hcju/isabelleps huggingface dataset:
real_polynomial_function_tendsto: fixes p :: "'a \<Rightarrow> real"   and x :: "'a"   and X :: "'a set" assumes "real_polynomial_function p" shows "(p \<longlongrightarrow> p x) (at x within X)"
incoming_alt: fixes u :: "nat" shows "incoming u = (\<lambda>v. (v, u)) ` E\<inverse> `` {u}"
compl: shows "\<forall>x\<in>\<P> outcomes. \<forall>y\<in>\<P> outcomes. x \<succeq> y \<or> y \<succeq> x"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C I\<^sub>C.map I\<^sub>C.cmp FG.map FG.cmp \<epsilon>'.map\<^sub>0 \<epsilon>'.map\<^sub>1"
PiE_eq_iff: fixes I :: "'f set"   and F :: "'f \<Rightarrow> 'g set"   and F' :: "'f \<Rightarrow> 'g set" shows "(Pi\<^sub>E I F = Pi\<^sub>E I F') = ((\<forall>i\<in>I. F i = F' i) \<or> (\<exists>i\<in>I. F i = {}) \<and> (\<exists>i\<in>I. F' i = {}))"
elims_1: fixes x :: "('fsym, 'tp) efsym trm"   and y :: "bool" assumes "TE.wt x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y"   | f :: "('fsym, 'tp) efsym" and Tl :: "('fsym, 'tp) efsym trm list" where "x = Fn f Tl" and "y = (TE_wtFsym f \<and> list_all TE.wt Tl \<and> TE_arOf f = map TE.tpOf Tl)"
is_natural_transformation: shows "natural_transformation (\<cdot>) (\<cdot>) L local.map \<ll>.map"
os_reverse_def: fixes p :: "'c node ref option" shows "os_reverse p = os_reverse_aux None p"
\<theta>\<^sub>f_def: shows "\<theta>\<^sub>f = (g \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (\<a>[g, tab\<^sub>0 g, f\<^sub>0g\<^sub>1.p\<^sub>0] \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> ((g.tab \<star> f\<^sub>0g\<^sub>1.p\<^sub>0) \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> (f\<^sub>0g\<^sub>1.\<phi> \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[tab\<^sub>0 f, f\<^sub>0g\<^sub>1.p\<^sub>1, fg\<^sub>0h\<^sub>1.p\<^sub>1]"
liftAll_and: fixes phi :: "'a \<Rightarrow> bool"   and chi :: "'a \<Rightarrow> bool"   and inp :: "'b \<Rightarrow> 'a option" shows "liftAll (\<lambda>x. phi x \<and> chi x) inp = (liftAll phi inp \<and> liftAll chi inp)"
hyps_3: assumes "\<not> lnull (ltl xs)" shows "llast (ltl xs) \<in> lset (ltl xs)"
rel_G_eq_refl: fixes x :: "('n, 'o, 'p, 'q, 'r, 's, 't) G" shows "rel_G (=) (=) (=) (=) (=) (=) x x"
conj_chaos_left: fixes a :: "'b::refinement_lattice" shows "chaos \<iinter> (a::'b::refinement_lattice) = a"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> w"
IH: shows "poly p x = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"
inject_4: fixes x41 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and x42 :: "'j"   and x43 :: "('g \<Rightarrow> 'h option) \<Rightarrow> ('g \<Rightarrow> 'h option) \<Rightarrow> 'g \<Rightarrow> 'h option"   and y41 :: "('g \<Rightarrow> 'h option) \<times> 'i \<Rightarrow> bool"   and y42 :: "'j"   and y43 :: "('g \<Rightarrow> 'h option) \<Rightarrow> ('g \<Rightarrow> 'h option) \<Rightarrow> 'g \<Rightarrow> 'h option" shows "(x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 = y41\<hookleftarrow>\<^bsub>y42\<^esub>y43) = (x41 = y41 \<and> x42 = y42 \<and> x43 = y43)"
wfvarsoccs\<^sub>s\<^sub>t_proj_union: fixes A :: "('f strand_label \<times> ('e, 'd) strand_step) list" shows "wfvarsoccs\<^sub>s\<^sub>t (unlabel A) = (\<Union>l. wfvarsoccs\<^sub>s\<^sub>t (proj_unl l A))"
simps_14: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Abs f) env = map_option (tm_abs prec I a) (approx_tm prec ord I a f env)"
