hcju/isabelleps huggingface dataset:
comm_monoid_cancel_def: fixes G :: "('c, 'd) monoid_scheme" shows "comm_monoid_cancel G \<equiv> monoid_cancel G \<and> Group.comm_monoid G"
dk: fixes k :: "nat" assumes "k < m" shows "g k \<in> carrier_mat N N"
conjugate_l_def: fixes x :: "'a"   and y :: "'a" shows "x \<lhd> y \<equiv> - (- x \<leftarrow> y)"
s11: shows "0 \<le> real_of_int (root_int_floor_pos p x) + 1"
update_triv: fixes al :: "('b \<times> 'a) list"   and k :: "'b"   and v :: "'a" assumes "map_of al k = Some v" shows "update k v al = al"
BS2': fixes f :: "'a"   and g :: "'a"   and r :: "'a" assumes "is_left_adjoint f"   and "is_left_adjoint g"   and "src f = src g"   and "g \<star> f\<^sup>* \<cong> r" shows "\<exists>\<rho>. tabulation (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> f g"
Gcd_fin_dvd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<in> A" shows "dvd.dvd times (semiring_gcd.Gcd_fin one zero gcd A) a"
minus_diff_eq: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.group_add minus plus zero uminus" shows "uminus (minus a b) = minus b a"
reduction: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f" assumes "fg_run A rho HOs SHOs coords" shows "CSHORun A (coarse_run rho) HOs SHOs coords"
module: fixes R :: "('g, 'h) ring_scheme"   and N :: "'i set"   and M :: "('g, 'i, 'j) module_scheme" assumes "LinearCombinations.submodule R N M" shows "module R M"
not_one_less_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
fls_inverse_shift_nonzero: fixes f :: "'a fls"   and m :: "int" assumes "f \<noteq> 0" shows "inverse (fls_shift m f) = fls_shift (- m) (inverse f)"
IH: fixes x2a :: "('fun, 'var) Term.term" assumes "x2a \<in> set T"   and "wf\<^sub>t\<^sub>r\<^sub>m x2a" shows "(f \<in> funs_term (\<Gamma> x2a)) = (f \<in> funs_term x2a \<or> (\<exists>x\<in>fv x2a. f \<in> funs_term (\<Gamma> (Var x))))"
nabla_coinduction: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "d y \<le>  |x\<rangle> y + d z" shows "d y \<le> \<nabla>x +  |x\<^sup>\<star>\<rangle> z"
results_gpv_Fail: fixes \<Gamma> :: "('b, 'c) \<I>" shows "results_gpv \<Gamma> Fail = {}"
split_at_def: fixes n :: "nat"   and xs :: "'c list" shows "split_at n xs = (take n xs, drop n xs)"
less_eq_quasinorm_def: fixes N1 :: "'a quasinorm"   and N2 :: "'a quasinorm" shows "(N1 \<le> N2) = (\<exists>C\<ge>0. \<forall>f. eNorm N2 f \<le> ennreal C * eNorm N1 f)"
W5a: fixes a :: "'a"   and b :: "'a" shows "(-l a r\<rightarrow> -l b) l\<rightarrow> b l\<rightarrow> a = (1::'a)"
szip_eq: fixes u :: "'a stream"   and v :: "'b stream"   and r :: "'a stream"   and s :: "'b stream" shows "(u ||| v = r ||| s) = (u = r \<and> v = s)"
span_INT_subset: fixes f :: "'d \<Rightarrow> 'b set"   and A :: "'d set" shows "span (\<Inter> (f ` A)) \<subseteq> (\<Inter>a\<in>A. span (f a))"
bounded_by_mult: fixes P :: "'c \<Rightarrow> real"   and a :: "real"   and Q :: "'c \<Rightarrow> real"   and b :: "real" assumes "sound P"   and "bounded_by a P"   and "sound Q"   and "bounded_by b Q" shows "bounded_by (a * b) (\<lambda>s. P s * Q s)"
summable_exp_generic: fixes x :: "'b" shows "summable (\<lambda>n. x ^ n /\<^sub>R fact n)"
SUP_sup_distrib: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'c" shows "\<Squnion> (f ` A) \<squnion> \<Squnion> (g ` A) = (\<Squnion>a\<in>A. f a \<squnion> g a)"
3: fixes s1 :: "'a"   and sc1 :: "'a scheduler"   and s2 :: "'a"   and sc2 :: "'a scheduler"   and t :: "'a" assumes "X (s1, sc1) (s2, sc2)"   and "t \<in> set_pmf (action_sch sc2)" shows "X (t, cont_sch sc1 t) (t, cont_sch sc2 t)"
