hcju/isabelleps huggingface dataset:
omega_fix: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "omega x = inf (times x (omega x)) one"
ipresOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'i, 'j, 'opSym, 'k, 'g, 'h, 'l) model_scheme" shows "ipresOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = igOp MOD delta (lift h inp) (lift hA binp)"
of_nat_le_iff: 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"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
Inf_spec_def: fixes X :: "'c set" shows "Inf_spec X = SPEC (\<lambda>r. \<forall>x\<in>X. r \<le> x)"
active_minimal: fixes l2 :: "'l" assumes "l2 \<noteq> active" shows "active \<sqsubset>L l2"
T_adv: fixes l :: "nat"   and A :: "(nat list \<Rightarrow> 'b) \<times> (nat list \<times> 'b \<Rightarrow> nat \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and n :: "nat" assumes "l \<noteq> 0" shows "T_off (adv A) [0..<l] (cruel A ([0..<l], fst A [0..<l]) (Suc n)) \<le> l\<^sup>2 + l + 1 + (l + 1) * n div 2"
nat_power_1_right: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat" assumes "class.nat_power one times plus zero nat_power" shows "nat_power n one = semiring_1.of_nat one plus zero n"
intro: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "dpo le"   and "dlo_axioms le" shows "dlo le"
True: shows "z \<in> \<real>\<^sub>\<ge>\<^sub>0"
smap_simps_1: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d stream" shows "shd (smap f a) = f (shd a)"
induct: fixes P :: "('c, 'd) SkewBinomialTree list \<Rightarrow> bool"   and a0 :: "('c, 'd) SkewBinomialTree list" assumes "\<And>t. P [t]"   and "\<And>t v va. \<lbrakk>P (v # va); \<not> prio t \<le> prio (getMinTree (v # va)) \<Longrightarrow> P (v # va)\<rbrakk> \<Longrightarrow> P (t # v # va)"   and "P []" shows "P a0"
p_Suc': fixes x :: "'s"   and y :: "'s"   and n :: "nat" shows "p x y (Suc n) = measure_pmf.expectation (K x) (\<lambda>x'. p x' y n)"
assms_2: shows "X \<in> obj\<^bsub>NTCatDom \<eta>1\<^esub>"
exhaust_sel: fixes list :: "'c list" obtains "list = []"   | "list = hd list # tl list"
simps_2: fixes A :: "'c \<times> ('d option \<Rightarrow> 'c \<Rightarrow> 'c set) \<times> ('c \<Rightarrow> bool)"   and a :: "'d"   and w :: "'d list" shows "NAe.steps A (a # w) = (eps A)\<^sup>* O step A (Some a) O NAe.steps A w"
cfg_onD_state: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.cfg_on s" shows "state cfg = s"
qItem_simps_9: fixes xs :: "'varSort"   and x :: "'var" shows "qGood (qVar xs x) = True"
authKeys_insert: fixes A :: "agent"   and K :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and evs :: "event list" shows "authKeys (Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key K, Agent Peer, Number Ta\<rbrace>, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K, Number Ta\<rbrace>\<rbrace> # evs) = insert K (authKeys evs)"
lprefix_lappend2I: fixes xs :: "'a llist"   and ys :: "'a llist"   and zs :: "'a llist" assumes "lprefix xs ys" shows "lprefix xs (lappend ys zs)"
left_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "a \<in> R"
dim_col_matrix_A_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "dim_col (M_mat_R signs subsets) = length signs"
map_of_permute: fixes \<sigma> :: "'c \<Rightarrow> 'c"   and xs :: "('c \<times> 'd) list" assumes "\<sigma> permutes fst ` set xs" shows "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x, y). (inv \<sigma> x, y)) xs)"
override_on_emptyset: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "override_on f g {} = f"
monic_factor: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic (p * q)"   and "monic p" shows "monic q"
minus_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier S"   and "y \<in> carrier S" shows "x \<ominus>\<^bsub>S\<^esub> y \<in> carrier S"
fds_truncate_truncate': fixes m :: "nat"   and f :: "'b fds" shows "fds_truncate m (fds_truncate m f) = fds_truncate m f"
arith_simps_44: shows "\<bar>1::'a\<bar> = (1::'a)"
