hcju/isabelleps huggingface dataset:
monad_state_alt_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state_alt return bind get put alt" shows "monad_state_alt (return_option return) (bind_option return bind) (get_option get) (put_option put) (alt_option alt)"
CHAIN: fixes i :: "nat" shows "(f i, f (Suc i)) \<in> S"
unconst_tm_def: fixes t :: "('c, 'd) tm" shows "unconst t = (case t of Hd (hd.Var x) \<Rightarrow> Map.empty x | Hd (Sym a) \<Rightarrow> Some (to_name a) | tm.App tm1 x \<Rightarrow> Map.empty x)"
fmdom'_fmsubset_restrict_set: fixes X1 :: "'d set"   and X2 :: "'d set"   and s :: "('d, bool) fmap" assumes "X1 \<subseteq> X2"   and "fmdom' s = X2" shows "fmdom' (fmrestrict_set X1 s) = X1"
lm006: fixes P :: "('a \<times> 'b) set"   and X :: "'a set" shows "Outside' X (Outside' X P) = Outside' X P"
eqv_refl: fixes x :: "'b" assumes "class.equiv TYPE('b)" shows "x \<sim> x"
col_dim: fixes A :: "'c mat"   and i :: "nat" shows "col A i \<in> carrier_vec (dim_row A)"
opt_MatchAny_match_expr_def: fixes m :: "'c match_expr" shows "opt_MatchAny_match_expr m \<equiv> repeat_stabilize 2 opt_MatchAny_match_expr_once m"
prems_2: shows "\<forall>V\<in>rv S n. state_val s V = state_val s' V"
ltake_ltl: fixes n :: "enat"   and xs :: "'a llist" shows "ltake n (ltl xs) = ltl (ltake (eSuc n) xs)"
lossless_spmf_inline1: fixes gpv :: "('c, 'call, 'ret) gpv"   and s :: "'s" assumes "plossless_gpv \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<And>s x. \<lbrakk>x \<in> outs_\<I> \<I>; I s\<rbrakk> \<Longrightarrow> lossless_spmf (the_gpv (callee s x))"   and "I s" shows "lossless_spmf (inline1 callee gpv s)"
CNV_meqD: fixes x :: "'e"   and y :: "'e" assumes "CNV x y" shows "x \<equiv> y"
ci_length: fixes ns :: "nat list"   and n :: "nat"   and ai :: "abc_inst" shows "length (ci ns n ai) div 2 = length_of ai"
icPar2B: fixes Q' :: "pi" assumes "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "F (P \<parallel> Q')"
equals_cod_if_value_is_ide: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" assumes "constant_transformation A B g"   and "partial_magma.ide B g" shows "constant_transformation.map A B g = constant_functor.map A B (partial_magma.cod B g)"
partial_disconnectTE: fixes s :: "'c net_state"   and i :: "nat"   and i' :: "nat"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, disconnect(i, i'), s') \<in> pnet_sos S T" obtains sa :: "'c net_state" and s'a :: "'c net_state" and t :: "'c net_state" and t' :: "'c net_state" where "s = SubnetS sa t" and "s' = SubnetS s'a t'" and "(sa, disconnect(i, i'), s'a) \<in> S" and "(t, disconnect(i, i'), t') \<in> T"
app2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and vs :: "v list"   and env' :: "v sem_env"   and e :: "exp0" assumes "evaluate_list ck env s1 (rev es) (s2, Rval vs) \<and> do_opapp (rev vs) = Some (env', e) \<and> clock s2 = 0 \<and> ck" shows "evaluate ck env s1 (App Opapp es) (s2, Rerr (Rabort Rtimeout_error))"
sum_group_def: fixes I :: "'b set"   and G :: "'b \<Rightarrow> ('c, 'd) monoid_scheme" shows "sum_group I G \<equiv> subgroup_generated (product_group I G) {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}"
prems_2: shows "non_volatile_owned_or_read_only pending_write \<S> \<O> (x # sb)"
w_in_hom_2: shows "\<guillemotleft>w' : w' \<Rightarrow> w'\<guillemotright>"
star_of_nat_def: fixes n :: "nat" shows "of_nat n = star_of (of_nat n)"
aform_err_trunc_bound_pdevsI: fixes y :: "real"   and e :: "nat \<Rightarrow> real"   and c :: "real"   and xs :: "real pdevs"   and d :: "real"   and p :: "nat" assumes "y \<in> aform_err e ((c, xs), d)"   and "e \<in> UNIV \<rightarrow> {- 1..1}" shows "y \<in> aform_err e ((c, fst (trunc_bound_pdevs p xs)), snd (trunc_bound_pdevs p xs) + d)"
pathVertices_singleton_iff: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and u :: "nat" shows "(pathVertices s p = [u]) = (p = [] \<and> s = u)"
A92_NSources_L1: shows "\<forall>C\<in>AbstrLevel level1. sA92 \<notin> Sources level1 C"
