hcju/isabelleps huggingface dataset:
FAcc: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and v :: "val" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l, sh)\<rangle>"   and "h a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,NonStatic:t in D"   and "fs (F, D) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"
eventually_at_right_less: fixes x :: "'c" shows "eventually ((<) x) (at_right x)"
zero_aux: shows "liouville_lambda 0 = (0::'b)"
mult_col_div_row_carrier_2: fixes a :: "'b"   and k :: "nat"   and A :: "'b mat" shows "dim_col (mult_col_div_row a k A) = dim_col A"
run_owner_iv_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_owner_iv n s)"
CM_def: shows "CM \<equiv> M \<and> CM_axioms"
mult_when: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b"   and P :: "bool" assumes "class.mult_zero times zero" shows "times a (zero.when zero b P) = zero.when zero (times a b) P"
simps_1: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "lock_actions_ok l t [] = True"
power_set_0: fixes A :: "'a set" shows "A *^ 0 = {1::'a}"
ord_twopow_aux: fixes k :: "nat"   and x :: "nat" assumes "3 \<le> k"   and "odd x" shows "[x ^ 2 ^ (k - 2) = 1] (mod 2 ^ k)"
NEG_elim_2: fixes N :: "ntrm"   and B :: "ty" assumes "N \<in> \<parallel>(B)\<parallel>" shows "N \<in> NEGn B (\<parallel><B>\<parallel>)"
arbitrary_intersection_of_idempot: fixes P :: "'b set \<Rightarrow> bool" shows "arbitrary intersection_of arbitrary intersection_of P = arbitrary intersection_of P"
uniformly_convergent_improper_integral': fixes a :: "real"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and A :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "\<And>x. a \<le> x \<Longrightarrow> (G has_real_derivative g x) (at x within {a..})"   and "\<And>a' b x. \<lbrakk>x \<in> A; a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> f x integrable_on {a'..b}"   and "convergent (\<lambda>x. G (real x))"   and "\<forall>\<^sub>F x in at_top. \<forall>y\<in>A. norm (f y x) \<le> g x" shows "uniformly_convergent_on A (\<lambda>x xa. integral {a..real x} (f xa))"
plus_monotone: fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s \<preceq> t" shows "s + u \<preceq> t + u"
bisimulation: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" assumes "Times r s = Times s t" shows "Times (Star r) s = Times s (Star t)"
bal_def2: fixes n :: "nat"   and xs :: "'b list" shows "bal n xs = (if n = 0 then (\<langle>\<rangle>, xs) else let m = n div 2; (l, ys) = bal m xs; (r, zs) = bal (n - 1 - m) (tl ys) in (\<langle>l, hd ys, r\<rangle>, zs))"
ccpo_Sup_least: fixes A :: "'a set"   and z :: "'a" assumes "Complete_Partial_Order.chain orda A"   and "\<And>x. x \<in> A \<Longrightarrow> orda x z" shows "orda (luba A) z"
HSem_redo: fixes ESem :: "'f \<Rightarrow> ('d \<Rightarrow> 'e) \<rightarrow> 'e"   and \<Gamma> :: "('d \<times> 'f) list"   and \<rho> :: "'d \<Rightarrow> 'e"   and \<Delta> :: "('d \<times> 'f) list" shows "has_ESem.HSem ESem \<Gamma>\<cdot>(has_ESem.HSem ESem (\<Gamma> @ \<Delta>)\<cdot>\<rho> f|` (edom \<rho> \<union> domA \<Delta>)) = has_ESem.HSem ESem (\<Gamma> @ \<Delta>)\<cdot>\<rho>"
strict_trans: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" assumes "b \<subset># a"   and "c \<subset># b" shows "c \<subset># a"
card_UNIV_length_enum: shows "CARD('b) = length enum_class.enum"
IH: fixes s :: "'a trm"   and t :: "'a trm" assumes "x \<in> vars_of s"   and "replace_subterm t q v s" shows "x \<in> vars_of t \<union> vars_of v"
length_concat_merge_pairs: fixes key :: "'b \<Rightarrow> 'c"   and xss :: "'b list list" shows "length (concat (merge_pairs key xss)) = length (concat xss)"
counit_simps_3: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (counit a) = G.map\<^sub>0 a"
dagger_prod_unfold: fixes y :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" shows "Id \<union> y ; ((x ; y)\<^sup>* ; x) = (y ; x)\<^sup>*"
