hcju/isabelleps huggingface dataset:
Join_commute: fixes F :: "'b program"   and G :: "'b program" shows "F \<squnion> G = G \<squnion> F"
that: fixes zs :: "'a list" assumes "ys = xs @ zs" shows "thesis"
enres_monad_laws_1: fixes x :: "'e"   and f :: "'e \<Rightarrow> ('c + 'd) nres" shows "ebind (ERETURN x) f = f x"
InitNone: fixes sh :: "sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "sh C = None"   and "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh(C \<mapsto> (sblank P C, Prepared)))\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh)\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
rel_simps_28: fixes n :: "Num.num" shows "\<not> numeral n \<le> (0::'a)"
ga_automaton_def: shows "ga_automaton \<equiv> \<lparr>g_V = UNIV, g_E = ga_step, g_V0 = {pid_init}, sa_L = pid_label\<rparr>"
UP_a_comm: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "q \<in> carrier (UP Zp)" shows "p \<oplus>\<^bsub>UP Zp\<^esub> q = q \<oplus>\<^bsub>UP Zp\<^esub> p"
right_inverse: fixes a :: "'b" shows "a + - a = (0::'b)"
M: shows "M = {ev. eigen_value cA ev \<and> cmod ev = sr}"
mk_dimvec_inj: shows "inj_on mk_dimvec {xs. length xs = CARD('b)}"
maxsimpchain_CConsI: fixes X :: "'b set set"   and w :: "'b set"   and x :: "'b set"   and xs :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimp X w"   and "SimplicialComplex.maxsimpchain X (x # xs)"   and "w \<sim> x" shows "SimplicialComplex.maxsimpchain X (w # x # xs)"
lagrange: fixes H :: "'a set" assumes "subgroup H G" shows "card (rcosets H) * card H = order G"
B_scale: fixes x :: "(real, 'n) vec"   and a :: "real" assumes "x \<in> B"   and "a \<in> {0..1}" shows "a *\<^sub>R x \<in> B"
Card_order_cmax: fixes r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order s" shows "Card_order (cmax r s)"
simps_2: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "lock_action_ok l t Unlock = has_lock l t"
pdivmod2_def: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.pdivmod2 m f g = (if poly_mod.Mp m g = 0 then (0, f) else let ilc = inverse_p m (lead_coeff (poly_mod.Mp m g)); h = Polynomial.smult ilc (poly_mod.Mp m g); (q, r) = pseudo_divmod (poly_mod.Mp m f) (poly_mod.Mp m h) in (Polynomial.smult ilc q, r))"
pivot_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "let s' = pivot x\<^sub>i x\<^sub>j s in \<V> s' = \<V> s \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U> s' = \<U> s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"
dec_above_inc_above: fixes b :: "nat"   and i :: "nat"   and x :: "nat" shows "dec_above b i (inc_above b i x) = x"
sorted_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sort_key less_eq (\<lambda>x. x) xs)"
GB_adds_lt: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_Groebner_basis G"   and "f \<in> local.punit.pmdl G"   and "f \<noteq> 0" obtains g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "g \<in> G" and "g \<noteq> 0" and "local.punit.adds_term (lpp g) (lpp f)"
Nats_add: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_1 one times plus zero"   and "a \<in> semiring_1.Nats one plus zero"   and "b \<in> semiring_1.Nats one plus zero" shows "plus a b \<in> semiring_1.Nats one plus zero"
non_empty_cycle_root_point_exchange: fixes r :: "'a"   and x :: "'a"   and p :: "'a" assumes "non_empty_cycle_root r x"   and "point p" shows "(r \<le> x\<^sup>\<star> ; p) = (p \<le> x\<^sup>\<star> ; r)"
termination_is_local: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and dec :: "'g \<Rightarrow> 'h option"   and m :: "nat"   and p :: "'f"   and v :: "'h" assumes "locally_similar \<sigma> \<tau>"   and "dec (\<sigma> m p) = Some v" shows "\<exists>n. dec (\<tau> n p) = Some v"
connectedin_iff_connected: fixes S :: "'a set" shows "connectedin euclidean S = connected S"
