hcju/isabelleps huggingface dataset:
error_free_LVar_lemma: fixes s :: "state"   and vn :: "lname"   and w :: "val" assumes "error_free s" shows "error_free (assign (\<lambda>v. supd lupd(vn\<mapsto>v)) w s)"
assms_4: shows "\<forall>(k, \<gamma>)\<in>one_chain. valid_path \<gamma>"
beta_eta_appL: fixes s :: "dB"   and s' :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* s'" shows "s \<degree> t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* s' \<degree> t"
pts_symbolic_c_preserves_wf_prot: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>" shows "wf\<^sub>s\<^sub>t\<^sub>s' \<S>' \<A>'"
in_prime_factors_imp_prime: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and x :: "'c" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "p \<in># factorial_semiring.prime_factorization zero times one normalize x" shows "normalization_semidom.prime zero times one normalize p"
CE_sound: fixes var :: "'sc"   and \<phi> :: "('sf, 'sc, 'sz) formula"   and \<psi> :: "('sf, 'sc, 'sz) formula" shows "CE_holds var \<phi> \<psi>"
Abstract_def: shows "A! \<equiv> \<^bold>\<lambda>x. \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<^sup>P\<rparr>"
vcons_eq_iff: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence xs\<^sub>1"   and "vfsequence xs\<^sub>2" shows "(xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2) = (xs\<^sub>1 = xs\<^sub>2 \<and> x\<^sub>1 = x\<^sub>2)"
hassoc_map_of_filter_all: fixes p :: "('a \<times> 'b) list"   and k :: "'a" shows "map_of p |` (- {k}) = map_of (filter (\<lambda>(a, b). a \<noteq> k) p)"
1: shows "|pick ` XS| <o |UNIV| \<or> finite (pick ` XS)"
support_on_def: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "support_on S f = {x \<in> S. f x \<noteq> (0::'e)}"
sq_mtx_diag_simps_2: fixes i :: "'a"   and j :: "'a"   and f :: "'a \<Rightarrow> real" assumes "i \<noteq> j" shows "sq_mtx_diag f $$ i $ j = 0"
comp_ide_arr: fixes b :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "ide b"   and "seq b f" shows "b \<cdot> f = f"
compat: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "delta_gt \<delta> y z" shows "delta_gt \<delta> x z"
rep_eq: fixes x :: "'a dual"   and xa :: "'a dual" shows "(x < xa) = (undual xa < undual x)"
store_sub2_result: fixes instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and asi :: "word8"   and address :: "word32"   and s :: "'b sparc_state" shows "snd (store_sub2 instr curr_win rd asi address s) = False"
UN_mono: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> 'd set"   and g :: "'c \<Rightarrow> 'd set" assumes "A \<subseteq> B"   and "\<And>x. s2p A x \<Longrightarrow> f x \<subseteq> g x" shows "\<mu> (\<P> f A) \<subseteq> \<mu> (\<P> g B)"
ord_eq_le_subst: fixes a :: "'c"   and f :: "'d \<Rightarrow> 'c"   and b :: "'d"   and c :: "'d" assumes "a = f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
inv_state: fixes f :: "nat \<Rightarrow> 'o"   and n :: "nat" shows "inv (state f n)"
example_test_tools: shows "test_tools M\<^sub>S M\<^sub>I FAIL\<^sub>I PM\<^sub>I V\<^sub>I \<Omega>\<^sub>I"
plus_nth_eucl: fixes x :: "'b"   and y :: "'b"   and i :: "nat" shows "(x + y) $\<^sub>e i = x $\<^sub>e i + y $\<^sub>e i"
2: shows "lselect s w = w ?! least s % lselect (s - {least s}) w"
neq_no_wait_locksE: fixes ln :: "'a \<Rightarrow>f nat" assumes "ln \<noteq> no_wait_locks" obtains l :: "'a" where "0 < ln $ l"
induct: fixes P :: "nat \<Rightarrow> term \<Rightarrow> term \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "term"   and a2 :: "term" assumes "\<And>lev i t. P lev (Bound i) t"   and "\<And>lev t\<^sub>1 t\<^sub>2 t. \<lbrakk>P lev t\<^sub>1 t; P lev t\<^sub>2 t\<rbrakk> \<Longrightarrow> P lev (t\<^sub>1 $ t\<^sub>2) t"   and "\<And>lev u t. P (lev + 1) u t \<Longrightarrow> P lev (\<Lambda> u) t"   and "\<And>uu_ v uv_. P uu_ (Const v) uv_"   and "\<And>uu_ v uv_. P uu_ (Free v) uv_" shows "P a0 a1 a2"
