hcju/isabelleps huggingface dataset:
r_into_trancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r" shows "(a, b) \<in> r\<^sup>+"
a_mul_d': fixes x :: "('b \<times> 'b) set" shows "rel_antirange_kleene_algebra.ardual.apd_d x O rel_ar x = {}"
distr_bij_count_space: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "distr (count_space A) (count_space B) f = count_space B"
results_gpv_assert_gpv: fixes \<Gamma> :: "('d, 'e) \<I>"   and b :: "bool" shows "results_gpv \<Gamma> (assert_gpv b) = (if b then {()} else {})"
is_empty_map_entry: fixes k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and m :: "('b, 'c) mapping" shows "is_empty (map_entry k f m) = is_empty m"
mono_hide_T: fixes P :: "'a process"   and Q :: "'a process"   and A :: "'a set" assumes "P \<sqsubseteq>\<^sub>T Q" shows "P \ A \<sqsubseteq>\<^sub>T Q \ A"
pred_Suc_mono: fixes m :: "nat"   and n :: "nat" assumes "m < Suc n"   and "0 < m" shows "m - Suc 0 < n"
prj_simps_2: shows "C.seq \<mu>\<nu>_\<pi>.prj\<^sub>0 \<p>\<^sub>1[\<pi>.leg0 \<cdot> \<mu>\<nu>_\<pi>.prj\<^sub>0, \<rho>.leg1]"
show_law_def: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and x :: "'a" shows "show_law s x = (\<forall>p y z. s p x (y @ z) = s p x y @ z)"
in_proj_in_GSMP: fixes t :: "('fun, 'var) Term.term"   and n :: "'b"   and A :: "('b strand_label \<times> ('fun, 'var) strand_step) list" assumes "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))" shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)"
sets_not_eq: fixes A :: "'c set"   and B :: "'c set" assumes "A \<noteq> B"   and "B \<subseteq> A" shows "\<exists>a\<in>A. a \<notin> B"
strong_low_bisim_mm_def: fixes \<R> :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "conc.strong_low_bisim_mm \<R> \<equiv> sym \<R> \<and> conc.closed_glob_consistent \<R> \<and> (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>C, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> conc.low_mds_eq mds mem\<^sub>1 mem\<^sub>2 \<and> (\<forall>c\<^sub>1' mds' mem\<^sub>1'. \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>\<^sub>C \<longrightarrow> (\<exists>c\<^sub>2' mem\<^sub>2'. \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>\<^sub>C \<and> (\<langle>c\<^sub>1', mds', mem\<^sub>1'\<rangle>\<^sub>C, \<langle>c\<^sub>2', mds', mem\<^sub>2'\<rangle>\<^sub>C) \<in> \<R>)))"
wt_sym_ge\<^sub>h: fixes f :: "'s" shows "\<epsilon>\<^sub>h - \<delta>\<^sub>h * arity_sym\<^sub>h f \<le> wt_sym f"
deg_pm_minus: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" assumes "s adds t" shows "deg_pm (t - s) = deg_pm t - deg_pm s"
cil_set_induct_fix\<alpha>: fixes w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list"   and P :: "'d list \<Rightarrow> ('d \<Rightarrow> 'e set \<times> 'e set) \<Rightarrow> 'd list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2"   and "P [] \<alpha> [] []"   and "\<And>e w' w1' w2. \<lbrakk>w' \<in> w1' \<otimes>\<^bsub>\<alpha>\<^esub> w2; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w2) = {}; P w' \<alpha> w1' w2\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> (e # w1') w2"   and "\<And>e w' w2' w1. \<lbrakk>w' \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2'; fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w1) = {}; P w' \<alpha> w1 w2'\<rbrakk> \<Longrightarrow> P (e # w') \<alpha> w1 (e # w2')" shows "P w \<alpha> w1 w2"
of_int_sum: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> int"   and A :: "'b set" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. ring_1.of_int one plus zero minus (f x)) A"
uninst_trivial_simps_4: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and u :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and phis :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option"   and g :: "'d"   and p :: "'d \<Rightarrow> 'e \<times> 'c \<Rightarrow> 'c list option" shows "CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs u (phis(g := p g)) g = CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs u p g"
