hcju/isabelleps huggingface dataset:
classes_changed_sym: fixes P :: "'b prog"   and P' :: "'b prog" shows "classes_changed P P' = classes_changed P' P"
ordLeq_finite_Field: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "r \<le>o s"   and "finite (Field s)" shows "finite (Field r)"
andalso_thms_5: fixes y :: "tr" shows "(y andalso y) = y"
obj_fields_marked_locs_subseteq_hs_in_sync_locs: shows "obj_fields_marked_locs \<subseteq> hs_in_sync_locs"
nvalid_to_nvalid_strip: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and n :: "nat"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and F' :: "'g set" assumes "\<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "F' \<subseteq> - F" shows "strip F' \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
kac_formula_nonergodic: fixes A :: "'a set" assumes "A \<in> sets M" shows "\<integral>\<^sup>+ x. of_nat (return_time_function A x) \<partial>M = emeasure M (\<Union>n. (T ^^ n) --` A)"
normalize_quot_aux_1: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "a = fst (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
fps_divide_unit_factor_both': fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g \<le> subdegree f" shows "unit_factor f / unit_factor g = fps_shift (subdegree f - subdegree g) (f / g)"
Polygamma_approx_def: fixes j :: "nat"   and m :: "nat" shows "Polygamma_approx j m = (deriv ^^ j) (\<lambda>x. (x - (1::'a) / (2::'a)) * ln x - x + of_real (ln (2 * pi)) / (2::'a) + stirling_sum 0 m x)"
irreflI: fixes R :: "('d \<times> 'd) set" assumes "\<And>a. (a, a) \<notin> R" shows "irrefl R"
assignNIhighlemma: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and s1 :: "char list \<Rightarrow> val option"   and s2 :: "char list \<Rightarrow> val option"   and V :: "char list"   and s1' :: "char list \<Rightarrow> val option"   and e :: "expr"   and s2' :: "char list \<Rightarrow> val option" assumes "\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2"   and "\<Gamma> V = Some High"   and "s1' = s1(V := \<lbrakk>e\<rbrakk>s1)"   and "s2' = s2(V := \<lbrakk>e\<rbrakk>s2)" shows "\<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'"
utp_theory_lattice_axioms: shows "utp_theory_lattice \<H>"
subst_atm_list_length: fixes As :: "'a list"   and \<sigma> :: "'s" shows "length (As \<cdot>al \<sigma>) = length As"
iso_is_section: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.section f"
fds_nth_zero: shows "fds_nth 0 = (\<lambda>_. 0::'b)"
of_nat_neq_0: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc n) \<noteq> zero"
impl_partition: fixes implem :: "chan \<Rightarrow> msg"   and NI :: "msg set"   and I :: "msg set"   and K :: "msg set" assumes "semivalid_implem implem"   and "NI \<subseteq> payload"   and "I \<subseteq> basic_implem.valid implem" shows "I \<subseteq> semivalid_implem.extractable implem K I \<union> {implem (Confid A B M) |A B M. (A, B) \<in> UNIV - broken K \<and> M \<in> payload} \<union> {implem (Secure A B M) |A B M. (A, B) \<in> UNIV - broken K \<and> M \<in> payload}"
P11_b: fixes a :: "'a"   and b :: "'a" shows "a \<le> b r\<rightarrow> a"
antimono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'b" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
map_of_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "('b \<times> 'd) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
pax3: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> \<circle>\<box>F"
fBall_mono: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and S :: "'c fset" assumes "P \<le> Q" shows "fBall S P \<le> fBall S Q"
d\<mu>_impl_def: fixes fs :: "int vec list" shows "d\<mu>_impl fs = dmu_array (IArray fs) (length fs) (IArray []) 0"
isomorphic_symmetric: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.isomorphic f g" shows "Maps.isomorphic g f"
