hcju/isabelleps huggingface dataset:
intro: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" assumes "to_tree_order = l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_to_tree_order get_child_nodes" shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes to_tree_order"
sup_absorb2: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x \<sqinter> y = y"
runit_agreement: fixes a :: "'c local.term set" assumes "ide a" shows "runit a = \<r>[a]"
take_indices_def: fixes subsets :: "'a list"   and indices :: "nat list" shows "take_indices subsets indices = map ((!) subsets) indices"
bigo_mult: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "O(f) * O(g) \<subseteq> O(f * g)"
exe_sig_conds_def: fixes \<Sigma> :: "exesignature" shows "exe_sig_conds \<Sigma> \<equiv> alist_conds (execonst_type_of \<Sigma>) \<and> alist_conds (exetyp_arity_of \<Sigma>) \<and> exe_osig_conds (exesorts \<Sigma>)"
singletonE: fixes b :: "'c"   and a :: "'c" assumes "b \<in> {a}" obtains "b = a"
adjuster_already_in_span: fixes w :: "complex vec"   and us :: "complex vec list" assumes "w \<in> carrier_vec d"   and "set us \<subseteq> carrier_vec d"   and "corthogonal us"   and "w \<in> local.span (set us)" shows "adjuster d w us + w = 0\<^sub>v d"
prod_list_dvd: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "x dvd prod_list xs"
ball_insert: fixes a :: "'d"   and B :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "(\<forall>x\<in>insert a B. P x) = (P a \<and> (\<forall>x\<in>B. P x))"
ofilter_def: fixes A :: "('a \<times> 'c) set" shows "rt.ofilter A \<equiv> A \<subseteq> Field (r *o t) \<and> (\<forall>a\<in>A. rt.under a \<subseteq> A)"
app_subst_TVar: fixes S :: "nat \<Rightarrow> typ"   and n :: "nat" shows "$ S (TVar n) = S n"
add_nonpos_nonpos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less_eq b zero" shows "less_eq (plus a b) zero"
pullback_topology_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d topology" shows "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f -` U \<inter> A)"
delete_simps_2: fixes k :: "'a"   and p :: "'a \<times> 'b"   and ps :: "('a \<times> 'b) list" shows "AList.delete k (p # ps) = (if fst p = k then AList.delete k ps else p # AList.delete k ps)"
bind_lub_spmf2: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "'c set"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'e spmf"   and x :: "'d spmf" assumes "Complete_Partial_Order.chain ord Y"   and "\<And>y. monotone ord (ord_spmf (=)) (g y)" shows "x \<bind> (\<lambda>y. lub_spmf (g y ` Y)) = lub_spmf ((\<lambda>p. x \<bind> (\<lambda>y. g y p)) ` Y)"
presAtm_def: fixes atm :: "atom" shows "Example_PL.presAtm atm \<equiv> \<forall>s. Example_PL.indisAbbrev s (aval atm s)"
add: shows "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"
subset_eq_mset_impl: fixes xs :: "'k list"   and ys :: "'k list" shows "(subset_eq_mset_impl xs ys = None) = (\<not> mset xs \<subseteq># mset ys) \<and> (subset_eq_mset_impl xs ys = Some True) = (mset xs \<subset># mset ys) \<and> (subset_eq_mset_impl xs ys = Some False \<longrightarrow> mset xs = mset ys)"
assms_2: shows "lookup p v \<noteq> (0::'b)"
Inf_prod_def: fixes X :: "('c \<times> 'd) set" shows "Inf_prod X = (\<Sqinter>{fst x |x. x \<in> X}, \<Sqinter>{snd x |x. x \<in> X})"
multiplicity_dvd': fixes n :: "nat"   and p :: "'a"   and x :: "'a" assumes "n \<le> multiplicity p x" shows "p ^ n dvd x"