unlift_vars_frees: fixes n :: "nat"   and x :: "nat itrm" shows "\<forall>i\<in>frees (unlift_vars n x). i \<in> set (opaque x) \<or> n \<le> i"
ext_inject: fixes annul :: "bool"   and resett :: "bool"   and exe :: "bool"   and reset :: "bool"   and err :: "bool"   and ticc :: "word7"   and itrpt_lvl :: "word3"   and st_bar :: "bool"   and atm_ldst_byte :: "word32 \<Rightarrow> bool"   and atm_ldst_word :: "word32 \<Rightarrow> bool"   and more :: "'b"   and annul' :: "bool"   and resett' :: "bool"   and exe' :: "bool"   and reset' :: "bool"   and err' :: "bool"   and ticc' :: "word7"   and itrpt_lvl' :: "word3"   and st_bar' :: "bool"   and atm_ldst_byte' :: "word32 \<Rightarrow> bool"   and atm_ldst_word' :: "word32 \<Rightarrow> bool"   and more' :: "'b" shows "(\<lparr>annul = annul, resett = resett, exe = exe, reset = reset, err = err, ticc = ticc, itrpt_lvl = itrpt_lvl, st_bar = st_bar, atm_ldst_byte = atm_ldst_byte, atm_ldst_word = atm_ldst_word, \<dots> = more\<rparr> = \<lparr>annul = annul', resett = resett', exe = exe', reset = reset', err = err', ticc = ticc', itrpt_lvl = itrpt_lvl', st_bar = st_bar', atm_ldst_byte = atm_ldst_byte', atm_ldst_word = atm_ldst_word', \<dots> = more'\<rparr>) = (annul = annul' \<and> resett = resett' \<and> exe = exe' \<and> reset = reset' \<and> err = err' \<and> ticc = ticc' \<and> itrpt_lvl = itrpt_lvl' \<and> st_bar = st_bar' \<and> atm_ldst_byte = atm_ldst_byte' \<and> atm_ldst_word = atm_ldst_word' \<and> more = more')"
diff_funE: fixes x :: "'a" assumes "x \<in> manifold_eucl.carrier" obtains c1 :: "('a, 'a) chart" where "c1 \<in> manifold_eucl.atlas k" and "x \<in> domain c1" and "k-smooth_on (codomain c1) (g \<circ> inv_chart c1)"
to_tree_order_wf_get_root_node_wf_is_l_to_tree_order_wf_get_root_node_wf: shows "l_to_tree_order_wf_get_root_node_wf DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_root_node heap_is_wellformed"
disjoint_Nv1_inter_Delta1_inter_E2: shows "N\<^bsub>\<V>1\<^esub> \<inter> \<Delta>\<^bsub>\<Gamma>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"
is_Terminates_def: fixes behaviour :: "'a behaviour" shows "is_Terminates behaviour = (\<exists>x1. behaviour = Terminates x1)"
that: fixes C :: "('a, 'b) ctxt" assumes "s = C\<langle>t\<rangle>" shows "thesis"
isLubD1a: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isLub R S x" shows "x \<in> R"
shadow_root_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_shadow_root_ptr_kind ptr" shows "cast (the (cast ptr)) = ptr"
set_iterator_finite: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set" assumes "set_iterator it S0" shows "finite S0"
VLambda_def: fixes A :: "V"   and b :: "V \<Rightarrow> V" shows "VLambda A b \<equiv> ZFC_in_HOL.set ((\<lambda>x. \<langle>x, b x\<rangle>) ` elts A)"
cobounded2: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<inter># b \<subseteq># b"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and dir :: "dir" shows "h (case dir of L \<Rightarrow> f1 | R \<Rightarrow> f2) = (case dir of L \<Rightarrow> h f1 | R \<Rightarrow> h f2)"
that: fixes P' :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x x'. P x x' = \<up> (P' x x')" shows "thesis"
right_neutral: fixes a :: "'e multiset" shows "a \<union># {#} = a"
Q_def: fixes name :: "name"   and ts :: "value list"   and name' :: "name"   and us :: "value list" shows "(\<turnstile>\<^sub>v Vconstr name ts \<approx> Vconstr name' us) = (name = name' \<and> list_all2 vrelated ts us)"
is_lub_set: fixes S :: "'d set set" shows "S <<| \<Union> S"
mIH: shows "\<exists>m n. as = m @ n \<and> (\<forall>e\<in>set m. f (hd as) = f e) \<and> (\<forall>e\<in>set n. f e < f (hd as))"
map_fun_apply: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'f"   and x :: "'d" shows "(f ---> g) h x = g (h (f x))"
pcp_lim_def: fixes C :: "'b set set"   and S :: "'b set" shows "pcp_lim (C::'b set set) (S::'b set) \<equiv> \<Union> {pcp_seq C S n |n::nat. True}"
tanh_real_eq_iff: fixes x :: "real"   and y :: "real" shows "(tanh x = tanh y) = (x = y)"