Lp_I_1: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> borel_measurable M"   and "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)" shows "f \<in> space\<^sub>N (\<LL> (ennreal p) M)"
L'_eq_L: shows "H.L = (\<otimes>) \<I>"
fact: shows "factorization_m (q.Mp (Polynomial.smult ilc f)) (a, mset b)"
prime_imp_power_coprime_nat: fixes p :: "nat"   and a :: "nat"   and m :: "nat" assumes "normalization_semidom_class.prime p"   and "\<not> p dvd a" shows "coprime a (p ^ m)"
int_e: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and parOf :: "'b \<Rightarrow> 'c list"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'a trm" assumes "UM_Struct wtFsym wtPsym arOf parOf D intF intP" shows "U.Struct.int intF \<xi> T = M.Struct.int intF \<xi> T"
exe_all_normalized_and_ex_tcsigs_def: fixes cs :: "(String.literal \<times> String.literal) list"   and arss :: "('a \<times> ('b \<times> String.literal set list) list) list" shows "exe_all_normalized_and_ex_tcsigs cs arss \<equiv> \<forall>ars\<in>snd ` set arss. \<forall>ss\<in>snd ` set ars. \<forall>s\<in>set ss. exewf_sort cs s"
groupWIs3_default_policy_groupWIs2: fixes rs :: "'a simple_rule list"   and c :: "parts_connection" assumes "has_default_policy rs" shows "groupWIs2 c rs = groupWIs3_default_policy c rs"
supp_5: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "supp ([x3\<frown>x2]x1) = supp x3 \<union> (supp x2 \<union> supp x1)"
wfactors_perm_cong_l: fixes fs :: "'a list"   and a :: "'a"   and fs' :: "'a list" assumes "wfactors G fs a"   and "fs <~~> fs'"   and "set fs \<subseteq> carrier G" shows "wfactors G fs' a"
exists_split_characterisations_3: shows "(\<exists>x. linear_orderable_1 x) = (\<exists>x. linear_orderable_3a x)"
card_bijections_domain_and_range_permutation_eq_1: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A = card B" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B) = 1"
dom_def: fixes f :: "'a1" shows "C1.dom f = (if C1.domains f \<noteq> {} then SOME a. a \<in> C1.domains f else C1.null)"
adjoint_pair_antipar_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.adjoint_pair V H \<a> \<i> src trg f g" shows "src g = trg f"
fis_singleton_def: shows "fis_singleton \<equiv> id (\<lambda>A. is_singleton (fset A))"
inject: fixes x21 :: "'a"   and x22 :: "'a llist"   and y21 :: "'a"   and y22 :: "'a llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
conn: fixes nid :: "'nid"   and k :: "cnf" assumes "\<parallel>nid\<parallel>\<^bsub>k\<^esub>"   and "honest nid" shows "pin (\<sigma>\<^bsub>nid\<^esub>k) = (\<Union>nid'\<in>actHn k. {pout (\<sigma>\<^bsub>nid'\<^esub>k)})"
mon_iff_hom_one: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group H" shows "(f \<in> mon G H) = (f \<in> hom G H \<and> (\<forall>x. x \<in> carrier G \<and> f x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>))"
complement_involution: fixes x :: "i" assumes "x \<noteq> u" shows "x = \<midarrow> (\<midarrow> x)"
ras_is_empty_def: fixes s :: "'b array \<times> nat" shows "ras_is_empty s \<equiv> ras_length s = 0"
add_mset_le_left_le: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset" assumes "a \<le> b" shows "add_mset a A \<le> add_mset b A"
less_linear: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b \<or> a = b \<or> b \<prec> a"
cases_Total3: fixes a :: "'a"   and b :: "'a"   and phi :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "{a, b} \<subseteq> Field r"   and "(a, b) \<in> r - Id \<or> (b, a) \<in> r - Id \<Longrightarrow> phi a b"   and "a = b \<Longrightarrow> phi a b" shows "phi a b"
nonzero_mult_div_cancel_left: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)" shows "a * b / a = b"
space_bind: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> sets (f x) = sets N"   and "space M \<noteq> {}" shows "space (M \<bind> f) = space N"
least_not_less: fixes k :: "'b"   and A :: "'b set" assumes "k < least A" shows "k \<notin> A"
env_restr_evalHeap_same: fixes h :: "('a \<times> 'c) list"   and eval :: "'c \<Rightarrow> 'b" shows "evalHeap h eval f|` domA h = evalHeap h eval"