subfieldE_1: fixes K :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "subfield K R" shows "subring K R"
Ndet_is_CSP: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "is_CSP_process (action.relation_of P \<or> action.relation_of Q)"
bind_commute: fixes m :: "'m"   and m' :: "'m"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'm" shows "bind m (\<lambda>x. bind m' (f x)) = bind m' (\<lambda>y. bind m (\<lambda>x. f x y))"
sup_continuous_mono: fixes F :: "'c \<Rightarrow> 'd" assumes "sup_continuous F" shows "mono F"
cl_inD_aux: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> cl_in \<E> X" shows "x \<in> \<E>.cl X"
GDPR'_def: shows "GDPR' \<equiv> {gdpr_scenario'}"
insert_inter_insert: fixes a :: "'a"   and A :: "'a set"   and B :: "'a set" shows "insert a A \<inter> insert a B = insert a (A \<inter> B)"
adv_P1_def: fixes funct :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and P1_real_view :: "'d \<Rightarrow> 'e \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'g) spmf"   and x :: "'d"   and y :: "'e"   and z :: "'h"   and S :: "('i \<Rightarrow> 'd \<Rightarrow> 'h \<Rightarrow> ('d \<times> 'k) spmf) \<times> ('i \<Rightarrow> 'd \<Rightarrow> 'h \<Rightarrow> 'f \<Rightarrow> 'k \<Rightarrow> 'j spmf)"   and \<A> :: "'i"   and D :: "'j \<times> 'g \<Rightarrow> bool spmf" shows "malicious_base.adv_P1 funct P1_real_view x y z S \<A> D = \<bar>spmf (P1_real_view x y z \<A> \<bind> D) True - spmf (malicious_base.ideal_view_1 funct x y z S \<A> \<bind> D) True\<bar>"
swap_args2_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'c"   and y :: "'b" shows "swap_args2 f x y \<equiv> f y x"
domain: shows "Domainp cr_st_vec = (\<lambda>v. non_neg_vec v \<and> stoch_vec v)"
iNextStrong_def: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextStrong t0 I P \<equiv> \<diamond> t {inext t0 I} \<down>> t0. P t"
"1_2": shows "x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta> = \<forall>X\<langle>\<or>\<noteq>: (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>)\<rangle>\<^sub>s\<^sub>t"
in_set_R: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x" shows "sign \<in> set (characterize_consistent_signs_at_roots p qs)"
AE_impI: fixes P :: "bool"   and Q :: "'c \<Rightarrow> bool"   and M :: "'c measure" assumes "P \<Longrightarrow> almost_everywhere M Q" shows "AE x in M. P \<longrightarrow> Q x"
Pa: shows "row_add_iterate (Q ** (P ** A)) (nrows A - 1) (mod_type_class.from_nat i) (mod_type_class.from_nat k) = Pa ** (Q ** (P ** A))"
ccBindsExtra_eq: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "CoCallAnalysis.ccBindsExtra ccExp \<Gamma>\<cdot>(ae, G) = G \<squnion> CoCallAnalysis.ccBinds ccExp \<Gamma>\<cdot>(ae, G) \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x G)"
mult_sheaf_spec_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set"   and s :: "'c set \<Rightarrow> ('c \<times> 'c) set"   and s' :: "'c set \<Rightarrow> ('c \<times> 'c) set" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.mult_sheaf_spec R addition multiplication zero U s s' \<equiv> \<lambda>\<pp>\<in>U. quotient_ring.mult_rel (R\<setminus>\<pp>) R addition multiplication zero (s \<pp>) (s' \<pp>)"
hbt_insert_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "hbt t" shows "hbt (local.insert x t)"
