hcju/isabelleps huggingface dataset:
bcomp_exec_n: fixes b :: "bexp"   and f :: "bool"   and j :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and n :: "nat"   and i :: "int"   and s' :: "char list \<Rightarrow> int"   and stk' :: "int list" assumes "bcomp b f j \<turnstile> (0, s, stk) \<rightarrow>^n (i, s', stk')"   and "size (bcomp b f j) \<le> i"   and "0 \<le> j" shows "i = size (bcomp b f j) + (if f = bval b s then j else 0) \<and> s' = s \<and> stk' = stk"
preserves_hom: fixes f :: "'j"   and a :: "'j"   and b :: "'j" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>" shows "\<guillemotleft>\<chi> f : \<chi>.A.map a \<rightarrow> D b\<guillemotright>"
integer_set_bit_def: fixes x :: "integer"   and n :: "integer"   and b :: "bool" shows "integer_set_bit x n b = (if n < 0 then undefined x n b else set_bit_class.set_bit x (nat_of_integer n) b)"
prec: shows "atomic_step_precondition s tid (SK_EV_SIGNAL EV_SIGNAL_FINISH partner)"
uniformly_on_def: fixes S :: "'b set"   and l :: "'b \<Rightarrow> 'c" shows "uniformly_on S l = (INF e\<in>{0<..}. principal {f. \<forall>x\<in>S. dist (f x) (l x) < e})"
arrI: fixes f :: "'a" assumes "right (src f) f" shows "Right.arr f"
iptables_bigstep_add_match_and: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m1 :: "'c match_expr"   and m2 :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m1 (add_match m2 rs), s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchAnd m1 m2) rs, s\<rangle> \<Rightarrow> t"
crel_vs_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and v :: "'e"   and f :: "['f]" shows "crel_vs R v f \<equiv> \<forall>heap. P heap \<and> Q heap \<and> state_dp_consistency.cmem heap \<longrightarrow> (case execute f heap of None \<Rightarrow> False | Some (v', heap') \<Rightarrow> P heap' \<and> Q heap' \<and> R v v' \<and> state_dp_consistency.cmem heap')"
units_of_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>units_of G\<^esub> x = inv\<^bsub>G\<^esub> x"
scale_left_commute: fixes a :: "complex"   and b :: "complex"   and x :: "'b" shows "a *\<^sub>C b *\<^sub>C x = b *\<^sub>C a *\<^sub>C x"
bij_betw_sphere_usphere: fixes r :: "real"   and z0 :: "complex" assumes "0 < r" shows "bij_betw (\<lambda>x. complex_of_real r * x + z0) (sphere 0 1) (sphere z0 r)"
poly_eval_poly_eval_focus: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'d \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'d set"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (poly_eval b (focus X p)) = poly_eval (\<lambda>x. if x \<in> X then poly_eval a (b x) else a x) p"
R_mult_left_mono: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<le> b"   and "0 \<le> c" shows "c * a \<le> c * b"
that: fixes x :: "real" assumes "0 < x"   and "poly p x = 0" shows "thesis"
monofunI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" shows "monofun f"
assms: shows "(0::'b) \<notin> snd ` set xs"
Axioms_def: shows "Axioms = {z. \<exists>p vs. z = ([FAtom Pos p vs, FAtom Neg p vs], {})}"
splitAtRec_ram: fixes ram :: "'c"   and vs :: "'c list"   and a :: "'c list"   and b :: "'c list"   and us :: "'c list" assumes "ram \<in> set vs"   and "(a, b) = splitAtRec ram us vs" shows "us @ vs = a @ [ram] @ b"
assign_local_def: fixes upd :: "('d list \<Rightarrow> 'd list) \<Rightarrow> 'c control_state_scheme \<Rightarrow> 'c control_state_scheme"   and rhs :: "'c control_state_scheme \<Rightarrow> 'd" shows "assign_local upd rhs = assign (\<lambda>\<sigma>. (upd \<circ> map_hd) (\<lambda>_. rhs \<sigma>) \<sigma>)"
verts_def: fixes G :: "graph" shows "verts G = {i. i < Dijkstra.size G}"
ucast_bl: fixes w :: "'c word" shows "ucast w = of_bl (to_bl w)"
set_lebesgue_integral_def: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