diff_chain_at: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and g :: "'d \<Rightarrow> 'e"   and g' :: "'d \<Rightarrow> 'e" assumes "(f has_derivative f') (at x)"   and "(g has_derivative g') (at (f x))" shows "(g \<circ> f has_derivative g' \<circ> f') (at x)"
H_2: shows "\<not> (m \<le> M \<and> poly [:c, b, a:] m = 0)"
union_inter_neutral: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "\<forall>x\<in>A \<inter> B. g x = (0::'c)" shows "sum g (A \<union> B) = sum g A + sum g B"
leadsETo_Diff: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set"   and CC :: "'d set set"   and C :: "'d set" assumes "F \<in> (A - B) leadsTo[CC] C"   and "F \<in> B leadsTo[CC] C" shows "F \<in> A leadsTo[CC] C"
fset_both_sides: fixes s :: "'b set"   and f :: "'b fset" shows "(Abs_fset s = f) = (fset (Abs_fset s) = fset f)"
mset_le_not_sym: fixes M :: "'k multiset"   and N :: "'k multiset" assumes "M < N" shows "\<not> N < M"
finite_Nlist: fixes A :: "'a set"   and N :: "nat set" assumes "finite A"   and "finite N" shows "finite (NList A N)"
simps_5: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Exists \<phi>) = (\<exists>z. MFOTL.sat \<sigma> (z # v) i \<phi>)"
assms_2: assumes "\<gamma> x t n" shows "\<gamma>' t n"
homologous_rel_trans: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "homologous_rel p X S b c"   and "homologous_rel p X S a b" shows "homologous_rel p X S a c"
SSpec_strong: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes " \<Turnstile> c : A " shows "\<forall>s t. SSpec c s t \<longrightarrow> A s t"
dsj_I1: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A" shows "H \<turnstile> A OR B"
iffb_inv2_aux: fixes \<phi> :: "'c set \<Rightarrow> 'd set"   and x :: "'d"   and Y :: "'c set" assumes "Inf_pres \<phi>"   and "\<Inter> {X. s2p (\<phi> X) x} \<subseteq> Y" shows "s2p (\<phi> Y) x"
sum_gp_offset: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "sum ((^) x) {m..m + n} = (if x = (1::'a) then of_nat n + (1::'a) else x ^ m * ((1::'a) - x ^ Suc n) / ((1::'a) - x))"
reach: fixes x :: "Value" shows "(\<Squnion>n. \<psi>\<^sup>D\<^bsub>n\<^esub>\<cdot>x) = x"
matrix_to_iarray_echelon_form_of: fixes A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "matrix_to_iarray (echelon_form_of A bezout) = echelon_form_of_iarrays (matrix_to_iarray A) bezout"
full_deriv: shows "full_chain (\<leadsto>\<^sub>w) Sts"
scaleR: fixes r :: "real"   and b :: "'a" shows "f' (r *\<^sub>R b) = r *\<^sub>R f' b"
indRel_respects_pred_iff_indRelR_respects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "rel_respects_pred indRel Pred = rel_respects_pred indRelR Pred"
kleisli_def: fixes f :: "'b \<Rightarrow> 'b option"   and g :: "'b \<Rightarrow> 'b option" shows "f \<rhd> g \<equiv> \<lambda>x. f x \<bind> g"
assms_1: shows "(XS, X) \<in> \<langle>A\<rangle>list_set_rel"
surjective_conv_total: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.surjective x = matrix_bounded_idempotent_semiring.total (x\<^sup>t)"
tabulation_data_axioms: shows "tabulation_data (\<cdot>) (\<star>) \<a> \<i> src trg r \<rho> f g"
cb: assumes "carrier_vec n \<subseteq> local.span (set G)" shows "basis (set G)"
simps_2: fixes C :: "char list"   and e :: "'a exp" shows "max_vars (Cast C e) = max_vars e"
prod3_intro: fixes A :: "'b \<times> 'c \<times> 'd"   and a :: "'b"   and b :: "'c"   and c :: "'d" assumes "fst A = a"   and "fst (snd A) = b"   and "snd (snd A) = c" shows "A = (a, b, c)"