elims_3: fixes x :: "'b comparator"   and xa :: "'b list" assumes "\<not> Sorting_Algorithms.sorted x xa" obtains cmp :: "'b comparator" and y :: "'b" and x :: "'b" and xs :: "'b list" where "xa__ = cmp" and "xa = y # x # xs" and "\<not> (compare cmp y x \<noteq> Greater \<and> Sorting_Algorithms.sorted cmp (x # xs))"
assms_2: shows "0 < zcount (c_pts c loc) t"
X: shows "topspace X \<subseteq> \<Union> \<C>"
wlp_skip_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> SKIP Q s = Q s"
case_eq_if: fixes f :: "float poly \<Rightarrow> float interval \<Rightarrow> 'a"   and taylor_model :: "taylor_model" shows "(case taylor_model of TaylorModel x xa \<Rightarrow> f x xa) = f (tm_poly taylor_model) (tm_bound taylor_model)"
greaterThanAtMost_def: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and l :: "'g"   and u :: "'g" shows "ord.greaterThanAtMost less_eq less l u \<equiv> ord.greaterThan less l \<inter> ord.atMost less_eq u"
IHP_3: fixes csp :: "'a list multiset" assumes "c' = csp + c"   and "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p" shows "P"
perm_rows_transpose: fixes M :: "('c, 'd) sq_matrix"   and p :: "'d \<Rightarrow> 'd" shows "perm_rows (Square_Matrix.transpose M) p = Square_Matrix.transpose (perm_cols M p)"
times_matrix_I_1: fixes x :: "(nat \<times> nat) \<times> nat \<times> nat" shows "I \<otimes> x = x"
eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x = y" shows "le x y"
Q: fixes A :: "fmla"   and H :: "fmla set" assumes "A \<in> Q_axioms" shows "H \<turnstile> A"
msg_fresh_4: fixes \<sigma> :: "nat \<Rightarrow> state"   and d :: "nat"   and dip :: "nat" shows "msg_fresh \<sigma> (Newpkt d dip) = True"
limit: fixes F :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a"   and fs :: "nat \<Rightarrow> real \<Rightarrow> 'a"   and a :: "int"   and N :: "nat"   and C :: "'a"   and Y :: "real set" assumes "euler_maclaurin_int F f fs a N C Y" shows "((\<lambda>b. (\<Sum>k = a..b. f (real_of_int k)) - F (real_of_int b) - (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real_of_int b))) \<longlongrightarrow> C) at_top"
map_ident: fixes t :: "'e set['f]" shows "map_bset (\<lambda>x. x) t = t"
I: shows "j < offs_next ns ub (y # xs) index key mi ma i"
pred_bool_Id: shows "(=) = (\<lambda>x y. (x, y) \<in> bool_rel)"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "bst_of_list (x # xs) = \<langle>bst_of_list (filter (\<lambda>y. y < x) xs), x, bst_of_list (filter ((<) x) xs)\<rangle>"
finsum_cong2: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "A = B"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i"   and "f \<in> B \<rightarrow> carrier V" shows "finsum V f A = finsum V g B"
zero_n2: shows "is_zero_row_upt_k n (mod_type_class.to_nat (mod_type_class.from_nat k)) A"
disc_156: fixes x13 :: "'c" shows "is_fun_call (Ubx.instr.ICall x13)"
strict_order_def: fixes a :: "'b G"   and b :: "'b G" shows "a < b \<equiv> a \<le> b \<and> \<not> b \<le> a"
Least_equality: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq x y" shows "ord.Least less_eq P = x"
ring_nsum_zeroTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<forall>j\<le>n. f j \<in> carrier A) \<and> (\<forall>j\<le>n. f j = \<zero>) \<longrightarrow> \<Sigma>\<^sub>e A f n = \<zero>"
combine_add: fixes Ks :: "'a list"   and Us :: "'a list"   and Ks' :: "'a list" assumes "length Ks = length Us"   and "length Ks' = length Us"   and "set Ks \<subseteq> carrier R"   and "set Ks' \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "combine Ks Us \<oplus> combine Ks' Us = combine (map2 (\<oplus>) Ks Ks') Us"