degree_Poly_list_of_vec: fixes x :: "'b vec"   and n :: "nat" assumes "x \<in> carrier_vec n"   and "0 < n" shows "degree (Poly (list_of_vec x)) < n"
rel_pred_mono: fixes A' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "A' \<le> A" shows "rel_pred A \<le> rel_pred A'"
object_ptr_kinds_eq2_h2: shows "object_ptr_kinds h2 = object_ptr_kinds h' |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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 shadow_root_ptr|}"
pathfinish_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "pathfinish (subpath u v g) = g v"
\<tau>move2Val: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and v :: "'addr val" shows "\<tau>move2 P h stk (Val v) 0 None"
alpha_Tree_relE: fixes x :: "('b, 'c, 'd, 'e) Tree"   and y :: "('b, 'c, 'd, 'e) Tree" assumes "(x, y) \<in> FL_Validity.alpha_Tree_rel" obtains "x =\<^sub>\<alpha> y"
flopped_half_chamber_systems_gf: 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 "ChamberComplex.chamber_system X - g \<turnstile> ChamberComplex.chamber_system X = f \<turnstile> ChamberComplex.chamber_system X"
rep_eq: fixes x :: "'a freeword" shows "freeword (- x) = rev (map flip_signed (freeword x))"
convex_hull_set_plus: fixes S :: "'c set"   and T :: "'c set" shows "convex hull (S + T) = convex hull S + convex hull T"
assms: shows "(E, E') \<in> unif\<^sup>*"
subring_degree_one_imp_pirreducible: fixes K :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subring K R"   and "a \<in> Units (R\<lparr>carrier := K\<rparr>)"   and "b \<in> K" shows "pirreducible K [a, b]"
K_bind_def: shows "K_bind \<equiv> \<lambda>x y. x"
subrel: shows "\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q x y"
cons_pre_rule: fixes P :: "assn"   and P' :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn" assumes "P \<Longrightarrow>\<^sub>A P'"   and "<P'> c <Q>" shows "<P> c <Q>"
U_Model: shows "CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> (InfModel.intTF any) local.M.intFF local.M.intPF"
Union_rel_br: fixes l :: "'a \<Rightarrow> 'd set"   and lI :: "'a \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'c set"   and sI :: "'d \<Rightarrow> bool" shows "\<langle>br l lI, br s sI\<rangle>Union_rel = br (\<lambda>a. \<Union> (s ` l a)) (\<lambda>a. lI a \<and> (\<forall>c\<in>l a. sI c))"
locally_respects_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "locally_respects P I D R \<equiv> \<forall>u\<in>range D. \<forall>xs x. (D x, u) \<notin> I \<and> x \<in> next_events P xs \<longrightarrow> (xs, xs @ [x]) \<in> R u"
tendsto_def: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
ordermap_bij: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf r"   and "total_on A r"   and "small A" shows "bij_betw (ordermap A r) A (elts (ordertype A r))"
switch_trans: fixes v :: "view"   and c :: "cars"   and u :: "view"   and d :: "cars"   and w :: "view" shows "v = c > u \<and> u = d > w \<longrightarrow> v = d > w"
anycast_simps_6: fixes P :: "'a \<Rightarrow> bool"   and d :: "nat" shows "anycast P (deliver d) = True"
oddE: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'b" where "a = plus (times (numeral.numeral one plus (num.Bit0 num.One)) b) one"
drop_bit_rec: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = (if n = 0 then a else drop_bit (n - 1) (divide a (numeral.numeral one plus (num.Bit0 num.One))))"
freeIn_list_disj: fixes L :: "atom fm list"   and var :: "nat" assumes "\<forall>f\<in>set L. freeIn var f" shows "freeIn var (list_disj L)"
abc_only_cba_4: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<not> [[c a b]]"
leg0_uniquely_isomorphic_1: shows "tab\<^sub>0 (cod \<tau>) \<star> \<tau>.chine \<cong> tab\<^sub>0 (local.dom \<tau>)"
Dx_def: shows "Dx_pair = {(x, y). y \<in> cbox a b \<and> x \<in> cbox (g2 y) (g1 y)}"
that_1: assumes "x = \<bottom>"   and "y = \<bottom>" shows "thesis"