image_replicate_mset: fixes f :: "'b \<Rightarrow> 'a"   and n :: "nat"   and a :: "'b" shows "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
less_eq_pnat_def: fixes x :: "pnat"   and y :: "pnat" shows "x \<le> y \<equiv> x + y = y"
less_add: fixes N :: "'k multiset"   and a :: "'k"   and M0 :: "'k multiset"   and r :: "('k \<times> 'k) set" assumes "(N, add_mset a M0) \<in> mult1 r" shows "(\<exists>M. (M, M0) \<in> mult1 r \<and> N = add_mset a M) \<or> (\<exists>K. (\<forall>b. b \<in># K \<longrightarrow> (b, a) \<in> r) \<and> N = M0 + K)"
callee_invariant_on_def: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and I :: "'e \<Rightarrow> bool"   and \<I> :: "('f, 'g) \<I>" shows "callee_invariant_on callee I \<I> \<equiv> (\<forall>s x y s'. (y, s') \<in> set_spmf (callee s x) \<longrightarrow> I s \<longrightarrow> x \<in> outs_\<I> \<I> \<longrightarrow> I s') \<and> (\<forall>s. I s \<longrightarrow> \<I> \<turnstile>c callee s \<surd>)"
prod_of_pdevs_def: fixes x :: "'d pdevs"   and y :: "'e pdevs" shows "prod_of_pdevs x y = binop_pdevs Pair x y"
norm_minus_commute: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) = norm (minus b a)"
orderE: fixes a :: "'r"   and b :: "'r" assumes "a \<le> b" obtains "a = min a b"
shext_new: fixes sh :: "sheap"   and C :: "char list"   and x :: "sfields \<times> init_state" assumes "sh C = None" shows "sh \<unlhd>\<^sub>s sh(C \<mapsto> x)"
function_ring_not_car: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "a \<notin> carrier R"   and "f \<in> carrier (function_ring (carrier R) R)" shows "f a = undefined"
simps_14: fixes e :: "'a expr1"   and D :: "String.literal"   and F :: "String.literal"   and e' :: "'a expr1"   and e'' :: "'a expr1" shows "compE2 (e\<bullet>compareAndSwap(D\<bullet>F, e', e'')) = compE2 e @ compE2 e' @ compE2 e'' @ [CAS F D]"
mr_is_allow: shows "applied_rule_rev C x p = \<lfloor>AllowPortFromTo a b po\<rfloor>"
Proj2_pmf_def: fixes D :: "('f + 'e) pmf" shows "Proj2_pmf D = map_pmf (\<lambda>a. case a of Inr e \<Rightarrow> e) (cond_pmf D {f. \<exists>e. Inr e = f})"
linorder_on_axioms: shows "linorder_on D1 cmp1"
homogeneous_subterm: fixes t :: "('fun, 'var) Term.term"   and \<A> :: "('a strand_label \<times> ('fun, 'var) strand_step) list"   and Sec :: "('fun, 'var) Term.term set"   and t' :: "('fun, 'var) Term.term" assumes "homogeneous\<^sub>l\<^sub>s\<^sub>t t \<A> Sec"   and "t' \<sqsubseteq> t" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t t' \<A> Sec"
domains_char: fixes f :: "'a arrow_of_spans_data" shows "V.domains f = (if V.arr f then {V.dom f} else {})"
list_all2_Cons1: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and xs :: "'e list"   and ys :: "'f list" shows "list_all2 P (x # xs) ys = (\<exists>z zs. ys = z # zs \<and> P x z \<and> list_all2 P xs zs)"
prod_list_ge1: fixes x :: "real list" assumes "\<forall>a\<in>set x. 1 \<le> a" shows "1 \<le> prod_list x"
closed_sum_assoc: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A +\<^sub>M (B +\<^sub>M C) = A +\<^sub>M B +\<^sub>M C"
trans_less_add2_hmset: fixes i :: "hmultiset"   and j :: "hmultiset"   and m :: "hmultiset" assumes "i < j" shows "i < m + j"
sym_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "sym_digraph G \<equiv> wf_digraph G \<and> sym_digraph_axioms G"
PolysD_1: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "p \<in> P[X]" shows "keys p \<subseteq> .[X]"
prj_in_hom_4: shows "\<guillemotleft>\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>1 : H\<mu>H\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<mu>.apex\<guillemotright>"