run_asset_iii_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_asset_iii n s))) \<notin> used s"
ir_While_backwards_frontier_both: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<And>n. ir_hoare (\<lambda>s s'. P n s s' \<and> bval b s \<and> bval b' s') c c' (P (Suc n))"   and "ir_hoare (\<lambda>s s'. \<exists>n. P n s s') (WHILE b DO c) (WHILE b' DO c') Q" shows "ir_hoare (P 0) (WHILE b DO c) (WHILE b' DO c') Q"
Goto: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and pc :: "int"   and S :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "get_ins M l = Some (goto pc)" shows "(M, l, S, 1, pc, S) \<in> Step"
int_mf: shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>join M = \<integral>\<^sup>+ M'. \<integral>\<^sup>+ x. ennreal (- f x) \<partial>M' \<partial>M"
compat_label_def: fixes c :: "('b com option list \<times> 'b) list"   and clist :: "('b com option \<times> 'b) list list" shows "compat_label c clist \<equiv> \<forall>j. Suc j < length c \<longrightarrow> c ! j -pc\<rightarrow> c ! Suc j \<and> (\<exists>i<length clist. clist ! i ! j -c\<rightarrow> clist ! i ! Suc j \<and> (\<forall>l<length clist. l \<noteq> i \<longrightarrow> clist ! l ! j -e\<rightarrow> clist ! l ! Suc j)) \<or> c ! j -pe\<rightarrow> c ! Suc j \<and> (\<forall>i<length clist. clist ! i ! j -e\<rightarrow> clist ! i ! Suc j)"
closed_def: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
pair_sym_digraph_axioms: shows "pair_sym_digraph sG"
imirror_eq_imirror_bounds: fixes I :: "nat set" shows "imirror I = imirror_bounds I (iMin I) (Max I)"
sort_eq_Nil_iff: fixes xs :: "'a list" shows "(sort xs = []) = (xs = [])"
deriv0: fixes x :: "(nat \<times> form) list"   and y :: "(nat \<times> form) list" shows "((0, x) \<in> deriv y) = (x = y)"
is_red_addsE': fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_red F p" shows "\<exists>f\<in>F. \<exists>v\<in>keys p. f \<noteq> 0 \<and> local.punit.adds_term (lpp f) v"
set_update_subset_insert: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" shows "set (xs[i := x]) \<subseteq> insert x (set xs)"
carrier_local_ring_at_def: shows "carrier_local_ring_at \<equiv> cq.carrier_quotient_ring"
set_map_column: fixes G :: "(('a, 'c) vec, 'b) vec"   and b :: "'c" shows "set (map (\<lambda>n. column (mod_type_class.from_nat n) G) [0..<mod_type_class.to_nat b]) = {column i G |i. i < b}"
assms_5: fixes x :: "'b" assumes "x \<in> U\<^sub>2"   and "x \<in> B" shows "f x \<le>\<^sub>o\<^sub>w g x"
assms_2: shows "ipresIGCons h hA MOD MOD'"
comp_arr_dom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "CC.arr f"   and "CC.dom f = a" shows "CC.comp f a = f"
sqn_if_updated: fixes rip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat" shows "sqn (\<lambda>x. if x = rip then Some v else rt x) ip = (if ip = rip then \<pi>\<^sub>2 v else sqn rt ip)"
dom_uses_in_graph: fixes g :: "'g" shows "Mapping.keys (uses g) \<subseteq> set (\<alpha>n g)"
app_hyps_2: shows "size t2 = n div 2 \<and> heap t2 \<and> braun t2 \<and> mset_tree t2 = mset (take (n div 2) ys1)"
uninitAborts_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "uninitAborts s \<equiv> {d. \<exists>ivs\<in>initSets s. \<exists>rs\<in>pendingSeqs s. d = \<Sqinter> ivs \<star> rs}"
