hcju/isabelleps huggingface dataset:
Gets_imp_knows_Spy: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> set_cr" shows "X \<in> knows Spy evs"
ext_inject: fixes NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a"   and NonceS' :: "rat option"   and IntMapK' :: "rat option"   and ExtMapK' :: "rat option"   and IntAgrK' :: "rat option"   and ExtAgrK' :: "rat option"   and more' :: "'a" shows "(\<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = \<lparr>NonceS = NonceS', IntMapK = IntMapK', ExtMapK = ExtMapK', IntAgrK = IntAgrK', ExtAgrK = ExtAgrK', \<dots> = more'\<rparr>) = (NonceS = NonceS' \<and> IntMapK = IntMapK' \<and> ExtMapK = ExtMapK' \<and> IntAgrK = IntAgrK' \<and> ExtAgrK = ExtAgrK' \<and> more = more')"
gap_measure: fixes k :: "nat" assumes "gap_precond l k" shows "sum_heights_measure (gap_effect l k) \<le> sum_heights_measure l"
enumerator_length: fixes ls :: "nat list"   and m :: "nat"   and n :: "nat" assumes "ls \<in> set (enumerator m n)" shows "2 \<le> |ls|"
inconsist_bounded: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>" shows "the (inconsist i e) \<le> e_length e"
1: shows "list_all (bddh n) (map (\<lambda>(i, j). bdd_binop (\<lambda>k l. the (fst (prod_dfs A B (0, 0)) ! k ! l)) (fst A ! i) (fst B ! j)) (snd (prod_dfs A B (0, 0))))"
2: fixes x :: "'a" assumes "x \<in> S" shows "x \<notin> set pre'"
iUntil_def: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "iUntil I P Q \<equiv> \<diamond> t I. Q t \<and> (\<box> t' I \<down>< t. P t')"
simps_2: fixes f :: "POPLmarkRecord.type \<Rightarrow> POPLmarkRecord.type"   and T :: "POPLmarkRecord.type" shows "mapB f (TVarB T) = TVarB (f T)"
proj_stoch_singleton_set: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat"   and w :: "bool stream" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X" shows "proj_stoch_proc X n -` {proj_stoch_proc X n w} \<in> sets (F n)"
arr_dom: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "VV.arr (VV.dom f)"
insort_map_mono: fixes f :: "'a \<Rightarrow> 'b"   and n :: "'a"   and ns :: "'a list" assumes "\<forall>n m. n < m \<longrightarrow> f n < f m" shows "map f (insort n ns) = insort (f n) (map f ns)"
HPair_def: fixes X :: "msg"   and Y :: "msg" shows "Hash[X] Y \<equiv> \<lbrace>Hash \<lbrace>X, Y\<rbrace>, Y\<rbrace>"
fdia_mult: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "\<langle>y \<cdot> x| z = \<langle>x| \<langle>y| z"
proj_lnull: fixes c :: "'id"   and t :: "cnf llist" shows "(\<pi>\<^bsub>c\<^esub>t = []\<^sub>l) = (\<forall>k\<in>lset t. \<not> \<parallel>c\<parallel>\<^bsub>k\<^esub>)"
assms: shows "f - fps_cutoff n f \<noteq> 0"
obtain_idx: fixes x :: "'a extended" assumes "x \<in> set (ext_ids s)" shows "\<exists>i. idx s x = Inr i"
UniquePre: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and a1 :: "'c" assumes "Worder D"   and "a \<in> carrier D"   and "ExPre D a"   and "a1 \<in> carrier D \<and> a1 \<prec>\<^bsub>D\<^esub> a \<and> \<not> (\<exists>y\<in>carrier D. a1 \<prec>\<^bsub>D\<^esub> y \<and> y \<prec>\<^bsub>D\<^esub> a)" shows "Pre D a = a1"
pn_map_def: fixes nds :: "('c, 'd) node_scheme set" shows "pn_map nds \<equiv> FOREACH nds (\<lambda>nd m. pn_props (old nd) \<bind> (\<lambda>PN. RETURN (m(name nd \<mapsto> PN)))) Map.empty"
lm002: fixes R :: "('c \<times> 'd) set" assumes "runiq R" shows "graph (Domain R) (toFunction R) \<subseteq> R"
wrap_not_empty: fixes M :: "'b set" shows "(M \<noteq> {}) = (wrap M = {M})"
fmlookup_drop: fixes a :: "'b"   and m :: "('b, 'a) fmap"   and x :: "'b" shows "fmdrop a m $$ x = (if x \<noteq> a then m $$ x else None)"
assms: fixes x :: "'a" assumes "x \<in> space M" shows "0 \<le> f x"
