hcju/isabelleps huggingface dataset:
C_eq_Until_InsertDeniesQ: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" assumes "DenyAll \<in> set (policy2list p)"   and "all_in_list (policy2list p) l"   and "allNetsDistinct (policy2list p)" shows "C (list2FWpolicy (insertDenies (separate (qsort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)))) = C p"
ifTrue: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e2 :: "('a, 'b, 'c) expr" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])" shows "substitution.revision_step subst r s (s(r \<mapsto> (\<sigma>, \<tau>, \<E> [e1])))"
Disjoint_Eats_E2: fixes z :: "tm"   and x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (Disjoint z x) (insert (Neg (y IN z)) H) \<turnstile> A" shows "insert (Disjoint z (Eats x y)) H \<turnstile> A"
\<ll>_ide_simp: fixes f :: "'a" assumes "B.ide f" shows "B.\<ll> f = B.lunit f"
inverse_arrows: fixes \<mu> :: "'a arrow_of_spans_data" assumes "V.arr \<mu>"   and "C.iso (Chn \<mu>)" shows "V.inverse_arrows \<mu> \<lparr>Chn = C.inv (Chn \<mu>), Dom = Cod \<mu>, Cod = Dom \<mu>\<rparr>"
A: shows "(\<lambda>n. \<Sum>k<n. (- 1) ^ k / real (Suc k)) \<longlonglongrightarrow> (\<Sum>k. (- 1) ^ k / real (Suc k))"
exec_instr_CM_change: fixes ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and xcp' :: "'addr option"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and C :: "String.literal"   and M :: "String.literal"   and pc' :: "nat"   and frs :: "'addr frame list"   and ins :: "'addr instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and C' :: "String.literal"   and M' :: "String.literal" assumes "(ta, xcp', h', (stk', loc', C, M, pc') # frs) \<in> exec_instr ins P t h stk loc C M pc frs" shows "(ta, xcp', h', (stk', loc', C', M', pc') # frs) \<in> exec_instr ins P t h stk loc C' M' pc frs"
bintrunc_Suc_numeral_2: fixes n :: "nat" shows "take_bit (Suc n) (- 1) = 1 + 2 * take_bit n (- 1)"
bounds_of_interval_eq_lower_upper: fixes ivl :: "'a interval" assumes "lower ivl \<le> upper ivl" shows "bounds_of_interval ivl = (lower ivl, upper ivl)"
is_natural: fixes f :: "'a" assumes "arr f" shows "\<l>[cod f] \<cdot> T (\<I>, f) = local.map f \<cdot> \<l>[local.dom f]"
assms_1: shows "x \<Turnstile>\<^sub>c\<^sub>s set (x_mat_eq_vec b A)"
Lam_eq_same_var: fixes y :: "Vars.var"   and e :: "exp"   and e' :: "exp" shows "(Lam [y]. e = Lam [y]. e') = (e = e')"
A_def: shows "A = {x \<in> space M. f x \<le> f (T x)}"
equiintegrable_sum_real: fixes F :: "(real \<Rightarrow> 'a) set"   and a :: "real"   and b :: "real" assumes "F equiintegrable_on {a..b}" shows "(\<Union>I\<in>Collect finite. \<Union>c\<in>{c. (\<forall>i\<in>I. 0 \<le> c i) \<and> sum c I = 1}. \<Union>f\<in>I \<rightarrow> F. {\<lambda>x. \<Sum>i\<in>I. c i *\<^sub>R f i x}) equiintegrable_on {a..b}"
add_less_cancel_right: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" shows "(a + c \<subset># b + c) = (a \<subset># b)"
toTree_def: shows "toTree \<equiv> rec_list Empty (\<lambda>a xs. (\<lhd>) a)"
nth_sshiftr: fixes w :: "'a word"   and m :: "nat"   and n :: "nat" shows "(w >>> m) !! n = (n < size w \<and> (if size w \<le> n + m then w !! (size w - 1) else w !! (n + m)))"
isPrefixOf_strict_2: fixes x :: "'b"   and xs :: "['b]" shows "isPrefixOf\<cdot>(x : xs)\<cdot>\<bottom> = \<bottom>"
normal_simps_7: fixes \<phi> :: "'a fm" shows "normal (ExQ \<phi>) = normal \<phi>"
pq: fixes x :: "'a"   and y :: "'b" assumes "(x, y) \<in> set_pmf pq" shows "R x y"
