hcju/isabelleps huggingface dataset:
ord_less_eq_trans: fixes a :: "'b Word.word"   and b :: "'b Word.word"   and c :: "'b Word.word" assumes "a <s b"   and "b = c" shows "a <s c"
iASSUME_def: fixes return :: "unit \<Rightarrow> 'b"   and \<Phi> :: "bool" shows "iASSUME return \<Phi> \<equiv> if \<Phi> then return () else bot"
unset_bit_def: 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 bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.unset_bit one push_bit and not n a = and a (not (push_bit n one))"
nprv_eq_eqlI: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set" assumes "t1 = t2"   and "F \<subseteq> fmla"   and "finite F"   and "t1 \<in> trm" shows "nprv F (eql t1 t2)"
pw_eq_on_suffix: fixes S :: "'b set"   and w :: "nat \<Rightarrow> 'b set"   and w' :: "nat \<Rightarrow> 'b set"   and k :: "nat" assumes "pw_eq_on S w w'" shows "pw_eq_on S (suffix k w) (suffix k w')"
simps_12: fixes v :: "tm" shows "qfree (Eq v) = True"
check_int: fixes t :: "MIR.num"   and bs :: "real list" assumes "check_int t" shows "isint t bs"
right_edge_smooth: shows "(\<lambda>y. twoC (y, 1)) piecewise_C1_differentiable_on {0..1}"
cont2monofunE: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "cont f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
set_restriction_in_imp: fixes f :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and A :: "'a set" assumes "set_restriction f"   and "x \<in> f A" shows "x \<in> A"
irreducible_smult_field: fixes c :: "'c"   and p :: "'c poly" shows "irreducible (smult c p) = (c \<noteq> (0::'c) \<and> irreducible p)"
zero_less_dist_iff: fixes x :: "'a"   and y :: "'a" shows "(0 < dist x y) = (x \<noteq> y)"
simps: fixes fmask :: "tcp_flag set"   and c :: "tcp_flag set"   and flags :: "tcp_flag set" shows "match_tcp_flags (TCP_Flags fmask c) flags = (flags \<inter> fmask = c)"
frac_unique_iff: fixes x :: "'a"   and a :: "'a" shows "(frac x = a) = (x - a \<in> \<int> \<and> (0::'a) \<le> a \<and> a < (1::'a))"
rep_inverse: fixes x :: "('b, 'c) mapping" shows "Mapping (rep x) = x"
Q': shows "Q' \<in> {Q'. partition_on (B \<union> C) Q'}"
IH: assumes "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}" shows "setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>) = (\<lambda>p. (fst p, snd p \<cdot>\<^sub>p \<theta>)) ` setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t S"
gr0I: fixes n :: "nat" assumes "n = 0 \<Longrightarrow> False" shows "0 \<sqsubset> n"
induced_arrow_cones_map_2: fixes a' :: "'c"   and f :: "'c" assumes "C.ide a'"   and "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>" shows "induced_arrow a' (D.cones_map f \<chi>) = f"
K_def: shows "K = (if conv_radius (fps_nth f) = \<infinity> then r + 1 else (real_of_ereal (conv_radius (fps_nth f)) + r) / 2)"
returns_result_eq: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and y :: "'z"   and y' :: "'z" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y"   and "h \<turnstile> f \<rightarrow>\<^sub>r y'" shows "y = y'"
is_zero_row_utp_0': fixes A :: "(('c, 'd) vec, 'b) vec" shows "\<forall>m. is_zero_row_upt_k m 0 A"
function_ring_car_eqI: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier (function_ring (carrier R) R)"   and "g \<in> carrier (function_ring (carrier R) R)"   and "\<And>a. a \<in> carrier R \<Longrightarrow> f a = g a" shows "f = g"
tdghm_NTMap_vrange_vifunion: shows "\<R>\<^sub>\<circ> (\<NN>\<lparr>NTMap\<rparr>) \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>). \<Union>\<^sub>\<circ> (VLambda (\<R>\<^sub>\<circ> (\<GG>\<lparr>ObjMap\<rparr>)) (Hom \<BB> a) `\<^sub>\<circ> \<R>\<^sub>\<circ> (\<GG>\<lparr>ObjMap\<rparr>))) `\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>))"
Quotient3_metric_completion: shows "Quotient3 (\<lambda>u v. Cauchy u \<and> Cauchy v \<and> (\<lambda>n. dist (u n) (v n)) \<longlonglongrightarrow> 0) abs_metric_completion rep_metric_completion"
empty_sasp_action_def: shows "empty_sasp_action \<equiv> \<rho>"