rtrancl_edges_pathE: fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> (edges g)\<^sup>*" obtains p :: "('b \<times> 'b) list" where "path g u p v"
assms_4: fixes x :: "V"   and y :: "V" assumes "x \<in> A"   and "y \<in> A"   and "x < y" shows "\<pi> x < \<pi> y"
simps_11: fixes v :: "nat"   and va :: "Formula.agg_type \<times> event_data"   and vb :: "nat"   and vc :: "Formula.trm"   and vd :: "Formula.formula" shows "remove_neg (formula.Agg v va vb vc vd) = formula.Agg v va vb vc vd"
II: fixes xa :: "'a" shows "xa = - (- (xa \<squnion> - \<rho> \<squnion> \<bottom>\<bottom>) \<squnion> - (xa \<squnion> \<rho>))"
preorder_class_def: shows "OFCLASS('a, preorder_class) \<equiv> (OFCLASS('a, ord_class) &&& class.preorder (\<le>) (<))"
AE_prob_1: fixes A :: "'a set" assumes "prob A = 1" shows "AE x in M. x \<in> A"
TryCatch\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and a :: "nat"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>1 :: "val list"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and i :: "nat"   and e\<^sub>2 :: "nat exp"   and e\<^sub>2' :: "nat exp"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>2 :: "val list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw a,(h\<^sub>1, ls\<^sub>1)\<rangle>"   and "h\<^sub>1 a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C"   and "i < length ls\<^sub>1"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,(h\<^sub>1, ls\<^sub>1[i := Addr a])\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, ls\<^sub>2)\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>try e\<^sub>1 catch(C i) e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e\<^sub>2',(h\<^sub>2, ls\<^sub>2)\<rangle>"
assms_2: shows "SN (q2\<^sup>* O s O q2\<^sup>*)"
continuous_closedin_preimage_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'd"   and a :: "'d" assumes "continuous_on S f" shows "closedin (top_of_set S) {x \<in> S. f x = a}"
lt_imp_ex_count_lt: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "M < N" shows "\<exists>y. count M y < count N y"
sch_to_pr_srj: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "\<exists>sch. f = sch_to_pr sch"
Int_Union2: fixes B :: "'a set set"   and A :: "'a set" shows "\<Union> B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
word_cat_inj: fixes a :: "'a word"   and b :: "'b word"   and c :: "'a word"   and d :: "'b word" assumes "LENGTH('a) + LENGTH('b) \<le> LENGTH('c)" shows "(word_cat a b = word_cat c d) = (a = c \<and> b = d)"
univalent_finite_2: fixes R :: "('d \<times> 'e) set" assumes "univalent R" shows "finite (Domain R) = finite R"
valid_mbind''_mt: fixes \<sigma> :: "'d"   and f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option"   and P :: "'e list \<Rightarrow> bool" shows "(\<sigma> \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e [] f) (unit\<^sub>S\<^sub>E (P s))) = P []"
partition_split_body_def: fixes r :: "'a \<times> 'a \<Rightarrow> bool"   and x :: "'a" shows "partition_split_body r x \<equiv> \<lambda>y (X', xc). if r (x, y) then (X', List.insert y xc) else (List.insert y X', xc)"
bisim1_Invoke_stkD: fixes P :: "'addr J1_prog"   and h :: "'heap"   and e :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and M :: "String.literal"   and n' :: "nat" assumes "P,e,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, None)"   and "pc < length (compE2 e)"   and "compE2 e ! pc = Invoke M n'" shows "\<exists>vs v stk'. stk = vs @ v # stk' \<and> length vs = n'"