Z''_2: shows "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto> \<langle>l'', \<Z>\<rangle>"
prefix_sdrop_slength: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "prefix\<cdot>xs\<cdot>ys = TT" shows "xs :@ sdrop\<cdot>(slength\<cdot>xs)\<cdot>ys = ys"
nsumElmTail: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "aGroup A"   and "\<forall>i. f i \<in> carrier A" shows "\<Sigma>\<^sub>e A f Suc n = \<Sigma>\<^sub>e A (\<lambda>i. f (Suc i)) n \<plusminus>\<^bsub>A\<^esub> f 0"
vsv: fixes a :: "V"   and b :: "V"   and A :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" shows "b = c"
spheremap_lemma3: fixes S :: "'a set"   and U :: "'b set" assumes "bounded S"   and "convex S"   and "subspace U"   and "aff_dim S \<le> int (dim U)" obtains T :: "'b set" where "subspace T" and "T \<subseteq> U" and "S \<noteq> {} \<Longrightarrow> aff_dim T = aff_dim S" and "rel_frontier S homeomorphic sphere (0::'b) 1 \<inter> T"
unfold2_o_dtor2: shows "unfold2 (F1map id dtor1 dtor2) (F2map id dtor1 dtor2) \<circ> dtor2 = id"
simE_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "subject"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>[Rel] Q"   and "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel"
gen_ge_dim: fixes A :: "'c set" assumes "finite A"   and "A \<subseteq> carrier V"   and "gen_set A" shows "local.dim \<le> card A"
assms_2: shows "\<forall>n. f n \<in> C"
map_pmf_eq_return_pmf_iff: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d pmf"   and x :: "'c" shows "(map_pmf f p = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = x)"
that: fixes p :: "'a"   and n' :: "'a"   and k :: "nat" assumes "prime p"   and "p dvd n"   and "\<not> p dvd n'"   and "0 < k"   and "n = p ^ k * n'" shows "thesis"
comp_arr_inv': fixes f :: "'a" assumes "A.iso f" shows "f \<cdot>\<^sub>A A.inv f = A.cod f"
onlyGoodsAreAllocatedAuxiliary: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b))) \<in> soldAllocations N (set \<Omega>)"
finite_edges: fixes E :: "'b set set"   and e :: "'b set" assumes "Graph_E E"   and "e \<in> E" shows "finite e"
critical_set_oSuc_lemma: fixes A :: "ordinal set"   and n :: "ordinal" shows "critical_set A (oSuc n) = critical_set A n \<inter> range (oDeriv (OrdinalVeblen.ordering (critical_set A n)))"
pp_sup_p: fixes x :: "'a" shows "- - (x \<squnion> - x) = top"
\<gamma>': shows "\<gamma> = \<delta> @ \<gamma>' \<and> LeftDerivationFix \<beta>' (i + ix - length \<delta>) (derivation_shift D (length \<delta>) 0) (j - length \<delta>) \<gamma>'"
sign_simps_19: fixes aj :: "'a"   and bj :: "'a"   and ci :: "'a" shows "aj * (bj - ci) = aj * bj - aj * ci"
foldr_cong: fixes a :: "'c"   and b :: "'c"   and l :: "'d list"   and k :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a" shows "foldr f l a = foldr g k b"
Nil_is_rev_conv: fixes xs :: "'c list" shows "([] = rev xs) = (xs = [])"
dfa_reach_is_node: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and p :: "nat"   and w :: "bool list list"   and q :: "nat" assumes "wf_dfa A n"   and "dfa_reach A p w q"   and "dfa_is_node A p"   and "list_all (is_alph n) w" shows "dfa_is_node A q"
sigma_sets_eq: fixes M :: "'d measure" shows "sigma_sets (space M) (sets M) = sets M"
proj_def: fixes r :: "('f \<times> 'e) set"   and x :: "'f" shows "Equiv_Relations.proj r x = r `` {x}"
dvd_mult_cancel_right: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.idom times one plus zero minus uminus" shows "dvd.dvd times (times a c) (times b c) = (c = zero \<or> dvd.dvd times a b)"
dcs_def: fixes a :: "'a"   and b :: "'a" shows "(a \<rightarrow>\<^sub>\<S> b) = (\<exists>s t. s \<in> \<S>\<^sub>a \<and> t \<in> \<S>\<^sub>a \<and> t \<le>\<^sub>\<S> \<lambda> (s, a) \<and> t \<circ> b \<noteq> b)"
Nprops_1: shows "{w \<in> space M. c \<noteq> d} \<subseteq> N"