point_indices_elem_in: fixes ps :: "'b set"   and t :: "nat" assumes "ps \<subseteq> \<V>'"   and "card ps = t" shows "int (\<B>' index ps) \<in> target.point_indices (int t)"
snoc: fixes g :: "'g"   and ns :: "'node list"   and s :: "'val \<Rightarrow> 'node option"   and m :: "'node"   and i :: "nat" assumes "g \<turnstile> ns\<Down>\<^sub>s s"   and "last ns = old.predecessors g m ! i"   and "m \<in> set (\<alpha>n g)"   and "i < length (old.predecessors g m)" shows "g \<turnstile> ns @ [m]\<Down>\<^sub>s ssaStep g m i s"
viewL_correct_2: fixes t :: "('a, 'b) FingerTree" assumes "t \<noteq> FingerTree.empty" shows "\<exists>a s. FingerTree.viewL t = Some (a, s) \<and> FingerTree.toList t = a # FingerTree.toList s"
lookup_update_neq: fixes k :: "'b"   and k' :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" assumes "k \<noteq> k'" shows "Mapping.lookup (Mapping.update k v m) k' = Mapping.lookup m k'"
udvd_nat_alt: fixes a :: "'b word"   and b :: "'b word" shows "(a udvd b) = (\<exists>n. unat b = n * unat a)"
CSP_is_rd: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" shows "P = R (\<not> P\<^sup>f\<^sub>f \<turnstile> P\<^sup>t\<^sub>f)"
r_not: fixes x :: "nat" shows "eval r_not [x] \<down>= (if x = 0 then 1 else 0)"
simps_5: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x1 :: "bool" shows "(case Det x1 of Det x \<Rightarrow> f1 x | Indet x \<Rightarrow> f2 x) = f1 x1"
eval_MulNum_correct: fixes xs :: "dynamic list" assumes "length xs = 2" shows "eval_MulNum xs = eval_Mul xs"
shift_3: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "exact_decomp m (local.shift k m qs)"
dim_image_invertible_mat: fixes P :: "(('a, 'b) vec, 'b) vec"   and W :: "('a, 'b) vec set" assumes "invertible P"   and "vec.subspace W" shows "vec.dim ((*v) P ` W) = vec.dim W"
mset_swap: fixes i :: "nat"   and ls :: "'b list"   and j :: "nat" assumes "i < length ls"   and "j < length ls" shows "mset (ls[j := ls ! i, i := ls ! j]) = mset ls"
simps_1: fixes xs :: "'x list" shows "deg_le_sect_pp_aux xs 0 = 1"
decode_r: shows "digit_decode (nths (dims T) rmodes) (nths is rmodes) < prod_list (nths (dims T) rmodes)"
diff_chain_within: fixes f :: "'e \<Rightarrow> 'f"   and f' :: "'e \<Rightarrow> 'f"   and x :: "'e"   and s :: "'e set"   and g :: "'f \<Rightarrow> 'g"   and g' :: "'f \<Rightarrow> 'g" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x) within f ` s)" shows "(g \<circ> f has_derivative g' \<circ> f') (at x within s)"
boundOutputBindObject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and y :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "y mem bn \<alpha>" shows "y \<in> supp (object \<alpha>)"
linear_diff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "clinear f" shows "f (x - y) = f x - f y"
heuristic: fixes n :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and var :: "nat" assumes "heuristic n L F = var" shows "var \<le> n"
intros_3: fixes r :: "dB"   and s :: "dB"   and ss :: "dB list"   and t :: "dB" assumes "r[s/0] \<degree>\<degree> ss \<rightarrow>\<^sub>s t" shows "Abs r \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
strict_mono_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "strict_mono f" shows "inj_on f A"
select_convs_1: fixes heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "heap \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = heap"
cyc_free_not_empty: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "[M]\<^bsub>v,n\<^esub> \<noteq> {}"
l_coset_carrier: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "y \<in> x <#\<^bsub>G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H G" shows "y \<in> carrier G"
assms_1: fixes k :: "nat" assumes "k < n" shows "f k \<in> carrier_mat d d"
poly_1: fixes x :: "'a" shows "poly 1 x = (1::'a)"
