hcju/isabelleps huggingface dataset:
monoid_def: fixes G :: "('b, 'c) monoid_scheme" shows "Group.monoid G \<equiv> ((\<forall>x y. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G) \<and> (\<forall>x y z. x \<in> carrier G \<longrightarrow> y \<in> carrier G \<longrightarrow> z \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z))) \<and> \<one>\<^bsub>G\<^esub> \<in> carrier G \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x) \<and> (\<forall>x. x \<in> carrier G \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = x)"
card_eq_SucD: fixes A :: "'b set"   and k :: "nat" assumes "card A = Suc k" shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
hyps: fixes ptr :: "(_) object_ptr" assumes "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
MakeCatDom: fixes f :: "'g"   and C :: "('h, 'g, 'i) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> f = dom\<^bsub>MakeCat C\<^esub> f"
suffixes_not_Nil: fixes xs :: "'c list" shows "suffixes xs \<noteq> []"
assms: shows "\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q)"
greaterThanAtMost_empty_iff: fixes k :: "'a multiset"   and l :: "'a multiset" shows "(subset_mset.greaterThanAtMost k l = {}) = (\<not> k \<subset># l)"
Exe1E: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" assumes "[\<lparr>F,x\<rparr> in v]" shows "\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> ex1 r v"
widens_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xs :: "ty list" shows "P \<turnstile> xs [\<le>] xs"
minus_zero_iarray: fixes A :: "'a iarray" shows "A - 0 = A"
CollectI: fixes P :: "'f \<Rightarrow> bool"   and a :: "'f" assumes "P a" shows "a \<in> {x. P x}"
INV_from_Refinement_basic: fixes R :: "('a \<times> 'b) set"   and pi :: "'c \<Rightarrow> 'd"   and Sa :: "('a, 'd) spec"   and Sc :: "('b, 'c) spec"   and I :: "'b set" assumes "refines R pi Sa Sc"   and "Range R \<subseteq> I" shows "reach Sc \<subseteq> I"
strategyproofness: shows "pmf (sds R21) c + (pmf (sds R21) b + pmf (sds R21) a) \<le> pmf (sds R35) c + (pmf (sds R35) b + pmf (sds R35) a)"
is_process4_S: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set"   and s :: "'a event list"   and Y :: "'a event set"   and X :: "'a event set" assumes "is_process P"   and "(s, Y) \<in> FAILURES P"   and "X \<subseteq> Y" shows "(s, X) \<in> FAILURES P"
is_K2_isometry_def: fixes J :: "cltn2" shows "is_K2_isometry J \<equiv> \<forall>p. (apply_cltn2 p J \<in> S) = (p \<in> S)"
bij_betw1: fixes N :: "nat"   and e :: "nat"   and d :: "nat" assumes "((N, e), d) \<in> set_spmf I" shows "bij_betw (F (N, e)) (local.range (N, e)) (local.range (N, e))"
path_targetnode: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" shows "last (targetnodes as) = n'"
eventually_removed_from_P: fixes D :: "'a literal multiset"   and i :: "nat" assumes "D \<in> P_of_state (lnth Sts i)"   and "fair_state_seq Sts"   and "enat i < llength Sts" shows "\<exists>l. D \<in> P_of_state (lnth Sts l) \<and> D \<notin> P_of_state (lnth Sts (Suc l)) \<and> i \<le> l \<and> enat (Suc l) < llength Sts"
hyps_2: fixes s :: "'a"   and n :: "nat"   and t :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile> \<langle>merge_guards c2,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> t"
lsb_odd: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and lsb :: "'a \<Rightarrow> bool" assumes "class.lsb plus minus zero times one divide modulo bit lsb" shows "lsb = (\<lambda>a. \<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
single_valued_lfp: fixes f :: "((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set \<Rightarrow> ((char list \<Rightarrow> int) \<times> (char list \<Rightarrow> int)) set" assumes "cont f"   and "\<And>r. single_valued r \<Longrightarrow> single_valued (f r)" shows "single_valued (lfp f)"
