hcju/isabelleps huggingface dataset:
ring_isom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "ring_isom hom \<equiv> inj_ring_hom hom \<and> surjective hom"
mult_comm: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" shows "a \<otimes> b = b \<otimes> a"
Fourier_sum_offset_Dirichlet_kernel_half: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and n :: "nat"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) - l = (LINT x|lebesgue_on {0..pi}. Dirichlet_kernel n x * (f (t + x) + f (t - x) - 2 * l)) / pi"
f_ind: fixes ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "rec_ci f = (ap, arity, fp)" shows "{\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec f (map (\<lambda>g. rec_exec g xs) gs) # 0 \<up> (fp - Suc arity) @ anything}"
erase_mat_def: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and i :: "'f"   and j :: "'e" shows "erase_mat A i j = (\<chi>i' j'. if i' = i \<or> j' = j then 0::'d else A $h i' $h j')"
inf_bot_right: fixes x :: "'e multiset" shows "x \<inter># {#} = {#}"
phiArg_trancl_same_var: fixes g :: "'g"   and \<phi> :: "'val"   and n :: "'val" assumes "(phiArg g)\<^sup>+\<^sup>+ \<phi> n" shows "var g \<phi> = var g n"
bounded_diff: fixes S :: "'a set"   and T :: "'a set" assumes "bounded S" shows "bounded (S - T)"
lm117: fixes P :: "'b \<Rightarrow> bool"   and l :: "'b list" assumes "P -` {True} \<inter> set l \<noteq> {}" shows "\<exists>n\<in>{0..<length l}. P (l ! n)"
rPar2: fixes \<Psi>\<^sub>P :: "'b"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and A\<^sub>Q :: "name list"   and \<Psi>\<^sub>Q :: "'b" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> M\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>P"   and "distinct A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P\<^sub>Q = A\<^sub>P @ A\<^sub>Q"   and "\<Psi>\<^sub>P\<^sub>Q = \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q" shows "Prop (P \<parallel> Q')"
assms_2: shows "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>"
disj_le: fixes x :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "x \<le> a \<or> x \<le> b \<or> x \<le> c"   and "0 \<le> a"   and "0 \<le> b"   and "0 \<le> c" shows "x \<le> a + b + c"
append_one_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "prefix xs ys"   and "length xs < length ys" shows "prefix (xs @ [ys ! length xs]) ys"
M5exist_var: fixes x :: "'b"   and y :: "'b"   and z :: "'b"   and w :: "'b" assumes "x \<parallel> y"   and "y \<parallel> z"   and "z \<parallel> w" shows "\<exists>t. x \<parallel> t \<and> t \<parallel> w"
arr_Rel_converse_Rel_id_Rel: fixes c :: "V" assumes "c \<in>\<^sub>\<circ> Vset \<alpha>" shows "arr_Rel \<alpha> ((id_Rel c)\<inverse>\<^sub>R\<^sub>e\<^sub>l)"
snd_echelon_form_of_upt_k_det_eq: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and n :: "'e"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "snd (echelon_form_of_upt_k_det bezout (n, A) k) = echelon_form_of_upt_k A k bezout"
axioms_1: fixes x :: "'d" shows "[] \<bullet> x = x"
fct_gen_subalgebra_sets_mem: fixes B :: "'b set"   and N :: "'b measure"   and X :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "B \<in> sets N" shows "X -` B \<inter> space M \<in> sets (fct_gen_subalgebra M N X)"
assms_1: shows "pps_to_list S ! i \<prec>\<^sub>t pps_to_list S ! j"
set_mset_add_mset_insert: fixes a :: "'b"   and A :: "'b multiset" shows "set_mset (a, A) = a \<triangleright> set_mset A"
mod_div_mult_eq: 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 a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times (divide a b) b) = a"
bij_betw_id_iff: fixes A :: "'a set"   and B :: "'a set" shows "bij_betw id A B = (A = B)"