step_elim_cases_17: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Throw # cs, css, s) \<rightarrow> u" obtains f :: "'f" where "s = Fault f" and "u = (cs, css, Fault f)"   | sa :: "'e" where "s = Abrupt sa__" and "u = (cs, css, Abrupt sa__)"   | "s = Stuck" and "u = (cs, css, Stuck)"   | sa :: "'e" where "s = Normal sa" and "u = (cs, css, Abrupt sa)"
sorted_filter: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq (map f xs)" shows "linorder.sorted less_eq (map f (filter P xs))"
coerce_llist_abs: fixes x :: "one \<oplus> 'b\<^sub>\<bottom> \<otimes> ('b\<cdot>llist)\<^sub>\<bottom>" shows "COERCE('b\<cdot>llist, 'a\<cdot>llist)\<cdot>(llist_abs\<cdot>x) = llist_abs\<cdot>(COERCE(one \<oplus> 'b\<^sub>\<bottom> \<otimes> ('b\<cdot>llist)\<^sub>\<bottom>, one \<oplus> 'a\<^sub>\<bottom> \<otimes> ('a\<cdot>llist)\<^sub>\<bottom>)\<cdot>x)"
x_nin_S: shows "x \<notin> {xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1}"
complete_interval: 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 a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "less a b"   and "P a"   and "\<not> P b" shows "\<exists>c. less_eq a c \<and> less_eq c b \<and> (\<forall>x. less_eq a x \<and> less x c \<longrightarrow> P x) \<and> (\<forall>d. (\<forall>x. less_eq a x \<and> less x d \<longrightarrow> P x) \<longrightarrow> less_eq d c)"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<preceq> b" obtains "a = a \<sqinter> b"
case_2: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case \<lfloor>x2\<rfloor> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
subseq_Cons': fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "subseq (x # xs) ys" shows "subseq xs ys"
transfer_path: fixes p :: "(nat \<times> nat) list"   and c' :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" assumes "set p \<inter> E \<subseteq> Graph.E c'"   and "isPath u p v" shows "Graph.isPath c' u p v"
foldseq_zerostart: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "\<forall>x. f (0::'b) (f (0::'b) x) = f (0::'b) x"   and "\<forall>i\<le>n. s i = (0::'b)" shows "foldseq f s (Suc n) = f (0::'b) (s (Suc n))"
that: fixes p :: "(name \<times> name) list" assumes "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "distinctPerm p"   and "yvec = p \<bullet> xvec" shows "thesis"
select_result_I: fixes h :: "'ap"   and f :: "('ap, 'aq, 'ar) prog"   and P :: "'ar \<Rightarrow> bool" assumes "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> P x" shows "P |h \<turnstile> f|\<^sub>r"
intro: fixes as :: "real list"   and k :: "nat"   and bs :: "real list" assumes "length as = k"   and "length bs = k"   and "\<And>a. a \<in> set as \<Longrightarrow> 0 \<le> a"   and "\<exists>a\<in>set as. 0 < a"   and "\<And>b. b \<in> set bs \<Longrightarrow> b \<in> {0<..<1}" shows "akra_bazzi_params_nonzero k as bs"
reindex: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'e" assumes "inj_on h A" shows "sum g (h ` A) = sum (g \<circ> h) A"
rtrancl_refl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" shows "r\<^sup>*\<^sup>* a a"
mod: shows "\<forall>e f g. e,f,g,ps \<Turnstile> p"
fst_sum: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set" shows "fst (sum f A) = (\<Sum>x\<in>A. fst (f x))"
composable_implies_arr_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.arr V \<mu>"
permutations_of_list_impl_aux_correct: fixes xs :: "'a list" shows "permutations_of_list_impl_aux [] xs = map rev (permutations_of_list_impl xs)"
x_plus_helpers_13: fixes x :: "nat" shows "16 + x = Suc (x + 15)"
powr_diff: fixes w :: "'a"   and z1 :: "'a"   and z2 :: "'a" shows "w powr (z1 - z2) = w powr z1 / w powr z2"
sets_F_mono: fixes i :: "'t"   and j :: "'t" assumes "i \<le> j" shows "sets (F i) \<subseteq> sets (F j)"
cnvalids_def: fixes A :: "triple set"   and C :: "triple set" shows "A |\<Turnstile> C \<equiv> \<forall>n. |\<Turnstile>n: A \<longrightarrow> |\<Turnstile>n: C"
2: shows "f\<^sup>\<star> \<le> forest_components (forest_components f \<sqinter> w)"