bisimilar_is_bisimulation: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool" assumes "nominal_ts satisfies transition" shows "nominal_ts.is_bisimulation satisfies transition (nominal_ts.bisimilar satisfies transition)"
optimal_for_men_def: fixes \<mu> :: "(M \<times> W) set" shows "Non_Strict.optimal_for_men \<mu> = (Non_Strict.stable \<mu> \<and> (\<forall>\<mu>''. Non_Strict.stable \<mu>'' \<longrightarrow> Non_Strict.weakly_preferred_by_men \<mu>'' \<mu>))"
inv_upOne_eq: fixes n :: "int" shows "inv upOne n = n - 1"
swap_commute: fixes a :: "atom"   and b :: "atom" shows "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
open_contains_ball_eq: fixes S :: "'c set"   and x :: "'c" assumes "open S" shows "(x \<in> S) = (\<exists>e>0. ball x e \<subseteq> S)"
simps_1: fixes x :: "'b" shows "RBT_Set2.del x \<langle>\<rangle> = \<langle>\<rangle>"
ll2f_LNil: fixes x :: "nat" shows "LNil !! x = None"
adapt_aexp_vars: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "Aexp.vars (adapt_aexp e s) = (\<lambda>v. symvar v s) ` Aexp.vars e"
lm090: fixes x :: "'c"   and X :: "'c set"   and Y :: "'d set" assumes "x \<notin> X" shows "(\<Union>f\<in>injections X Y. {f +< {(x, y)} |y. y \<in> Y - Range f}) = injections ({x} \<union> X) Y"
simps_2: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_vec_Tensor f (x # xs) ys = mult.times f x ys @ mult.vec_vec_Tensor f xs ys"
rvpeq_trans: fixes u :: "thread_id_t"   and s1 :: "rstate_t"   and s2 :: "rstate_t"   and s3 :: "rstate_t" assumes "rvpeq u s1 s2"   and "rvpeq u s2 s3" shows "rvpeq u s1 s3"
complement_universe_difference: fixes x :: "i" assumes "CMUD"   and "\<exists>y. \<not> O y x" shows "\<midarrow> x = u \<ominus> x"
name_rev_per: fixes pi :: "(name \<times> name) list"   and x :: "'a" shows "rev pi \<bullet> pi \<bullet> x = x"
intro: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool" assumes "equivp eq"   and "(\<sim>\<^sub>C) \<le> eq"   and "eq \<le> (\<sim>\<^sub>L)" shows "ltl_equivalence eq"
M: fixes i :: "'n"   and j :: "'n"   and k :: "'n"   and l :: "'n" shows "to_fun (minor (\<^bold>X - \<^bold>C A) i j) k l = (if k = i \<and> l = j then 1 else if k = i \<or> l = j then 0 else if k = l then [:- to_fun A k l, 1::'b:] else [:- to_fun A k l:])"
FOREACHci_weaken: fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and I' :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and \<sigma>0 :: "'d" assumes "\<And>it \<sigma>. \<lbrakk>it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> I' it \<sigma>" shows "FOREACH\<^sub>C\<^bsup>I'\<^esup> S c f \<sigma>0 \<le> FOREACH\<^sub>C\<^bsup>I\<^esup> S c f \<sigma>0"
frameImpAlphaEnt: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b"   and A\<^sub>F' :: "name list"   and \<Psi>\<^sub>F' :: "'b"   and \<phi> :: "'c" assumes "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> = \<langle>A\<^sub>F', \<Psi>\<^sub>F'\<rangle>"   and "A\<^sub>F \<sharp>* \<phi>"   and "A\<^sub>F' \<sharp>* \<phi>"   and "\<Psi>\<^sub>F' \<turnstile> \<phi>" shows "\<Psi>\<^sub>F \<turnstile> \<phi>"
prems_1: shows "graph_homomorphism (translation (A_Lbl l)) G f"
lessThan_borel: fixes a :: "'c" shows "{..<a} \<in> sets borel"
3: shows "w\<^sup>T\<^sup>\<star> \<le> forest_components (w \<sqinter> - q \<squnion> (w \<sqinter> q)\<^sup>T)"
subdistl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z \<cdot> x \<le> z \<cdot> (x + y)"
not_ord_strict_pI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lookup p v \<noteq> lookup q v"   and "lookup p v \<noteq> (0::'c)"   and "\<And>u. lt_of_nat_term_order cmp_term v u \<Longrightarrow> lookup p u = lookup q u" shows "\<not> ord_strict_p p q"
Figure_4: fixes k :: "nat"   and i :: "nat"   and js :: "nat list" assumes "foldl1 f1 (map (h1 k i) js) = One"   and "js \<noteq> []" shows "i \<in> set js"