uminus_uexpr_def: fixes u :: "('c, 'd) uexpr" shows "- u = uop uminus u"
pt3: fixes ta :: "'c event list"   and P :: "'c process"   and u :: "'c event list"   and Q :: "'c process"   and s :: "'c event list"   and t :: "'c event list"   and A :: "'c set" assumes "ta \<in> T P"   and "u \<in> T Q"   and "s @ t setinterleaves ((ta, u), insert tick (ev ` A))" shows "\<exists>t u X. (t, X) \<in> F P \<and> (\<exists>Y. (u, Y) \<in> F Q \<and> s setinterleaves ((t, u), insert tick (ev ` A)) \<and> tick \<notin> X \<and> tick \<notin> Y \<and> (X \<union> Y) \<inter> ev ` A = {} \<and> X \<inter> Y = {})"
dom_augment_rels_shared_eq: fixes S :: "nat set"   and R :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and S' :: "nat set" shows "dom (augment_rels S R \<R>) = dom (augment_rels S' R \<R>)"
eq2: shows "g \<equiv> C2 (ccpo.fixp (fun_lub lubb) (fun_ord ordb) (\<lambda>g. U2 (G (C2 g))))"
integrable_cong: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integrable M f = integrable N g"
carrier_mono: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "paths t \<subseteq> paths t'" shows "carrier t \<subseteq> carrier t'"
equal_eq: fixes equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.equal equal" shows "equal x y = (x = y)"
ord_eq_le_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "c \<le> b" shows "c \<le> a"
inf_chain_unique: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "[f[f 0 ..]X]"   and "[g[g 0 ..]X]" shows "\<forall>i. f i = g i"
ucast_up_ucast_id: fixes w :: "'b word" assumes "is_up ucast" shows "ucast (ucast w) = w"
thd1: shows "det (\<chi>i. row i A) = det A"
padd_Cons: fixes h :: "'a"   and t :: "'a list"   and l2 :: "'a list" shows "h # t +++ l2 = (if l2 = [] then h # t else (h + hd l2) # t +++ tl l2)"
vector_space_ow_axioms: shows "vector_space_ow U\<^sub>M\<^sub>_\<^sub>2 (+) (0::'c) (-) uminus (*s\<^sub>2)"
fold_append_concat_rev: fixes xss :: "'g list list" shows "fold (@) xss = (@) (concat (rev xss))"
splice_Nil2: fixes xs :: "'b list" shows "splice xs [] = xs"
weak_arrow_of_homs_def: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" shows "weak_arrow_of_homs C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F \<equiv> (horizontal_homs C src\<^sub>C trg\<^sub>C \<and> horizontal_homs D src\<^sub>D trg\<^sub>D) \<and> functor C D F \<and> weak_arrow_of_homs_axioms C src\<^sub>C trg\<^sub>C D src\<^sub>D trg\<^sub>D F"
WFSS: fixes S :: "(ZF, ZF) Signature"   and n :: "nat" shows "WellFormedToSet S (Suc n) = WellFormedToSet S n \<union> {f E@ e |f e. f \<in> Fn\<^bsub>S\<^esub> \<and> e \<in> WellFormedToSet S n}"
set1_FGcontra_def: fixes x :: "('a, 'c, 'd, 'c, 'e, 'f, ('g, 'h, 'i, 'j, 'c, 'k, 'l) G, 'g, 'm, 'n) F" shows "set1_FGcontra x = set1_F x"
I_2: shows "typ_of (mk_all_list vs B) = Some propT"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = []\<^sub>\<circ>" shows "\<FF>\<lparr>ArrMap\<rparr> = []\<^sub>\<circ>"
eq_Maybe_simps_4: fixes x :: "'b"   and y :: "'b" shows "eq\<cdot>(Just\<cdot>x)\<cdot>(Just\<cdot>y) = eq\<cdot>x\<cdot>y"
interior_of_subset_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "subtopology X S interior_of T \<subseteq> S"
coeffs_plus_eq_plus_coeffs: fixes p :: "'b poly"   and q :: "'b poly" shows "coeffs (p + q) = plus_coeffs (coeffs p) (coeffs q)"
perm_fresh_fresh: fixes a :: "name"   and x :: "'b"   and b :: "name" assumes "a \<sharp> x"   and "b \<sharp> x" shows "[(a, b)] \<bullet> x = x"