that: fixes \<C> :: "'a set set" assumes "\<C> \<subseteq> \<D>"   and "disjoint \<C>"   and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) / 3 ^ DIM('a) \<le> Sigma_Algebra.measure lebesgue (\<Union> \<C>)" shows "thesis"
assms_4: shows "\<pi> n pd\<rightarrow> \<pi> k"
not_outside: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "preference carrier relation"   and "x \<succeq>[relation] y" shows "x \<in> carrier"
fps_X_power_mult_right_nth: fixes f :: "'a fps"   and k :: "nat"   and n :: "nat" shows "fps_nth (f * fps_X ^ k) n = (if n < k then 0::'a else fps_nth f (n - k))"
less_literal_def: fixes L :: "'a literal"   and M :: "'a literal" shows "(L < M) = (atm_of L < atm_of M \<or> atm_of L \<le> atm_of M \<and> is_neg L < is_neg M)"
vimage_fst: fixes A :: "'b set" shows "fst -` A = A \<times> UNIV"
update_rule: fixes p :: "'d ref"   and y :: "'d"   and x :: "'d" shows "<p \<mapsto>\<^sub>r y> p := x <\<lambda>r. p \<mapsto>\<^sub>r x>"
simps_7: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and e1' :: "'b aexp"   and e2' :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Eq e1 e2) (Eq e1' e2') = (e1 < e1' \<or> e1 = e1' \<and> e2 < e2')"
star_rtc_least_eq: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" assumes "Id \<union> x \<union> y O y = y" shows "x\<^sup>* \<subseteq> y"
croot_principal: fixes n :: "nat"   and y :: "complex"   and x :: "complex" assumes "n \<noteq> 0"   and "y ^ n = x"   and "y \<noteq> croot n x" shows "Re y < Re (croot n x) \<or> Re y = Re (croot n x) \<and> Im y < Im (croot n x)"
bind_rule: fixes P :: "assn"   and f :: "'c Heap"   and Q :: "'c \<Rightarrow> assn"   and g :: "'c \<Rightarrow> 'd Heap"   and R :: "'d \<Rightarrow> assn" assumes "<P> f <Q>"   and "\<forall>x. <Q x> g x <R>" shows "<P> f \<bind> g <R>"
simps_2: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and U :: "'e set"   and x :: "'f"   and xs :: "'f list" shows "c_sources_aux I D U (x # xs) = (if \<exists>v\<in>c_sources_aux I D U xs. (D x, v) \<in> I then insert (D x) (c_sources_aux I D U xs) else c_sources_aux I D U xs)"
e2ls_0_iff_Nil: fixes e :: "nat" shows "(Partial_Recursive.list_decode (e2ls e) = []) = (e2ls e = 0)"
hom_merge': fixes lhs :: "'b heap list"   and h :: "'b heap"   and x :: "'b" assumes "no_Emptys lhs"   and "is_root' h" shows "hom (merge' (Hp x lhs) h) = link \<langle>homs lhs, x, hom h\<rangle>"
qfree_min_inf: fixes \<phi> :: "atom fm" assumes "nqfree \<phi>" shows "qfree (inf\<^sub>- \<phi>)"
xeq1_sym: fixes x :: "Value"   and s :: "Store"   and t :: "Store" shows "s \<equiv>[x] t = t \<equiv>[x] s"
span_clauses_1: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> More_Modules.ideal S"
1: shows "i < length xs \<longrightarrow> local.times a xs ! i = a * xs ! i"
cases: fixes \<I> :: "('d, 'e) \<I>"   and callee :: "'d \<Rightarrow> ('e \<times> 'f) spmf" assumes "\<I> \<turnstile>c callee \<surd>" obtains "\<And>call ret s. \<lbrakk>call \<in> outs_\<I> \<I>; (ret, s) \<in> set_spmf (callee call)\<rbrakk> \<Longrightarrow> ret \<in> responses_\<I> \<I> call"
map_is_Nil_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "(map f xs = []) = (xs = [])"
aligned\<^sub>d_0: fixes base :: "nat"   and M :: "hmultiset" assumes "2 \<le> base" shows "aligned\<^sub>d base 0 M"
wp_under_wlp_intros_8: fixes S :: "'a \<Rightarrow> 'b set"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> wp_under_wlp (f a)"   and "\<And>s a. a \<in> S s \<Longrightarrow> healthy (wp (f a))"   and "\<And>s. S s \<noteq> {}" shows "wp_under_wlp (SetDC f S)"