intros_1: fixes f :: "'c \<Rightarrow> real" assumes "bounded_linear f" shows "real_polynomial_function f"
rel_interior_EpigraphI: fixes f :: "'b \<Rightarrow> ereal" assumes "Lower_Semicontinuous.convex_on UNIV f" shows "rel_interior (Epigraph UNIV f) = {(x, z) |x z. x \<in> rel_interior (domain f) \<and> f x < ereal z}"
len_lexext_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "len_lexext gt ys xs" shows "len_lexext gt (map f ys) (map f xs)"
map_of_append: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" shows "map_of (xs @ ys) = map_of ys ++ map_of xs"
Says_imp_parts_knows_Spy: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X \<in> set evs" obtains "X \<in> parts (knows Spy evs)"
long_divisionI: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list"   and b :: "'a list list"   and r :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "q \<noteq> []"   and "UP.long_divides p q (b, r)" shows "(b, r) = (UP.pdiv p q, UP.pmod p q)"
pure_rel_eq_false_iff: fixes R :: "('b \<times> 'd) set"   and x :: "'d"   and y :: "'b" shows "(pure R x y = false) = ((y, x) \<notin> R)"
nrename_OrL': fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and M :: "trm"   and w :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = OrL v.M w.N u"   and "v \<sharp> (R, N, u, x, y)"   and "w \<sharp> (R, M, u, x, y)"   and "x \<noteq> y" shows "(\<exists>M' N'. R = OrL v.M' w.N' u \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N) \<or> (\<exists>M' N'. R = OrL v.M' w.N' x \<and> y = u \<and> M'[x\<turnstile>n>y] = M \<and> N'[x\<turnstile>n>y] = N)"
subst_const: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name"   and env :: "(name, 'b) fmap" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "subst (const name) env = const name"
vec_mult_inner: fixes A :: "((real, 'b) vec, 'b) vec"   and v :: "(real, 'b) vec"   and w :: "(real, 'b) vec" shows "(A *v v) \<bullet> w = v \<bullet> (Finite_Cartesian_Product.transpose A *v w)"
map_snd_zip: fixes xs :: "'k list"   and ys :: "'l list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
Y_nz: fixes y :: "'a" assumes "y \<in> Y" shows "y \<notin> S \<or> s y \<noteq> 0"
coprime_add_one_left: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (plus a one) a"
nn_integral_add: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and g :: "'d \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x + g x \<partial>M = integral\<^sup>N M f + integral\<^sup>N M g"
alw_snth: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream"   and n :: "nat" assumes "alw (\<lambda>xs. P (shd (stl xs)) (shd xs)) xs" shows "P (xs !! Suc n) (xs !! n)"
rev_below_trans: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and z :: "'c"   and x :: "'c" assumes "class.po below"   and "below y z"   and "below x y" shows "below x z"
alpha_other_1: fixes v :: "('e, 'f, 'g) Tree"   and va :: "('e, 'f, 'g) Tree set['e]" shows "(tNot v =\<^sub>\<alpha> tConj va) = False"
k_dvd: shows "\<k> dvd \<v> * (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
add_less_cancel_left0: fixes x :: "V"   and z :: "V" shows "(x < x + z) = (0 < z)"
assms_2: assumes "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> x \<in>\<^sub>\<circ> B" shows "P"
intros: fixes m :: "nat \<Rightarrow> nat"   and ts\<^sub>s\<^sub>b :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "m = flush_all_until_volatile_write ts\<^sub>s\<^sub>b m\<^sub>s\<^sub>b"   and "\<S> = share_all_until_volatile_write ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"   and "length ts\<^sub>s\<^sub>b = length ts"   and "\<forall>i<length ts\<^sub>s\<^sub>b. let (p, is\<^sub>s\<^sub>b, \<theta>, sb, \<D>\<^sub>s\<^sub>b, \<O>, \<R>) = ts\<^sub>s\<^sub>b ! i; suspends = dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb in \<exists>is \<D>. instrs suspends @ is\<^sub>s\<^sub>b = is @ prog_instrs suspends \<and> \<D>\<^sub>s\<^sub>b = (\<D> \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}) \<and> ts ! i = (hd_prog p suspends, is, \<theta> |` (dom \<theta> - read_tmps suspends), (), \<D>, acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<O>, release (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) (dom \<S>\<^sub>s\<^sub>b) \<R>)" shows "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"