factors_unique: fixes fs :: "'a list"   and a :: "'a"   and gs :: "'a list" assumes "comm_semiring_1_class.factors fs a"   and "comm_semiring_1_class.factors gs a"   and "a \<noteq> (0::'a)"   and "\<not> a dvd (1::'a)" shows "rel_mset (ddvd) (mset fs) (mset gs)"
subspace_0: fixes S :: "('c \<Rightarrow>\<^sub>0 'b) set" assumes "phull.subspace S" shows "0 \<in> S"
fthe_felem_eq: fixes x :: "'b" shows "fthe_elem {|x|} = x"
arccos_le_arccos: fixes x :: "real"   and y :: "real" assumes "- 1 \<le> x"   and "x \<le> y"   and "y \<le> 1" shows "arccos y \<le> arccos x"
distinct_13: fixes x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x24 :: "nat set"   and x25 :: "nat set"   and x26 :: "nat set"   and x27 :: "nat set"   and x51 :: "nat set"   and x52 :: "nat set"   and x53 :: "nat set"   and x54 :: "nat set" shows "Write x21 x22 x23 x24 x25 x26 x27 \<noteq> Ghost x51 x52 x53 x54"
minus_equation_iff: fixes a :: "'a"   and b :: "'a" shows "(- a = b) = (- b = a)"
local_order_prefix_closed_last: fixes history :: "nat \<Rightarrow> 'c list"   and x :: "'c"   and i :: "nat"   and y :: "'c"   and xs :: "'c list" assumes "node_histories history"   and "node_histories.history_order history x i y"   and "node_histories.prefix_of_node_history history (xs @ [y]) i" shows "x \<in> set xs"
c2_d: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> y = x \<cdot> 1\<^sub>\<pi> \<parallel> y"
wf_unify: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and f :: "'b"   and X :: "('b, 'a) Term.term list"   and S' :: "('b, 'a) strand_step list"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "wf\<^sub>s\<^sub>t V (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S')"   and "t \<in> ik\<^sub>s\<^sub>t S"   and "Unifier \<delta> (Fun f X) t"   and "bvars\<^sub>s\<^sub>t (S @ send\<langle>Fun f X\<rangle>\<^sub>s\<^sub>t # S') \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "wf\<^sub>s\<^sub>t (fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` V)) (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>)"
ccSUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and u :: "'c" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "Sup (f ` A) \<le> u"
weakenTransition: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and \<Psi>' :: "'b" assumes "\<Psi> \<rhd> P \<longmapsto> Rs" shows "\<Psi> \<otimes> \<Psi>' \<rhd> P \<longmapsto> Rs"
rev_prikey_secret: fixes s :: "(agent \<times> msg) set" assumes "s\<^sub>0 \<Turnstile> s" shows "Rev_PriKey \<notin> spied s"
eball_ereal: fixes z :: "'a"   and r :: "real" shows "eball z (ereal r) = ball z r"
STEPFMT_5: shows "(([entry fg q], ceh), wws, [vt], cet) \<in> trcl (trss fg)"
plus_u_strict_2: fixes x :: "'a\<^sub>\<bottom>" shows "\<bottom> + x = \<bottom>"
lookup_None_iff_find_None: fixes P :: "'b \<Rightarrow> bool"   and xs :: "('b \<times> 'a) list" shows "(lookup P xs = None) = (List.find (\<lambda>(k, uu_). P k) xs = None)"
leg1_in_hom_1: shows "\<guillemotleft>s\<^sub>1 : src s\<^sub>0 \<rightarrow> trg s\<guillemotright>"
transpose_invertible: fixes A :: "((real, 'a) vec, 'a) vec" assumes "invertible A" shows "invertible (Finite_Cartesian_Product.transpose A)"
pt_tvar2: fixes pi1 :: "(W.tvar \<times> W.tvar) list"   and pi2 :: "(W.tvar \<times> W.tvar) list"   and x :: "'a" shows "(pi1 @ pi2) \<bullet> x = pi1 \<bullet> pi2 \<bullet> x"
id_ring_hom: fixes R :: "('c, 'd) ring_scheme" shows "id \<in> ring_hom R R"
simps_2: fixes T :: "ty"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (newA T\<lfloor>e\<rceil>) A = \<D> e A"
S_def: fixes a :: "bool"   and b :: "bool" shows "S a b = a"
numeral_ne_\<omega>\<^sub>z: fixes n :: "num" shows "numeral n \<noteq> \<omega>\<^sub>z"
simps_4: fixes x2 :: "(val \<times> val) fset"   and x1 :: "nat" shows "VFun x2 \<noteq> VNat x1"
integral_distr: fixes g :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and f :: "'d \<Rightarrow> 'e" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integral\<^sup>L (distr M N g) f = LINT x|M. f (g x)"