incl: fixes A :: "'d set"   and B :: "'d set set"   and b :: "'d set" assumes "Congruence.partition A B"   and "b \<in> B" shows "b \<subseteq> A"
wpt_def: 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 "mbt_algebra.wpt one times inf top x = inf (times x top) one"
assms_2: shows "f \<in> state_measure V' \<Gamma> \<rightarrow>\<^sub>M subprob_algebra N"
zariski_open_is_subset: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "U \<subseteq> comm_ring.spectrum R addition multiplication zero unit"
assms: fixes u :: "nat \<Rightarrow> 'a" assumes "u \<longlonglongrightarrow> a"   and "\<forall>n. u n \<in> s" shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
circ_mult_1: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<star> = Rf (x * (y * x)\<^sup>\<star>) y"
simps: fixes b :: "bool"   and a1 :: "'a"   and a2 :: "'a" shows "sel2 b (a1, a2) = (if b then a2 else a1)"
subspace_def: fixes S :: "('b \<times> 'c) set" shows "p.subspace S = (0 \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. scale c x \<in> S))"
from_Gromov_completion_def: shows "from_Gromov_completion = inv to_Gromov_completion"
the_\<theta>_simps_5: shows "cod \<omega>.the_\<theta> = r\<^sub>0"
has_integral_cong: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and i :: "'b" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(f has_integral i) s = (g has_integral i) s"
simps_6: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<mu> = (if \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2 \<in> Y then true\<^sub>n else \<psi>\<^sub>1[Y]\<^sub>\<mu> U\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>)"
sets_distr: fixes M :: "'e measure"   and N :: "'d measure"   and f :: "'e \<Rightarrow> 'd" shows "sets (distr M N f) = sets N"
coeff_sum: fixes p :: "'d \<Rightarrow> 'c poly"   and A :: "'d set"   and i :: "nat" shows "coeff (sum p A) i = (\<Sum>x\<in>A. coeff (p x) i)"
Pxy_nn: fixes x :: "'b \<times> 'c" assumes "x \<in> space (S \<Otimes>\<^sub>M T)" shows "0 \<le> Pxy x"
level_all_eq: fixes p :: "(nat \<times> int) list"   and p' :: "(nat \<times> int) list" assumes "\<And>d. d < length p \<Longrightarrow> lv p d = lv p' d"   and "length p = length p'" shows "level p' = level p"
orient_lit_vars: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L :: "'b literal"   and t :: "'b trm"   and s :: "'b trm"   and p :: "sign" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.orient_lit trm_ord L t s p" shows "vars_of t \<subseteq> vars_of_lit L \<and> vars_of s \<subseteq> vars_of_lit L"
f_def: fixes v :: "(real, 'n) Finite_Cartesian_Product.vec" shows "local.f v = (1 / norm1 (local.B *v v)) *\<^sub>R (local.B *v v)"
abs_update_map_of_hceq: fixes l :: "('a \<times> 'b) list list"   and x :: "'a"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l"   and "x \<noteq> k"   and "bounded_hashcode_nat (length l) x = bounded_hashcode_nat (length l) k" shows "map_of (concat (abs_update k v l)) x = map_of (concat l) x"
homotopic_paths_loop_parts: fixes S :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and a :: "'b" assumes "homotopic_loops S (p +++ reversepath q) (linepath a a)"   and "path q" shows "homotopic_paths S p q"
intro: fixes f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. f (x + y) = f x + f y" shows "additive f"
K_subset: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "QI c_k c w"   and "i \<le> j" shows "QI.K c_k i j \<in> {i..j}"
enab_beta1: shows "\<turnstile> $pc1 = #b \<longrightarrow> Enabled beta1"
lm095: fixes bids :: "'d \<times> 'c set \<Rightarrow> nat"   and pair :: "'d \<times> 'c set" shows "summedBidSecond (real \<circ> bids) pair = real (summedBidSecond bids pair)"
simps: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and S :: "('b, 'c, 'a) formula list \<times> ('b, 'c, 'a) formula list" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.start_proof S = ([S], S)"