extremum: fixes a :: "'a multiset" shows "{#} \<subseteq># a"
coinduct: fixes X :: "('a, 'b, 'c, 'd) converter \<Rightarrow> bool"   and x :: "('a, 'b, 'c, 'd) converter"   and \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>conv. x = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> plossless_gpv \<I>' (run_converter conv x)) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> X xb \<or> plossless_converter \<I> \<I>' xb)" shows "plossless_converter \<I> \<I>' x"
a_coset_add_inv1: fixes M :: "'a set"   and x :: "'a"   and y :: "'a" assumes "M +> x \<oplus> \<ominus> y = M"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "M \<subseteq> carrier R" shows "M +> x = M +> y"
not_in_set_insert: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> list.set xs" shows "List.insert x xs = x # xs"
inconsist_converg_1: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>" shows "inconsist i e = (if \<exists>x<e_length e. \<phi> i x \<down>\<noteq> e_nth e x then Some (LEAST x. x < e_length e \<and> \<phi> i x \<down>\<noteq> e_nth e x) else Some (e_length e))"
AllN_5: shows "insertion (nth_default 0 ((drop z L)[var := B / C])) b = B"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow> g\<guillemotright>" shows "trg g = trg f"
match_4: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b"   and h :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C h g)" shows "partial_magma.arr C (C h (C g f))"
factors_equal: fixes a :: "int vec"   and b :: "int vec"   and c :: "int vec"   and d :: "int vec" assumes "a = b"   and "c = d" shows "a + c = b + d"
clm_vu: shows "1 \<le> |restrict vu (clm ts) c|"
size_Diff_singleton_if: fixes A :: "'a multiset"   and x :: "'a" shows "size (remove1_mset x A) = (if x \<in># A then size A - 1 else size A)"
intros_9: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"   and "G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<rightarrow> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
arr_inv: fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.arr (C.VV.inv f)"
negligible_setdiff: fixes T :: "'a set"   and S :: "'a set" assumes "T \<subseteq> S" shows "negligible (T - S)"
configuration_step_eq: fixes q :: "'b"   and n :: "nat" assumes "q \<noteq> q\<^sub>0" shows "configuration q (Suc n) = \<Union> {configuration q' n |q'. q = \<delta> q' (w n)}"
Dynkin_Basic: fixes A :: "'c set"   and M :: "'c set set"   and \<Omega> :: "'c set" assumes "A \<in> M" shows "A \<in> Dynkin \<Omega> M"
assms_1: shows "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)"
is_processT3_ST_pref: fixes t :: "'a event list"   and P :: "'a process"   and s :: "'a event list" assumes "t \<in> T P"   and "s \<le> t" shows "s \<in> T P"
Q'_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'b option"   and u :: "'a" shows "Prim2.Q' w g r Q \<pi> u \<equiv> (Prim2.Qinter w g r Q \<pi> u)(u := \<infinity>)"
iso_tuple_update_accessor_eq_assist_idI: fixes v' :: "'a"   and f :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "v' = f v" shows "Record.iso_tuple_update_accessor_eq_assist id id v f v' v"
subgroup_generated_subset_carrier_subset: fixes S :: "'a set" assumes "S \<subseteq> carrier B" shows "S \<subseteq> carrier (subgroup_generated B S)"
inj_on_image_subset_iff: fixes f :: "'c \<Rightarrow> 'd"   and C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "inj_on f C"   and "A \<subseteq> C"   and "B \<subseteq> C" shows "(f ` A \<subseteq> f ` B) = (A \<subseteq> B)"
pdevs_apply_filter_pdevs: fixes I :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d pdevs"   and i :: "nat" shows "pdevs_apply (filter_pdevs I x) i = (if I i (pdevs_apply x i) then pdevs_apply x i else (0::'d))"
oexp_Well_order: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "Well_order (r ^o s)"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Eq of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f1"
uncountable_minus_countable: fixes A :: "'f set"   and B :: "'f set" assumes "uncountable A"   and "countable B" shows "uncountable (A - B)"