hyps: shows "(0::'a) \<le> (\<Sum>mc\<leftarrow>p. max (0::'a) (snd mc))"
cmod_diff_squared: fixes r1 :: "real"   and r2 :: "real"   and \<theta>1 :: "real"   and \<theta>2 :: "real" assumes "0 \<le> r1"   and "0 \<le> r2" shows "(cmod (complex_of_real r1 * exp (\<i> * complex_of_real \<theta>1) - complex_of_real r2 * exp (\<i> * complex_of_real \<theta>2)))\<^sup>2 = r1\<^sup>2 + r2\<^sup>2 - 2 * r1 * r2 * cos (\<theta>1 - \<theta>2)"
domA_map_transform: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "domA (map_transform t ae \<Gamma>) = domA \<Gamma>"
add_less_same_cancel1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and b :: "'a"   and a :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus b a) b = less a zero"
rep_eq: fixes x :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and xa :: "'f cset"   and xb :: "'g cset" shows "rel_cset x xa xb = rel_set x (rcset xa) (rcset xb)"
simulation1: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and tl1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "'x1 \<times> 'm1" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s1 -1-tl1\<rightarrow> s1'"   and "\<not> \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. r2.silent_moves t s2 s2' \<and> t \<turnstile> s2' -2-tl2\<rightarrow> s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> t \<turnstile> s1' \<approx> s2'' \<and> tl1 \<sim>m tl2"
Upper_is_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L" shows "is_closed\<^bsub>L\<^esub> (Upper L A)"
red_external_conf_extRet: fixes wf_md :: "'addr J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'addr expr) mdecl \<Rightarrow> bool"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and U :: "ty" assumes "wf_prog wf_md P"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "P,h \<turnstile> a\<bullet>M(vs) : U"   and "hconf h"   and "preallocated h"   and "P,h \<turnstile> t \<surd>t" shows "conf_extRet P h' va U"
binomial_Suc_Suc: fixes n :: "nat"   and k :: "nat" shows "Suc n choose Suc k = n choose k + (n choose Suc k)"
assms_2: shows "g \<in> \<Union> (range \<Sigma>n) - \<C>"
subspaceI: fixes S :: "'d set" assumes "(0::'d) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S" shows "subspace S"
Nonce_supply3: fixes evs :: "event list"   and evs' :: "event list"   and evs'' :: "event list" shows "\<exists>N N' N''. Nonce N \<notin> used evs \<and> Nonce N' \<notin> used evs' \<and> Nonce N'' \<notin> used evs'' \<and> N \<noteq> N' \<and> N' \<noteq> N'' \<and> N \<noteq> N''"
prv_imp_imp_triv: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp \<phi> (imp \<chi> \<phi>))"
le_fininf_lproject_visible: fixes u :: "'action list"   and v :: "'action stream" assumes "u \<preceq>\<^sub>F\<^sub>I v" shows "lproject visible (llist_of u) \<le> lproject visible (llist_of_stream v)"
LIM': shows "lim (the_state \<sigma>') \<le> lim (the_state \<sigma>'')"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_state return bind get put" shows "monad_state_axioms bind get put"
abs_eq: fixes y :: "'b" shows "finfun_const y = Abs_finfun (\<lambda>x. y)"
Throw: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref r,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw r,s\<^sub>1\<rangle>"
q'_def: shows "q' = (\<lambda>k. local.punit.monom_mult (1::'a) (monomial k y) q)"
append_self_conv2: fixes xs :: "'f list"   and ys :: "'f list" shows "(xs @ ys = ys) = (xs = [])"
simps_1: fixes s :: "('a, 'b option) insert_message list"   and m :: "('a, 'b) insert_message" shows "integrate s (Insert m) = integrate_insert m s (P m) (S m)"
e_if_intro: fixes n :: "nat"   and e1 :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v :: "val"   and e3 :: "exp"   and e2 :: "exp" assumes "VNat n \<in> E e1 \<rho>"   and "n = 0 \<longrightarrow> v \<in> E e3 \<rho>"   and "n \<noteq> 0 \<longrightarrow> v \<in> E e2 \<rho>" shows "v \<in> E (EIf e1 e2 e3) \<rho>"