case_2: fixes f1 :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b"   and f2 :: "bool \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f3 :: "nat \<Rightarrow> nat \<Rightarrow> nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat) \<Rightarrow> ((nat \<Rightarrow> nat option) \<Rightarrow> bool) \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and f4 :: "'b"   and f5 :: "nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> nat set \<Rightarrow> 'b"   and 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" shows "(case Write x21 x22 x23 x24 x25 x26 x27 of Read x xa xb \<Rightarrow> f1 x xa xb | Write x xa xb xc xd xe xf \<Rightarrow> f2 x xa xb xc xd xe xf | RMW x xa xb xc xd xe xf xg xh \<Rightarrow> f3 x xa xb xc xd xe xf xg xh | Fence \<Rightarrow> f4 | Ghost x xa xb xc \<Rightarrow> f5 x xa xb xc) = f2 x21 x22 x23 x24 x25 x26 x27"
wf_mdecl_def: fixes wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and 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 C :: "char list" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, Ts, T, mb). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> wf_md P C (M, Ts, T, mb)"
distinct_19: shows "cospan_shape.BB \<noteq> cospan_shape.TT"
sympartp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "sympartp r x y \<equiv> r x y \<and> r y x"
vertical_chop_assoc1: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and v3 :: "view"   and v4 :: "view" shows "v=v1--v2 \<and> v2=v3--v4 \<longrightarrow> (\<exists>v'. v=v'--v4 \<and> v'=v1--v3)"
SUP_escape_pos: shows "0 < \<Squnion> range escape_rate"
BinOpRed2: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v\<^sub>1 :: "val"   and bop :: "bop" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e,s,b\<rangle> \<rightarrow> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> e',s',b'\<rangle>"
simps_1: fixes uu :: "value option" shows "value_eq None uu = invalid"
subst_lsst_singleton: fixes l :: "'d strand_label"   and s :: "('e, 'f) stateful_strand_step"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "[(l, s)] \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = [(l, s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)]"
assms_4: shows "p' \<in> sterms \<Gamma> p"
times_vec_def: shows "(*) \<equiv> \<lambda>x y. \<chi>i. x $ i * y $ i"
lunit_ide_simp: fixes f :: "'d" assumes "D.ide f" shows "D'.lunit f = \<r>\<^sub>D[f]"
encode_frame: fixes f :: "recf"   and xs :: "nat list"   and ls :: "nat list" shows "encode_frame (f, xs, ls) = triple_encode (encode f) (Partial_Recursive.list_encode xs) (Partial_Recursive.list_encode ls)"
select_convs_1: fixes Obj :: "'g set"   and Mor :: "'h set"   and Dom :: "'h \<Rightarrow> 'g"   and Cod :: "'h \<Rightarrow> 'g"   and Id :: "'g \<Rightarrow> 'h"   and Comp :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and more :: "'i" shows "obj\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Obj"
simps: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "('c, 'e) resource"   and a2 :: "('d, 'f) resource" shows "rel_resource A B a1 a2 = (\<exists>res1 res2. a1 = res1 \<and> a2 = res2 \<and> rel_fun A (rel_spmf (rel_prod B (rel_resource A B))) (run_resource res1) (run_resource res2))"
message_must_be_delivered_2_trace: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and m :: "'m message"   and i :: "nat"   and q :: "'p"   and p :: "'p" assumes "trace c t c'"   and "m \<in> set (msgs c i)"   and "m \<notin> set (msgs c' i)"   and "channel i = Some (q, p)" shows "\<exists>ev\<in>set t. (\<exists>p q. ev = RecvMarker i p q \<and> m = Marker) \<or> (\<exists>p q s s' m'. ev = Recv i q p s s' m' \<and> m = Msg m')"
type_definition_atMost: shows "type_definition Rep_atMost Abs_atMost {..<LENGTH('c)}"