NotInDom_Zero: fixes H :: "fm set"   and t :: "tm" shows "H \<turnstile> NotInDom t Zero"
remove_vertex_E: fixes G :: "('a, 'b) Graph_scheme"   and x :: "'a" assumes "Digraph G" shows "E\<^bsub>Digraph.remove_vertex G x\<^esub> = Restr E\<^bsub>G\<^esub> (V\<^bsub>G\<^esub> - {x})"
games_lfp_option_stable: fixes g :: "ZF"   and opt :: "ZF" assumes "g \<in> games_lfp"   and "left_option g opt \<or> right_option g opt" shows "opt \<in> games_lfp"
fun_comp_eq_conv: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and fg :: "'b \<Rightarrow> 'c" shows "(f \<circ> g = fg) = (\<forall>x. f (g x) = fg x)"
axioms_2: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" assumes "module R M" shows "abelian_group M"
dist_le_1_imp_domain_eq: fixes P :: "'c \<Rightarrow>\<^sub>F 'd"   and Q :: "'c \<Rightarrow>\<^sub>F 'd" assumes "dist P Q < 1" shows "domain P = domain Q"
set_Eq: shows "{0..<IArray.length A} = set [0..<IArray.length A]"
cnst_slp_of_fa_induction_step: fixes Unop :: "floatarith"   and Unop' :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M' :: "(floatarith, nat) mapping"   and slp' :: "floatarith list"   and xs :: "real list"   and unop :: "real list \<Rightarrow> real" assumes "slp_of_fa_cnst Unop Unop' M slp = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs"   and "\<And>a b. subterms Unop = {Unop}"   and "interpret_floatarith Unop xs = unop xs"   and "interpret_floatarith Unop' (interpret_slp slp xs) = unop xs"   and "unop (interpret_slp slp xs) = unop xs" shows "{Unop} \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
d2: shows "(sterm_sem J \<theta>2 has_derivative frechet J \<theta>2 (fst \<nu>)) (at (fst \<nu>))"
jkbpC_prefix_closed: fixes t :: "'s Trace"   and s :: "'s" assumes "t \<leadsto> s \<in> SPR.jkbpC" shows "t \<in> SPR.jkbpC"
completely_multiplicative_function_divide_nat_power: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "completely_multiplicative_function f" shows "completely_multiplicative_function (\<lambda>n. f n / nat_power n s)"
integrable_on_exp_minus_to_infinity: fixes a :: "real"   and c :: "real" assumes "0 < a" shows "(\<lambda>x. exp (- a * x)) integrable_on {c..}"
9: shows "taq {x. poly p x = 0} q \<equiv> (\<Sum>y\<in>{x. poly p x = 0 \<and> 0 < poly q x}. 1) + (\<Sum>y\<in>{x. poly p x = 0 \<and> poly q x < 0}. - 1)"
execn_sequence_appE: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and ys :: "('e, 'd, 'f) com list"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> =n\<Rightarrow> t" obtains s' :: "('e, 'f) xstate" where "\<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> =n\<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> =n\<Rightarrow> t"
sumlist_Cons: fixes x :: "rat vec"   and xs :: "rat vec list" shows "gs.M.sumlist (x # xs) = x + gs.M.sumlist xs"
gvs: shows "\<forall>f\<in>\<F> g. \<V> f \<inter> set vs = {}"
ln_smallo_powr: fixes e :: "real" assumes "0 < e" shows "ln \<in> o(\<lambda>x. x powr e)"
PPf_def2: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "PPf t1 t2 = subst (subst Pf t1 yy) t2 xx"
poly_div: fixes q :: "'a poly"   and p :: "'a poly"   and x :: "'a"   and y :: "'a" assumes "q \<noteq> 0"   and "poly p x = (0::'a)"   and "poly q y = (0::'a)"   and "y \<noteq> (0::'a)" shows "poly (poly_div p q) (x / y) = (0::'a)"