mHom_add: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and m :: "'a"   and n :: "'a" assumes "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "f (m \<plusminus> n) = f m \<plusminus>\<^bsub>N\<^esub> f n"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "suffix b a"   and "suffix c b" shows "suffix c a"
lcm_1_right: fixes a :: "'a" shows "lcm a (1::'a) = normalize a"
ih2: assumes "\<Gamma> \<turnstile> t \<Leftrightarrow> v : T\<^sub>1" shows "\<Gamma> \<turnstile> s \<Leftrightarrow> v : T\<^sub>1"
simps_6: fixes uw :: "pdf_dist"   and e :: "expr" shows "free_vars (Random uw e) = free_vars e"
mono_def: fixes f :: "'a \<Rightarrow> 'c" shows "order.mono (\<lambda>x y. y \<le> x) f = (\<forall>x y. y \<le> x \<longrightarrow> f x \<le> f y)"
closed_orbit_global_existence: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.closed_orbit f X x" shows "auto_ll_on_open.existence_ivl0 f X x = UNIV"
simple_function_eq_measurable: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" shows "simple_function M f = (finite (f ` space M) \<and> f \<in> M \<rightarrow>\<^sub>M count_space UNIV)"
ev: fixes f :: "'a fds" assumes "fds_abs_converges f s"   and "fds_nth f (Suc 0) = (0::'a)" shows "eval_fds (fds_exp f) s = exp (eval_fds f s)"
axioms_3: 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\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
q_def: shows "q = [:a, - (1::'a):] * p"
stc_eq_approx: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> HFinite"   and "y \<in> HFinite"   and "stc x = stc y" shows "x \<approx> y"
assms: shows "x + y \<notin> \<int>\<^sub>\<le>\<^sub>0"
assoc_in_hom: fixes a :: "'c local.term set"   and b :: "'c local.term set"   and c :: "'c local.term set" assumes "ide a"   and "ide b"   and "ide c" shows "\<guillemotleft>\<a>[a, b, c] : (a \<otimes> b) \<otimes> c \<rightarrow> a \<otimes> b \<otimes> c\<guillemotright>"
INF_Lim: fixes X :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "decseq X"   and "X \<longlonglongrightarrow> l" shows "Inf (range X) = l"
add_mset_diff_bothsides: fixes a :: "'a"   and M :: "'a multiset"   and A :: "'a multiset" shows "(a, M) - (a, A) = M - A"
b: shows "b = (f ^ n) \<langle>$\<rangle> a"
subgroup_self: shows "subgroup (carrier (relative_homology_group (int n) (nsphere n) (equator n))) (relative_homology_group (int n) (nsphere n) (equator n))"
rk_\<alpha>\<beta>\<gamma>_special_case_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {A', P} = 1" shows "rk {\<alpha>, \<beta>, \<gamma>} \<le> 2"
len_space_right: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "left (ext v) \<le> right (space ts v c) \<longrightarrow> right (len v ts c) \<le> right (space ts v c)"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b list \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
psubsetI: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subseteq> B"   and "A \<noteq> B" shows "A \<subset> B"
cong_exp_iff_simps_9: 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus m) (numeral.numeral one plus q) = modulo (numeral.numeral one plus n) (numeral.numeral one plus q))"
vnm_utility_imp_reational_prefs: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real" assumes "vNM_utility outcomes relation u" shows "rational_preference (\<P> outcomes) relation"
mkArr_restrict_eq': fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B (restrict F A))" shows "S'.mkArr A B (restrict F A) = S'.mkArr A B F"
coprime_hom: fixes x :: "'b"   and y' :: "'a" shows "coprime (inv hom x) y' = coprime x (inv (inv hom) y')"
groups_of_size_alt_def_count: fixes g :: "nat" shows "groups_of_size g = count (image_mset card (mset_set \<G>)) g"
finite_cs_2: shows "\<forall>C\<in>Cs. finite C"
simps_3: fixes n :: "nat" shows "Stirling (Suc n) 0 = 0"