trans: fixes P :: "'b"   and Q :: "'b"   and Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus"   and R :: "'b" assumes "P \<R>\<mapsto><Rel,Cal> Q"   and "Q \<R>\<mapsto><Rel,Cal> R" shows "P \<R>\<mapsto><Rel,Cal> R"
hyperbola_method_semiring: fixes A :: "real"   and B :: "real"   and x :: "real"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "0 \<le> A"   and "0 \<le> B"   and "A * B = x" shows "sum_upto (dirichlet_prod f g) x + sum_upto f A * sum_upto g B = sum_upto (\<lambda>n. f n * sum_upto g (x / real n)) A + sum_upto (\<lambda>n. sum_upto f (x / real n) * g n) B"
enabled_R: fixes s :: "'state" assumes "s \<in> S" shows "\<exists>r\<in>R. \<exists>sl. eff r s sl"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.arr f = (B.arr (fst f) \<and> B.VV.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (fst (snd f)))"
iso_is_arr: fixes f :: "'c" assumes "C.iso f" shows "C.arr f"
BMax_rec_eq2: fixes R :: "nat \<Rightarrow> bool"   and x :: "nat" shows "BMax_rec R x = Max ({z. z \<le> x \<and> R z} \<union> {0})"
up_zero_top: fixes x :: "Arity\<^sub>\<bottom>" shows "x \<sqsubseteq> up\<cdot>0"
span: shows "span ((+) (- z) ` S) = (+) (- z) ` S"
eq0: fixes i :: "nat"   and j :: "nat" shows "(A * adjoint A) $$ (i, j) = (adjoint A * A) $$ (i, j)"
smooth_compat_compose_chart: fixes c :: "('c, 'd) chart"   and c' :: "('c, 'd) chart"   and p :: "'d \<Rightarrow> 'd"   and p' :: "'d \<Rightarrow> 'd" assumes "k-smooth_compat c c'"   and "diffeomorphism k UNIV UNIV p p'" shows "k-smooth_compat (compose_chart p p' c) c'"
first_edge_first: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and w' :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H_x\<^esub> w"   and "(v, w') \<in> H.edges_of_walk xs" shows "H.first_edge_of_walk xs = (v, w')"
simps_5: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x1 :: "'d" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
I_1: shows "x \<in> SAMEQUOT (any, replicate n False) (map \<pi> ` enc (w, I))"
integral_add: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "integrable M f"   and "integrable M g" shows "LINT x|M. f x + g x = integral\<^sup>L M f + integral\<^sup>L M g"
l_get_child_nodes_axioms: shows "l_get_child_nodes ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs"
tight: shows "tighten_poly_bounds_for_x q 0 l r (sgn (ipoly q r)) = (l', r', sr')"
stable_imp_bit_iff_odd: fixes a :: "'a"   and n :: "nat" assumes "a div (2::'a) = a" shows "bit a n = odd a"
normFaces_distinct: fixes fl :: "face list" assumes "distinct (normFaces fl)" shows "distinct fl"
ts_ord_anti_iff: fixes x :: "'a"   and y :: "'a" shows "(n x \<sqsubseteq> n y) = (t y \<sqsubseteq> t x)"
progressing_ConsD: fixes e :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" assumes "progressing (e # es)" shows "progressing es"
\<MM>_expand: shows "\<MM> = (\<lambda>x. ln x + c + r x)"
mod_nat_eqI: fixes r :: "nat"   and n :: "nat"   and m :: "nat" assumes "r < n"   and "r \<le> m"   and "n dvd m - r" shows "m mod n = r"
reduc_saturated_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "calculus.reduc_saturated Inf Red_I Red_F N = (inference_system.Inf_from Inf (N - Red_F N) \<subseteq> Red_I N)"
is_zero_iarray_eq_iff: fixes A :: "('a, 'b) vec" shows "(A = 0) = is_zero_iarray (vec_to_iarray A)"
Un: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "ring_of_sets \<Omega> M"   and "a \<in> M"   and "b \<in> M" shows "a \<union> b \<in> M"
in_carrier: shows "set (map (\<lambda>u'. u' \<otimes> u) Us) \<subseteq> carrier R"
interior_subset: fixes S :: "'b set" shows "interior S \<subseteq> S"
map_simp_ide: fixes a :: "'d \<times> 'c" assumes "DopxC.ide a" shows "\<Phi>.map a = \<Phi>o a"
update_convs_6: fixes getQ' :: "Literal list \<Rightarrow> Literal list"   and getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getQ_update getQ' \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ' getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr>"