assms_2: fixes x :: "nat" assumes "Suc x \<in> A" shows "f (Suc x) = g (Suc x)"
elims: fixes x :: "('b btree \<times> 'b) list"   and y :: "('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list" assumes "split_half x = y" obtains xs :: "('b btree \<times> 'b) list" where "x = xs" and "y = (take (length xs div 2) xs, drop (length xs div 2) xs)"
sorted01: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "length xs \<le> 1" shows "linorder.sorted less_eq xs"
sign_extended_iff_sign_extend: fixes n :: "nat"   and w :: "'b word" shows "sign_extended n w = (sign_extend n w = w)"
linepath_0': fixes a :: "'a"   and b :: "'a" shows "linepath a b 0 = a"
inf_mono: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset"   and d :: "'a multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
ide_in_hom_1: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>f : src\<^sub>B f \<rightarrow>\<^sub>B trg\<^sub>B f\<guillemotright>"
quicksort_permutes: fixes xs :: "'a list" shows "mset (quicksort xs) = mset xs"
sign_simps_3: fixes x :: "'a" assumes "x < (0::'a)" shows "sign x = - 1"
keys_update: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.update k v m) = insert k (Mapping.keys m)"
that: fixes ns :: "nat list" assumes "DirProds (\<lambda>n. Z (ns ! n)) {..<length ns} \<cong> G"   and "\<forall>n\<in>set ns. n = 0 \<or> (\<exists>p k. normalization_semidom_class.prime p \<and> 0 < k \<and> n = p ^ k)" shows "thesis"
elim: assumes "a \<notin> Units G"   and "\<forall>b. b \<in> carrier G \<and> properfactor G b a \<longrightarrow> b \<in> Units G" shows "P"
eval_monom_eq: fixes es1 :: "real list"   and es2 :: "real list"   and basis :: "(real \<Rightarrow> real) list" assumes "length es1 = length es2"   and "length es2 = length basis"   and "basis_wf basis"   and "compare_lists es1 es2 = EQ" shows "eval_monom (1, es1) basis = eval_monom (1, es2) basis"
lsu_inj_on: shows "inj_on \<nu>\<^sup>\<natural> (Fix \<nu>\<^sup>\<natural>)"
carrierI: fixes t :: "'a set"   and x :: "'a" assumes "t open"   and "x \<in> t" shows "x \<in> carrier"
cscc_outer_invar_loc_def: fixes G :: "('a, 'b) graph_rec_scheme"   and it :: "'a set"   and l :: "'a set list"   and D :: "'a set" shows "cscc_outer_invar_loc G it l D \<equiv> outer_invar_loc G it D \<and> cscc_invar_ext G l D"
l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_set_disconnected_nodes_get_disconnected_nodes_wf_axioms heap_is_wellformed get_disconnected_nodes set_disconnected_nodes \<equiv> \<forall>h ptr disc_nodes node_ptr h' disc_nodes'. heap_is_wellformed h \<longrightarrow> h \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes \<longrightarrow> h \<turnstile> set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes' \<longrightarrow> node_ptr \<notin> set disc_nodes'"
chain_iterate: fixes F :: "'c \<rightarrow> 'c" shows "chain (\<lambda>i. iterate i\<cdot>F\<cdot>\<bottom>)"
sqrt_int_ceiling: fixes x :: "int" shows "sqrt_int_ceiling x = \<lceil>sqrt (real_of_int x)\<rceil>"
transitive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "transitive B (\<sqsubseteq>)"
prod_simps_3: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "cod (f0 \<otimes> f1) = cod f0 \<otimes> cod f1"
assms: fixes \<rho>' :: "'var \<Rightarrow> 'value" shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub> = \<lbrakk> e' \<rbrakk>\<^bsub>\<rho>'\<^esub>"
q65_def: shows "q65 = {0, 1, 4, 10, 14, 9, 16, 26, 30, 25, 29, 40, 56, 36, 49, 61, 35, 51, 39, 55, 64}"
assms_2: fixes a :: "'a set" assumes "a \<in> M" shows "0 \<le> f a"
wp1_aux1aa: fixes p :: "('b, 'c) Combinators list" assumes "wellformed_policy1_strong p" shows "DenyAll \<in> set p"
pdevs_apply_update_slist: fixes n :: "nat"   and x :: "'d"   and xs :: "(nat, 'd) slist"   and i :: "nat" shows "pdevs_apply (Pdevs (update_slist n x xs)) i = (if i = n then x else pdevs_apply (Pdevs xs) i)"
permute_zero: fixes x :: "'b" shows "permute_atom 0 x = x"
sym_group_one: fixes n :: "nat" shows "\<one>\<^bsub>sym_group n\<^esub> = id"
