hcju/isabelleps huggingface dataset:
node_ptr_casts_commute2: fixes node_ptr :: "(_) node_ptr" shows "cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) = Some node_ptr"
comm_groupE_3: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
setdist_gt_0_compact_closed: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closed T" shows "(0 < setdist S T) = (S \<noteq> {} \<and> T \<noteq> {} \<and> S \<inter> T = {})"
order_refl: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.preorder less_eq less" shows "less_eq x x"
sum_norm_allsubsets_bound: fixes P :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and e :: "real" assumes "finite P"   and "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e" shows "(\<Sum>x\<in>P. norm (f x)) \<le> 2 * real DIM('b) * e"
config_config_set: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and init :: "'a list"   and qs :: "'a list" shows "set (fst (config\<^sub>p A init qs)) = set init"
pre_mono: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and C :: "acom"   and s :: "char list \<Rightarrow> int" assumes "\<forall>s. P s \<longrightarrow> P' s"   and "pre C P s" shows "pre C P' s"
post_decomposable: fixes res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res" shows "POST self a1 res = (POST' self a1 and (res \<triangleq> BODY self a1))"
ideal_deg_shifts: fixes X :: "'x set"   and d :: "nat"   and fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" assumes "finite X" shows "ideal (set (pp_pm.deg_shifts X d fs)) = ideal (set fs)"
scaleR_left_normalize: fixes a :: "real"   and b :: "'a"   and c :: "'a" shows "(a *\<^sub>R b = c) = (if a = 0 then c = (0::'a) else b = c /\<^sub>R a)"
r: shows "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
rel_generat_Pure1: fixes A :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'g" shows "rel_generat A B C (Pure x) = (\<lambda>r. \<exists>y. r = Pure y \<and> A x y)"
comp_assoc_tuple_2: fixes f0 :: "'a"   and a :: "'a"   and b0 :: "'a"   and f1 :: "'a"   and b1 :: "'a"   and f2 :: "'a"   and b2 :: "'a" assumes "\<guillemotleft>f0 : a \<rightarrow> b0\<guillemotright>"   and "\<guillemotleft>f1 : a \<rightarrow> b1\<guillemotright>"   and "\<guillemotleft>f2 : a \<rightarrow> b2\<guillemotright>" shows "\<a>\<^sup>-\<^sup>1[b0, b1, b2] \<cdot> \<langle>f0, \<langle>f1, f2\<rangle>\<rangle> = \<langle>\<langle>f0, f1\<rangle>, f2\<rangle>"
Gauss_Jordan_in_ij_unfold: fixes A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and i :: "'a" assumes "\<exists>n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n" obtains n :: "'a" and interchange_A :: "(('b, 'c) vec, 'a) vec" and A' :: "(('b, 'c) vec, 'a) vec" where "(LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n) = n" and "A $ n $ j \<noteq> (0::'b)" and "i \<le> n" and "interchange_A = interchange_rows A i n" and "A' = mult_row interchange_A i ((1::'b) / interchange_A $ i $ j)" and "Gauss_Jordan_in_ij A i j = (\<chi>s. if s = i then A' $ s else row_add A' s i (- interchange_A $ s $ j) $ s)"
profile_def: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set" shows "profile A Is P \<equiv> Is \<noteq> {} \<and> (\<forall>i\<in>Is. rpr A (P i))"
cod_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "cod (local.dom f) = local.dom f"
polymap_polysmult: fixes a :: "'f"   and p :: "'f poly" shows "polymap (smult a p) = a \<cdot>\<cdot> polymap p"
apply_power_eq_iff: fixes f :: "'a perm"   and m :: "nat"   and a :: "'a"   and n :: "nat" shows "((f ^ m) \<langle>$\<rangle> a = (f ^ n) \<langle>$\<rangle> a) = (m mod order f a = n mod order f a)"
cond_prob_def: fixes M :: "'a measure"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
P_def: shows "P = (\<not> (LIM x at_bot. poly q x / poly p x :> at_bot) \<and> \<not> (LIM x at_bot. poly q x / poly p x :> at_top))"