Q2''_a_def: shows "Q2''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . root \<noteq> nil \<and> p' = root \<and> t' = nil \<and> lnk' = lnk \<and> lbl' = lbl \<and> mrk' = mrk \<union> {root} :]"
nsgTr1: fixes N :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<triangleright> N"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "b \<cdot> \<rho> a \<in> N" shows "\<rho> b \<cdot> a \<in> N"
vec_hom_inj: fixes v :: "int vec"   and w :: "int vec" assumes "of_int_hom.vec_hom v = of_int_hom.vec_hom w" shows "v = w"
bot_assn_def: shows "false \<equiv> Abs_assn (\<lambda>_. False)"
add_cf_len: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d" shows "fst (add_cf S c d) = max (fst c) (fst d)"
sparse_def: fixes \<delta> :: "real"   and ps :: "(int \<times> int) set" shows "sparse \<delta> ps = (\<forall>p\<^sub>0\<in>ps. \<forall>p\<^sub>1\<in>ps. p\<^sub>0 \<noteq> p\<^sub>1 \<longrightarrow> \<delta> \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)))"
one_OrdSum_def: shows "1 \<equiv> Abs_OrdSum (1::'b, 1::'c)"
map_iterator_genord_dom_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "map_iterator_genord it m R"   and "\<And>k v k' v'. \<lbrakk>m k = Some v; m k' = Some v'; R (k, v) (k', v')\<rbrakk> \<Longrightarrow> R' k k'" shows "set_iterator_genord (map_iterator_dom it) (dom m) R'"
assms_5: shows "h \<turnstile> get_element_by_id ptr' idd \<rightarrow>\<^sub>r Some result"
universal_domain_def: fixes A :: "'c set"   and Is :: "'d set"   and P :: "'d \<Rightarrow> ('c \<times> 'c) set" shows "universal_domain A Is P \<equiv> profile A Is P"
Longest_common_prefix_def: fixes L :: "'c list set" shows "Longest_common_prefix L = (ARG_MAX length ps. \<forall>xs\<in>L. prefix ps xs)"
dismantle_fresh: fixes x :: "name"   and t :: "trm"   and k :: "stack" shows "x \<sharp> t \<star> k = (x \<sharp> t \<and> x \<sharp> k)"
mult_smult_left: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "Polynomial.smult a p * q = Polynomial.smult a (p * q)"
c1c: shows "path_image c1 \<inter> path_image c = {a, b}"
inj_on_fun_updD: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'c"   and A :: "'b set" assumes "inj_on (f(x := y)) A"   and "x \<notin> A" shows "inj_on f A"
strip_while_def: fixes P :: "'b \<Rightarrow> bool" shows "strip_while P = rev \<circ> dropWhile P \<circ> rev"
of_real_poly_map_power: fixes p :: "real poly"   and n :: "nat" shows "map_poly of_real (p ^ n) = map_poly of_real p ^ n"
sorted_word_upt: fixes l :: "'b list" assumes "length l \<le> unat (- 1)" shows "sorted_descending (map word_of_nat (rev [0..<Suc (length l)]))"
integrable_eq: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f integrable_on s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "g integrable_on s"
typeof_addr_loc_def: fixes P :: "'b prog"   and h :: "'heap"   and a :: "'addr"   and al :: "addr_loc" shows "h.typeof_addr_loc P h a al = (THE T. h.addr_loc_type P h a al T)"
NN_def: shows "NN \<equiv> \<lambda>t1 t2. psubst N [(t1, xx), (t2, yy)]"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
degree_of_X_minus: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R"   and "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>" shows "deg R (X_poly_minus R a) = 1"
intros: fixes A :: "fm"   and i :: "name"   and x :: "tm" shows "special_axiomsp (A(i::=x) IMP SyntaxN.Ex i A)"
that: fixes q :: "rat poly" assumes "p = map_poly of_rat q" shows "thesis"
\<R>_regions_distinct: fixes R :: "('c \<Rightarrow> real) set"   and v :: "'c \<Rightarrow> real"   and R' :: "('c \<Rightarrow> real) set" assumes "R \<in> \<R>"   and "v \<in> R"   and "R' \<in> \<R>"   and "R \<noteq> R'" shows "v \<notin> R'"