takeWhile_tail: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list"   and l :: "'g list" assumes "\<not> P x" shows "takeWhile P (xs @ x # l) = takeWhile P xs"
fds_nth_deriv: fixes f :: "'b fds"   and n :: "nat" shows "fds_deriv f $ n = - ln (real n) *\<^sub>R f $ n"
lhd_def: fixes llist :: "'b llist" shows "lhd llist = (case llist of LCons x21 x22 \<Rightarrow> x21)"
kstar_def: fixes f :: "'a \<Rightarrow> 'a set" shows "kstar f = \<Squnion> range (kpower f)"
tfr_stp_all_split_5: fixes S :: "('fun, 'var) strand_step list"   and x :: "('fun, 'var) strand_step"   and S' :: "('fun, 'var) strand_step list" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ x # S')" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (S @ S')"
Red1CastFail: fixes s :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and U :: "ty"   and P :: "'addr J1_prog"   and T :: "ty"   and uf :: "bool"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "\<not> P \<turnstile> U \<le> T" shows "uf,P,t \<turnstile>1 \<langle>Cast T (Val v),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt ClassCast),s\<rangle>"
degree_add_eq_left: fixes q :: "'a poly"   and p :: "'a poly" assumes "degree q < degree p" shows "degree (p + q) = degree p"
space_T: fixes s :: "'s" shows "space (K.T s) = space (stream_space (count_space UNIV))"
mono_Ndet2: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P |-| S) \<longrightarrow> Ra (P |-| S) s = Ra (Q |-| S) s"
c: shows "abc_steps_l (0, lm) (A [+] B) na = (sa, lma)"
UNpart_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "UNpart cl P \<equiv> \<Union>n<length cl. local.shift cl n ` P n"
cut_less_subset_mono: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "A \<subseteq> B" shows "A \<down>< t \<subseteq> B \<down>< t"
tranclp_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "less\<^sup>+\<^sup>+ = less"
FAccNull: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
pos_least: fixes n :: "nat"   and rs :: "'a stream"   and r :: "'a" assumes "shd (sdrop n rs) = r" shows "pos rs r \<le> n"
axioms_1: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "max_lideal I R addition multiplication zero unit" shows "lideal I R addition multiplication zero unit"
assms: shows "\<pi> =\<^bsub>k\<^esub> \<pi>'"
trcl_pair_induct: fixes xc1 :: "'a"   and xc2 :: "'b"   and xb :: "'c list"   and xa1 :: "'a"   and xa2 :: "'b"   and t :: "(('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c list \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "((xc1, xc2), xb, xa1, xa2) \<in> trcl t"   and "\<And>a b. P a b [] a b"   and "\<And>a b aa ab ba w ac bb. \<lbrakk>((a, b), aa, ab, ba) \<in> t; ((ab, ba), w, ac, bb) \<in> trcl t; P ab ba w ac bb\<rbrakk> \<Longrightarrow> P a b (aa # w) ac bb" shows "P xc1 xc2 xb xa1 xa2"
l1_inv1I: fixes x :: "l1_state" assumes "\<And>Ra A B K. 0 < l1_state.signals x (Commit A B \<langle>epubKF (Ra $ kE), K\<rangle>) \<Longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress x Ra = Some {xpkE, xskE, xsk} \<and> guessed_frame Ra xsk = Some K" shows "x \<in> l1_inv1"
inverse_incidence_sys_iso: shows "incidence_system_isomorphism \<V>' \<B>' \<V> \<B> (inv_into \<V> \<pi>)"
divides_ff_mult_cancel_left: fixes z :: "'b fract"   and x :: "'b fract"   and y :: "'b fract" shows "divides_ff (z * x) (z * y) = (z = 0 \<or> divides_ff x y)"
eq_nextl_def: shows "eq_nextl \<equiv> {(u, v). nextl (init M) u = nextl (init M) v}"
core_closure_diff_empty: fixes A :: "'b set"   and T :: "'b set set" assumes "A \<subseteq> carrier\<^bsub>T\<^esub>" shows "core\<^bsub>T\<^esub> (closure\<^bsub>T\<^esub> A - A) = {}"