r_neg: fixes x :: "'a set" assumes "x \<in> carrier (R Quot J)" shows "x \<oplus>\<^bsub>R Quot J\<^esub> \<ominus>\<^bsub>R Quot J\<^esub> x = \<zero>\<^bsub>R Quot J\<^esub>"
subsequence_append: fixes w :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" shows "prefix (i + j) w = prefix i w @ w [i \<rightarrow> i + j]"
C_def: shows "C = (\<lambda>x. x mod int p) ` B"
nat_mult_min_right: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m * min n q = min (m * n) (m * q)"
stoch_procs_def: fixes M :: "'d measure"   and N :: "'e measure" shows "stoch_procs M N = {X. \<forall>t. X t \<in> M \<rightarrow>\<^sub>M N}"
awalk_verts_conv: fixes u :: "'a"   and p :: "('a \<times> 'a) list" shows "pawalk_verts u p = (if p = [] then [u] else map fst p @ [snd (last p)])"
envObs_def_raw: fixes a :: "'a" shows "envObs a = (\<lambda>s. (envObsC (es s), ODList.lookup (ps s) a))"
star_of_approx_iff: fixes x :: "'c"   and y :: "'c" shows "(star_of x \<approx> star_of y) = (x = y)"
sub_associative': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (- y \<squnion> - z) = - x \<squnion> - y \<squnion> - z"
AbstrLevels_S11_A72: fixes i :: "AbstrLevelsID" assumes "sS11 \<in> AbstrLevel i" shows "sA72 \<notin> AbstrLevel i"
mod_minus_cong: fixes divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and euclidean_size :: "'k \<Rightarrow> nat"   and uminus :: "'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and a' :: "'k" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a b = modulo a' b" shows "modulo (uminus a) b = modulo (uminus a') b"
simps_4: shows "inPorts' Helper = [plain_ant anyP]"
irreflp_on_map: fixes Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "irreflp_on Q B"   and "h ` A \<subseteq> B" shows "irreflp_on (\<lambda>x y. Q (h x) (h y)) A"
assms: shows "\<exists>a. atom a \<sharp> (h, h $$ a)"
repeatAmountOfQuantifiers_helper_eval: fixes step :: "atom fm \<Rightarrow> atom fm"   and F :: "atom fm"   and xs :: "real list"   and n :: "nat" assumes "\<And>xs F. eval F xs = eval (step F) xs" shows "eval F xs = eval (repeatAmountOfQuantifiers_helper step n F) xs"
srelated_subst: fixes penv :: "(name, pterm) fmap"   and senv :: "(name, sterm) fmap"   and t :: "sterm" assumes "srelated.P_env penv senv" shows "subst (sterm_to_pterm t) penv = sterm_to_pterm (subst t senv)"
mat_cnj_cnj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_cnj (mat_cnj A) = A"
inverse_eq_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(inverse x = one) = (x = one)"
mk_acc_impl_def: fixes Flist :: "'b set list" shows "mk_acc_impl Flist \<equiv> let acc = Map.empty in nfoldli Flist (\<lambda>_. True) (\<lambda>A (i, acc). FOREACH\<^bsup>\<lambda>it acc'. acc' = (\<lambda>v. if v \<in> A - it then Some (insert i (the_default {} (acc v))) else acc v)\<^esup> A (\<lambda>v acc. RETURN (acc(v \<mapsto> insert i (the_default {} (acc v))))) acc \<bind> (\<lambda>acc. RETURN (Suc i, acc))) (0, acc) \<bind> (\<lambda>(uu_, acc). RETURN (\<lambda>x. the_default {} (acc x)))"
filterlim_uminus_at_top: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "filterlim f at_top F = (LIM x F. - f x :> at_bot)"
reds_preserves_lconf: fixes es :: "char list exp list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and es' :: "char list exp list"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,(h, l, sh),b\<rangle> [\<rightarrow>] \<langle>es',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> es [:] Ts"   and "P,h \<turnstile> l (:\<le>) E" shows "P,h' \<turnstile> l' (:\<le>) E"
vec_lambda_unique: fixes f :: "('e, 'd) vec"   and g :: "'d \<Rightarrow> 'e" shows "(\<forall>i. f $ i = g i) = (vec_lambda g = f)"
aforms_of_ivls_def: fixes ls :: "real list"   and us :: "real list" shows "aforms_of_ivls ls us = map2 (\<lambda>i (l, u). ((l + u) / 2, scaleR_pdevs ((u - l) / 2) (coord_pdevs i))) [0..<length ls] (zip ls us)"