GDD_axioms: shows "GDD \<V> \<B> \<G> 1"
slice_kind_Call: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "slice_kind S a = \<lambda>cf. False:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
and_forall_thm: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "((\<forall>x. P x) \<and> (\<forall>x. Q x)) = (\<forall>x. P x \<and> Q x)"
insertI: fixes A :: "'c set"   and a :: "'c" assumes "finite A" shows "finite ({a} \<union> A)"
pathfinish_join: fixes g1 :: "real \<Rightarrow> 'c"   and g2 :: "real \<Rightarrow> 'c" shows "pathfinish (g1 +++ g2) = pathfinish g2"
gcd_dvd2: fixes a :: "'b"   and b :: "'b" shows "normalization_euclidean_semiring_class.gcd a b dvd b"
optimize_matches_def: fixes f :: "'c match_expr \<Rightarrow> 'c match_expr"   and rs :: "'c rule list" shows "optimize_matches f rs = optimize_matches_option (\<lambda>m. if matcheq_matchNone (f m) then None else Some (f m)) rs"
split_min': fixes P :: "'a \<Rightarrow> bool"   and i :: "'a"   and j :: "'a" shows "P (min i j) = ((min i j = i \<longrightarrow> P i) \<and> (min i j = j \<longrightarrow> P j))"
coboundedI1: fixes c :: "'b \<times> 'c"   and a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" assumes "less_eq_prod c a" shows "less_eq_prod c (sup_prod a b)"
that: fixes zs :: "'a list" assumes "xs = zs @ ys"   and "\<And>z. z \<in> set zs \<Longrightarrow> P z"   and "no_leading P ys" shows "thesis"
set_return_pmf: fixes x :: "'a" shows "set_pmf (return_pmf x) = {x}"
less_def: fixes A :: "'b matrix"   and B :: "'b matrix" shows "(A < B) = (A \<le> B \<and> \<not> B \<le> A)"
act_act_inv: fixes g :: "'a"   and b :: "'b" assumes "g \<in> carrier G" shows "b <o g <o inv g = b"
cfs_closed: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "f n \<in> carrier R"
insert_Id_Acts: fixes F :: "'c program" shows "insert Id (Acts F) = Acts F"
\<delta>_def: shows "\<delta> = radiant_of (D - center)"
ir_Skip: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com" assumes "\<And>t t'. Q t t' \<Longrightarrow> \<exists>s'. P t s' \<and> (c', s') \<Rightarrow> t'" shows "ir_hoare P SKIP c' Q"
tranclp_trans: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
unit_ball_vol_odd: fixes n :: "nat" shows "unit_ball_vol (real (2 * n + 1)) = 2 ^ (2 * Suc n) * fact (Suc n) / fact (2 * Suc n) * pi ^ n"
quant_eqI: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" assumes "\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q" shows "x = y"
scaleR_scaleR: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "real"   and b :: "real"   and x :: "'d" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (scaleR b x) = scaleR (a * b) x"
expr_typing_opE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and oper :: "pdf_operator"   and e :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> oper $$ e : t" obtains t :: "pdf_type" where "\<Gamma> \<turnstile> e : t" and "op_type oper t = Some ta__"
l_m: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c" assumes "aGroup A"   and "a \<in> carrier A" shows "-\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>"
norm_power_ineq: fixes x :: "'a"   and n :: "nat" shows "norm (x ^ n) \<le> norm x ^ n"
unroll: shows "(c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) = (nil \<sqinter> c ; c\<^sup>\<star>) ; d"
brec_cond_def: shows "brec_cond \<equiv> \<lambda>(Q, W, rcm, qwit). \<not> fifo.isEmpty W \<and> qwit = None"
fun_left_comm: fixes z :: "('a, 'b) finfun"   and x :: "'a"   and b' :: "'b"   and y :: "'a" shows "finfun_update (finfun_update z x b') y b' = finfun_update (finfun_update z y b') x b'"