size_fsubset: fixes x2 :: "'a fset"   and x1 :: "'a fset" assumes "x2 |\<subset>| x1" shows "size x2 < size x1"
arr_Rel_length: shows "vcard T = 3\<^sub>\<nat>"
uminus_l_inv_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A + A = 0\<^sub>m nr nc"
map_zip_map: fixes f :: "'b \<times> 'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "'d list"   and ys :: "'c list" shows "map f (zip (map g xs) ys) = map2 (\<lambda>x y. f (g x, y)) xs ys"
NSCauchy_NSBseq: fixes X :: "nat \<Rightarrow> 'a" assumes "NSCauchy X" shows "NSBseq X"
assms_4: shows "alpha_rp.more r = alpha_rp.more r'"
uint_power_lower: fixes n :: "nat" assumes "n < LENGTH('b)" shows "uint (2 ^ n) = 2 ^ n"
UnCI: fixes c :: "'d"   and B :: "'d set"   and A :: "'d set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
exec_step_ind_Invokestatic_Done: fixes ps :: "val list"   and n :: "nat"   and stk :: "val list"   and D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and m :: "jvm_method"   and ics :: "init_call_status"   and sh :: "sheap"   and sfs :: "sfields"   and f' :: "frame"   and h :: "heap"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "ps = take n stk"   and "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D"   and "\<forall>Cs. ics \<noteq> Called Cs"   and "sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "f' = ([], rev ps @ replicate mxl\<^sub>0 undefined, D, M, 0, No_ics)" shows "exec_step_ind (StepI (Invokestatic C M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, f' # (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
assoc_simps_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.arr \<a>\<^sub>B[f, g, h]"
CptnModOne: fixes P :: "'b com option"   and s :: "'b" shows "[(P, s)] \<in> cptn_mod"
roofed_greaterI: fixes x :: "'k"   and S :: "'k set"   and G :: "('k, 'l) graph_scheme"   and B :: "'k set" assumes "x \<in> S" shows "x \<in> roofed_gen G B S"
Gateway_L8a: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and x :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "\<forall>j\<le>2 * d. lose (t + j) = [False]"   and "ack t = [sending_data]"   and "\<forall>t3\<le>t + d. a t3 = []"   and "x \<le> d + d" shows "ack (t + x) = [sending_data]"
supporters_auxdef: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" shows "supporters R x y = {i. R i y x}"
drinks_rejects_future: fixes d :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" shows "\<not> recognises_execution drinks 2 d ((l, i) # t)"
iT_Minus_Plus_assoc: fixes I :: "nat set"   and k :: "nat"   and n :: "nat" assumes "finite I"   and "Max I \<le> k" shows "k \<ominus> I \<oplus> n = k + n \<ominus> I"
resolution_classD2: fixes S :: "'a set set"   and bl :: "'a set" assumes "resolution_class S"   and "bl \<in> S" shows "bl \<in># \<B>"
mono: shows "A $$ (k, i) * B $$ (i, j) \<succ> A $$ (k, i) * C $$ (i, j)"
S_eq: shows "S = count_space (X ` space M \<times> Y ` space M)"
spr_trans_def: fixes agents :: "'b odlist"   and jkbp :: "'b \<Rightarrow> ('b, 'e, 'f) GC list"   and envAction :: "('b, 'c, 'd) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'f) \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> ('b, 'c, 'd) BEState"   and envVal :: "('b, 'c, 'd) BEState \<Rightarrow> 'e \<Rightarrow> bool"   and envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'h \<times> 'd option" shows "spr_trans agents jkbp envAction envTrans envVal envObs \<equiv> \<lambda>cec X. concat (map (\<lambda>(initialS, finalS). concat (map (\<lambda>eact. map (Pair initialS) (map (\<lambda>aact. envTrans eact aact finalS) (listToFuns (spr_jAction jkbp envVal envObs cec (initialS, finalS)) (toList agents)))) (envAction finalS))) (toList X))"