span_mono: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "A \<subseteq> B" shows "pmdl A \<subseteq> pmdl B"
scalprod_real: fixes z1 :: "complex"   and z2 :: "complex" shows "is_real (scalprod z1 z2)"
subcs_news: fixes c :: "'a"   and z :: "('a, 'b) form list"   and s :: "'a FOL_Fitting.term" assumes "news c z" shows "subcs c s z = z"
in_Def_valid_SDG_node: fixes V :: "'var"   and n :: "'node SDG_node" assumes "V \<in> Def\<^bsub>SDG\<^esub> n" shows "valid_SDG_node n"
simps: fixes i :: "nat"   and j :: "nat" shows "\<mu> i j = (if j < i then fs ! i \<bullet> gso j / \<parallel>gso j\<parallel>\<^sup>2 else if i = j then 1::'a else (0::'a))"
simps_5: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a"   and s :: "'b"   and s' :: "'b"   and msg :: "'c"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Recv i p q s s' msg) c' = (distributed_system.can_occur channel trans send recv (Recv i p q s s' msg) c \<and> (states c p = s \<and> states c' p = s') \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c i = Msg msg # msgs c' i \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> (\<forall>r. ps c r = ps c' r) \<and> (\<forall>i'. i' \<noteq> i \<longrightarrow> cs c i' = cs c' i') \<and> (if snd (cs c i) = Recording then cs c' i = (fst (cs c i) @ [msg], Recording) else cs c i = cs c' i))"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "ElementClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
bind_def: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'd list" shows "List.bind xs f = concat (map f xs)"
Sigma_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'd set"   and D :: "'b \<Rightarrow> 'd set" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sigma A C = Sigma B D"
sorted_Cons_Min: fixes x :: "'a"   and xs :: "'a list" assumes "sorted (x # xs)" shows "Min (insert x (set xs)) = x"
extended_distance_def: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded" shows "extended_distance x y = extend_distance_fp (max2 Bonk_Schramm_extension_unfolded_wo x y) x y"
all_not_fin_conv: fixes A :: "'c fset" shows "(\<forall>x. x |\<notin>| A) = (A = {||})"
predicate2D: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "P \<le> Q"   and "P x y" shows "Q x y"
nat_pow_inv: fixes x :: "'a"   and i :: "nat" assumes "x \<in> carrier R" shows "[i] \<cdot> (\<ominus> x) = \<ominus> ([i] \<cdot> x)"
sup_pres_multl: fixes x :: "'b" shows "sup_pres ((\<cdot>) x)"
YMapInvNatTransP: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and x :: "ZF" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "x |\<in>| (F @@ X)" shows "NatTransP (YMapInv' C X F x)"
bconf_Seq: fixes P :: "'a prog"   and sh :: "sheap"   and e1 :: "'b exp"   and e2 :: "'b exp"   and b :: "bool" shows "(P,sh \<turnstile>\<^sub>b (e1;; e2,b) \<surd>) = (case val_of e1 of None \<Rightarrow> case lass_val_of e1 of None \<Rightarrow> P,sh \<turnstile>\<^sub>b (e1,b) \<surd> | \<lfloor>p\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (e2,b) \<surd> | \<lfloor>v\<rfloor> \<Rightarrow> P,sh \<turnstile>\<^sub>b (e2,b) \<surd>)"
vcpower_two_ftimesD: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> A ^\<^sub>\<times> 2\<^sub>\<nat>" shows "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> A"
mult_left_le_one_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq zero x"   and "less_eq zero y"   and "less_eq y one" shows "less_eq (times y x) x"
mhomom3Tr: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "submodule R (mdl M (H \<minusplus> K)) K"
asubg_zero: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "A +> {\<zero>\<^bsub>A\<^esub>}"
monotone_fst: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "monotone (rel_prod orda ordb) orda fst"
base: fixes out :: "'b"   and c :: "('a, 'b, 'c) rpv"   and gpv :: "('a, 'b, 'c) gpv"   and input :: "'c"   and \<I> :: "('b, 'c) \<I>" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "c input \<in> sub_gpvs \<I> gpv"