sum_lepoll_mono: fixes A :: "'a set"   and C :: "'b set"   and B :: "'c set"   and D :: "'d set" assumes "A \<lesssim> C"   and "B \<lesssim> D" shows "A <+> B \<lesssim> C <+> D"
simps_4: fixes e :: "('f, 'g, 'h) exp"   and T :: "ty" shows "\<D> (e instanceof T) = \<D> e"
finite_GSgame_axioms: shows "GaleStewartDeterminedGames.finite_GSgame A N"
set_of_power_mono: fixes a :: "'a"   and A :: "'a interval"   and n :: "nat" assumes "a \<in>\<^sub>i A" shows "a ^ n \<in>\<^sub>i A ^ n"
to_fract_add: fixes x :: "'a"   and y :: "'a" shows "to_fract (x + y) = to_fract x + to_fract y"
abscissa: fixes z :: "ereal" assumes "fds_nth fds_zeta 1 \<noteq> 0"   and "abs_conv_abscissa fds_zeta < z" shows "abs_conv_abscissa (fds_deriv fds_zeta / fds_zeta) < z"
Swap: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "(X, Y) \<in> swapped" shows "(X, Y #[x \<and> y]_zs) \<in> swapped"
Inf_dres_def: fixes S :: "'b dres set" shows "Inf S \<equiv> if S \<subseteq> {dFAILi} then dFAILi else if dSUCCEEDi \<in> S then dSUCCEEDi else if \<exists>a b. a \<noteq> b \<and> dRETURN a \<in> S \<and> dRETURN b \<in> S then dSUCCEEDi else dRETURN (THE x. dRETURN x \<in> S)"
not_g_secure_implies_c_secure: shows "\<not> (g_secure (c_process g_step g_out g_s\<^sub>0) (c_dom g_D) \<longrightarrow> c_secure g_step g_out g_s\<^sub>0 g_I g_D)"
satisfies_bounded_rderiv: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and x :: "'x" assumes "local.wf (#\<^sub>V \<AA>) \<phi>"   and "#\<^sub>V \<AA> = size x" shows "(\<AA> \<Turnstile>\<^sub>b rderiv x \<phi>) = (SNOC x \<AA> \<Turnstile>\<^sub>b \<phi>)"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<FF>\<lparr>ArrMap\<rparr>"
is_acc_def: fixes G :: "('b, 'c) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "gb_graph G" shows "gb_graph.is_acc G r \<equiv> \<forall>A\<in>gbg_F G. \<exists>\<^sub>\<infinity>i. r i \<in> A"
Seq: fixes P\<^sub>1 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and e2' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and c\<^sub>1 :: "com"   and e1 :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and P\<^sub>2 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e2 :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and c\<^sub>2 :: "com"   and P\<^sub>3 :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" assumes "\<turnstile>\<^sub>1 {\<lambda>l s. P\<^sub>1 l s \<and> l x = e2' s} c\<^sub>1 { e1 \<Down> \<lambda>l s. P\<^sub>2 l s \<and> e2 s \<le> l x}"   and "\<turnstile>\<^sub>1 {P\<^sub>2} c\<^sub>2 { e2 \<Down> P\<^sub>3}"   and "x \<notin> support P\<^sub>1"   and "x \<notin> support P\<^sub>2"   and "\<And>l s. P\<^sub>1 l s \<Longrightarrow> e1 s + e2' s \<le> e s" shows "\<turnstile>\<^sub>1 {P\<^sub>1} c\<^sub>1;; c\<^sub>2 { e \<Down> P\<^sub>3}"
deg_q_prop: assumes "fst (factorize_polys qs) = []" shows "\<forall>q\<in>set qs. degree q = 0"
respect_1: shows "\<forall>v\<in>C. \<phi> (f v) = \<phi> v"
typeII_cube_explicit_spec: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" assumes "typeII_twoCube twoC" shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> cubeImage twoC = {(y, x). x \<in> {a..b} \<and> y \<in> {g2 x..g1 x}} \<and> twoC = (\<lambda>(y, x). ((1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b), (1 - x) * a + x * b)) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b} \<and> (\<lambda>x. twoC (0, x)) = (\<lambda>x. (g2 (a + (b - a) * x), a + (b - a) * x)) \<and> (\<lambda>y. twoC (y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b)) \<and> (\<lambda>x. twoC (1, x)) = (\<lambda>x. (g1 (a + (b - a) * x), a + (b - a) * x)) \<and> (\<lambda>y. twoC (y, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
DVD_def: fixes i :: "int"   and c :: "int"   and t :: "MIR.num" shows "DVD i c t = (if i = 0 then eq c t else if c = 0 then Dvd i t else if 0 < c then DVDJ \<bar>i\<bar> c t else DVDJ \<bar>i\<bar> (- c) (Neg t))"