roots_of_3_impl_def: fixes p :: "int poly" shows "roots_of_3_impl p = real_alg_2_list_convert (roots_of_2_list p)"
phiDefs_in_allDefs: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> phiDefs g n" shows "v \<in> allDefs g n"
induct: fixes x1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and x2 :: "'e"   and x3 :: "('f \<Rightarrow> 'g) set"   and x4 :: "'e"   and x5 :: "('f \<Rightarrow> 'g) set"   and P :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint) \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) set \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) set \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto> \<langle>x4, x5\<rangle>"   and "\<And>A l Z. P A l Z l ((Z \<inter> {u. u \<turnstile> inv_of A l})\<^sup>\<up> \<inter> {u. u \<turnstile> inv_of A l})"   and "\<And>A l g a r l' Z. A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l' \<Longrightarrow> P A l Z l' ((Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'})" shows "P x1 x2 x3 x4 x5"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "Left_a.arr f"   and "Left_a.dom f = a" shows "Left_a.comp f a = f"
leq_trans: fixes x :: "'b spmf"   and y :: "'b spmf"   and z :: "'b spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y z" shows "ord_spmf (=) x z"
simps_5: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> and\<^sub>n \<psi> = (\<xi> \<Turnstile>\<^sub>n \<phi> \<and> \<xi> \<Turnstile>\<^sub>n \<psi>)"
analz_Un_implAuthSet_aux_aux: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz (implAuthSet_aux G \<union> H) \<subseteq> implAuthSet_aux G \<union> synth (analz (G \<union> H))"
top_empty_eq: shows "top_class.top = (\<lambda>x. x \<in> UNIV)"
nb\<^sub>1: shows "dom (\<Pi>\<^sub>I) = set (\<Pi>\<^sub>\<V>)"
IH1_perm: assumes "lo \<le> p - Suc 0"   and "p - Suc 0 < length xs'" shows "set (WB_Sort.sublist xs'' lo (p - Suc 0)) = set (WB_Sort.sublist xs' lo (p - Suc 0))"
sort_oalist_def: fixes x :: "'o"   and xs :: "('a \<times> 'd) list" shows "sort_oalist x xs = foldr (update_by_pair x) xs []"
properfactor_mult_lI: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "monoid_cancel G"   and "properfactor G a b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G (c \<otimes>\<^bsub>G\<^esub> a) (c \<otimes>\<^bsub>G\<^esub> b)"
mult_less_cancel_left2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times c a) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
conv_radius_eqI: fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "0 \<le> R"   and "\<And>r. \<lbrakk>0 < r; ereal r < R\<rbrakk> \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)"   and "\<And>r. \<lbrakk>0 < r; R < ereal r\<rbrakk> \<Longrightarrow> \<exists>z. norm z = r \<and> \<not> summable (\<lambda>n. norm (f n * z ^ n))" shows "conv_radius f = R"
pos_fun_distr: fixes R :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and R' :: "'g \<Rightarrow> 'e \<Rightarrow> bool"   and S' :: "'h \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun R S OO rel_fun R' S' \<le> rel_fun (R OO R') (S OO S')"
pos_id_def: fixes k :: "int" shows "pos_id k = (if k < 0 then 0 else k)"
min_galleryI_chamber_distance_betw: fixes C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "codomain.gallery (C # Cs @ [D])"   and "Suc (length Cs) = codomain.chamber_distance C D" shows "codomain.min_gallery (C # Cs @ [D])"
prems: shows "\<forall>a\<in>set (f # fs). is_lit_plus a"
suminf_le_const: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "summable f"   and "\<And>n. sum f {..<n} \<le> x" shows "suminf f \<le> x"
eq_sym: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>L \<psi>" shows "\<psi> \<sim>\<^sub>L \<phi>"