sel'E: fixes m :: "'s"   and u :: "'u"   and v :: "'v"   and P :: "'u \<times> 'v \<Rightarrow> bool" assumes "invar m"   and "\<alpha> m u = Some v"   and "P (u, v)" obtains u :: "'u" and v :: "'v" where "sel' m P = Some (u, v)" and "\<alpha> m u = Some v" and "P (u, v)"
card_nth_least: fixes n :: "nat"   and A :: "'b set" assumes "enat n < esize A" shows "card {k \<in> A. k < nth_least A n} = n"
Abs_preal_inverse: fixes y :: "rat set" assumes "y \<in> Collect Dedekind_Real.cut" shows "Rep_preal (Abs_preal y) = y"
iso_class_memb_is_ide: fixes F :: "'a set"   and f :: "'a" assumes "is_iso_class F"   and "f \<in> F" shows "ide f"
Hkm1: shows "H_k m * 1\<^sub>m stm2.d1 = H_k m"
map_comp: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
simps: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" shows "normalization_euclidean_semiring.gcd zero modulo normalize a b = (if b = zero then normalize a else normalization_euclidean_semiring.gcd zero modulo normalize b (modulo a b))"
enat_min: fixes n' :: "nat"   and m :: "enat"   and n :: "nat" assumes "enat n' \<le> m"   and "enat n < m - enat n'" shows "enat n + enat n' < m"
throwET_def: shows "throwET = (\<Lambda> e. ErrorT\<cdot>(return\<cdot>(Err\<cdot>e)))"
div_poly_def: fixes a :: "'b"   and p :: "'b poly" shows "div_poly a p = map_poly (\<lambda>c. c div a) p"
outside_reduces_domain: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "Domain (P outside X) = Domain P - X"
fminus_iff: fixes c :: "'b"   and A :: "'b fset"   and B :: "'b fset" shows "(c |\<in>| A |-| B) = (c |\<in>| A \<and> c |\<notin>| B)"
stutter_equivalent_entails: fixes u :: "nat \<Rightarrow> 'interpretation"   and v :: "nat \<Rightarrow> 'interpretation" assumes "u \<approx> v" shows "u \<Turnstile>\<^sub>p \<phi> = v \<Turnstile>\<^sub>p \<phi>"
assms: shows "m\<^sub>2 \<subseteq>\<^sub>f m\<^sub>1"
comp_char: fixes f :: "'c local.term set"   and g :: "'c local.term set" shows "f \<cdot> g = (if seq f g then mkarr (rep f \<^bold>\<cdot> rep g) else null)"
collapse: fixes kripke :: "('a, 'b) kripke" shows "Kripke (\<pi> kripke) (\<K> kripke) = kripke"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "set (map f xs) = f ` set xs"
not_0_cCons_eq: fixes p :: "'c poly"   and a :: "'c" assumes "p \<noteq> 0" shows "cCons a (Polynomial.coeffs p) = a # Polynomial.coeffs p"
convergence_norm_F: fixes x :: "'s"   and y :: "'s"   and z :: "'a" assumes "convergence_G x y z" shows "summable (\<lambda>n. f x y n * norm z ^ n)"
noVal2_def: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and K :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'d" shows "BD_Security.noVal2 istate step \<phi> f T K v \<equiv> \<forall>s a ou s'. BD_Security.reachNT istate step T s \<and> K s v \<and> step s a = (ou, s') \<and> \<phi> (Trans s a ou s') \<longrightarrow> f (Trans s a ou s') \<noteq> v"
adjunct_Chain: fixes X :: "'a set"   and b :: "'a" assumes "Chain D X"   and "b \<in> carrier D"   and "\<forall>x\<in>X. x \<preceq> b" shows "Chain D (insert b X)"
inf_disj_index: fixes n :: "nat"   and nS :: "nat \<Rightarrow> nat \<Rightarrow> 'a list"   and k :: "nat"   and t :: "nat" assumes "inf_disj n nS"   and "nS k t \<noteq> []"   and "k < n" shows "(SOME i. i < n \<and> nS i t \<noteq> []) = k"
Domain_converse: fixes r :: "('f \<times> 'e) set" shows "Domain (r\<inverse>) = Range r"
permutes_compose: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and q :: "'b \<Rightarrow> 'b" assumes "p permutes S"   and "q permutes S" shows "q \<circ> p permutes S"
homomorphism: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "Applicative.ap (return_spmf f) (return_spmf x) = return_spmf (f x)"
order_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "ordb x y"   and "ordb y z" shows "ordb x z"
hyps_3: fixes p :: "'a \<Rightarrow> 'a" assumes "bij p"   and "\<forall>x. x \<notin> F \<longrightarrow> p x = x" shows "permutation p"
FDG: shows "(f\<^sub>1, f\<^sub>2, g') = splitFace g u v f vs"