save_restore_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and result :: "word32"   and new_cwp :: "word5"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (save_retore_sub1 result new_cwp rd s1)) \<and> t2 = snd (fst (save_retore_sub1 result new_cwp rd s2))" shows "low_equal t1 t2"
add_increasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less_eq b c" shows "less_eq b (plus a c)"
vdiff_vintersection_left: fixes A :: "V"   and B :: "V"   and C :: "V" shows "A \<inter>\<^sub>\<circ> B -\<^sub>\<circ> C = (A -\<^sub>\<circ> C) \<inter>\<^sub>\<circ> (B -\<^sub>\<circ> C)"
intro: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and a :: "'d" assumes "digraph_map G M"   and "bidel_arc_axioms G a" shows "bidel_arc G M a"
iwf_outPort: fixes fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and ent :: "'form fset \<times> 'form"   and "is" :: "nat list" assumes "iwf fc t ent"   and "is \<in> it_paths t" shows "Reg (iOutPort (tree_at t is)) |\<in>| outPorts (iNodeOf (tree_at t is))"
M_def: shows "M = sylvester_mat (poly_x_mult_y p) (poly_lift (reflect_poly q))"
lucas_lehmer_to_real_power: fixes x :: "int \<times> int"   and n :: "nat" shows "lucas_lehmer_to_real (x [^]\<^bsub>lucas_lehmer_ring\<^esub> n) = lucas_lehmer_to_real x ^ n"
indep_sets_sigma_sets_iff: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'a set set" assumes "\<And>i. i \<in> I \<Longrightarrow> Int_stable (F i)" shows "indep_sets (\<lambda>i. sigma_sets (space M) (F i)) I = indep_sets F I"
sumc: shows "matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<in> carrier_mat d d"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b" assumes "yoneda_lemma C S \<phi> F a"   and "yoneda_lemma_fixed_e_axioms S F a e" shows "yoneda_lemma_fixed_e C S \<phi> F a e"
jumpF_polyR_smult_1: fixes c :: "real"   and q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jumpF_polyR (smult c q) p x = sgn c * jumpF_polyR q p x"
on_triple_ID_restrict: fixes G :: "('e, 'f) labeled_graph" shows "on_triple (Id_on (vertices G)) `` edges G = edges (restrict G)"
preserves_cod: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.cod (fst f \<star>\<^sub>B snd f) = fst (B.VV.cod f) \<star>\<^sub>B snd (B.VV.cod f)"
nexts''_def: shows "nexts'' = (j, f tj v) # nexts"
hcomp_def: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" shows "\<mu> \<star> \<nu> \<equiv> if arr \<mu> \<and> arr \<nu> \<and> src \<mu> = trg \<nu> then MkArr (Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>) (Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>) (B.can (Cod \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod \<nu>) (Cod \<mu> \<^bold>\<star> Cod \<nu>) \<cdot>\<^sub>B (Map \<mu> \<star>\<^sub>B Map \<nu>) \<cdot>\<^sub>B B.can (Dom \<mu> \<^bold>\<star> Dom \<nu>) (Dom \<mu> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom \<nu>)) else null"
sgn_eq_0_iff: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "(sgn a = zero) = (a = zero)"
nth_length_takeWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "length (takeWhile P xs) < length xs" shows "\<not> P (xs ! length (takeWhile P xs))"
replace_id: fixes x :: "'a"   and xs :: "'a list" shows "replace x [x] xs = xs"
eq_refl: fixes x :: "'b" assumes "eq\<cdot>x\<cdot>x \<noteq> \<bottom>" shows "eq\<cdot>x\<cdot>x = TT"
l_get_owner_document_wf_axioms: shows "l_get_owner_document_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs get_disconnected_nodes Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_child_nodes"
assms: shows "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s"
delete_block_design: fixes bl :: "'a set" shows "design \<V> (del_block bl)"