simps_3: fixes t :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)" shows "valid_sops_expr t (Tmp sop) = ((\<forall>t'\<in>fst sop. t' < t) \<and> valid_sop sop)"
fpslsts_cases: fixes r :: "'b llist"   and A :: "'b set" assumes "r \<in> A\<^sup>\<clubsuit>" obtains a :: "'b" and rs :: "'b llist" where "r = a ## rs" and "a \<in> A" and "rs \<in> A\<^sup>\<star>"
trivial_relative_homology_group_empty: fixes X :: "'a topology"   and p :: "int"   and S :: "'a set" assumes "topspace X = {}" shows "trivial_group (relative_homology_group p X S)"
14: fixes x :: "'a" shows "x \<sqinter> - x = bot"
neg_one_not_one: shows "\<ominus>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub> \<noteq> \<one>\<^bsub>S\<^esub>"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>B) F'"
compatible_conv: fixes NS :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "NS O S \<subseteq> S" shows "(NS \<union> S)\<^sup>* O S O (NS \<union> S)\<^sup>* = S O S\<^sup>* O NS\<^sup>*"
inheritedPropagate: fixes P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and subs :: "'b \<Rightarrow> 'b set"   and delta :: "'b" assumes "\<not> P (tree subs delta)"   and "inherited subs P"   and "fans subs"   and "\<not> terminal subs delta" shows "\<exists>sigma\<in>subs delta. \<not> P (tree subs sigma)"
scale_minus_right: fixes a :: "'a"   and x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "a *s - x = - (a *s x)"
dual_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "dual x = dual y" shows "x = y"
ae_geq: shows "(AE w in M. 0 \<le> cls_val_process Mkt pf1 m w) = (AE w in M. 0 \<le> cls_val_process Mkt pf2 m w)"
elims: fixes g :: "('a, 'b) generator"   and x :: "'b"   and y :: "'a list" assumes "list.unfoldr g x = y" obtains s :: "'b" where "x = s" and "y = (if list.has_next g s then let (a, s') = list.next g s in a # list.unfoldr g s' else [])"
Cons_4: shows "A, n \<turnstile> (ps, a) # block # branch"
build_rel_SPEC: fixes M :: "'b nres"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<alpha> :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" assumes "M \<le> SPEC (\<lambda>x. \<Phi> (\<alpha> x) \<and> I x)" shows "M \<le> \<Down> (br \<alpha> I) (SPEC \<Phi>)"
primes_omega_primorial'_asymp_equiv: shows "(\<lambda>x. real (primes_omega (primorial' x))) \<sim>[sequentially] (\<lambda>k. ln (real (primorial' k)) / ln (ln (real (primorial' k))))"
hyps: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) \<times> 'x set) list" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp m qs" shows "(exact_aux k m qs = []) = (qs = [])"
B: fixes zs :: "'a list" shows "\<not> xs @ [x] \<cong> {[], zs, P}"
o2: shows "n = 3 ^ 3 * 5 * 23 * 37 * 53"
backward_reduction_P_6: shows "\<forall>k. (C, k) \<in># wP_of_wstate (N, P + {#(C' + {#L#}, i')#}, Q, n) \<longrightarrow> i \<le> k"
split_spec_params_split_spec_param: fixes d :: "nat" assumes "d = DIM('c)" shows "(split_spec_params d, split_spec_param) \<in> nat_rel \<rightarrow> \<langle>lv_rel\<rangle>set_rel \<rightarrow> \<langle>\<langle>lv_rel\<rangle>set_rel \<times>\<^sub>r \<langle>lv_rel\<rangle>set_rel\<rangle>nres_rel"
assms: fixes x :: "'state"   and y :: "'state"   and z :: "'state" assumes "x \<rightarrow> y"   and "x \<rightarrow> z" shows "y = z"
simps_2: fixes v :: "'b"   and va :: "'b"   and l :: "'b set list" shows "in_list (DenyAllFromTo v va) l = (bothNet (DenyAllFromTo v va) \<in> set l)"