csmset: shows "fmset G cs = fmset G as - fmset G bs + fmset G bs"
cr_option_le_gpv_def: fixes x :: "'d option"   and gpv :: "('d, 'e, 'f) gpv" shows "cr_option_le_gpv x gpv = (gpv = (lift_spmf \<circ> return_pmf) x \<or> x = None)"
poly_pos_between_leq_less: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "(\<forall>x. a \<le> x \<and> x < b \<longrightarrow> 0 < poly p x) = ((b \<le> a \<or> 0 < poly p a) \<and> (\<forall>x. a \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0))"
blocks_nempty: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and bl :: "'c set" assumes "design point_set block_collection"   and "bl \<in># block_collection" shows "bl \<noteq> {}"
idx_build_is_index: fixes m_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and s_ops :: "('e, 'b, 'f) set_ops_scheme"   and t_ops :: "('e, 'g, 'h) set_ops_scheme"   and t :: "'g"   and f :: "'e \<Rightarrow> 'a" assumes "build_index_loc m_ops s_ops t_ops"   and "set_op_invar t_ops t" shows "index_loc.is_index m_ops s_ops f (set_op_\<alpha> t_ops t) (build_index_loc.idx_build m_ops s_ops t_ops f t)"
le_supE: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and x :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq (sup a b) x" obtains "less_eq a x" and "less_eq b x"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t :: "taylor_model" shows "tm_lower_order prec ord I a t = tm_lower_order_of_normed prec ord I a (tm_norm_poly t)"
operative_axioms_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b set \<Rightarrow> 'a" shows "operative_axioms f z g \<equiv> (\<forall>a b. box a b = {} \<longrightarrow> g (cbox a b) = z) \<and> (\<forall>a b c k. k \<in> Basis \<longrightarrow> g (cbox a b) = f (g (cbox a b \<inter> {x. x \<bullet> k \<le> c})) (g (cbox a b \<inter> {x. c \<le> x \<bullet> k})))"
lossless_return_spmf: fixes x :: "'b" shows "lossless_spmf (return_spmf x)"
non0: fixes x :: "'a" shows "g x \<noteq> 0"
zy: shows "\<omega>_limit_point y z \<or> \<alpha>_limit_point y z"
quo_is_div: fixes y :: "nat"   and x :: "nat" assumes "0 < y" shows "quo [x, y] = x div y"
Assoc': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" shows "(\<Psi>', S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel'"
vimageD: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and A :: "'b set" assumes "a \<in> f -` A" shows "f a \<in> A"
3: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "t \<in> set M"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "comp_timpl_closure {t} (set TI) \<subseteq> timpl_closure_set (set M) (set TI)"
get_element_by_id_def: shows "get_element_by_id = (\<lambda>ptr iden. first_in_tree_order ptr (\<lambda>ptr. case cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> return None | Some element_ptr \<Rightarrow> Heap_Error_Monad.bind (get_attribute element_ptr ''id'') (\<lambda>id_opt. if id_opt = Some iden then return (Some element_ptr) else return None)))"
success_lookupI: fixes r :: "'b ref"   and h :: "heap" shows "success (!r) h"
assms: shows "rho xs (x @xs[z1 \<and> z2]_zs) = Some X"
infinite_imp_elem: fixes A :: "'a set" assumes "infinite A" shows "\<exists>x. x \<in> A"
simps_5: fixes as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" shows "transform_alts as (Upd v # S) = Upd v # transform_alts as S"
val_def: fixes H :: "(('b, 'c) fm list \<times> 'c) set"   and "is" :: "'c set"   and x :: "'b" shows "val H is x \<equiv> \<exists>i\<in>is. Pro x at i in' H"
assms: shows "\<forall>x\<in>E. f x = g x"
hyps: assumes "invar bq"   and "\<forall>t\<in>set bq. rank t < rank t'"   and "tree_invar t'" shows "invar (bq @ [t'])"