tm2: fixes x :: "'a" shows "!x \<cdot> t x = !(1::'a)"
Un_infinite: fixes S :: "'b set"   and T :: "'b set" assumes "infinite S" shows "infinite (S \<union> T)"
d_OUT_mono: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "\<And>y. f (x, y) \<le> g (x, y)" shows "d_OUT f x \<le> d_OUT g x"
permutation_insert_delete: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {0..<Suc n}"   and "i < Suc n" shows "permutation_insert i (p i) (permutation_delete p i) = p"
preserves_seq: fixes g :: "'c"   and f :: "'c" assumes "Cop.seq g f" shows "S.seq (F g) (F f)"
insi: shows "insert (Var i IN t) H \<turnstile> Fls"
ncoplanar_perm_11: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet B D C A"
mult_zero_circ: fixes x :: "'a" shows "(x * bot)\<^sup>\<circ> = Rf x bot"
measure_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and A :: "'c set" assumes "\<Omega> \<inter> space M \<in> sets M"   and "A \<subseteq> \<Omega>" shows "Sigma_Algebra.measure (restrict_space M \<Omega>) A = Sigma_Algebra.measure M A"
assms_3: shows "S \<subseteq> U\<^sub>M\<^sub>_\<^sub>1"
atMost_iff: fixes i :: "'t"   and k :: "'t" shows "(i \<in> ord_term_lin.atMost k) = (i \<preceq>\<^sub>t k)"
Hoare_haltI: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and p :: "(action \<times> nat) list"   and Q :: "cell list \<times> cell list \<Rightarrow> bool" assumes "\<And>l r. P (l, r) \<Longrightarrow> \<exists>n. is_final (steps0 (1, l, r) p n) \<and> Q holds_for steps0 (1, l, r) p n" shows "{P} p {Q}"
primes_M_nat_cancel: fixes x :: "int" shows "\<MM> (real (nat x)) = \<MM> (real_of_int x)"
starfun_ln_exp: fixes x :: "real star" shows "(*f* real_ln) ((*f* exp) x) = x"
l_zero: fixes a :: "'a" assumes "a \<in> carrier M" shows "\<zero> \<plusminus> a = a"
of_type_trans: fixes T1 :: "PDDL_STRIPS_Semantics.type"   and T2 :: "PDDL_STRIPS_Semantics.type"   and T3 :: "PDDL_STRIPS_Semantics.type" assumes "of_type T1 T2"   and "of_type T2 T3" shows "of_type T1 T3"
vector_space_pair_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" shows "vector_space_pair_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
consistent_eq1: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>\<^sub>s\<^sub>y\<^sub>m :: "'c \<times> nat \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m s = (\<forall>sv\<in>symvars s. \<sigma>\<^sub>s\<^sub>y\<^sub>m sv = \<sigma> (fst sv))"
indRelRTPO_impl_TRel_is_weak_reduction_simulation_rev: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)" shows "weak_reduction_simulation ((TRel\<^sup>+)\<inverse>) Target"
homeomorphic_maps_id: fixes X :: "'b topology"   and Y :: "'b topology" shows "homeomorphic_maps X Y id id = (Y = X)"
iff_simps_3: fixes p :: "fm" shows "Ferrack.iff (NOT p) p = F"
FaultProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f"   and n :: "nat" shows "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> =n\<Rightarrow> Fault f"
mset: fixes f :: "'a list \<Rightarrow> 'a"   and xss :: "'a list list" shows "mset (sort1 f xss) = mset xss"
cases: fixes a1 :: "('d, 'e) Term.term"   and a2 :: "('d, 'e) Term.term" assumes "a1 \<unrhd> a2" obtains t :: "('d, 'e) Term.term" where "a1 = t__" and "a2 = t__"   | u :: "('d, 'e) Term.term" and ss :: "('d, 'e) Term.term list" and t :: "('d, 'e) Term.term" and f :: "'d" where "a1 = Fun f ss" and "a2 = t" and "u \<in> set ss" and "u \<unrhd> t"