isUnitClauseRemoveAllUnitLiteralIsFalse: fixes uClause :: "Literal list"   and uLiteral :: "Literal"   and valuation :: "Literal list" assumes "isUnitClause uClause uLiteral valuation" shows "clauseFalse (removeAll uLiteral uClause) valuation"
finsubsetall: fixes x :: "'b llist"   and A :: "'b set" assumes "x \<in> A\<^sup>\<star>" shows "x \<in> A\<^sup>\<infinity>"
M: shows "(\<rightarrow>\<^sub>M) M = (\<rightarrow>\<^sub>M) (embed_measure N f)"
in_diff_zcount: fixes a :: "'a"   and M :: "'a zmultiset"   and N :: "'a zmultiset" shows "(a \<in>#\<^sub>z M - N) = (zcount N a \<noteq> zcount M a)"
simps_1: fixes c :: "color" shows "paint c Leaf = Leaf"
relcycle_group_0_eq_chain_group: fixes X :: "'b topology" shows "relcycle_group 0 X {} = chain_group 0 X"
a_group: fixes G :: "('e, 'f) ring_scheme" assumes "abelian_group G" shows "Group.group (add_monoid G)"
cases: fixes x :: "'a \<times> 'a" obtains l :: "'a" and u :: "'a" where "x = (l, u)"
conga_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C CongA D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E"
bind_SE'_def: fixes f :: "'d \<Rightarrow> ('f \<times> 'd) option"   and g :: "'d \<Rightarrow> ('e \<times> 'd) option" shows "f ;- g = _bind_SE _ f g"
delta_Nil: fixes A :: "'c \<times> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> bool)"   and s :: "'c" shows "delta A [] s = s"
cinner_eq_flip: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(cinner x y = cinner z w) = (cinner y x = cinner w z)"
expand: fixes option :: "\<langle>'d\<rangle>\<^sub>\<bottom>"   and option' :: "\<langle>'d\<rangle>\<^sub>\<bottom>" assumes "(option = None) = (option' = None)"   and "\<lbrakk>option \<noteq> None; option' \<noteq> None\<rbrakk> \<Longrightarrow> the option = the option'" shows "option = option'"
kruskal2_def: shows "kruskal2 \<equiv> obtain_sorted_carrier' \<bind> (\<lambda>l. let initial_union_find = per_init V in nfoldli l (\<lambda>_. True) (\<lambda>ce (uf, T). ASSERT (\<alpha> ce \<in> E) \<bind> (\<lambda>_. endpoints ce \<bind> (\<lambda>(a, b). ASSERT (a \<in> V \<and> b \<in> V \<and> a \<in> Domain uf \<and> b \<in> Domain uf) \<bind> (\<lambda>_. if \<not> per_compare uf a b then let uf = per_union uf a b in ASSERT (ce \<notin> set T) \<bind> (\<lambda>_. RETURN (uf, T @ [ce])) else RETURN (uf, T))))) (initial_union_find, []) \<bind> (\<lambda>(per, spanning_forest). RETURN spanning_forest))"
P2: shows "Bet A' C' E' \<and> Cong C' E' C E"
rec_noteq_def: shows "rec_noteq = Cn (Suc (Suc 0)) rec_not [Cn (Suc (Suc 0)) rec_eq [recf.id (Suc (Suc 0)) 0, recf.id (Suc (Suc 0)) (Suc 0)]]"
cobounded1: fixes a :: "'e"   and b :: "'e" shows "a \<le> sup a b"
spmf_map_inj: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c spmf"   and x :: "'c" assumes "inj_on f (set_spmf M)"   and "x \<in> set_spmf M" shows "spmf (map_spmf f M) (f x) = spmf M x"
in_\<iota>: shows "\<iota> \<in> Inf"
set_of_fin: fixes b :: "'b" shows "finite (set_of b)"
llength_lmap: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h llist" shows "llength (lmap f xs) = llength xs"
uminus_ow_def: fixes U :: "'b set"   and uminus :: "'b \<Rightarrow> 'b" shows "uminus_ow U uminus \<equiv> \<forall>a. a \<in> U \<longrightarrow> uminus a \<in> U"
ordinal_oLogI: fixes b :: "ordinal"   and y :: "ordinal"   and x :: "ordinal" assumes "b ** y \<le> x"   and "x < b ** y * b" shows "oLog b x = y"
AbstrLevels_A7_A71: fixes i :: "AbstrLevelsID" assumes "sA7 \<in> AbstrLevel i" shows "sA71 \<notin> AbstrLevel i"