rbt_eq_iff: fixes t1 :: "('a, 'b) RBT.rbt"   and t2 :: "('a, 'b) RBT.rbt" shows "(t1 = t2) = (impl_of t1 = impl_of t2)"
restrict_to_iff: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" shows "restrict_to P A x y = (x \<in> A \<and> y \<in> A \<and> P x y)"
gmft_correct2: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t" shows "FingerTreeStruc.gmft t = sum_list (map snd (FingerTreeStruc.toList t))"
star_square: fixes x :: "'a" shows "x\<^sup>\<star> = ((1::'a) \<squnion> x) * (x * x)\<^sup>\<star>"
cbelow_on_transitive: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "cbelow_on A ord z rel a b"   and "cbelow_on A ord z rel b c" shows "cbelow_on A ord z rel a c"
sturm_firsttwo_signs: fixes p :: "real poly"   and x\<^sub>0 :: "real" assumes "rsquarefree p"   and "poly p x\<^sub>0 = 0" shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly (p * sturm p ! 1) x) = (if x\<^sub>0 < x then 1 else - 1)"
vars_subst_var_supset: fixes subst_var :: "nat \<Rightarrow> linear_poly \<Rightarrow> linear_poly \<Rightarrow> linear_poly"   and lp :: "linear_poly"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly" assumes "SubstVar TYPE('a) subst_var" shows "vars lp - {x\<^sub>j} - vars lp' \<subseteq> vars (subst_var x\<^sub>j lp' lp)"
L_infinity_pos_measure: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and C :: "real" assumes "f \<in> borel_measurable M"   and "ennreal C < eNorm (\<LL> \<infinity> M) f" shows "0 < emeasure M {x \<in> space M. C < \<bar>f x\<bar>}"
ZFTFst_def: shows "ZFTFst = Fst \<circ> Fst"
finfun_left_compose: fixes y :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "y \<in> finfun" shows "g \<circ> y \<in> finfun"
induced_automorphism_fixespointwise_C0_int_D0: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "fixespointwise (OpposedThinChamberComplexFoldings.induced_automorphism X f g) (C0 \<inter> g ` C0)"
swap_apply_3: fixes c :: "'d"   and a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'e" assumes "c \<noteq> a"   and "c \<noteq> b" shows "Fun.swap a b f c = f c"
sublocale_sas_plus_finite_domain_representation_ii_5: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "dom (\<Psi>\<^sub>G\<^sub>+) \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"
ipset_from_netmask_def: fixes addr :: "'b word"   and netmask :: "'b word" shows "ipset_from_netmask addr netmask \<equiv> let network_prefix = addr && netmask in {network_prefix..network_prefix || ~~ netmask}"
disjoint_union: fixes orb1 :: "'c set"   and orb2 :: "'c set" assumes "orb1 \<in> orbits G E \<phi>"   and "orb2 \<in> orbits G E \<phi>" shows "orb1 = orb2 \<or> orb1 \<inter> orb2 = {}"
j0_allin: shows "\<forall>k>n. enat k < llength Ns \<longrightarrow> prems_of \<iota> ! j0 \<in> active_subset (lnth Ns k)"
singleton_insert_inj_eq: fixes b :: "'e"   and a :: "'e"   and A :: "'e set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
vchop_def: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<equiv> N_Chop(lan v,lan u,lan w) \<and> ext v = ext u \<and> ext v = ext w \<and> own v = own u \<and> own v = own w \<and> more v = more w \<and> more v = more u"
assms: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and eP' :: "Syntax.term"   and eV' :: "Syntax.term" assumes "\<Gamma> \<turnstile> e, eP, eV : \<tau>"   and "\<Gamma> \<turnstile> e, eP', eV' : \<tau>" shows "eP = eP' \<and> eV = eV'"
more_arith_simps_7: fixes a :: "'b"   and b :: "'b" shows "- a * b = - (a * b)"
assms_2: shows "tendsto_in\<^sub>N (\<LL> (ennreal p) M) f g"
unlex_inverse: fixes x :: "'b lexlist" shows "Lex (unlex x) = x"
vector_smult_lid: fixes x :: "('a, 'b) vec" shows "(1::'a) *s x = x"
atom_def: fixes a :: "('U, 'C, 'Out) Noninterference.aprop"   and i :: "nat"   and \<pi>l :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream list" shows "atom a i \<pi>l \<equiv> a \<in> apropsOf (\<pi>l ! i)"