scast_scast_b: fixes a :: "'b word" assumes "is_up SCAST('b \<rightarrow> 'c)" shows "SCAST('c \<rightarrow> 'd) (SCAST('b \<rightarrow> 'c) a) = SCAST('b \<rightarrow> 'd) a"
ineq: shows "\<bar>(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2\<bar> \<le> (\<bar>l\<bar> + sum_list (map abs ks) + \<bar>(l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) mod 2\<bar>) div 2"
b_assn_subtyping_match: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P' :: "'a \<Rightarrow> bool" assumes "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt A' x y"   and "\<lbrakk>vassn_tag (hn_ctxt A x y); vassn_tag (hn_ctxt A' x y); P x\<rbrakk> \<Longrightarrow> P' x" shows "hn_ctxt (b_assn A P) x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn A' P') x y"
restrict_times: fixes ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and ls :: "'c list"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and ms :: "'c list" shows "ks \<langle>ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms\<rangle> ms = ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms"
s_max: fixes a :: "real"   and v :: "real"   and s0 :: "real"   and x :: "real" assumes "braking_movement a v" shows "movement.s a v s0 x \<le> movement.s a v s0 (movement.t_stop a v)"
le: fixes n :: "nat" shows "a n + ln (real n) \<le> a (Suc n) + ln (real (Suc n))"
is_unit_gauss_i: shows "is_unit \<i>\<^sub>\<int>"
VerduennungA: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<DD> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Theta>" shows "\<DD> # \<Gamma> \<Rightarrow> \<Theta>"
linepath_translate_right: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "linepath (a + c) (b + c) = (\<lambda>x. x + c) \<circ> linepath a b"
real_0_less_add_iff: fixes x :: "real"   and y :: "real" shows "(0 < x + y) = (- x < y)"
hyps: fixes w :: "'a stream"   and thesis :: "bool" assumes "w !! 0 < wa__ !! 0"   and "\<And>k. sdrop k w = sconst (w !! k) \<Longrightarrow> thesis"   and "sdescending w" shows "thesis"
next_snapshot: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and c' :: "('a, 'b, 'c) configuration" shows "c \<turnstile> Snapshot p \<mapsto> c' = (can_occur (Snapshot p) c \<and> ps c' p = Some (states c p) \<and> (\<forall>p'. states c p' = states c' p') \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> ps c' p' = ps c p') \<and> (\<forall>i. (\<exists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i @ [Marker]) \<and> (\<forall>i. (\<exists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = (fst (cs c i), Recording)) \<and> (\<forall>i. (\<nexists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i) \<and> (\<forall>i. (\<nexists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = cs c i))"
comparable: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" shows "less_eq x y \<or> less_eq y x"
gs_bound: shows "\<bar>fs.gs.gso i $ j\<bar> \<le> rat_of_nat N"
pathVertices_append: fixes u :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list" shows "pathVertices u (p1 @ p2) = butlast (pathVertices u p1) @ pathVertices (last (pathVertices u p1)) p2"
ak_gt_max: shows "a ! k * maxne0 ys b < a ! k * xs ! k"
path_image_subpath_subset: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" assumes "u \<in> {0..1}"   and "v \<in> {0..1}" shows "path_image (subpath u v g) \<subseteq> path_image g"