top_times_top: shows "top_pres ((\<sqinter>) \<top>)"
two_comparisons_into_case_order_12: fixes x :: "'b"   and y :: "'b"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if x = y then P else if c1.le x y then Q else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
ba_lim_leq: shows "lim (\<lambda>n. real_of_int (b (n + 1) - b n) / real_of_int (a n)) \<le> 0"
ide_map\<^sub>0_obj: fixes a :: "'a" assumes "C.obj a" shows "D.ide (\<rho>\<^sub>0 a)"
exp_2dist_1_equal: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "exp_2dist a b = 1" shows "a = b"
smap_strict_scons: fixes f :: "'b \<rightarrow> 'a"   and x :: "'b"   and xs :: "[:'b:]" assumes "f\<cdot>\<bottom> = \<bottom>" shows "smap\<cdot>f\<cdot>(x :# xs) = f\<cdot>x :# smap\<cdot>f\<cdot>xs"
lscan_simps_2: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c llist"   and a :: "'b" shows "lscan f (x % xs) a = a % lscan f xs (f x a)"
case_2: fixes f1 :: "fdecl \<Rightarrow> 'a"   and f2 :: "mdecl \<Rightarrow> 'a"   and x2 :: "mdecl" shows "(case mdecl x2 of fdecl x \<Rightarrow> f1 x | mdecl x \<Rightarrow> f2 x) = f2 x2"
combine_eq_foldr: fixes Ks :: "'a list"   and Us :: "'a list" shows "combine Ks Us = foldr (\<lambda>(k, u). (\<oplus>) (k \<otimes> u)) (zip Ks Us) \<zero>"
drop_bit_int_def: fixes n :: "nat"   and k :: "int" shows "drop_bit n k = k div 2 ^ n"
set_intros_7: fixes ym :: "('d, 'e) Term.term"   and x24 :: "('d, 'e) Term.term list"   and yo :: "'e"   and x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x23 :: "('d, 'e) ctxt" assumes "ym \<in> set x24"   and "yo \<in> vars_term ym" shows "yo \<in> vars_ctxt (More x21 x22 x23 x24)"
inverse_le_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse x) one = (less_eq x zero \<or> less_eq one x)"
assms_2: shows "\<gamma>2 piecewise_C1_differentiable_on {0..1}"
diff_fun_space_def: fixes k :: "enat" shows "manifold_eucl.diff_fun_space k = {f. diff_fun k charts_eucl f \<and> extensional0 manifold_eucl.dest.carrier f}"
mahler_measure_def: fixes p :: "int poly" shows "mahler_measure p = mahler_measure_poly (of_int_poly p)"
ideD_2: fixes a :: "(bool, bool list) concrete_category.arr" assumes "ide a" shows "local.dom a = a"
xcarr: shows "lincomb l1 X \<in> carrier_vec n"
erHomTr1: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g"   and a :: "'a"   and b :: "'a" assumes "PolynRg A B Y"   and "h \<in> rHom S B"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<noteq> \<zero>"   and "b \<noteq> \<zero>"   and "a \<plusminus> b \<noteq> \<zero>"   and "deg_n R S X a = deg_n R S X b" shows "erH R S X A B Y h (a \<plusminus> b) = erH R S X A B Y h a \<plusminus>\<^bsub>A\<^esub> erH R S X A B Y h b"
dvd_triv_left: fixes a :: "'a"   and b :: "'a" shows "a dvd a * b"
SIod_Torder: fixes D :: "'c Order"   and X :: "'c set" assumes "Torder D"   and "X \<subseteq> carrier D" shows "Torder (SIod D X)"
add_cancr: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "D x z"   and "D y z"   and "x \<oplus> z = y \<oplus> z" shows "x = y"
that_1: assumes "h1 \<in> o[F](h2)" shows "thesis"
raWrite_def: fixes arr :: "nat \<times> (nat \<Rightarrow> 'b)"   and i :: "nat"   and "value" :: "'b"   and fillValue :: "'b" shows "raWrite arr i value fillValue \<equiv> let len = fst arr; f = snd arr in if i < len then (len, f(i := value)) else (i + 1, fillAndUpdate len f i value fillValue)"
execute: fixes p :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme"   and a :: "('e \<times> 'f) \<times> 'd" assumes "p \<in> nodes A"   and "snd a \<in> succ A (fst a) p" shows "snd a \<in> nodes A"
sup_le1: fixes a :: "'a"   and b :: "'a" shows "a \<le> pseudo_hoop_algebra.sup1 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) a b"