is_arg_min_def: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "is_arg_min f P x = (P x \<and> (\<nexists>y. P y \<and> f y < f x))"
berlekamp_resulting_mat_constant: fixes u :: "'b mod_ring poly" assumes "degree u = 0" shows "berlekamp_resulting_mat u = 1\<^sub>m 0"
that: fixes ns :: "'node SDG_node list" assumes "CFG_node (_Entry_) cc-ns\<rightarrow>\<^sub>d* n" shows "thesis"
sup1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> k \<sqinter> a" shows "A \<sqinter> b \<equiv> k \<sqinter> (a \<sqinter> b)"
in_state_set_imp_eq_exec_prefix: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and slist :: "('a, 'b) fmap list"   and s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set" assumes "as \<noteq> []"   and "slist \<noteq> []"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "slist \<in> state_set (state_list s as)" shows "\<exists>as'. prefix as' as \<and> exec_plan s as' = last slist \<and> length slist = Suc (length as')"
image_mono: fixes A :: "'d set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<subseteq> B" shows "f ` A \<subseteq> f ` B"
set_map: fixes a :: "'c"   and b :: "'c" assumes "C.ide a"   and "C.ide b" shows "S.set (Hom.map (b, a)) = Hom.set (b, a)"
Ord_in_Ord: fixes k :: "hf"   and m :: "hf" assumes "Ord k"   and "m \<^bold>\<in> k" shows "Ord m"
finprod_infinite: fixes G :: "('e, 'f) monoid_scheme"   and A :: "'g set"   and f :: "'g \<Rightarrow> 'e" assumes "Group.comm_monoid G"   and "infinite A" shows "finprod G f A = \<one>\<^bsub>G\<^esub>"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "B (B h g) f = B h (B g f)"
simps: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val" shows "sound_mode_use (cms, mem) = (list_all (\<lambda>cm. locally_sound_mode_use (cm, mem)) cms \<and> globally_sound_mode_use (cms, mem))"
assms_5: shows "(\<Sum>x\<in>I. \<beta> x * exp (\<alpha> x)) = 0"
add_mset_not_empty: fixes a :: "'k"   and A :: "'k multiset" shows "add_mset a A \<noteq> {#}"
ec: shows "P,shp s2 \<turnstile>\<^sub>b (es2,b2) \<surd>"
PQe: fixes x :: "'a" shows "P x \<tturnstile> Q x"
sums_group: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a"   and k :: "nat" assumes "f sums s"   and "0 < k" shows "(\<lambda>n. sum f {n * k..<n * k + k}) sums s"
RSpan_contains_RSpans_append_left: fixes ns :: "'m list"   and ms :: "'m list" assumes "set ns \<subseteq> M" shows "RSpan ms \<subseteq> RSpan (ms @ ns)"
simps: fixes it :: "'a rose_tree" shows "iAnts it = children it"
simps: fixes ord :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'c option"   and a2 :: "'d option" shows "ord_option ord a1 a2 = ((\<exists>x. a1 = None \<and> a2 = x) \<or> (\<exists>x y. a1 = Some x \<and> a2 = Some y \<and> ord x y))"
assms: shows "\<not> list_all (\<lambda>x. x = dir.Left) p"
vec_nth_cbox_1_eq: fixes u :: "('c, 1) vec"   and v :: "('c, 1) vec" shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u $ 1) (v $ 1)"
code: fixes r :: "'a language"   and s :: "'a language" shows "StarLR r s = Lang (\<oo> r) (\<lambda>a. StarLR (\<dd> (Times r (Plus One s)) a) s)"
A1: shows "A\<^sub>1 \<in> carrier_mat nr\<^sub>1 n"
fminusI: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A"   and "c |\<notin>| B" shows "c |\<in>| A |-| B"
valid_lists_many_calls: fixes x :: "Vars.var"   and p :: "Vars.var list"   and S :: "Vars.var set"   and G :: "CoCalls" assumes "\<not> one_call_in_path x p"   and "p \<in> valid_lists S G" shows "x--x\<in>G"
option_relE: fixes x :: "'b option"   and x' :: "'c option"   and R :: "('b \<times> 'c) set" assumes "(x, x') \<in> \<langle>R\<rangle>option_rel" obtains "x = None" and "x' = None"   | a :: "'b" and a' :: "'c" where "x = Some a" and "x' = Some a'" and "(a, a') \<in> R"
r: shows "set (map snd pp) = p ` {0..<nr} - {nc}"