Norm_uminus: fixes N :: "'c quasinorm"   and x :: "'c" shows "Norm N (- x) = Norm N x"
lim: shows "(f \<longlongrightarrow> f0) F"
andfold_MatchExp_normalized_n_primitive: fixes ms :: "'c match_expr list"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> bool" assumes "\<forall>m\<in>set ms. normalized_n_primitive (disc, sel) f m" shows "normalized_n_primitive (disc, sel) f (andfold_MatchExp ms)"
scalar_eq_scaleR: fixes c :: "real"   and x :: "(real, 'b) vec" shows "c *s x = c *\<^sub>R x"
diff_diff_eq: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m"   and "k \<le> n" shows "m - k - (n - k) = m - n"
cindex_pathE_point: fixes a :: "complex"   and b :: "complex" shows "cindex_pathE (linepath a a) b = 0"
basic_trans_rules_21: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y < z" shows "x < z"
bind_nondet_def: fixes bind :: "'h \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and merge :: "'i \<Rightarrow> ('g \<Rightarrow> 'h) \<Rightarrow> 'h"   and m :: "('g, 'h) nondetT"   and f :: "'g \<Rightarrow> ('g, 'h) nondetT" shows "nondetM_base.bind_nondet bind merge m f = NondetT (bind (run_nondet m) (\<lambda>A. merge A (run_nondet \<circ> f)))"
InvariantUniqAfterDecide: fixes M :: "(Literal \<times> bool) list"   and literal :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantUniq M"   and "var literal \<notin> vars (elements M)"   and "M' = M @ [(literal, True)]" shows "InvariantUniq M'"
fpxs_compose_power_mult: fixes f :: "'b fpxs"   and g :: "'b fpxs"   and r :: "rat" shows "fpxs_compose_power (f * g) r = fpxs_compose_power f r * fpxs_compose_power g r"
powsum_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.dioid_one_zero plus times one zero less_eq less" shows "dioid_one_zero.powsum plus times one zero x m n = comm_monoid_add.sum plus zero (power.power one times x) {m..n + m}"
weak_convexD: fixes Pr :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and u :: "real"   and v :: "real" assumes "weak_convex_pref Pr"   and "x \<succeq>[Pr] y"   and "0 < u"   and "0 < v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<succeq>[Pr] y"
pow_mult_distrib: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and n :: "nat" assumes "Group.monoid G"   and "x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y) [^]\<^bsub>G\<^esub> n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> y [^]\<^bsub>G\<^esub> n"
assms: shows "h \<turnstile> find_slot open_flag slotable \<rightarrow>\<^sub>r slot_opt"
Dirichlet_L_Re_ge_1_nonzero_nonprincipal: fixes u :: "complex" assumes "\<chi> \<noteq> \<chi>\<^sub>0"   and "1 \<le> Re u" shows "Dirichlet_L n \<chi> u \<noteq> 0"
zabs_def: fixes i :: "Int.int" shows "\<bar>i\<bar> = (if i < 0 then - i else i)"
is_strongly_dom_component_safe_def: shows "is_strongly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe get_root_node to_tree_order"
simps_4: fixes v :: "'a rexp"   and va :: "'a rexp" shows "pnPlus (Plus v va) Zero = Plus v va"
assms: shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f s0 \<noteq> FAIL"
is_aligned_no_overflow_mask: fixes x :: "'b word"   and n :: "nat" assumes "is_aligned x n" shows "x \<le> x + mask n"
iso_fgraph_def: fixes g\<^sub>1 :: "'a list list"   and g\<^sub>2 :: "'a list list" shows "g\<^sub>1 \<simeq> g\<^sub>2 \<equiv> \<exists>\<phi>. is_iso \<phi> g\<^sub>1 g\<^sub>2"
filtercomap_sup: fixes f :: "'a \<Rightarrow> 'b"   and F1 :: "'b filter"   and F2 :: "'b filter" shows "sup (filtercomap f F1) (filtercomap f F2) \<le> filtercomap f (sup F1 F2)"
gen_Union: fixes I :: "'a set"   and P :: "'a set set" assumes "I \<in> P" shows "gen P I = \<Union> {J \<in> P. J \<subseteq> gen P I}"
semiring_norm_135: shows "(0::'a) < (1::'a)"
Q\<^sub>L_reach: fixes \<Sigma> :: "'b list"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" assumes "finite (reach (set \<Sigma>) \<delta> q\<^sub>0)" shows "Q\<^sub>L \<Sigma> \<delta> q\<^sub>0 = reach (set \<Sigma>) \<delta> q\<^sub>0"