simps: fixes a1 :: "nat \<Rightarrow> 'a"   and a2 :: "nat"   and prob :: "('a, 'b, 'c, 'd) ifc_problem" shows "((a1, a2) \<in> IFC_def.scp prob) = ((\<exists>h \<sigma> n. a1 = IFC_def.path prob \<sigma> \<and> a2 = n \<and> h \<in> IFC_def.hvars prob \<and> h \<in> IFC_def.reads prob (IFC_def.path prob \<sigma> n) \<and> (\<forall>k<n. h \<notin> IFC_def.writes prob (IFC_def.path prob \<sigma> k))) \<or> (\<exists>\<pi> m n. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> IFC_def.scp prob \<and> IFC_def.is_cdi prob n \<pi> m) \<or> (\<exists>\<pi> m n v. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> IFC_def.scp prob \<and> IFC_def.is_ddi prob n \<pi> v m) \<or> (\<exists>\<pi> m \<pi>' m' n v. a1 = \<pi> \<and> a2 = n \<and> (\<pi>, m) \<in> IFC_def.scp prob \<and> (\<pi>', m') \<in> IFC_def.scp prob \<and> IFC_def.is_dcdi_via prob n \<pi> v m \<pi>' m'))"
less_eq_neg_nonpos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus a) = less_eq a zero"
filter_M_pure_I: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> ('b, 'c, bool) prog"   and h :: "'b" assumes "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "pure (filter_M P xs) h"
p: shows "p = (\<Sum>b\<in>B. q0 b * b)"
normal_quotient_add_well_defined: fixes H :: "'g set"   and g :: "'g"   and g' :: "'g" assumes "Group H \<and> order.greater_eq G H"   and "normal H"   and "g \<in> G"   and "g' \<in> G" shows "LCoset_rel H `` {g} + LCoset_rel H `` {g'} = LCoset_rel H `` {g + g'}"
c_def: shows "c = (\<lambda>(x, y). if x = s \<and> y \<le> n then f y else if x \<le> n \<and> n < y \<and> y \<le> 2 * n + 1 \<and> (x, y - n - 1) \<in> R' then LARGE else if n < x \<and> x \<le> 2 * n + 1 \<and> y = t then g (x - n - 1) else 0)"
sum_upto_plus1: fixes x :: "real"   and f :: "nat \<Rightarrow> 'b" assumes "0 \<le> x" shows "sum_upto f (x + 1) = sum_upto f x + f (Suc (nat \<lfloor>x\<rfloor>))"
subword_split2: fixes w :: "'c list"   and i :: "nat"   and j :: "nat"   and l :: "'c list"   and r :: "'c list" assumes "subword w i j = l \<cdot> r"   and "i + j \<le> length w"   and "0 < length l"   and "0 < length r" shows "l = subword w i (length l) \<and> r = subword w (i + length l) (j - length l)"
f_rephrased: fixes x :: "nat"   and y :: "nat" shows "f x y = x + (2 * x + 2 * y)"
frt_def: fixes T :: "'b set set"   and s :: "'b set" shows "frontier\<^bsub>T\<^esub> s = closure\<^bsub>T\<^esub> s - core\<^bsub>T\<^esub> s"
summable_comparison_test_ev: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> real" assumes "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> g n"   and "summable g" shows "summable f"
prime_factors_list: fixes n :: "nat" assumes "3 < n"   and "\<not> prime n" shows "\<exists>qs. prime_factors n = set qs \<and> prod_list qs = n \<and> 2 \<le> length qs"
authTicket_authentic: fixes A :: "agent"   and authK :: "nat"   and Ta :: "msg"   and evs :: "event list" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbV" shows "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Ta\<rbrace>\<rbrace> \<in> set evs"
sg1: shows "inf_disj n nSend \<longrightarrow> Broadcast n nSend recv"
lookup_mult_scalar_monomial_right: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and p :: "'b \<Rightarrow>\<^sub>0 'd"   and v :: "'a"   and c :: "'d"   and u :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.mult_scalar pair_of_term term_of_pair p (monomial c v)) u = (lookup p (term_powerprod.pp_of_term pair_of_term u - term_powerprod.pp_of_term pair_of_term v) * c when term_powerprod.adds_term pair_of_term v u)"
generate_eq_imp_subset: fixes A :: "'a set"   and B :: "'a set" assumes "generate G A = generate G B" shows "A \<subseteq> generate G B"
sum_SDG_edge_SDG_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt False n'" shows "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt n'"
ide_char: fixes a :: "'a" shows "S.ide a = (S.arr a \<and> B.ide a)"