Argument_exists_unique: fixes z :: "complex"   and R :: "real set"   and r :: "real" assumes "z \<noteq> 0"   and "R = {r - pi<..r + pi}" obtains s :: "real" where "is_Arg z s" and "s \<in> R" and "\<And>t. \<lbrakk>is_Arg z t; t \<in> R\<rbrakk> \<Longrightarrow> s = t"
meet_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z"
min_term_min: fixes v :: "'a" shows "punit.min_term \<preceq> v"
quasi_inverses_some_quasi_inverse_1: fixes f :: "'c" assumes "C.equivalence_map f" shows "C.quasi_inverses f (C.some_quasi_inverse f)"
real_cond_exp_mult: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable F"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. f x * g x)" shows "AE x in M. real_cond_exp M F (\<lambda>x. f x * g x) x = f x * real_cond_exp M F g x"
ide_def: fixes a :: "'a" shows "A.ide a \<equiv> a \<cdot>\<^sub>A a \<noteq> A.null \<and> (\<forall>f. (f \<cdot>\<^sub>A a \<noteq> A.null \<longrightarrow> f \<cdot>\<^sub>A a = f) \<and> (a \<cdot>\<^sub>A f \<noteq> A.null \<longrightarrow> a \<cdot>\<^sub>A f = f))"
rtrancl_on_converseD: fixes y :: "'a"   and x :: "'a"   and F :: "'a set"   and r :: "('a \<times> 'a) set" assumes "(y, x) \<in> rtrancl_on F (r\<inverse>)" shows "(x, y) \<in> rtrancl_on F r"
assms_3: shows "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])"
gval_Le: fixes a1 :: "'a aexp"   and a2 :: "'a aexp"   and s :: "'a \<Rightarrow> value option" shows "gval (Le a1 a2) s = \<not>? value_gt (aval a1 s) (aval a2 s)"
bij_lists_succ: fixes xss :: "'b list list" assumes "distincts xss" shows "bij (lists_succ xss)"
myless: fixes a :: "'a"   and b :: "'a" shows "(a < b) = ((0::'a) < b - a)"
sel_4: fixes x1 :: "'b" shows "ttl (TNil x1) = TNil x1"
coeff_add: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and q :: "'a mpoly" shows "MPoly_Type.coeff p m + MPoly_Type.coeff q m = MPoly_Type.coeff (p + q) m"
iMODb_iMOD_iTILL_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "[ r, mod m, c ] = [ r, mod m ] \<inter> [\<dots>r + m * c]"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case L of L \<Rightarrow> f1 | R \<Rightarrow> f2) = f1"
xor_def: fixes x :: "'b"   and y :: "'b" shows "x \<oplus> y = (x \<squnion> y) \<sqinter> - (x \<sqinter> y)"
lift00: fixes a :: "real mpoly" shows "liftPoly 0 0 a = a"
sf_1: shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
bisim: shows "P,es,h \<turnstile> (map Val vs @ Throw a # es'', loc) [\<leftrightarrow>] (stk, loc, pc, \<lfloor>a\<rfloor>)"
S_def: fixes x :: "'a" shows "S x = (if \<forall>\<omega>\<in>streams (space M). T (x ## \<omega>) = 0 then lim_stream x else L (\<lambda>\<omega>. epred (T (x ## \<omega>))) x)"
coprime_absorb_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times x y" shows "algebraic_semidom.coprime times one x y = dvd.dvd times x one"
hyps_1: shows "\<Gamma> \<turnstile> (c\<^sub>1, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
prebicategory_with_homs_and_units_def: 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" shows "prebicategory_with_homs_and_units V H \<a> \<i> src trg \<equiv> prebicategory_with_units V H \<a> \<i> \<and> prebicategory_with_homs V H \<a> src trg"
eq_iff: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less" shows "(x = y) = (less_eq x y \<and> less_eq y x)"
5: shows "\<forall>\<^sub>F n in sequentially. (\<Sum>j. real_of_int (b (n + j)) / real_of_int (a (n + j))) \<le> real_of_int (2 * b n) / real_of_int (a n)"
unit_char_3: fixes a :: "'c" assumes "C.obj a" shows "F.unit a \<cdot>\<^sub>D \<i>\<^sub>D[F.map\<^sub>0 a] = (local.map \<i>\<^sub>C[a] \<cdot>\<^sub>D cmp (a, a)) \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.unit a)"
index_mat_addcol_4: fixes a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "dim_row (addcol a k l A) = dim_row A"