l_cancel_one': fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier R"   and "a \<in> carrier R" shows "(x = x \<oplus> a) = (a = \<zero>)"
T_append: fixes xs :: "'a list"   and ys :: "'a list" shows "T_append xs ys = length xs + 1"
hg: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "h (g x) = x"
dvd_associated: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> (0::'b)" shows "(a dvd b \<and> b dvd a) = (\<exists>u. u dvd (1::'b) \<and> a = u * b)"
reflexive: fixes a :: "'a" assumes "a \<in> G" shows "(a, a) \<in> kernel.Congruence"
AND_def: fixes x2 :: "ty"   and x1 :: "ty" shows "x2 AND x1 \<equiv> Abs_ty (AND_Rep (Rep_ty x2) (Rep_ty x1))"
arith: fixes b :: "real"   and a :: "real" assumes "0 \<le> b"   and "0 \<le> a"   and "b * 100 \<le> 751"   and "a * 25 + b * 25 \<le> 203" shows "norm a + norm b \<le> 20"
PpathE: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool"   and gamma :: "'b"   and n :: "nat"   and R :: "bool" assumes "inherited subs P"   and "fans subs"   and "\<not> P (tree subs gamma)"   and "P (tree subs (path subs gamma P n))" shows "R"
case_2: fixes f1 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f2 :: "('d, 'e) Term.term \<Rightarrow> 'c"   and f3 :: "poscheckvariant \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> ('d, 'e) Term.term \<Rightarrow> 'c"   and f4 :: "'e list \<Rightarrow> (('d, 'e) Term.term \<times> ('d, 'e) Term.term) list \<Rightarrow> 'c"   and x2 :: "('d, 'e) Term.term" shows "(case receive\<langle>x2\<rangle>\<^sub>s\<^sub>t of send\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f1 x | receive\<langle>x\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f2 x | \<langle>x: xa \<doteq> xb\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f3 x xa xb | \<forall>x\<langle>\<or>\<noteq>: xa\<rangle>\<^sub>s\<^sub>t \<Rightarrow> f4 x xa) = f2 x2"
order_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a = sup a b)"
eq_right_lang_finite_index: shows "finite (states M // eq_right_lang)"
vNM_utilityD_2: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real"   and p :: "'b pmf"   and q :: "'b pmf" assumes "vNM_utility outcomes relation u"   and "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "p \<succeq>[relation] q = (measure_pmf.expectation q u \<le> measure_pmf.expectation p u)"
neg_closed: fixes g :: "'m" assumes "g \<in> M" shows "- g \<in> M"
pochhammer_double: fixes z :: "'b"   and n :: "nat" shows "pochhammer ((2::'b) * z) (2 * n) = of_nat (2 ^ (2 * n)) * pochhammer z n * pochhammer (z + (1::'b) / (2::'b)) n"
u_3: shows "x1 = (1 - u) *\<^sub>R a + u *\<^sub>R b"
downsetI: fixes xss :: "'b list set" assumes "\<And>xs. \<lbrakk>xs \<in> xss; xs \<noteq> []\<rbrakk> \<Longrightarrow> butlast xs \<in> xss" shows "downset xss"
add_minus_left: fixes x :: "'a" assumes "x \<in> V" shows "- x + x = (0::'a)"
simps: fixes xs :: "(int poly \<times> nat) list" shows "create_factor_tree_balanced xs = (if length xs \<le> 1 then Factor_Leaf () (fst (hd xs)) else case partition_factors xs of (l, r) \<Rightarrow> Factor_Node () (create_factor_tree_balanced l) (create_factor_tree_balanced r))"
l_p_1: shows "l \<in> C\<^sub>1'"
subset_is_non_overlapping: fixes P :: "'b set set"   and Q :: "'b set set" assumes "P \<subseteq> Q"   and "is_non_overlapping Q" shows "is_non_overlapping P"
depth_max_heightE_left: fixes t\<^sub>1 :: "'a tree"   and a :: "'a"   and t\<^sub>2 :: "'b tree" assumes "depth t\<^sub>1 a = max (height t\<^sub>1) (height t\<^sub>2)" obtains "depth t\<^sub>1 a = height t\<^sub>1" and "height t\<^sub>2 \<le> height t\<^sub>1"