c_nc_comp2: shows "1\<^sub>\<pi> \<inter> NC = {}"
typeof_and_norm_unboxed_imp_cast_and_box: fixes x' :: "('dyn, 'ubx1, 'ubx2) unboxed"   and \<tau> :: "Unboxed.type"   and x :: "'dyn" assumes "typeof x' = Some \<tau>"   and "Subx.norm_unboxed x' = x" shows "Subx.cast_and_box \<tau> x' = Some x"
u: shows "isLub UNIV {x. \<exists>n. X n = x} u"
coin: shows "l = STR ''coin'' \<longrightarrow> P (Some 1, [value_plus (r $ 2) (Some (hd i))], r(2 $:= value_plus (r $ 2) (Some (i ! 0))))"
real_divide_square_eq: fixes r :: "real"   and a :: "real" shows "r \<cdot> a / (r \<cdot> r) = a / r"
assms_3: shows "nprv (insert \<phi>2 F) \<psi>"
that: fixes r :: "recf" assumes "recfn 1 r"   and "\<forall>x. eval r [x] = f x" shows "thesis"
f: shows "has_bochner_integral (lebesgue_on {a..b}) f i"
monomial_set_is_GB: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "is_monomial_set G" shows "punit.is_Groebner_basis G"
carrier_chain_group: fixes p :: "nat"   and X :: "'b topology" shows "carrier (chain_group p X) = singular_chain_set p X"
CutAux: fixes H :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and P :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and D :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "H \<rhd> c : A"   and "H = insert P D"   and "G \<rhd> Call : P"   and "G \<subseteq> D" shows "D \<rhd> c : A"
find_None_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(find P xs = None) = (\<nexists>x. x \<in> set xs \<and> P x)"
free_group_def: fixes gens :: "'e set" shows "\<F>\<^bsub>gens\<^esub> \<equiv> \<lparr>carrier = {l \<in> lists (UNIV \<times> gens). canceled l}, mult = \<lambda>x y. Cancelation.normalize (x @ y), one = []\<rparr>"
seqI: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f"   and "J_C.arr g"   and "J_C.dom g = J_C.cod f" shows "J_C.seq g f"
elims: fixes x :: "'b \<times> 'b"   and xa :: "'b"   and y :: "'b" assumes "swp_apply x xa = y" obtains a :: "'b" and b :: "'b" and xb :: "'b" where "x = (a, b)" and "xa = xb" and "y = (if xb = a then b else if xb = b then a else xb)"
gba_to_idx_ext_impl_def: fixes def_size :: "nat"   and eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and bhc :: "nat \<Rightarrow> 'e \<Rightarrow> nat" shows "gba_to_idx_ext_impl def_size eq bhc \<equiv> \<lambda>x. gbg_to_idx_ext_impl eq bhc def_size (\<lambda>xa. \<lparr>igbai_L = gbai_L xa, \<dots> = x xa\<rparr>)"
respects_labels: fixes B :: "'c set"   and \<phi> :: "'a \<Rightarrow> 'c"   and v :: "'a" assumes "folding_g.label_wrt B \<phi>"   and "v \<in> \<Union> X" shows "\<phi> (f v) = \<phi> v"
nth_akiyama_tanigawa_row: fixes i :: "nat"   and u :: "nat"   and l :: "nat"   and n :: "nat" assumes "i < u - l" shows "akiyama_tanigawa_row n l u ! i = akiyama_tanigawa n (i + l)"
that: shows "y \<in> aform_err e (c, sum_list ds)"
cont_right_D: fixes Y :: "nat \<Rightarrow> 'c process"   and S :: "'c process" assumes "chain Y" shows "(S `;` (\<Squnion>i. Y i)) = (\<Squnion>i. S `;` Y i)"
pderivs_simps_3: fixes s :: "'a list"   and r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pderivs s (Plus r1 r2) = (if s = [] then {Plus r1 r2} else pderivs s r1 \<union> pderivs s r2)"
MapOps_lookup_updateD: fixes \<alpha> :: "'c \<Rightarrow> 'b"   and k :: "'c"   and d :: "'b set"   and k' :: "'c"   and ops :: "('d, 'c, 'e) MapOps"   and e :: "'e"   and M :: "'d" assumes "\<alpha> k \<in> d"   and "\<alpha> k' \<in> d"   and "MapOps \<alpha> d ops" shows "lookup ops (update ops k e M) k' = (if \<alpha> k' = \<alpha> k then Some e else lookup ops M k')"