primepowI: fixes p :: "'a"   and k :: "nat"   and n :: "'a" assumes "prime p"   and "0 < k"   and "p ^ k = n" shows "primepow n \<and> aprimedivisor n = p"
sign_simps_29: fixes am :: "'d"   and bm :: "'d" shows "am + bm = bm + am"
C_def: shows "C \<equiv> {z \<in> \<int>. \<bar>z * e / 2 / real DIM('a)\<bar> \<le> B}"
hoare_fixpoint_mbt: fixes F :: "'b \<Rightarrow> 'b"   and x :: "'b"   and p :: "'c \<Rightarrow> 'b Assertion"   and q :: "'b Assertion"   and u :: "'c" assumes "F x = x"   and "\<And>w f. (\<And>v. v < w \<Longrightarrow> hoare (p v) f q) \<Longrightarrow> hoare (p w) (F f) q" shows "hoare (p u) x q"
assms_1: fixes xm :: "'a \<rightharpoonup>\<^sub>f 'b"   and ym :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "R (f xm) (f ym)" shows "strict_subtuple (\<lambda>x y. R x y \<or> x = y) xm ym"
stepE: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "DF_System.run s\<^sub>0 lstep s" obtains l :: "'b" where "lstep l (s i) (s (Suc i))"
prod_coprime_left: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one (f i) a" shows "algebraic_semidom.coprime times one (comm_monoid_mult.prod times one f A) a"
disc_filtr_def: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure" shows "disc_filtr M F = ((\<forall>n. subalgebra M (F n)) \<and> (\<forall>n m. n \<le> m \<longrightarrow> subalgebra (F m) (F n)))"
rcos_eq_homeq: fixes a :: "'a" assumes "a \<in> carrier R" shows "a_kernel R S h +> a = {x \<in> carrier R. h x = h a}"
LLL_inv_wD_2: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs" shows "length (map of_int_hom.vec_hom fs) = m"
img_is_subgroup: shows "subgroup (h ` carrier G) (H\<lparr>carrier := h ` carrier G\<rparr>)"
assms_1: shows "lift_instr instr \<Sigma>1 = Ok (instr', \<Sigma>2)"
sup_bot_right: fixes x :: "'a" shows "x \<sqinter> top = x"
CISK_control_spec: shows "\<forall>s d aseqs. case CISK_control s d aseqs of (a, aseqs', s') \<Rightarrow> int.thread_empty aseqs \<and> (a, aseqs') = (None, []) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> \<not> aborting s' d (the a) \<and> \<not> waiting s' d (the a) \<and> (a, aseqs') = (Some (hd (hd aseqs)), tl (hd aseqs) # tl aseqs) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> waiting s' d (the a) \<and> (a, aseqs', s') = (Some (hd (hd aseqs)), aseqs, s) \<or> (a, aseqs') = (None, tl aseqs)"
map_filter_def: fixes f :: "'h \<Rightarrow> 'g option"   and xs :: "'h list" shows "List.map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
enat_setp_enat_set_eq: shows "enat_setp = (\<lambda>x. x \<in> enat_set)"
assms: shows "subprob_cexpr V V' \<Gamma> e"
R3: shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
exhaust: fixes y :: "'b flowtable_behavior" obtains x1 :: "'b" where "y = Action x1"   | "y = NoAction"   | "y = Undefined"
apath_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "S.apath u p v \<equiv> S.awalk u p v \<and> distinct (pawalk_verts u p)"
greaterThanLessThan_empty: fixes l :: "'a"   and k :: "'a" assumes "l \<le> k" shows "{k<..<l} = {}"
delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_pointer_ptr_in_heap: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "delete\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr h = Some h'" shows "ptr |\<in>| object_ptr_kinds h"
assms_2: shows "Quotient3 R2 Abs2 Rep2"
not_less: fixes x :: "'t"   and y :: "'t" shows "(\<not> lt_of_nat_term_order cmp_term x y) = le_of_nat_term_order cmp_term y x"