Suc: fixes n :: "nat"   and b :: "bool"   and w :: "int" shows "bin_sc (Suc n) b w = of_bool (odd w) + 2 * bin_sc n b (w div 2)"
B: shows "gauss_jordan A (0\<^sub>m n 0) = (B, C)"
thetaSCh_def: fixes ch :: "'c"   and c1 :: "('a, 'b, 'c) cmd"   and c2 :: "('a, 'b, 'c) cmd"   and d1 :: "('d, 'e, 'c) cmd"   and d2 :: "('d, 'e, 'c) cmd" shows "thetaSCh ch c1 c2 d1 d2 \<equiv> {(Ch ch c1 c2, Ch ch d1 d2)}"
Dijkstra_Impl_axioms_def: fixes G_invar :: "'a \<Rightarrow> bool"   and g :: "'a" shows "Dijkstra_Impl_axioms G_invar g \<equiv> G_invar g"
cfcomp2: fixes f :: "'d \<rightarrow> 'b"   and g :: "'c \<rightarrow> 'd"   and x :: "'c" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
concurrent_set_def: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "happens_before hb_weak hb" shows "happens_before.concurrent_set hb x xs \<equiv> \<forall>y\<in>set xs. happens_before.concurrent hb x y"
wellformed: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>C" shows "b \<subseteq> \<V>"
E_finite_sum_fun: fixes X :: "'c pmf"   and f :: "'c \<Rightarrow> real" assumes "finite (set_pmf X)" shows "E (map_pmf f X) = (\<Sum>x\<in>set_pmf X. pmf X x * f x)"
hinsert_1: assumes "x \<noteq> 0"   and "\<And>i. i \<^bold>\<in> x \<Longrightarrow> Ord i" shows "\<Sqinter>x \<^bold>\<in> x"
apath_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "H.apath u p v \<equiv> H.awalk u p v \<and> distinct (pawalk_verts u p)"
fps_compose_divide_distrib: fixes g :: "'a fps"   and f :: "'a fps"   and h :: "'a fps" assumes "g dvd f"   and "h $ 0 = (0::'a)"   and "g oo h \<noteq> 0" shows "f / g oo h = (f oo h) / (g oo h)"
open_codomain: fixes c :: "('b, 'a) chart" shows "open (codomain c)"
lift2_def: fixes h :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and inp' :: "'f \<Rightarrow> 'h option"   and inp :: "'f \<Rightarrow> 'i option" shows "lift2 h inp' inp \<equiv> \<lambda>i. case (inp' i, inp i) of (None, b) \<Rightarrow> None | (Some v', None) \<Rightarrow> None | (Some v', Some v) \<Rightarrow> Some (h v' v)"
select_sets_def: fixes M :: "'b measure measure" shows "select_sets M = (SOME N. sets M = sets (subprob_algebra N))"
absorb_iff1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b) = (a \<inter># b = a)"
assms_3: fixes x :: "\<nu>" shows "[\<phi> (x\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) in v]"
ev_small_real: fixes rho :: "real" assumes "eigen_value (erase_mat A i i) rho" shows "\<bar>rho\<bar> < sr"
moebius_inv_cmat_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "moebius_inv_cmat M = mat_inv M"
disc_martingale_charact: fixes M :: "'c measure"   and X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and F :: "nat \<Rightarrow> 'c measure" assumes "\<forall>n. integrable M (X n)"   and "Filtration.filtration M F"   and "\<forall>m. sigma_finite_subalgebra M (F m)"   and "\<forall>m. X m \<in> borel_measurable (F m)"   and "\<forall>n. AEeq M (real_cond_exp M (F n) (X (Suc n))) (X n)" shows "martingale M F X"
is_And_def: fixes formula :: "formula" shows "is_And formula = (\<exists>x81 x82. formula = And x81 x82)"
simps: fixes a1 :: "(char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset"   and a2 :: "(char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset" shows "add_poly_p a1 a2 = ((\<exists>p x q r. a1 = (p, add_mset x q, r) \<and> a2 = (p, q, add_mset x r)) \<or> (\<exists>x p q r. a1 = (add_mset x p, q, r) \<and> a2 = (p, q, add_mset x r)) \<or> (\<exists>x n p q m r. a1 = (add_mset (x, n) p, q, add_mset (x, m) r) \<and> a2 = (p, q, add_mset (x, n + m) r)) \<or> (\<exists>p x n q m r. a1 = (p, add_mset (x, n) q, add_mset (x, m) r) \<and> a2 = (p, q, add_mset (x, n + m) r)) \<or> (\<exists>p q x r. a1 = (p, q, add_mset (x, 0) r) \<and> a2 = (p, q, r)))"