I_lift_dnf_qe_anormal: fixes qe :: "'a list \<Rightarrow> 'a fm"   and f :: "'a fm"   and xs :: "'b list" assumes "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<Longrightarrow> qfree (qe as)"   and "\<And>as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a \<Longrightarrow> normal (qe as)"   and "\<And>xs as. \<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a \<Longrightarrow> is_dnf_qe qe as"   and "normal f" shows "I (lift_dnf_qe qe f) xs = I f xs"
Rep_OrdSum: fixes x :: "('b, 'c) OrdSum" shows "Rep_OrdSum x \<in> OrdSum"
ltl_ltakeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "ltl (ltakeWhile P xs) = (if P (lhd xs) then ltakeWhile P (ltl xs) else LNil)"
rep_eq: fixes x :: "('a, 'c) urel"   and xa :: "('c, 'b) urel" shows "\<lbrakk>x ;; xa\<rbrakk>\<^sub>e = (\<lambda>b. b \<in> {p. \<lbrakk>x\<rbrakk>\<^sub>e p} O {q. \<lbrakk>xa\<rbrakk>\<^sub>e q})"
filter_diff_mset: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset"   and N :: "'a multiset" shows "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
LIM: shows "lim h \<le> lim (the_state \<sigma>')"
N: shows "N = density M (\<lambda>x. ennreal (D x))"
boundOutputDistinct: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and P :: "('d, 'e, 'f) psi"   and \<alpha> :: "'d action"   and P' :: "('d, 'e, 'f) psi" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> P (\<alpha> \<prec> P')" shows "distinct (bn \<alpha>)"
dom_eqI: fixes a :: "'d"   and f :: "'d" assumes "D.ide a"   and "D.seq f a" shows "D.dom f = a"
inject_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Match x1 = Match y1) = (x1 = y1)"
sublinear_intros_10: fixes p :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat" assumes "sublinear (wp p)"   and "healthy (wp p)" shows "sublinear (wp (repeat n p))"
floor_log_eq_powr_iff: fixes x :: "real"   and b :: "real"   and k :: "int" assumes "0 < x"   and "1 < b" shows "(\<lfloor>log b x\<rfloor> = k) = (b powr real_of_int k \<le> x \<and> x < b powr real_of_int (k + 1))"
minus_add_cancel: fixes a :: "'a"   and b :: "'a" shows "- a + (a + b) = b"
phantom_rel_def: fixes A :: "('e \<times> 'f) set" shows "\<langle>A\<rangle>phantom_rel = {(Some x, y) |x y. (x, y) \<in> A} \<union> {uu_. \<exists>x y. uu_ = (None, y) \<and> (x, y) \<in> A}"
next_def: fixes A :: "'c \<times> 'b \<times> 'd" shows "next A = fst (snd A)"
oinvariant_anyact: fixes A :: "('a \<times> 'b, 'c) automaton"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "A \<Turnstile> (act TT, U \<rightarrow>) P" shows "A \<Turnstile> (S, U \<rightarrow>) P"
finished_fin: fixes c :: "('a, 'b, 'c) cmd" assumes "finished c" shows "noWhile c"
assign\<^sub>1: fixes x :: "Eg1.var"   and \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and e :: "aexp"   and t :: "bexp set"   and P :: "bexp set"   and P' :: "bexp set"   and \<S> :: "Eg1.var set \<times> Eg1.var set" assumes "x \<notin> dom \<Gamma>"   and "x \<notin> \<C>"   and "type_aexpr \<Gamma> e t"   and "subtype t P (dma_type x)"   and "P' = restrict_preds_to_vars (assign_post P x e) {v. stable \<S> v}" shows "has_type \<Gamma> \<S> P (x \<leftarrow> e) \<Gamma> \<S> P'"
inner_Basis_0_2: fixes a :: "'a" shows "local.inner_Basis a (0::'a) = 0"
sum_fun_def: fixes f :: "'e \<Rightarrow> 'c \<Rightarrow> 'd"   and I :: "'e set" shows "sum f I = (\<lambda>x. \<Sum>i\<in>I. f i x)"
fup_Aexp_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "fup\<cdot>(CoCallArityAnalysis.Aexp cccExp e)\<cdot>a = fst (fup\<cdot>(cccExp e)\<cdot>a)"
composite_pseudofunctor_axioms: shows "composite_pseudofunctor (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G"
IH: fixes p :: "nat \<Rightarrow> 'a" assumes "\<And>q. \<lbrakk>q \<in> carrier P; deg R q < deg R p\<rbrakk> \<Longrightarrow> Q q"   and "p \<in> carrier P"   and "0 < deg R p" shows "Q p"
higher_lower_decomp: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "punit.higher p v + monomial (lookup p v) v + punit.lower p v = p"