var_order_rel_def: shows "var_order_rel \<equiv> lexord less_than_char"
\<chi>''_2: fixes y :: "'a" assumes "y \<in> adjoin G {\<one>} x" shows "\<chi>'' y = \<chi>' y"
simps_5: fixes f1 :: "'g option \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> ('i \<Rightarrow> ('g, 'h, 'i) Resumption.resumption) \<Rightarrow> 'f"   and x1 :: "'g option" shows "(case Resumption.resumption.Done x1 of Resumption.resumption.Done x \<Rightarrow> f1 x | Resumption.resumption.Pause x xa \<Rightarrow> f2 x xa) = f1 x1"
Act_Basic_9: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<A>\<phi> in v]"
RBT_impl_of: fixes t :: "('b, 'c) RBT.rbt" shows "RBT (rbt.impl_of t) = t"
rr: shows "rroots2 [:v, f, 1:] = [u, u']"
ltake_lappend2: fixes xs :: "'c llist"   and n :: "enat"   and ys :: "'c llist" assumes "llength xs \<le> n" shows "ltake n (lappend xs ys) = lappend xs (ltake (n - llength xs) ys)"
numeral_code_1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.numeral plus" shows "numeral.numeral one plus num.One = one"
carrier_free_Abelian_group_iff: fixes x :: "'b \<Rightarrow>\<^sub>0 int"   and S :: "'b set" shows "(x \<in> carrier (free_Abelian_group S)) = (Poly_Mapping.keys x \<subseteq> S)"
order_mult: fixes p :: "'b poly"   and q :: "'b poly"   and a :: "'b" assumes "p * q \<noteq> 0" shows "order a (p * q) = order a p + order a q"
strict_mono_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less" shows "order.strict_mono less f = (\<forall>x y. less x y \<longrightarrow> f x < f y)"
has_integral_neg: fixes f :: "'a \<Rightarrow> 'b"   and k :: "'b"   and S :: "'a set" assumes "(f has_integral k) S" shows "((\<lambda>x. - f x) has_integral - k) S"
If_cases: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite A" shows "(\<Prod>x\<in>A. if P x then h x else g x) = prod h (A \<inter> {x. P x}) * prod g (A \<inter> - {x. P x})"
omega_dom: fixes x :: "'a"   and y :: "'a" shows "d (\<Omega> x y) = \<Omega> x y"
h1: shows "all_list_constr (get_subsets (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (length (qs1 @ qs2))"
fref_def: fixes P :: "'e \<Rightarrow> bool"   and R :: "('b \<times> 'e) set"   and S :: "('d \<times> 'f) set" shows "[P]\<^sub>f R \<rightarrow> S \<equiv> {(f, g). \<forall>x y. P y \<and> (x, y) \<in> R \<longrightarrow> (f x, g y) \<in> S}"
1: shows "((\<lambda>w. f z / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
bit_list_induct: fixes P :: "bit list \<Rightarrow> bool"   and w :: "bit list" assumes "P []"   and "\<And>bs. P bs \<Longrightarrow> P (\<zero> # bs)"   and "\<And>bs. P bs \<Longrightarrow> P (\<one> # bs)" shows "P w"
Abs_poly_mapping: shows "Abs_poly_mapping (\<lambda>k. 0::'c) = 0"
mset_factorsD_2: fixes F :: "'a multiset"   and p :: "'a" assumes "mset_factors F p" shows "F \<noteq> {#}"
sel_2: fixes x2 :: "'a" shows "projr (Inr x2) = x2"
abs_eq: fixes x :: "int extended \<times> int extended" shows "below (abs_ivl x) = abs_ivl (below_rep x)"
strip_Step: fixes f :: "char list \<Rightarrow> aexp \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "bexp \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b"   and C :: "'b acom" shows "strip (Step f g S C) = strip C"
isFDERIV_imp_isFDERIV_FDERIV_n: fixes fas :: "floatarith list"   and ds :: "floatarith list"   and N :: "nat"   and xs :: "nat list"   and vs :: "real list"   and n :: "nat" assumes "length fas = length ds"   and "isFDERIV N xs fas vs"   and "isFDERIV N xs ds vs" shows "isFDERIV N xs (FDERIV_n_floatariths fas xs ds n) vs"
Iod_sub_sub: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T"   and "T \<subseteq> carrier D" shows "Iod (Iod D T) S = Iod D S"
conj_residuated: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "residuated (conjugate f)"
leg_simps_3: shows "C.arr Dom\<mu>_Dom\<nu>.leg1"
assms_2: shows "ws t = \<lfloor>w\<rfloor>"