jumpNestingOk_expr_list_simp: fixes jmps :: "jump set"   and es :: "expr list" shows "jumpNestingOk jmps (In3 es) = True"
SUP_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
subspace_diff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
\<Z>_csgrp_def: fixes \<alpha> :: "V"   and \<SS> :: "V" shows "\<Z>_csgrp \<alpha> \<SS> \<equiv> \<Z>_sgrp \<alpha> \<SS> \<and> \<Z>_csgrp_axioms \<SS>"
weakBisimBangPresAux: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q"   and "guarded P"   and "guarded Q" shows "\<Psi> \<rhd> R \<parallel> !P \<approx> R \<parallel> !Q"
simps_2: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and as :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted_descending less_eq (a # as) = ((\<forall>x\<in>set as. less_eq x a) \<and> linorder.sorted_descending less_eq as)"
coincidence_ode: fixes ODE :: "('d, 'e) ODE"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp"   and J :: "('d, 'f, 'e) interp" assumes "osafe ODE"   and "Vagree \<nu> \<nu>' (Inl ` FVO ODE)"   and "Iagree I J ({Inl x |x. Inl x \<in> SIGO ODE} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE})" shows "ODE_sem I ODE (fst \<nu>) = ODE_sem J ODE (fst \<nu>')"
Ch_range: shows "\<forall>h X. Ch h X \<subseteq> {x \<in> X. Xh x = h}"
P_path: shows "v0 \<leadsto>P\<leadsto> v1"
all: shows "\<forall>r\<in>lotteries_on outcomes. mix_pmf \<alpha> x r \<succeq>[\<R>] mix_pmf \<alpha> y r"
strip_guards_simps_5: fixes F :: "'f set"   and b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "strip_guards F (Cond b c\<^sub>1 c\<^sub>2) = Cond b (strip_guards F c\<^sub>1) (strip_guards F c\<^sub>2)"
function_ring_defs_3: shows "(\<oplus>\<^bsub>F\<^esub>) = function_add S R"
genPrefix_append_both: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "refl r"   and "(xs, ys) \<in> genPrefix r"   and "length xs = length ys" shows "(xs @ zs, ys @ zs) \<in> genPrefix r"
binrelchain_snoc: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and y :: "'b" assumes "binrelchain P (xs @ [x])"   and "P x y" shows "binrelchain P (xs @ [x, y])"
finite_same_card_bij: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>h. bij_betw h A B"
B_SMP_repr: shows "finite_SMP_representation arity Ana \<Gamma> B"
simps_1: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> Void) = (P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Void\<rfloor>)"
le_llist_conv_lprefix: shows "(\<le>) = lprefix"
pendingSeqs_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and s :: "('e, 'f, 'g) SLin_state" assumes "SLin \<delta> \<gamma> bot" shows "SLin.pendingSeqs s \<equiv> {rs. set rs \<subseteq> SLin.pendingReqs s}"
ovalid_property_def: fixes P :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option" shows "ovalid_property P x = (\<lambda>s f. \<forall>r. Some r = x s f \<longrightarrow> P r s)"
sup_top_left: fixes x :: "'a" shows "1 + x = 1"
1: shows "1 \<le> \<parallel>fs ! 0\<parallel>\<^sup>2"
apply_operations_def: fixes es :: "('a \<times> ('a, 'b) operation) event list" shows "apply_operations es \<equiv> hb.apply_operations (node_deliver_messages es) (\<lambda>x. ({}, {}))"
ground_sound_\<Gamma>_def: shows "ground_sound_\<Gamma> = {Infer CC D E |CC D E. \<forall>I. I \<Turnstile>m CC \<longrightarrow> I \<Turnstile> D \<longrightarrow> I \<Turnstile> E}"
compat_equiv_prec: fixes C1 :: "'f"   and D1 :: "'f"   and C2 :: "'f"   and D2 :: "'f" assumes "C1 \<doteq> D1"   and "C2 \<doteq> D2"   and "C1 \<prec>\<cdot> C2" shows "D1 \<prec>\<cdot> D2"
plossless_exec_gpv: fixes gpv :: "('e, 'a, 'b) gpv"   and s :: "'s" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (callee s x)"   and "I s" shows "lossless_spmf (exec_gpv callee gpv s)"