assms_1: shows "Cantor_sum \<alpha>s ms = Cantor_sum \<beta>s ns"
bounded_subset_box_symmetric: fixes S :: "'b set" assumes "bounded S" obtains a :: "'b" where "S \<subseteq> box (- a) a"
comp_apply: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and x :: "'d" shows "(f \<circ> g) x = f (g x)"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> otway"   and "KK \<subseteq> - range shrK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
eq2: shows "Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i) \<inter> {x. (axis m 1 - axis n 1) \<bullet> x \<le> a $ m}) + Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i) \<inter> {x. b $ m \<le> (axis m 1 - axis n 1) \<bullet> x}) = Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then a $ m + b $ n else b $ i))"
pre: shows "wf_matchP_aux \<sigma> V n R I r aux ne"
zip_eq_ConsE: fixes xs :: "'c list"   and ys :: "'d list"   and xy :: "'c \<times> 'd"   and xys :: "('c \<times> 'd) list" assumes "zip xs ys = xy # xys" obtains x :: "'c" and xs' :: "'c list" and y :: "'d" and ys' :: "'d list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'"
basisD: fixes x :: "'b list"   and G :: "'b list set" assumes "x \<in> \<BB> G" shows " x \<in>B G "
simps_1: fixes c :: "'b poly"   and p :: "'b poly" shows "isconstant (CN c 0 p) = False"
earlyBisimStructCong: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim>\<^sub>e Q"
netglobalsimp: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and s :: "'s net_state" shows "netglobal P s = P (fst (initmissing (netgmap sr s)))"
simps_1: fixes n :: "nat" shows "hmset_of_enat (enat n) = of_nat n"
asm: shows "(as, x) \<approx>set (=) supp p' (as', x')"
units_of_one: fixes G :: "('a, 'b) monoid_scheme" shows "\<one>\<^bsub>units_of G\<^esub> = \<one>\<^bsub>G\<^esub>"
midpoint_abs: fixes la :: "real"   and ka :: "real"   and ua :: "real" assumes "la \<le> ka"   and "ka \<le> ua" shows "\<bar>ka - (la / 2 + ua / 2)\<bar> \<le> \<bar>ua / 2 - la / 2\<bar>"
diff_right_commute: fixes a :: "'d"   and c :: "'d"   and b :: "'d" shows "a - c - b = a - b - c"
interpret_Unop: fixes a :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (Unop a) xs = unop (interpret_floatarith a xs)"
beforex_appendD2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt\<^sub>1 :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and xt\<^sub>2 :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and f :: "nat"   and t :: "nat"   and D :: "char list"   and h :: "nat"   and d :: "nat"   and I :: "nat set"   and J :: "nat set" assumes "P,C,M \<rhd> xt\<^sub>1 @ xt\<^sub>2 @ [(f, t, D, h, d)] / I,d"   and "pcs xt\<^sub>2 \<subseteq> J"   and "J \<subseteq> I"   and "J \<inter> pcs xt\<^sub>1 = {}" shows "P,C,M \<rhd> xt\<^sub>2 / J,d"
unit_factor_mult_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "dvd.dvd times a one" shows "unit_factor (times a b) = times a (unit_factor b)"
monotone_le_le: shows "monotone (\<le>) (\<le>) = mono"
distinct_1: fixes x2 :: "'b" shows "ext.Bot \<noteq> ext.Val x2"
\<S>_inv: fixes l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "(l, R) \<in> \<S>" shows "R \<subseteq> {v. v \<turnstile> PTA.inv_of A l}"
simps: fixes k :: "nat"   and x :: "'a"   and t :: "'a btree" shows "delete k x t = reduce_root (del k x t)"
l1_step3_refines_a0i_commit_skip: fixes A :: "agent"   and B :: "agent"   and Ra :: "fid_t"   and K :: "msg" shows "{R01ia \<inter> UNIV \<times> l1_inv3} (a0i_commit A B \<langle>epubKF (Ra $ kE), K\<rangle>)\<^sup>=, l1_step3 Ra A B K {> R01ia}"
list_slice2_list_slice_nth: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < length xs div k" shows "list_slice2 xs k ! m = list_slice xs k ! m"