simps_5: fixes ch :: "'choice"   and c1 :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd" shows "brn (Ch ch c1 c2) = 2"
sum_of_index_diff: fixes f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" shows "(\<Sum>i = a..<a + b. f (i - a)) = sum f {..<b}"
infsetsum_scaleR_right: fixes c :: "real"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>\<^sub>ax\<in>A. c *\<^sub>R f x) = c *\<^sub>R infsetsum f A"
lowlink_set_def: fixes s :: "('v, 'a) state_scheme"   and v :: "'v" shows "lowlink_set s v \<equiv> {w \<in> dom (discovered s). v = w \<or> (v, w) \<in> E\<^sup>+ \<and> (w, v) \<in> E\<^sup>+ \<and> (\<exists>p. lowlink_path s v p w)}"
divide_minus_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (uminus a) b = uminus (divide a b)"
lprefix_cons_def: fixes sg :: "'sg"   and sh :: "'sh" shows "local.lprefix_cons sg sh = lprefix (lunstream g sg) (lunstream h sh)"
l9_9_bis: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p" assumes "P Q OS A B" shows "\<not> P Q TS A B"
bi_total_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_total R = ((\<forall>x. \<exists>y. R x y) \<and> (\<forall>y. \<exists>x. R x y))"
insert_def: fixes a :: "'d"   and B :: "'d set" shows "insert a B = {x. x = a} \<union> B"
simp_dense: fixes x :: "'c dense" shows "dense (Rep_dense x)"
mem_vars_mset_subst_mset: fixes y :: "'c"   and x :: "'c"   and t :: "('d, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "y \<in> vars_mset (subst_mset (subst x t) M)"   and "y \<noteq> x"   and "y \<notin> vars_term t" shows "y \<in> vars_mset M"
mult_integer_mod_group: fixes n :: "nat" shows "(\<otimes>\<^bsub>integer_mod_group n\<^esub>) = (\<lambda>x y. (x + y) mod int n)"
mono_Follows_o: fixes h :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'b" assumes "mono h" shows "f Fols g \<subseteq> (h \<circ> f) Fols (h \<circ> g)"
trace_callee_eq_coinduct_strong: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and A :: "'c set"   and callee1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and callee2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf" assumes "S p q"   and "\<And>p q a. \<lbrakk>S p q; a \<in> A\<rbrakk> \<Longrightarrow> bind_spmf p (\<lambda>s. map_spmf fst (callee1 s a)) = bind_spmf q (\<lambda>s. map_spmf fst (callee2 s a))"   and "\<And>p q a res res' b s'' s'. \<lbrakk>S p q; a \<in> A; res \<in> set_spmf p; (b, s'') \<in> set_spmf (callee1 res a); res' \<in> set_spmf q; (b, s') \<in> set_spmf (callee2 res' a)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (bind_spmf p (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (bind_spmf q (\<lambda>s. callee2 s a)) b) \<or> trace_callee_eq callee1 callee2 A (cond_spmf_fst (bind_spmf p (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (bind_spmf q (\<lambda>s. callee2 s a)) b)" shows "trace_callee_eq callee1 callee2 A p q"
f': fixes x :: "real" assumes "x \<in> C" shows "(f has_real_derivative f' x) (at x)"
cop: shows "\<not> comm_monoid_mult_class.coprime b b'"
reflexivity_rule_def: fixes t :: "'d" shows "reflexivity_rule t = (LG {} {0}, LG {(t, 0, 0)} {0})"
strip_all_body_unchanged_imp_strip_all_single_body_unchanged: fixes B :: "Core.term" assumes "strip_all_body B = B" shows "strip_all_single_body B = B"
square_part_def: fixes n :: "'b" shows "square_part n = (if n = (0::'b) then 0::'b else normalize (\<Prod>p\<in>prime_factors n. p ^ (multiplicity p n div 2)))"
const_lt_component_SomeD2: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and cmp :: "'k" assumes "const_lt_component p = Some cmp" shows "component_of_term (lt p) = cmp"
domAllow: shows "dom (Cp (AllowPortFromTo a b po)) \<noteq> {}"
ne1: shows "1 + \<i> * complex_of_real x \<noteq> 0"
swaps_inv: fixes sws :: "nat list"   and xs :: "'b list" shows "set (swaps sws xs) = set xs \<and> length (swaps sws xs) = length xs \<and> distinct (swaps sws xs) = distinct xs"
filterlim_at_top: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "filterlim f at_top F = (\<forall>Z. \<forall>\<^sub>F x in F. Z \<le> f x)"