strictly_generalizes_def: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "(C \<prec>\<cdot> D) = strict generalizes C D"
le_diff_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "(a \<le> c - b) = (a + b \<le> c)"
geodesic_segment_param_5: fixes G :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_segment_between G x y" shows "geodesic_segment_param G x ` {0..dist x y} = G"
supp_supports: fixes x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "supp x supports x"
map_cong0: fixes x :: "'d list"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>z. z \<in> set x \<Longrightarrow> f z = g z" shows "map f x = map g x"
ContractsWithSubstitutesAndLAD_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithSubstitutesAndLAD Xd Xh Pd Ch \<equiv> ContractsWithSubstitutes Xd Xh Pd Ch \<and> ContractsWithLAD Xd Xh Pd Ch"
assms: fixes z :: "'a" assumes "z \<in> set1_FGco x" shows "l1 z = l1' z"
scaleR_scaleC: fixes r :: "real" shows "(*\<^sub>R) r = (*\<^sub>C) (complex_of_real r)"
steps_Abrupt: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "'a" shows "\<Gamma>\<turnstile> (c, Abrupt s) \<rightarrow>\<^sup>* (Skip, Abrupt s)"
simps_1: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and i :: "nat"   and s :: "'f"   and R :: "nat set" shows "netgmap sr (NodeS i s R) = ([i \<mapsto> fst (sr s)], NodeS i (snd (sr s)) R)"
assms: shows "rt ip = Some (dsn, dsk, flag, hops, nhip, pre)"
Suc_1: fixes A :: "'a mat" assumes "A \<in> carrier_mat n m"   and "I < n" shows "swap_row_to_front A I = mat n m (\<lambda>a. case a of (i, j) \<Rightarrow> if i = 0 then A $$ (I, j) else if i \<le> I then A $$ (i - 1, j) else A $$ (i, j))"
generate_check_filter_conv: fixes a :: "nat list"   and b :: "nat list" shows "generate_check a b = concat (map (\<lambda>ys. map (\<lambda>xs. (xs, ys)) (filter (suffs (C\<^sub>1 b (fst ys)) a) (alls (B\<^sub>1 b) a))) (filter (suffs C\<^sub>2 b) (alls B\<^sub>2 b)))"
assms_2: shows "pf stk (Suc n) w = 0"
assms_3: shows "\<turnstile> imply G (A \<^bold>\<or> B)"
true_clss_empty: fixes I :: "'b set" shows "I \<TTurnstile>s {}"
size_annos_same2: fixes C1 :: "'b acom"   and C2 :: "'c acom" assumes "strip C1 = strip C2" shows "(length (annos C1) = length (annos C2)) = True"
append_take_drop_id: fixes n :: "nat"   and xs :: "'b list" shows "xs \<down> n @ xs \<up> n = xs"
ereal_Liminf_uminus: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> ereal" shows "Liminf net (\<lambda>x. - f x) = - Limsup net f"
ci_def: shows "ci = (\<lambda>r. cindexE (- r) r (\<lambda>x. poly (map_poly Im p) x / poly (map_poly Re p) x))"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'a" shows "incseq f = (\<forall>n. f n \<le> f (Suc n))"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> yahalom"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> yahalom; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> yahalom; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> yahalom; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1)"   and "\<And>evs2 NB B A NA. \<lbrakk>evs2 \<in> yahalom; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB B NB A NA. \<lbrakk>evs3 \<in> yahalom; P evs3; Key KAB \<notin> used evs3; Gets Server \<lbrace>Agent B, Nonce NB, Crypt (shrK B) \<lbrace>Agent A, Nonce NA\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server A \<lbrace>Nonce NB, Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key KAB, Nonce NB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 A NB B K NA X. \<lbrakk>evs4 \<in> yahalom; P evs4; Gets A \<lbrace>Nonce NB, Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA\<rbrace>, X\<rbrace> \<in> set evs4; Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4)"   and "\<And>evso A NB B K NA X. \<lbrakk>evso \<in> yahalom; P evso; Says Server A \<lbrace>Nonce NB, Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA\<rbrace>, X\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