curryD: fixes f :: "'a \<times> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "curry f a b" shows "f (a, b)"
run_get_state: fixes f :: "'e \<Rightarrow> ('e, 'd) stateT"   and s :: "'e" shows "run_state (get_state f) s = run_state (f s) s"
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>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
prj_simps_6: shows "C.dom \<mu>\<nu>\<pi>.Prj\<^sub>0 = HH\<mu>\<nu>\<pi>.chine"
Max_Un: fixes A :: "'a word set"   and B :: "'a word set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "signed.Max (A \<union> B) = signed.max (signed.Max A) (signed.Max B)"
tendsto_mult_filterlim_at_infinity: fixes f :: "'b \<Rightarrow> 'c"   and c :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "(f \<longlongrightarrow> c) F"   and "c \<noteq> (0::'c)"   and "filterlim g at_infinity F" shows "LIM x F. f x * g x :> at_infinity"
cp0: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" shows "f X \<tau> = f (\<lambda>_. X \<tau>) \<tau>"
nodeAbs_subset_grot: fixes S :: "'b"   and x :: "'a"   and xs :: "'a list" assumes "invariant S"   and "isNode x"   and "list_all isNode xs"   and "\<not> memb x S" shows "nodeAbs ` {n. isNode n \<and> \<not> memb n (ins x S)} \<subset> nodeAbs ` {n. isNode n \<and> \<not> memb n S}"
simps_1: fixes x1 :: "'b equation"   and y1 :: "'b equation" shows "(Pos x1 = Pos y1) = (x1 = y1)"
Uniq_D: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "Uniq P"   and "P a"   and "P b" shows "a = b"
add_nonneg_pos: 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 a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less zero b" shows "less zero (plus a b)"
pointermap_p_valid_def: fixes p :: "nat"   and m :: "('c, 'd) pointermap_scheme" shows "pointermap_p_valid p m \<equiv> p < length (entries m)"
I_outer_def: fixes s :: "'b list"   and t :: "'b list" shows "I_outer s t \<equiv> \<lambda>(i, j, pos). (\<forall>ii<i. \<not> sublist_at s t ii) \<and> (case pos of None \<Rightarrow> (\<forall>jj<j. t ! (i + jj) = s ! jj) \<and> j < length s | Some p \<Rightarrow> p = i \<and> sublist_at s t i)"
select_convs_1: fixes edge_rev :: "'b \<Rightarrow> 'b"   and edge_succ :: "'b \<Rightarrow> 'b"   and more :: "'c" shows "edge_rev \<lparr>edge_rev = edge_rev, edge_succ = edge_succ, \<dots> = more\<rparr> = edge_rev"
cases: fixes x :: "('a + 'b) + 'c + 'd" obtains iadv_core :: "'a" where "x = Inl (Inl iadv_core)"   | iadv_rest :: "'b" where "x = Inl (Inr iadv_rest)"   | iusr_core :: "'c" where "x = Inr (Inl iusr_core)"   | iusr_rest :: "'d" where "x = Inr (Inr iusr_rest)"
assms_1: shows "ModalSequents.mset c = \<LM> Modal T Ts  \<RM>"
disjoint_add_mset_2: fixes A :: "'b multiset"   and b :: "'b"   and B :: "'b multiset" shows "({#} = A \<inter># (b, B)) = (b \<notin># A \<and> {#} = A \<inter># B)"
inorder_skew: fixes t :: "('b \<times> nat) tree" shows "Tree2.inorder (skew t) = Tree2.inorder t"
simps_2: fixes F :: "'a formula"   and Fs :: "'a formula list" shows "\<^bold>\<Or>F # Fs = F \<^bold>\<or> \<^bold>\<Or>Fs"
C1_differentiable_imp_continuous_on: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" assumes "f C1_differentiable_on S" shows "continuous_on S f"
M'_3: fixes l :: "nat" shows "closure (V l) \<subseteq> \<Union> (U ` M' l)"