case_two_not_learn: fixes j :: "nat"   and f :: "nat \<Rightarrow> nat option" assumes "\<exists>t. local.prefixes t j \<up>"   and "f \<in> \<R>"   and "\<And>x. x < longest_prefix j \<Longrightarrow> f x = \<psi> j x" shows "\<not> learn_total \<phi> {f} s"
getRel_homR: fixes y :: "'d"   and z :: "'d"   and l :: "'e"   and G :: "('e, 'd) labeled_graph"   and u :: "'f"   and f :: "('d \<times> 'f) set"   and v :: "'f" assumes "(y, z) \<in> getRel l G"   and "(y, u) \<in> f"   and "(z, v) \<in> f" shows "(u, v) \<in> getRel l (map_graph f G)"
eq1: shows "partial_state.encode1 st1.dims0 st1.vars1' = partial_state.encode2 st2.dims0 st2.vars1'"
lsu_Inf_pres: shows "Inf_pres \<nu>\<^sup>\<natural>"
cod_comp: fixes g :: "'j"   and f :: "'j" assumes "J.seq g f" shows "J.cod (g \<cdot>\<^sub>J f) = J.cod g"
sec_is_extensional: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set" assumes "s \<in> \<O> U" shows "s \<in> extensional U"
inserted_ids_exist: fixes oid :: "'c"   and ops :: "('c \<times> ('c, 'd) list_op) list" shows "(oid \<in> set (inserted_ids ops)) = (\<exists>ref val. (oid, Insert ref val) \<in> set ops)"
binding_subst_identity: fixes X :: "tyvrs"   and B :: "binding"   and U :: "ty" assumes "X \<sharp> B" shows "B[X \<mapsto> U]\<^sub>b = B"
override_on_insert: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and X :: "'c set" shows "f ++\<^bsub>insert x X\<^esub> g = (f ++\<^bsub>X\<^esub> g)(x := g x)"
simps_2: fixes eff :: "'b \<Rightarrow> 'a \<Rightarrow> 'a fset \<Rightarrow> bool"   and rs :: "'b stream"   and s :: "'a" shows "cont (RuleSystem_Defs.mkTree eff rs s) = RuleSystem_Defs.mkTree eff (stl (RuleSystem_Defs.trim eff rs s)) |`| RuleSystem_Defs.pickEff eff (shd (RuleSystem_Defs.trim eff rs s)) s"
borel_integrable_compact: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "compact S"   and "continuous_on S f" shows "integrable lborel (\<lambda>x. indicat_real S x *\<^sub>R f x)"
semiring_norm_138: shows "\<not> (1::'a) < (1::'a)"
less_1: fixes y :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "y < i"   and "\<exists>a. y cd\<^bsup>\<pi>\<^esup>\<rightarrow> a" shows "\<exists>a. y icd\<^bsup>\<pi>\<^esup>\<rightarrow> a"
preserves_arr: fixes f :: "'a" assumes "Left_a.arr f" shows "Left_a.arr (H\<^sub>L a f)"
csif: shows "s\<inverse> O f \<subseteq> ov\<inverse>"
groupF_empty: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "(groupF f xs \<noteq> []) = (xs \<noteq> [])"
DIAndSG2_def: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.DIAndSG2 vid1 pid2 pid4 = (Pc pid2 \<rightarrow> ([[$\<alpha> vid1]]Pc pid4) \<rightarrow> [[$\<alpha> vid1]]Pc pid2)"
USUP_is_forall: fixes P :: "'b \<Rightarrow> 'a upred" shows "(\<Squnion> x \<bullet> P x) = ushAll P"
fr_2: shows "\<forall>w\<in>cball z fr - {z}. f w = fp w * (w - z) powr of_int fn \<and> fp w \<noteq> 0"
intros_7: fixes b :: "bexp"   and s1 :: "char list \<Rightarrow> int option"   and c :: "com"   and x :: "nat"   and s2 :: "char list \<Rightarrow> int option"   and y :: "nat"   and s3 :: "char list \<Rightarrow> int option"   and z :: "nat" assumes "pbval b s1"   and "vars b \<subseteq> dom s1"   and "(c, s1) \<Rightarrow>\<^sub>A x \<Down> s2"   and "(WHILE b DO c, s2) \<Rightarrow>\<^sub>A y \<Down> s3"   and "1 + x + y = z" shows "(WHILE b DO c, s1) \<Rightarrow>\<^sub>A z \<Down> s3"
right_lang_def: fixes q :: "hf" shows "ST.right_lang q \<equiv> {u. ST.nextl q u \<in> dfa.final \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>}"