l2: shows "l\<^sub>2 \<equiv>\<^sub>L r2"
Pg_less_def: fixes G :: "Pg"   and H :: "Pg" shows "(G < H) = (G \<le> H \<and> G \<noteq> H)"
assms_1: shows "list_all (\<lambda>a. \<phi> a \<longrightarrow> \<psi> a) al"
A_def: shows "A = {x. p x \<in> - univ_ce}"
loop_init: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat" assumes "BinPacking_Complete U w c" shows "BinPacking_Complete.inv\<^sub>3 U w c {} {} {} {} {u \<in> U. w u \<le> real c / 2} {u \<in> U. real c / 2 < w u}"
ALC: shows "Domainp ALC = (\<lambda>x. x \<in> U)"
rel_funD2: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and x :: "'c" assumes "rel_fun A B f g"   and "A x x" shows "B (f x) (g x)"
assms: shows "Sig iS \<triangleright> Vx : A \<turnstile> e : B"
partial_equiv_sym: fixes eqv :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.partial_equiv eqv"   and "eqv x y" shows "eqv y x"
assms: shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
rt_graphE: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> rt_graph \<sigma> dip" shows "ip \<noteq> dip \<and> (\<exists>r. rt (\<sigma> ip) = r \<and> (\<exists>dsn dsk hops pre. r dip = Some (dsn, dsk, val, hops, ip', pre)))"
in_set_dropD: fixes x :: "'g"   and n :: "nat"   and xs :: "'g list" assumes "x \<in> set (drop n xs)" shows "x \<in> set xs"
odlist_Abs_inverse: fixes y :: "'c list" assumes "y \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)" shows "toList (odlist_Abs y) = y"
aug_ipp_closed0: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and A :: "'e set"   and x :: "'e" assumes "commute_bpp f (aug_pm_set z i A)"   and "assoc_bpp (aug_pm_set z i A) f"   and "ipp_cond1 A i"   and "ipp_cond2 z A i f"   and "ipp_cond3 z i"   and "x \<in> add_set f (aug_pm_set z i A) 0" shows "\<^sub>i- x \<in> add_set f (aug_pm_set z i A) 0"
chine_is_induced_map: shows "h.is_induced_map h.chine"
I_def: shows "I \<equiv> S \<bullet> K \<bullet> K"
of_complex_zero_iff: fixes x :: "complex" shows "(of_complex x = 0\<^sub>h) = (x = 0)"
prems_1: shows "set (i # xs) \<subseteq> {0..n}"
clock_simInit_def: fixes envInit :: "'d list"   and envObs :: "'b \<Rightarrow> 'd \<Rightarrow> 'c" shows "ClockView.clock_simInit envInit envObs \<equiv> \<lambda>a iobs. let cec = fromList envInit in (cec, ODList.filter (\<lambda>s. envObs a s = iobs) cec)"
mcont_monoD: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "mcont luba orda lubb ordb f"   and "orda x y" shows "ordb (f x) (f y)"
tdghm_NTMap_vdomain: 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 \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
add_diff_inverse: fixes a :: "'a"   and b :: "'a" assumes "\<not> a < b" shows "b + (a - b) = a"
G1_def: shows "G1 = (\<lambda>x. ln x * sum_upto (\<lambda>xa. \<psi> (x / real xa)) x)"
mult_of_image_mset: fixes R :: "('a \<times> 'a) set"   and R' :: "('c \<times> 'c) set"   and N :: "'a multiset"   and M :: "'a multiset"   and f :: "'a \<Rightarrow> 'c" assumes "trans R"   and "trans R'"   and "\<And>x y. \<lbrakk>x \<in># N; y \<in># M; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, f y) \<in> R'"   and "(N, M) \<in> mult R" shows "(image_mset f N, image_mset f M) \<in> mult R'"
arr_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VVV.arr f = (arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f)))"
lad_onD: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and C :: "'d set"   and B :: "'d set" assumes "lad_on A f"   and "C \<subseteq> B"   and "B \<subseteq> A" shows "card (f C) \<le> card (f B)"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and a :: "name" assumes "\<forall>y. P[x::=y] \<approx> Q[x::=y]" shows "a<x>.P \<approx> a<x>.Q"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Msg x2 = Msg y2) = (x2 = y2)"
fds_scale_const: fixes c :: "nat"   and c' :: "'b" assumes "0 < c" shows "fds_scale c (fds_const c') = fds_const c'"
assms_3: shows "arr_mset arr' l (m - 1) = arr_mset arr l m - {#Min_mset (arr_mset arr l m)#}"