psimps: fixes t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_trd_aux_dom (t, p)" shows "sig_trd_aux (t, p) = (let p' = case find_sig_reducer (map spp_of bs) (lt p) t 0 of None \<Rightarrow> p | Some i \<Rightarrow> p - monom_mult (lookup (rep_list p) t / punit.lc (rep_list (bs ! i))) (t - punit.lt (rep_list (bs ! i))) (bs ! i); p'' = punit.lower (rep_list p') t in if p'' = 0 then p' else sig_trd_aux (punit.lt p'', p'))"
Union_of_finite_supp_sets: fixes S :: "'a set" assumes "finite S" shows "finite (\<Union> (supp ` S))"
in_small_setD: fixes x :: "V"   and A :: "V set" assumes "x \<in>\<^sub>\<circ> set A"   and "small A" shows "x \<in> A"
ffb_fbd_galois: fixes f :: "'c \<Rightarrow> 'd set" shows "bd\<^sub>\<F> f \<stileturn> fb\<^sub>\<F> f"
Fvars_ground_fmla_aux: fixes A :: "fmla"   and B :: "name set" assumes "Fvars A \<subseteq> B" shows "ground_fmla_aux A (atom ` B)"
preserves_dom: fixes f :: "'b" assumes "arr f" shows "local.dom (\<rr>'.map f) = local.map (local.dom f)"
frac_gt_0_iff: fixes x :: "'a" shows "((0::'a) < frac x) = (x \<notin> \<int>)"
in_set_tree_if: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree"   and t :: "'b tree" assumes "\<langle>l, a, r\<rangle> \<in> subtrees t" shows "a \<in> set_tree t"
perm_swap_1: fixes a :: "name"   and b :: "name"   and x :: "'d" shows "[(a, b)] \<bullet> [(a, b)] \<bullet> x = x"
res: fixes Z :: "nat set"   and z :: "'a option list" assumes "wf_tuple n Z z"   and "A_x \<union> A_y \<subseteq> Z" shows "(restrict (A_x \<union> A_y) z \<in> join x True y) = (restrict A_x z \<in> x \<and> restrict A_y z \<in> y)"
two: shows "2\<^sub>\<nat> = ZFC_in_HOL.set {[]\<^sub>\<circ>, 1\<^sub>\<nat>}"
emeasure_measure_of_sigma: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal"   and A :: "'b set" assumes "sigma_algebra \<Omega> M"   and "positive M \<mu>"   and "countably_additive M \<mu>"   and "A \<in> M" shows "emeasure (measure_of \<Omega> M \<mu>) A = \<mu> A"
adjacent_opp_chamber: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> f \<turnstile> folding.\<C>"   and "D \<in> f \<turnstile> folding.\<C>"   and "C \<sim> D" shows "(THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C) \<sim> (THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = D)"
fold_deriv_FOr: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and xs :: "'x list"   and \<phi> :: "('a, 'k) aformula"   and \<psi> :: "('a, 'k) aformula" shows "fold (deriv d0) xs (FOr \<phi> \<psi>) = FOr (fold (deriv d0) xs \<phi>) (fold (deriv d0) xs \<psi>)"
intros_31: fixes \<C> :: "t_context"   and n :: "nat" assumes "memory \<C> = Some n" shows "\<C> \<turnstile> [Current_memory] : [] _> [T_i32]"
R10_bc_2: shows "pmf (sds R10) c = 0"
MVOKBA_def: fixes v :: "Vals" shows "MVOKBA v = (v \<in> MemVal \<or> v = OK \<or> v = BadArg)"
fin: shows "f \<in> pmdl (set (comp_red_basis xs))"
assoc: fixes a :: "'l list"   and b :: "'l list"   and c :: "'l list" shows "(a @ b) @ c = a @ b @ c"
simps_2: fixes n1 :: "'d net_state"   and n2 :: "'d net_state" shows "net_ips (SubnetS n1 n2) = net_ips n1 \<union> net_ips n2"
step_less: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and t :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "akra_bazzi_recursion x\<^sub>0 x\<^sub>1 k as bs ts"   and "t \<in> set ts"   and "x\<^sub>1 \<le> x" shows "t x < x"
simps: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a \<times> 'a" shows "support_flowp f a = (\<exists>e. a = e \<and> 0 < f e)"
uplus_def: fixes P :: "'b hrel" shows "P\<^sup>+ = P ;; P\<^sup>\<star>"
complex_of_real_add_Complex: fixes r :: "real"   and x :: "real"   and y :: "real" shows "complex_of_real r + Complex x y = Complex (r + x) y"