lower_split_interval1: fixes X :: "'a interval"   and m :: "'a" shows "lower (fst (split_interval X m)) = min (lower X) m"
llist_all2_lmap2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and g :: "'e \<Rightarrow> 'd"   and ys :: "'e llist" shows "llist_all2 P xs (lmap g ys) = llist_all2 (\<lambda>x y. P x (g y)) xs ys"
n_distrib2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "n x \<oplus> n y \<cdot> n z = (n x \<oplus> n y) \<cdot> (n x \<oplus> n z)"
restrict_def: fixes G :: "('a, 'b) labeled_graph" shows "restrict G = LG {(l, v1, v2). (l, v1, v2) \<in> edges G \<and> v1 \<in> vertices G \<and> v2 \<in> vertices G} (vertices G)"
pref_profile_wf_def: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" shows "pref_profile_wf agents alts R \<equiv> (agents \<noteq> {} \<and> alts \<noteq> {}) \<and> (\<forall>i. i \<in> agents \<longrightarrow> finite_total_preorder_on alts (R i)) \<and> (\<forall>i x y. i \<notin> agents \<longrightarrow> \<not> R i x y)"
compl_structD3: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "compl_struct compl"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "0 \<notin> fst ` set (fst (compl gs bs (ps -- sps) sps data))"
intro: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "po_on P A"   and "wfp_on P A" shows "minimal_element P A"
Sup_atMost: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 bot :: "'c"   and top :: "'c"   and y :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atMost less_eq y) = y"
generalizes_atm_def: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'a"   and B :: "'a" shows "substitution_ops.generalizes_atm subst_atm A B = (\<exists>\<sigma>. subst_atm A \<sigma> = B)"
Let: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and L :: "Vars.var list"   and body :: "exp"   and \<Theta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "atom ` domA \<Delta> \<sharp>* (\<Gamma>, L)"   and "\<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z" shows "\<Gamma> : Terms.Let \<Delta> body \<Down>\<^bsub>L\<^esub> \<Theta> : z"
eucl_rel_poly_smult_left: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and a :: "'b" assumes "eucl_rel_poly x y (q, r)" shows "eucl_rel_poly (smult a x) y (smult a q, smult a r)"
nonneg: fixes x :: "'alt" shows "0 \<le> pmf p x + \<epsilon> * (pmf q x - pmf p x)"
b_e_type_set_local_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Set_local i" shows "\<exists>t. ts = ts' @ [t] \<and> local \<C> ! i = t"
nths_weave: fixes xs :: "'m list"   and A :: "nat set"   and ys :: "'m list" assumes "length xs = card {a \<in> A. a < length xs + length ys}"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "nths (weave A xs ys) A = xs \<and> nths (weave A xs ys) (- A) = ys"
assms: fixes C :: "'a tensor" assumes "C \<in> set CS" shows "dims C = ds"
prime_factorization_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "(prime_factorization x = prime_factorization y) = (normalize x = normalize y)"
hypreal_of_real_le_add_Infininitesimal_cancel: fixes x :: "real"   and u :: "real star"   and y :: "real"   and v :: "real star" assumes "hypreal_of_real x + u \<le> hypreal_of_real y + v"   and "u \<in> Infinitesimal"   and "v \<in> Infinitesimal" shows "hypreal_of_real x \<le> hypreal_of_real y"
bind_mono1: fixes f :: "'c \<Rightarrow> 'd nres" shows "mono (\<lambda>M. M \<bind> f)"
countable_set_pmf: fixes p :: "'b pmf" shows "countable (set_pmf p)"
takeWhile_eq_all_conv: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "(takeWhile P xs = xs) = (\<forall>x\<in>set xs. P x)"
simps_1: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "send\<langle>t\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = send\<langle>t \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t"
uint_test_bit_def: fixes x :: "uint"   and n :: "integer" shows "uint_test_bit x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined bit x n else bit x (nat_of_integer n))"