preserves_reflects_arr: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "natural_transformation A B F G \<tau>" shows "partial_magma.arr B (\<tau> f) = partial_magma.arr A f"
antisympI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>r x y; r y x\<rbrakk> \<Longrightarrow> x = y" shows "antisymp r"
sorted_list_of_set_set_of: fixes l :: "'a list" assumes "strict_sorted l" shows "sorted_list_of_set (list.set l) = l"
bounded_check_eqv_soundness: fixes \<AA> :: "interp"   and idx :: "nat"   and \<phi> :: "(presb, unit) aformula"   and \<psi> :: "(presb, unit) aformula" assumes "#\<^sub>V \<AA> = idx"   and "Presb.bounded.check_eqv idx \<phi> \<psi>" shows "Presb.sat\<^sub>b \<AA> \<phi> = Presb.sat\<^sub>b \<AA> \<psi>"
Res: fixes P :: "pi"   and s :: "(name \<times> name) list"   and Q :: "pi"   and x :: "name" assumes "P[
] \<approx> Q[
]"   and "x \<sharp> s" shows "<\<nu>x>P[
] \<approx> <\<nu>x>Q[
]"
Ln_minus: fixes z :: "complex" assumes "z \<noteq> 0" shows "Ln (- z) = (if Im z \<le> 0 \<and> \<not> (Re z < 0 \<and> Im z = 0) then Ln z + \<i> * complex_of_real pi else Ln z - \<i> * complex_of_real pi)"
assms_2: shows "component_of_term (lt p) = component_of_term (lt q)"
gallery_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "gallery (xs @ ys)" shows "gallery xs"
HM_def: shows "rev.HM = \<lparr>edge_rev = perm_restrict (edge_rev M) (arcs G - {a', rev.a'}), edge_succ = perm_rem a' (perm_rem rev.a' (edge_succ M))\<rparr>"
intros: fixes base :: "nat"   and M :: "hmultiset" assumes "2 \<le> base"   and "\<forall>N\<in>#hmsetmset M. well_base\<^sub>h base N"   and "well_base base (hmsetmset M)" shows "well_base\<^sub>h base M"
inject: fixes x :: "'b"   and ya :: "'b" shows "(\<tau>\<^sub>c\<^sub>s\<^sub>t x = \<tau>\<^sub>c\<^sub>s\<^sub>t ya) = (x = ya)"
append_Cons: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" shows "(x # xs) @ ys = x # xs @ ys"
distinct_4: fixes name1 :: "name"   and name2 :: "name" shows "\<tau> \<noteq> name1<name2>"
simps_2: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option" shows "not_MT C [] = False"
simps_8: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (\<phi> U\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> or\<^sub>n (af_letter \<phi> \<nu> and\<^sub>n \<phi> U\<^sub>n \<psi>)"
listall_app: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "listall P (xs @ ys) = (listall P xs \<and> listall P ys)"
inf1: fixes A :: "'d"   and k :: "'d"   and a :: "'d"   and b :: "'d" assumes "A \<equiv> inf k a" shows "inf A b \<equiv> inf k (inf a b)"
reducible\<^sub>dI: fixes p :: "'b poly" assumes "0 < degree p \<Longrightarrow> \<exists>q r. degree q < degree p \<and> degree r < degree p \<and> p = q * r" shows "\<not> irreducible\<^sub>d p"
inf_sup_aci_5: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" shows "sup_prod x y = sup_prod y x"
ft_def: shows "ft = infiniteExecutionMsg cfg fStepCfg fStepMsg"
weakPsiCongI: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q"   and "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"   and "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P" shows "\<Psi> \<rhd> P \<doteq> Q"
Seq_starD: fixes c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and c' :: "com"   and t :: "char list \<Rightarrow> int" assumes "(c1;; c2, s) \<rightarrow>* (c', t)" shows "(\<exists>c1'. c' = c1';; c2 \<and> (c1, s) \<rightarrow>* (c1', t)) \<or> (\<exists>s2. (c1, s) \<rightarrow>* (SKIP, s2) \<and> (c2, s2) \<rightarrow>* (c', t))"
aezfun_setspan_proj_def: fixes A :: "'d set"   and x :: "('c, 'd) aezfun" shows "aezfun_setspan_proj A x \<equiv> Abs_aezfun (aezfun x \<down> A)"
norm_complex_def: fixes z :: "complex" shows "cmod z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
Gamma_eq_zero_iff: fixes z :: "'a" shows "(Gamma z = (0::'a)) = (z \<in> \<int>\<^sub>\<le>\<^sub>0)"
strengthen_pre: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile> {P} c {Q}" shows "\<turnstile> {P'} c {Q}"
1: shows "prv (imp \<phi> (PP \<langle>\<phi>\<rangle>))"