elims_2: fixes G :: "('a, 'b) graph"   and x :: "'a"   and xa :: "('a \<times> 'b \<times> 'a) list"   and xb :: "'a" assumes "valid_graph G"   and "valid_graph.is_path G x xa xb" obtains v :: "'a" and v' :: "'a" where "x = va__" and "xa = []" and "xb = v'__" and "va__ = v'__ \<and> v'__ \<in> nodes G"   | v :: "'a" and v1 :: "'a" and w :: "'b" and v2 :: "'a" and p :: "('a \<times> 'b \<times> 'a) list" and v' :: "'a" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> edges G \<and> valid_graph.is_path G v2 p v'"
identify_blocks_cong: fixes k :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "k < n"   and "local.same_diag A B"   and "local.same_upto k A B" shows "identify_blocks A k = identify_blocks B k"
unit_simps_4: shows "S.src (S.inv (S.cmp\<^sub>U\<^sub>P (g, f)) \<cdot>\<^sub>S S.UP \<eta> \<cdot>\<^sub>S UP.unit (src f)) = S.src (S.UP f)"
front_tickFree_dw_closed: fixes s :: "'c event list"   and t :: "'c event list" assumes "front_tickFree (s @ t)" shows "front_tickFree s"
split_sel: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and resumption :: "('g, 'h, 'i) Resumption.resumption" shows "P (case resumption of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = ((resumption = Resumption.resumption.Done (Resumption.resumption.result resumption) \<longrightarrow> P (f1 (Resumption.resumption.result resumption))) \<and> (resumption = Resumption.resumption.Pause (Resumption.resumption.output resumption) (Resumption.resumption.resume resumption) \<longrightarrow> P (f2 (Resumption.resumption.output resumption) (Resumption.resumption.resume resumption))))"
routing_decided: fixes rtbl :: "'i routing_rule list"   and p :: "('i, 'pkt_ext) tagged_packet_scheme" assumes "rtblo = Some rtbl" shows "output_iface (routing_table_semantics rtbl (p_dst p)) = p_oiface p"
AllAgree: shows "\<forall>cfg. reachable c cfg \<longrightarrow> agreement cfg"
inject_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Inl x1 = Inl y1) = (x1 = y1)"
assms_1: shows "\<rho> \<equiv> run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w"
filtration_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" shows "Filtration.filtration M F = ((\<forall>t. subalgebra M (F t)) \<and> (\<forall>s t. s \<le> t \<longrightarrow> subalgebra (F t) (F s)))"
cond_1r: assumes "u_max < s\<^sub>o" shows "no_collision_react {0..}"
list_update_beyond: fixes xs :: "'f list"   and i :: "nat"   and x :: "'f" assumes "length xs \<le> i" shows "xs[i := x] = xs"
filter_filter: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
val_Integer_def: fixes i :: "Integer" shows "\<lbrakk>i\<rbrakk> = (THE j. i = MkI\<cdot>j)"
ex_un_null: shows "\<exists>!n. \<forall>f. C.VV.comp n f = n \<and> C.VV.comp f n = n"
l_join: shows "revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3"
needle_def: fixes l :: "real"   and x :: "real"   and \<phi> :: "real" shows "needle l x \<phi> = closed_segment (x - l / 2 * sin \<phi>) (x + l / 2 * sin \<phi>)"
Sup_bot_dual: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_dual f" shows "bot_dual f"
inv_in_hom: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and a :: "'c"   and b :: "'c" assumes "category C"   and "category.iso C f"   and "partial_magma.in_hom C f a b" shows "partial_magma.in_hom C (category.inv C f) b a"
Dom_dom: fixes f :: "(bool, 'arr list) concrete_category.arr" assumes "arr f" shows "Dom (local.dom f) = Dom f"
split_face_edges_f12_f21: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []" shows "\<E> f12 \<union> \<E> f21 = \<E> f \<union> {(hd vs, ram1), (ram1, hd vs), (last vs, ram2), (ram2, last vs)} \<union> Edges vs \<union> Edges (rev vs)"
renaming_distr_completion: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and \<E> :: "('a, 'b, 'c) cntxt"   and e :: "('a, 'b, 'c) expr" shows "\<R>\<^sub>E \<alpha> \<beta> (\<E> [e]) = (\<R>\<^sub>C \<alpha> \<beta> \<E>) [\<R>\<^sub>E \<alpha> \<beta> e]"
