit0)"
rep_eq: shows "pmf local.pmf_of_set = (\<lambda>x. indicat_real S x / real (card S))"
lin: fixes x :: "'a"   and y :: "'a" shows "R (h x) (h y) \<or> R (h y) (h x)"
subgraph_subset_2: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2" shows "edges (restrict G\<^sub>1) \<subseteq> edges G\<^sub>2"
is_Var_imp_decode_Var: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "is_Var x" shows "x = \<lbrakk>\<guillemotleft>Var (decode_Var x)\<guillemotright>\<rbrakk>e"
push_down_senior_tokens: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x \<le> n"   and "y \<le> n"   and "senior x n = senior y n" shows "\<exists>q. x \<in> configuration q n \<and> y \<in> configuration q n"
mult_2_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "class.semiring_numeral one times plus" shows "times z (numeral.numeral one plus (num.Bit0 num.One)) = plus z z"
w\<^sub>\<mu>': shows "ide (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<and> is_left_adjoint (\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)"
x0_ge_1: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p" shows "1 \<le> x\<^sub>0"
prime_factorization_prime: fixes p :: "'a" assumes "prime p" shows "prime_factorization p = {#p#}"
ACI_norm_toplevel_summands_Zero: fixes r :: "'c rexp" assumes "toplevel_summands r \<subseteq> {Zero}" shows "\<guillemotleft>r\<guillemotright> = Zero"
simps: fixes xs :: "'a list" shows "list_of (IArray xs) = xs"
type_definition_mapping_rbt: shows "type_definition impl_of Mapping_RBT' {t. ord.is_rbt cless t \<or> ID ccompare = None}"
simps_2: fixes \<mu> :: "'a" shows "E.Trg \<^bold>\<langle>\<mu>\<^bold>\<rangle> = \<^bold>\<langle>trg\<^sub>B \<mu>\<^bold>\<rangle>\<^sub>0"
\<A>'_2: shows "\<I> \<Turnstile> \<langle>\<A>'\<rangle>"
mgu_subst_domain: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "mgu s t = Some \<sigma>" shows "subst_domain \<sigma> \<subseteq> vars_term s \<union> vars_term t"
before_in_index1: fixes l :: "'b list"   and x :: "'b"   and y :: "'b" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 0 else 1) = index l x"
semiring_normalization_rules_16: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and lx :: "'d"   and ly :: "'d"   and rx :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "times (times lx ly) rx = times (times lx rx) ly"
FG_advice_subst: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X]\<^sub>\<mu> = subst \<phi> (\<lambda>\<psi>. Some (\<psi>[X]\<^sub>\<mu>))"
biB: fixes i :: "nat" assumes "i < n" shows "b i \<in> B"
dual_minus_eq: fixes x :: "'d"   and y :: "'d" shows "dual (x - y) = - (dual y - dual x)"
awalk_verts_in_verts: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "u \<in> verts G"   and "set p \<subseteq> arcs G"   and "v \<in> set (pre_digraph.awalk_verts G u p)" shows "v \<in> verts G"
simps_14: fixes x41 :: "'d"   and x42 :: "('b, 'c, 'd) Tree"   and x2 :: "('b, 'c, 'd) Tree" shows "tAct x41 x42 \<noteq> tNot x2"
eval_poly_def: fixes h :: "'d \<Rightarrow> 'c"   and p :: "'d poly" shows "eval_poly h p = poly (map_poly h p)"
bounded_antilinear_o_bounded_antilinear: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_antilinear f"   and "bounded_antilinear g" shows "bounded_clinear (\<lambda>x. f (g x))"
smult_closed: fixes a :: "'a"   and x :: "'c" assumes "a \<in> carrier K"   and "x \<in> carrier V" shows "a \<odot>\<^bsub>V\<^esub> x \<in> carrier V"
child_level: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p" shows "level (child p dir d) = level p + 1"
propex\<^sub>1: fixes F :: "\<Pi>\<^sub>1" shows "\<exists>r. Some r = Semantics.d\<^sub>1 F"
inv_unique': fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "y = inv\<^bsub>G\<^esub> x"
hyps: assumes "a \<le> mod_type_class.from_nat i"   and "i \<le> j"   and "j < ncols A" shows "column a (Gram_Schmidt_upt_k A i) = column a (Gram_Schmidt_upt_k A j)"
simps_2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "successively P [x] = True"
assms_4: assumes "nsqn rt1 dip < nsqn rt2 dip" shows "P rt1 rt2 dip"
inv_is_inverse: fixes f :: "'a \<times> 'a" assumes "VoV.iso f" shows "VoV.inverse_arrows f (VoV.inv f)"
option_rule: fixes v :: "'b option"   and S1 :: "'c nres"   and \<Phi> :: "'c \<Rightarrow> bool"   and f2 :: "'b \<Rightarrow> 'c nres" assumes "v = None \<Longrightarrow> S1 \<le> SPEC \<Phi>"   and "\<And>x. v = Some x \<Longrightarrow> f2 x \<le> SPEC \<Phi>" shows "(case v of None \<Rightarrow> S1 | Some x \<Rightarrow> f2 x) \<le> SPEC \<Phi>"