distributed_def: fixes M :: "'b measure"   and N :: "'c measure"   and X :: "'b \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> ennreal" shows "distributed M N X f = (distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> M \<rightarrow>\<^sub>M N)"
simps_4: fixes f :: "'a \<Rightarrow> 'a fmUni"   and \<phi> :: "'a fmUni"   and \<psi> :: "'a fmUni" shows "map_atomUni f (AndUni \<phi> \<psi>) = AndUni (map_atomUni f \<phi>) (map_atomUni f \<psi>)"
set_iterator_linord_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'a set" shows "set_iterator_linord iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> sorted l0 \<and> iti = foldli l0)"
simps: fixes p :: "int"   and x :: "int"   and n :: "nat" shows "power_p p x n = (if n = 0 then 1 else let (d, r) = divmod_nat n 2; rec = power_p p (mult_p p x x) d in if r = 0 then rec else mult_p p rec x)"
simps_2: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and h :: "nat"   and r :: "('c \<times> nat) tree" shows "invar \<langle>l, (a, h), r\<rangle> = (invar l \<and> invar r \<and> h = lvl l + 1 \<and> (h = lvl r + 1 \<or> (\<exists>lr b rr. r = \<langle>lr, (b, h), rr\<rangle> \<and> h = lvl rr + 1)))"
int_prod: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set" shows "int (prod f A) = (\<Prod>x\<in>A. int (f x))"
continuous_on_compact_surface_projection_aux: fixes S :: "'a set"   and T :: "'a set"   and q :: "'a \<Rightarrow> 'a"   and p :: "'a \<Rightarrow> 'a" assumes "compact S"   and "S \<subseteq> T"   and "q ` T \<subseteq> S"   and "continuous_on T p"   and "\<And>x. x \<in> S \<Longrightarrow> q x = x"   and "\<And>x. x \<in> T \<Longrightarrow> q (p x) = q x"   and "\<And>x. x \<in> T \<Longrightarrow> p (q x) = p x" shows "continuous_on T q"
setmonotone_mk_regular1: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "setmonotone (mk_regular1 P F)"
eq: fixes x :: "real" assumes "1 \<le> x" shows "F2 x * of_real (ln x) + dirichlet_prod' mangoldt F2 x = dirichlet_prod' moebius_mu G2 x"
poly_zero: fixes p :: "'b poly"   and x :: "'b" assumes "poly p x = (0::'b)" shows "(p = 0) = (degree p = 0)"
x_2: shows "(i \<circ> g \<circ> r) x = x"
eq1: shows "of_weak_ranking (A # As) = restrict_relation (\<Union> (set (A # As))) le"
enum_ex_prod_def: fixes P :: "'a \<times> 'b \<Rightarrow> bool" shows "enum_ex P = enum_ex (\<lambda>x. enum_ex (\<lambda>y. P (x, y)))"
unit_factor_snd_normalize_quot: fixes x :: "'b \<times> 'b" shows "unit_factor (snd (normalize_quot x)) = (1::'b)"
sig_red_rtrancl_lt: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" shows "lt q \<preceq>\<^sub>t lt p"
lookup_bulkload: fixes xs :: "('a \<times> 'b) list" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.lookup (RBT_Mapping2.bulkload xs) = map_of xs"
sup_of_two_exists: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<exists>s. is_lub L s {x, y}"
u: shows "u\<^sup>2 + f * u + v = 0"
absorb_iff1: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (inf a b = a)"
setops\<^sub>s\<^sub>s\<^sub>t_member_iff: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and A :: "('a, 'b) stateful_strand_step list" shows "((t, s) \<in> setops\<^sub>s\<^sub>s\<^sub>t A) = (insert\<langle>t,s\<rangle> \<in> set A \<or> delete\<langle>t,s\<rangle> \<in> set A \<or> (\<exists>ac. \<langle>ac: t \<in> s\<rangle> \<in> set A) \<or> (\<exists>X F F'. \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> \<in> set A \<and> (t, s) \<in> set F'))"