final_edge: fixes prog :: "cmd"   and n :: "w_node" assumes "labels_nodes prog n Skip" shows "prog \<turnstile> n -\<Up>id\<rightarrow> (_Exit_)"
inverts_components: fixes a :: "'c" assumes "C.ide a" shows "C.inverse_arrows (\<epsilon> a) (\<epsilon>'.map a)"
ptr'_owner_document: shows "h \<turnstile> get_owner_document ptr' \<rightarrow>\<^sub>r ptr'_owner_document"
strictly_subsumes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "strictly_subsumes C D = strict subsumes C D"
tick_set_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock" assumes "dilating f sub r" shows "{k. hamlet (Rep_run r k c)} = f ` {k. hamlet (Rep_run sub k c)}"
hmem_not_sym: fixes x :: "hf"   and y :: "hf" shows "\<not> (x \<^bold>\<in> y \<and> y \<^bold>\<in> x)"
M_def: shows "M \<equiv> \<lambda>i j. min (A i j) (B i j)"
freshen_def: fixes n :: "nat" shows "freshen n = renameLCs (freshenLC n)"
prems_1: shows "not_shadows_vconsts_env \<Gamma>"
nth_filter: fixes i :: "nat"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and Q :: "'a \<Rightarrow> bool" assumes "i < length (filter P xs)"   and "\<And>i'. \<lbrakk>i' < length xs; P (xs ! i')\<rbrakk> \<Longrightarrow> Q (xs ! i')" shows "Q (filter P xs ! i)"
dependent_live_vars_Use_cases: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and dyn_control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> bool"   and V :: "'d"   and as' :: "'a list"   and as :: "'a list"   and n' :: "'b"   and n :: "'b" assumes "DynPDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit dyn_control_dependence"   and "(V, as', as) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "\<exists>nx as''. as = as' @ as'' \<and> CFG.path sourcenode targetnode valid_edge n as' nx \<and> DynPDG.DynPDG_path sourcenode targetnode valid_edge Def Use dyn_control_dependence nx as'' n' \<and> V \<in> Use nx \<and> (\<forall>n''\<in>set (CFG.sourcenodes sourcenode as'). V \<notin> Def n'')"
aGroup_def: fixes A :: "('e, 'f) aGroup_scheme" shows "aGroup A \<equiv> ((\<plusminus>\<^bsub>A\<^esub>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A \<and> (\<forall>a b c. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> c \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)) \<and> (\<forall>a b. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b = b \<plusminus>\<^bsub>A\<^esub> a)) \<and> (mop A \<in> carrier A \<rightarrow> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>)) \<and> \<zero>\<^bsub>A\<^esub> \<in> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> \<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a)"
cases: fixes a1 :: "'a list"   and a2 :: "'a list multiset"   and a3 :: "('b, 'c) label"   and a4 :: "'a list"   and a5 :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((a1, a2), a3, a4, a5) \<in> trss fg" obtains u :: "'a" and a :: "'c" and v :: "'a" and r :: "'a list" and c :: "'a list multiset" where "a1 = ub__ # r__" and "a2 = c__" and "a3 = LBase a" and "a4 = va__ # r__" and "a5 = c__" and "(ub__, Base a, va__) \<in> edges fg"   | u :: "'a" and p :: "'b" and v :: "'a" and c :: "'a list multiset" and r :: "'a list" where "a1 = uc__ # ra__" and "a2 = ca__" and "a3 = LCall pa__" and "a4 = entry fg pa__ # vb__ # ra__" and "a5 = ca__" and "(uc__, Call pa__, vb__) \<in> edges fg" and "mon fg pa__ \<inter> mon_c fg ca__ = {}"   | p :: "'b" and r :: "'a list" and c :: "'a list multiset" where "a1 = return fg pb__ # rb__" and "a2 = cb__" and "a3 = LRet" and "a4 = rb__" and "a5 = cb__"   | u :: "'a" and p :: "'b" and v :: "'a" and r :: "'a list" and c :: "'a list multiset" where "a1 = u # r" and "a2 = c" and "a3 = LSpawn p" and "a4 = v # r" and "a5 = add_mset [entry fg p] c" and "(u, Spawn p, v) \<in> edges fg"