hom_prod_list: fixes xs :: "rat poly list" shows "map_poly of_rat (prod_list xs) = prod_list (map (map_poly of_rat) xs)"
prems_2: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> insert S__ \<F>"   and "T \<in> insert S__ \<F>"   and "S \<noteq> T" shows "S \<inter> T \<subseteq> K"
inf_sup_ord_3: fixes x :: "'b"   and y :: "'b" shows "x \<le> sup x y"
prime_factor_nat: fixes n :: "nat" assumes "n \<noteq> 1" shows "\<exists>p. p dvd n \<and> prime p"
restrict_nested: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and A :: "nat set"   and B :: "nat set"   and x :: "'c option list" assumes "getIJ getIJ" shows "restrict A (restrict B x) = restrict (A \<inter> B) x"
multiplicative_function_of_natI: fixes f :: "nat \<Rightarrow> nat" assumes "multiplicative_function f" shows "multiplicative_function (\<lambda>n. of_nat (f n))"
desargues_config_not_col_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col C C' B'"
vfind_match_rel': fixes t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and cs :: "(Term_Class.term \<times> sterm) list" assumes "\<turnstile>\<^sub>v t\<^sub>1 \<approx> t\<^sub>2" shows "rel_option (rel_prod (fmrel vrelated) (=)) (vfind_match cs t\<^sub>1) (vfind_match cs t\<^sub>2)"
fds_perzeta_def: fixes q :: "real" shows "fds_perzeta q = fds (\<lambda>m. exp (complex_of_real (2 * pi) * \<i> * of_nat m * complex_of_real q))"
inorder_insert: fixes t :: "'b tree234"   and a :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (insert a t) = ins_list a (inorder t)"
well_order_induct_imp: fixes r :: "('d \<times> 'd) set"   and P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "wo_rel r"   and "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x" shows "a \<in> Field r \<longrightarrow> P a"
oid_simps: fixes C :: "CTypeId"   and a :: "nat" shows "oid (objV C a) = a"
d_1: fixes p :: "complex poly" shows "roots_of_complex_main p \<equiv> let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then croots2 p else complex_roots_of_rat_poly (map_poly to_rat p)"
simps_2: fixes s :: "dB"   and t :: "dB"   and k :: "nat" shows "lift (s \<degree> t) k = lift s k \<degree> lift t k"
assms_2: shows "v\<^sub>2 \<in> carrier_vec n"
bdd_above_is_intervalI: fixes I :: "real set"   and a :: "real"   and b :: "real" assumes "is_interval I"   and "a \<le> b"   and "a \<in> I"   and "b \<notin> I" shows "bdd_above I"
sum_defs_1: fixes A :: "hf"   and B :: "hf" shows "A + B = \<lbrace>0\<rbrace> * A \<squnion> \<lbrace>1\<rbrace> * B"
fv\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
map2_val_compat_map_val_pair: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "tc.map2_val_compat (map_val_pair f)"
irreducible_smult_field: fixes c :: "'b"   and p :: "'b poly" shows "irreducible (Polynomial.smult c p) = (c \<noteq> (0::'b) \<and> irreducible p)"
assms_8: fixes i :: "nat" shows "emeasure M (A i) \<noteq> \<infinity>"
extend_set_inverse: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and C :: "'d set" assumes "Extend h" shows "project_set h (extend_set h C) = C"
inject_2: fixes x2 :: "'l"   and y2 :: "'l" shows "(Ok x2 = Ok y2) = (x2 = y2)"
infer_complete: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and M :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> M : \<tau>" shows "infer \<Gamma> M = Some \<tau>"
mult1_lessE: fixes N :: "'a multiset"   and M :: "'a multiset" assumes "(N, M) \<in> mult1 {(x, y). x < y}" obtains a :: "'a" and M0 :: "'a multiset" and K :: "'a multiset" where "M = add_mset a M0" and "N = M0 + K" and "a \<notin># K" and "\<And>b. b \<in># K \<Longrightarrow> b < a"
simps_9: fixes f :: "'f"   and g :: "'d set"   and c :: "('d, 'e, 'f) com" shows "flatten (Guard f g c) = [Guard f g c]"
carT2_def: shows "carT2 = {(Kl, lab) |Kl lab. isTree Kl lab \<and> isNode2 Kl lab []}"
list_lens_def: fixes i :: "nat" shows "list_lens i = \<lparr>lens_get = \<lambda>xs. nth' xs i, lens_put = \<lambda>xs. list_augment xs i\<rparr>"