transfinite_induction: fixes D :: "('c, 'd) Order_scheme"   and s0 :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "Worder D"   and "minimum_elem D (carrier D) s0"   and "P s0"   and "\<forall>t\<in>carrier D. (\<forall>u\<in>segment D t. P u) \<longrightarrow> P t" shows "\<forall>x\<in>carrier D. P x"
simps_5: fixes v :: "'h val"   and A :: "'f set option" shows "\<D> (Val v) A = True"
last_nth_length: fixes x :: "'d list" assumes "x \<noteq> []" shows "x ! (length x - Suc 0) = last x"
Suc_2: shows "Suc n = nat ((int l - int k) mod int (length cs))"
path_component_of_eq: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" shows "(path_component_of X x = path_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x y)"
cases: fixes \<Gamma> :: "('k, 'l) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" obtains f :: "'k \<times> 'k \<Rightarrow> ennreal" where "hindrance_by \<Gamma> f \<epsilon>" and "current \<Gamma> f" and "wave \<Gamma> f"
coprime_1_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one one a"
echelon_form_of_column_k_def: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A' :: "(('a, 'b) vec, 'c) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k bezout A' k = (let (A, i) = A' in if (\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a)) \<or> i = nrows A then (A, i) else if \<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a) then (A, i + 1) else let n = LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n; interchange_A = interchange_rows A (mod_type_class.from_nat i) n in (bezout_iterate interchange_A (nrows A - 1) (mod_type_class.from_nat i) (mod_type_class.from_nat k) bezout, i + 1))"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1))))))) PC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s1))))))))))) \<and> t2 = snd (fst (write_cpu (get_addr (snd instr) (snd (fst (get_curr_win () s2)))) nPC (snd (fst (write_cpu (cpu_reg_val nPC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2))))))) PC (snd (fst (write_reg (cpu_reg_val PC (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) (get_operand_w5 (snd instr ! 3)) (snd (fst (get_curr_win () s2)))))))))))"
simps_3: fixes p :: "int \<times> int"   and \<delta> :: "real"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest p \<delta> (p\<^sub>0 # v # va) = (if \<delta> \<le> real_of_int (snd p\<^sub>0 - snd p) then p\<^sub>0 else let p\<^sub>1 = find_closest p (min \<delta> (dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))) (v # va) in if dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then p\<^sub>0 else p\<^sub>1)"
fold_congs_2: fixes r :: "('b, 'c) routing_action_scheme"   and r' :: "('b, 'c) routing_action_scheme"   and v' :: "'b word option"   and f :: "'b word option \<Rightarrow> 'b word option"   and f' :: "'b word option \<Rightarrow> 'b word option" assumes "r = r'"   and "next_hop r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "next_hop_update f r = next_hop_update f' r'"
prime_imp_coprime_nat: fixes p :: "nat"   and n :: "nat" assumes "normalization_semidom_class.prime p"   and "\<not> p dvd n" shows "coprime p n"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat" assumes "\<And>n p q r. P (Suc (Suc n)) p q r"   and "\<And>p q r. P 0 p q r"   and "\<And>p q r. P (Suc 0) p q r" shows "P a0 a1 a2 a3"
identity_pseudofunctor_axioms: shows "identity_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
unch_eq2: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'd" shows "Unchanged (x, y) \<equiv> Unchanged x \<and> Unchanged y"
prems_2: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>sequence Seq (x # xs),Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>sequence Seq ys \<down> s'"
infinite_growing: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "class.linorder less_eq less"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. less x y" shows "infinite X"
norm_conv_dist: fixes x :: "'b" shows "norm x = dist x (0::'b)"