ASSUME_simps_1: shows "cASSUME True = creturn ()"
assms_1: fixes i :: "nat" assumes "i \<le> n" shows "A i \<in> sigma_sets sp B"
inverse_le_1_iff: fixes x :: "'a" shows "(inverse x \<le> (1::'a)) = (x \<le> (0::'a) \<or> (1::'a) \<le> x)"
Map_axioms: shows "Map M_empty M_update M_delete M_lookup M_invar"
height_bal: fixes n :: "nat"   and xs :: "'a list"   and t :: "'a tree"   and zs :: "'a list" assumes "n \<le> length xs"   and "bal n xs = (t, zs)" shows "height t = nat \<lceil>log 2 (real (n + 1))\<rceil>"
refl: fixes x :: "'a" assumes "x \<in> A" shows "x \<bowtie> x"
c_e_incd_self: fixes f :: "'a \<Rightarrow> 'c"   and J :: "'c set" assumes "f \<in> rHom R R'"   and "ideal R' J" shows "i_extension f R R' (i_contract f R R' J) \<subseteq> J"
rep_eq: fixes x :: "('c, 'd) lifted_boolean_algebra"   and xa :: "('c, 'd) lifted_boolean_algebra" shows "Rep_lifted_boolean_algebra (x \<squnion> xa) = (\<lambda>f. Rep_lifted_boolean_algebra x f \<squnion> Rep_lifted_boolean_algebra xa f)"
bt_augment_alist_union_2: fixes r :: "('d, 'c) bintree"   and ts :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" assumes "distinct (bts_dfs val (r # ts))"   and "set (bts_dfs val (r # ts)) \<inter> set |q| = {}" shows "set (pq.alist_of (bts_augment ts (bt_augment r q))) = set (pq.alist_of (bts_augment ts q)) \<union> set (pq.alist_of (bt_augment r q))"
hlog_powhr_cancel: fixes a :: "real star"   and y :: "real star" assumes "0 < a"   and "a \<noteq> 1" shows "hlog a (a powhr y) = y"
simps_2: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree" shows "BinomialHeapStruc.rank_invar [t] = True"
is_type_nt: fixes P :: "'b prog" shows "is_type P NT = True"
simps_8: fixes vb :: "('b, 'c) Combinators"   and vc :: "('b, 'c) Combinators" shows "separate [vb \<oplus> vc] = [vb \<oplus> vc]"
REF_R_BRK: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "\<forall>x\<in>S - it. \<forall>y\<in>it. RR x y"   and "\<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "it \<noteq> {}"   and "it' \<noteq> {}"   and "\<not> c \<sigma>"   and "\<not> c' \<sigma>'" shows "(\<sigma>, \<sigma>') \<in> R'"
preserves_seq: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.seq (A.map g) (A.map f)"
plus_odd: shows "{j. j + length (take 2 xs) \<in> {a. odd a}} = {a. odd a}"
zero_order_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq n zero = (n = zero)"
abs_dvd_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "'a"   and k :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "dvd.dvd times (abs m) k = dvd.dvd times m k"
herbrand': fixes Cs :: "fterm literal set set" assumes "\<forall>T. \<exists>G. branch G T \<and> \<not> falsifies\<^sub>c\<^sub>s G Cs"   and "finite Cs"   and "\<forall>C\<in>Cs. finite C" shows "\<exists>G. eval\<^sub>c\<^sub>s HFun G Cs"
lmeasurable_iff_indicator_has_integral: fixes S :: "'c set"   and m :: "real" shows "(S \<in> lmeasurable \<and> m = Sigma_Algebra.measure lebesgue S) = (indicat_real S has_integral m) UNIV"
in_existence_between_zeroI: fixes t :: "real"   and x0 :: "'a"   and s :: "real" assumes "t \<in> existence_ivl t0 x0"   and "s \<in> {t0--t}" shows "s \<in> existence_ivl t0 x0"
scalar_product: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec"   and w :: "('d, 'e) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v \<bullet> Mod_Type_Connect.from_hma\<^sub>v w = scalar_product v w"