fst_normalize0_polynomial_subsetD: fixes a :: "char list list"   and b :: "int"   and p :: "(char list list \<times> int) list" assumes "(a, b) \<in> set (merge_coeffs0 p)" shows "a \<in> mononoms p"
mult_smult_right: fixes p :: "'a poly"   and a :: "'a"   and q :: "'a poly" shows "p * Polynomial.smult a q = Polynomial.smult a (p * q)"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C"   and "product_category C C" shows "CartesianMonoidalCategory.diagonal_functor C"
dom_None: fixes S :: "'d"   and F :: "'d \<Rightarrow> 'e option" assumes "S \<notin> dom F" shows "F S = None"
lexord_Nil_right: fixes x :: "'g list"   and r :: "('g \<times> 'g) set" shows "(x, []) \<notin> lexord r"
Exit_label_incr: fixes n :: "label"   and i :: "nat" assumes "Exit = n \<oplus> i" shows "n = Exit"
is_extensional: fixes f :: "('a S.E.term, 'a) S.arr" assumes "\<not> S.arr f" shows "S.trg f = S.null"
foldli_not_cond: fixes c :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d"   and xs :: "'e list"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<not> c \<sigma>" shows "foldli xs c f \<sigma> = \<sigma>"
in_ideal_gen_by_prod: fixes \<aa> :: "'a set"   and \<bb> :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal \<aa> R (+) (\<cdot>) \<zero> \<one>"   and "ideal \<bb> R (+) (\<cdot>) \<zero> \<one>"   and "a \<in> R"   and "b \<in> ideal_gen_by_prod \<aa> \<bb>" shows "a \<cdot> b \<in> ideal_gen_by_prod \<aa> \<bb>"
opnet_tau1: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and \<sigma>' :: "nat \<Rightarrow> 'a"   and s' :: "'b net_state"   and S :: "(((nat \<Rightarrow> 'a) \<times> 'b net_state) \<times> 'c node_action \<times> (nat \<Rightarrow> 'a) \<times> 'b net_state) set"   and t :: "'b net_state"   and T :: "(((nat \<Rightarrow> 'a) \<times> 'b net_state) \<times> 'c node_action \<times> (nat \<Rightarrow> 'a) \<times> 'b net_state) set" assumes "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> S" shows "((\<sigma>, SubnetS s t), \<tau>, \<sigma>', SubnetS s' t) \<in> opnet_sos S T"
Domain_mono: fixes r :: "('d \<times> 'e) set"   and s :: "('d \<times> 'e) set" assumes "r \<subseteq> s" shows "Domain r \<subseteq> Domain s"
encode_config_def: fixes cfg :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" shows "encode_config cfg \<equiv> prod_encode (Partial_Recursive.list_encode (map encode_frame (fst cfg)), encode_option (snd cfg))"
not_Some_eq: fixes x :: "'e option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
simps_87: fixes f :: "'b \<Rightarrow> 'a"   and x71 :: "'b poly"   and x72 :: "nat" shows "map_poly f (Pw x71 x72) = Pw (map_poly f x71) x72"
right_idem: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> b \<oplus> b = a \<oplus> b"
generates_max_ideal_def: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "generates_max_ideal X F = (ideal F \<noteq> UNIV \<and> (\<forall>F'\<subseteq>P[X]. ideal F \<subset> ideal F' \<longrightarrow> ideal F' = UNIV))"
vangle_eq_0D: fixes u :: "'b"   and v :: "'b" assumes "vangle u v = 0" shows "norm u *\<^sub>R v = norm v *\<^sub>R u"
startfin_2: shows "(a $ 1 \<le> pathfinish f $ 1 \<and> pathfinish f $ 1 \<le> b $ 1) \<and> a $ 2 \<le> pathfinish f $ 2 \<and> pathfinish f $ 2 \<le> b $ 2"
head_in_verts: fixes e :: "nat \<times> nat" assumes "e \<in> g_E G" shows "head G e \<in> set (g_V G)"
rev_image_eqI: fixes x :: "'h"   and A :: "'h set"   and b :: "'i"   and f :: "'h \<Rightarrow> 'i" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
subset_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (\<forall>t. s2p A t \<longrightarrow> s2p B t)"
rel_frontier_retract_of_punctured_affine_hull: fixes S :: "'a set"   and a :: "'a" assumes "bounded S"   and "convex S"   and "a \<in> rel_interior S" shows "rel_frontier S retract_of affine hull S - {a}"
unless_apply_cong: fixes C :: "bool"   and C' :: "bool"   and s :: "'c"   and s' :: "'c"   and m :: "'c \<Rightarrow> (('d + unit) \<times> 'c) set \<times> bool"   and m' :: "'c \<Rightarrow> (('d + unit) \<times> 'c) set \<times> bool" assumes "C = C'"   and "s = s'"   and "\<not> C' \<Longrightarrow> m s' = m' s'" shows "unlessE C m s = unlessE C' m' s'"