lossless_exec_gpv: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and I :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>"   and gpv :: "('g, 'e, 'f) gpv"   and s :: "'d" assumes "callee_invariant_on callee I \<I>"   and "lossless_gpv \<I> gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (callee s out)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "lossless_spmf (exec_gpv callee gpv s)"
vimage_insert: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and B :: "'b set" shows "f -` insert a B = f -` {a} \<union> f -` B"
cons_perm_imp_perm: fixes z :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "z # xs <~~> z # ys" shows "xs <~~> ys"
init_imp_inv_imps_work_sum: fixes c :: "('loc, 't) configuration" assumes "init_config c" shows "inv_imps_work_sum c"
subst_atm_mset_empty_iff: fixes AA :: "'a multiset"   and \<eta> :: "'s" shows "(AA \<cdot>am \<eta> = {#}) = (AA = {#})"
subset_setmonotone: fixes f :: "'a set \<Rightarrow> 'a set"   and X :: "'a set" assumes "setmonotone f" shows "X \<subseteq> f X"
semiring_of_sets_axioms: shows "semiring_of_sets \<Omega>' A'"
nat_dvd_1_iff_1: fixes m :: "nat" shows "gcd_nat.greater_eq 1 m = (m = 1)"
set_plus_intro: fixes a :: "'d"   and C :: "'d set"   and b :: "'d"   and D :: "'d set" assumes "a \<in> C"   and "b \<in> D" shows "a + b \<in> C + D"
assms_4: shows "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}"
frange_vunion: fixes r :: "V"   and s :: "V" shows "\<R>\<^sub>\<bullet> (r \<union>\<^sub>\<circ> s) = \<R>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> s"
case_eq_if: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm"   and vertex :: "'n bi.vertex" assumes "countable_bipartite_web \<Gamma>" shows "(case vertex of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = (if vertex = bi.SOURCE then f1 else if vertex = bi.SINK then f2 else f3 (bi.inner vertex))"
w_Ay: shows "w = Finite_Cartesian_Product.transpose A *v y"
eq_reflection: fixes x :: "'d"   and y :: "'d" assumes "x = y" shows "x \<equiv> y"
min_arg_not_ge_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a"   and n :: "'a" assumes "class.linorder less_eq less" shows "(\<not> less (ord.min less_eq m n) m) = (ord.min less_eq m n = m)"
non_empty_n2l: fixes xs :: "'b nel" shows "n2l xs \<noteq> []"
sin_mono_le_eq: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) \<le> x"   and "x \<le> pi / 2"   and "- (pi / 2) \<le> y"   and "y \<le> pi / 2" shows "(sin x \<le> sin y) = (x \<le> y)"
mult_sign_intros_1: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "(0::'b) \<le> b" shows "(0::'b) \<le> a * b"
progress_constraint: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and c :: "Formula.trm \<times> bool \<times> mconstraint \<times> Formula.trm"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula_of_constraint c) j = j"
frameImpResChainPres: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and F :: "'d frame"   and G :: "'d frame"   and xvec :: "name list" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "assertionAux.FrameStatImp SImp F G" shows "assertionAux.FrameStatImp SImp (\<lparr>\<nu>*xvec\<rparr>F) (\<lparr>\<nu>*xvec\<rparr>G)"
permut_aux_sound: fixes i :: "nat"   and as :: "'c list"   and f :: "nat \<Rightarrow> nat"   and bs :: "'c list" assumes "i < length as" shows "permut_aux as f bs ! i = bs ! f i"
splay_not_Leaf: fixes t :: "'a tree"   and a :: "'a" assumes "t \<noteq> \<langle>\<rangle>" shows "\<exists>l x r. splay a t = \<langle>l, x, r\<rangle>"
dist_eq: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list" assumes "is_shortest_path G m n p" shows "dist G m n = path_weight G p"