L: fixes s :: "'a" assumes "s \<in> cball t u \<inter> T" shows "L-lipschitz_on (cball x u \<inter> X) (f s)"
simps_12: fixes f :: "'b \<Rightarrow> 'a" shows "lmap f []\<^sub>l = []\<^sub>l"
ladder_n_minus_1_bound: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" assumes "is_ladder D L"   and "1 \<le> index"   and "index < length L" shows "ladder_n L (index - Suc 0) < length D"
strongly_confluent_E1n: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and n :: "nat" assumes "strongly_confluent r"   and "(x, y) \<in> r\<^sup>="   and "(x, z) \<in> r ^^ n" shows "\<exists>u. (y, u) \<in> r\<^sup>* \<and> (z, u) \<in> r\<^sup>="
higher_differentiable_on_cos: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. cos (f x)) n"
echelon_form_echelon_form_of_euclidean: fixes A :: "(('d, 'e) vec, 'f) vec" shows "echelon_form (echelon_form_of_euclidean A)"
has_lock_may_lock_t_eq: fixes l :: "('b \<times> nat) option"   and t :: "'b"   and t' :: "'b" assumes "has_lock l t"   and "may_lock l t'" shows "t = t'"
remove_child_def: shows "remove_child \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove_child get_child_nodes set_child_nodes get_owner_document get_disconnected_nodes set_disconnected_nodes"
i_cut_max_all_1: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "Max I \<le> t" shows "I \<down>\<le> t = I"
1: fixes y :: "'a" shows "y \<in> P \<longrightarrow> {x. (x = p \<or> x \<in> P) \<and> x \<sqsubset> y} = {x \<in> P. x \<sqsubset> y}"
bigI: fixes c :: "real"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "0 < c"   and "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)" shows "f \<in> \<Omega>[F](g)"
simps_7: fixes f1 :: "brick list \<Rightarrow> 'a"   and f2 :: "brick list \<Rightarrow> wall \<Rightarrow> 'a \<Rightarrow> 'a"   and x1 :: "brick list" shows "rec_wall f1 f2 (basic x1) = f1 x1"
r2_def: shows "r2 = (\<lambda>(x, y). flow0 x y) ` ({a<--<flow0 x t2} \<times> {- d<..<0})"
vecspace_cpx_vec: fixes n :: "nat" shows "vectorspace cpx_rng (module_cpx_vec n)"
m_o_h: fixes X :: "char list set"   and opt :: "'av st option" assumes "finite X" shows "m\<^sub>o opt X \<le> h * card X + 1"
drest_succ_proj_eq_drest_succ: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap"   and vs :: "'d set" assumes "fst a \<subseteq>\<^sub>f s" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s a)"
induct: fixes P :: "Unary_PCF.type \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "Unary_PCF.type"   and a1 :: "nat"   and a2 :: "nat" assumes "\<And>T i. P T i 0"   and "\<And>T i j. \<lbrakk>i < ar T \<and> j < ar (T !- i) \<Longrightarrow> P (T !- i) j 0; \<And>x. \<lbrakk>i < ar T \<and> j < ar (T !- i); x \<in> set [0..<ar (T !- i !- j)]\<rbrakk> \<Longrightarrow> P (T !- i) j (Suc x); i < ar T \<and> j < ar (T !- i) \<Longrightarrow> P T i 0\<rbrakk> \<Longrightarrow> P T i (Suc j)" shows "P a0 a1 a2"
eq_neg_iff_add_eq_0: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(x = \<ominus>\<^bsub>R\<^esub> y) = (x \<oplus>\<^bsub>R\<^esub> y = \<zero>\<^bsub>R\<^esub>)"
exhaust: fixes y :: "Sec" obtains "y = High"   | "y = Low"
intros_3: fixes l :: "'b list"   and S :: "'b set"   and Q :: "'b \<Rightarrow> 'b set set"   and n :: "'b" assumes "path l"   and "S \<in> Q (hd l)"   and "n \<in> S" shows "path (n # l)"
keys_lift_keys_subset: fixes i :: "nat"   and xs :: "(('a, 'b) pp \<times> nat, 'c) oalist_ntm" shows "keys (MP_oalist (lift_keys_pprod i xs)) \<subseteq> map_component_pprod ((+) i) ` keys (MP_oalist xs)"