map_upt_Suc: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "map f [0..<Suc n] = f 0 # map (\<lambda>i. f (Suc i)) [0..<n]"
self_finance_init: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and v :: "real"   and w :: "'b" assumes "\<forall>w. prices Mkt asset 0 w \<noteq> 0"   and "portfolio pf" shows "val_process Mkt (self_finance Mkt v pf asset) 0 w = v"
plus_uint32_def: shows "(+) \<equiv> map_fun Rep_uint32 (map_fun Rep_uint32 Abs_uint32) (+)"
map_def: fixes f :: "'a" shows "\<tau>'oF_\<tau>oF.map f = (if F.A.arr f then C ((\<tau>' \<circ> F) (F.A.cod f)) ((\<tau> \<circ> F) f) else \<tau>.B.null)"
kstar_unfoldl_eq: fixes f :: "'a \<Rightarrow> 'a set" shows "\<eta> \<squnion> f \<circ>\<^sub>K kstar f = kstar f"
linear_0: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f 0 = 0"
simps_8: fixes x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "sos (All1 x \<phi>) = sos \<phi>"
id: shows "ff (coeff q' i) / ff x' = ff (coeff q' i div x')"
assms: shows "ok (e\<lparr>a:X\<rparr>\<lparr>b:Y\<rparr>)"
local_steps_def: fixes T :: "((('h \<Rightarrow> 'i) \<times> 'j) \<times> 'k \<times> ('h \<Rightarrow> 'l) \<times> 'm) set"   and J :: "'h set" shows "local_steps T J \<equiv> \<forall>\<sigma> \<zeta> s a \<sigma>' s'. ((\<sigma>, s), a, \<sigma>', s') \<in> T \<and> (\<forall>j\<in>J. \<zeta> j = \<sigma> j) \<longrightarrow> (\<exists>\<zeta>'. (\<forall>j\<in>J. \<zeta>' j = \<sigma>' j) \<and> ((\<zeta>, s), a, \<zeta>', s') \<in> T)"
SUP_UNION: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e set"   and A :: "'f set" shows "\<Squnion>(f ` \<Union> (g ` A)) = (SUP y\<in>A. \<Squnion>(f ` g y))"
eq: shows "takeWhile (\<lambda>x. x = a) w \<cdot> [hd (dropWhile (\<lambda>x. x = a) w)] \<cdot> tl (dropWhile (\<lambda>x. x = a) w) = w"
bigo_elt_subset2: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> g +o O(h)" shows "O(f) \<subseteq> O(g) + O(h)"
set_iterator_genord_def: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "set_iterator_genord iti S0 R \<equiv> \<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
select_convs_5: fixes Functions :: "'b \<Rightarrow> (real, 'a) vec \<Rightarrow> real"   and Predicates :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> bool"   and Contexts :: "'c \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set"   and Programs :: "'a \<Rightarrow> (((real, 'a) vec \<times> (real, 'a) vec) \<times> (real, 'a) vec \<times> (real, 'a) vec) set"   and ODEs :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and ODEBV :: "'a \<Rightarrow> 'a set"   and more :: "'d" shows "ODEs \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = ODEs"
confluence_to_local_confluence: fixes y :: "'a"   and x :: "'a" assumes "y\<^sup>\<star> ; x\<^sup>\<star> \<le> x\<^sup>\<star> ; y\<^sup>\<star>" shows "y ; x \<le> x\<^sup>\<star> ; y\<^sup>\<star>"
simps_1: fixes x2 :: "'k"   and y2 :: "'k" shows "(Some x2 = Some y2) = (x2 = y2)"
w: shows "P w x \<and> P w y \<and> P w z"
f_def: shows "f = (\<lambda>x. \<bar>u 1 x\<bar>)"
trd_term_def: fixes d :: "'a \<Rightarrow> nat" shows "trd_term d = {(x, y). dgrad_p_set_le d (set (fst (snd x) # fst x)) (set (fst (snd y) # fst y)) \<and> fst (snd x) \<prec>\<^sub>p fst (snd y)}"
inext_imirror_iprev_conv': fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "n \<in> I" shows "inext (mirror_elem n I) (imirror I) = mirror_elem (iprev n I) I"
absolutely_integrable_on_subcbox: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" assumes "f absolutely_integrable_on S"   and "cbox a b \<subseteq> S" shows "f absolutely_integrable_on cbox a b"
prems_2: shows "\<I> \<Turnstile>\<^sub>c \<langle>S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>,\<theta> \<circ>\<^sub>s \<delta>\<rangle>"