B: fixes ff :: "'b list \<Rightarrow> 'b list"   and A :: "'a" shows "S (y_update ff A) = x_update ff (S A)"
d_absorb2: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> (d x + d y) = d x"
prod_right_def: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bounded_cbilinear prod" shows "bounded_cbilinear.prod_right prod \<equiv> map_fun id CBlinfun prod"
mult_commute: fixes a :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
words_infI_construct: fixes w :: "'action stream"   and p :: "'state" assumes "\<And>v. v \<le>\<^sub>F\<^sub>I w \<Longrightarrow> path v p" shows "run w p"
can_syncE: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap"   and LT :: "('addr + 'thread_id + 'thread_id) set" assumes "execd_mthr.can_sync P t x m LT" obtains ta :: "('addr, 'thread_id, 'heap) jvm_thread_action" and x' :: "'addr option \<times> 'addr frame list" and m' :: "'heap" where "execd_mthr.r_syntax P t x m ta x' m'" and "LT = collect_waits ta"
Init_extend: fixes h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program" shows "Init (extend h F) = extend_set h (Init F)"
finite_Sup: fixes X :: "'b set set" assumes "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<subseteq> z)" shows "finite (\<Union> X)"
LtK_Pair: fixes X :: "ltkey"   and X' :: "msg"   and Y' :: "msg" shows "LtK X \<noteq> \<langle>X', Y'\<rangle>"
assms: shows "learn_fin \<psi> U (\<phi> i)"
hyps_10: shows "A\<^sub>P \<sharp>* M"
dom_closed: fixes f :: "'a \<times> 'a" assumes "B.arr (fst f) \<and> B.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (snd f)" shows "B.arr (fst (B.VxV.dom f)) \<and> B.arr (snd (B.VxV.dom f)) \<and> src\<^sub>B (fst (B.VxV.dom f)) = trg\<^sub>B (snd (B.VxV.dom f))"
c': fixes x :: "real" shows "pbernpoly n x \<le> c"
sub_branch_id: fixes branch :: "(('b, 'c) fm list \<times> 'c) list" shows "sub_branch id branch = branch"
gHom_func: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "f \<in> carrier F \<rightarrow> carrier G"
degree_Poly: fixes xs :: "'a list" shows "degree (Poly xs) \<le> length xs"
diff_cancel: fixes a :: "'a" shows "a - a = (0::'a)"
adds_poly_mappingI: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "lookup s \<le> lookup t" shows "s adds t"
east_west_tangents_far_north: shows "proj2_intersection (polar east) (polar west) = far_north"
predicate_prob_positive: fixes P :: "complex mat"   and \<rho> :: "complex mat" assumes "0\<^sub>m d d \<le>\<^sub>L P"   and "\<rho> \<in> density_states" shows "0 \<le> trace (P * \<rho>)"
e2: shows "pdevs_val e (fst (trunc_bound_pdevs p (scaleR_pdevs alpha (snd X)))) = pdevs_val e (scaleR_pdevs alpha (snd X)) + e2"
simps_2: fixes x :: "char list"   and a :: "'b" shows "Malformed\<lparr>x:a\<rparr> = Malformed"
that: fixes bl2 :: "'a set" assumes "b = bl2 - bl"   and "bl2 \<in># remove1_mset bl \<B>" shows "thesis"
pratt_triple_extend: fixes q :: "nat"   and p :: "nat"   and a :: "nat"   and y :: "nat"   and x :: "nat" assumes "prime q"   and "pratt_triple p a y"   and "1 < p"   and "0 < x"   and "x = q * y"   and "[a ^ ((p - 1) div q) \<noteq> 1] (mod p)" shows "pratt_triple p a x"
knows_subset_knows_Says: fixes A :: "agent"   and evs :: "event list"   and A' :: "agent"   and B :: "agent"   and X :: "msg" shows "knows A evs \<subseteq> knows A (Says A' B X # evs)"
invertible_right_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "u \<cdot> inverse u = \<one>"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 r f"
defined_args_valid: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<delta> x->excludes\<^sub>S\<^sub>e\<^sub>t(y)) = (\<tau> \<Turnstile> \<delta> x \<and> \<tau> \<Turnstile> \<upsilon> y)"
commutative_square_def: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" shows "C.commutative_square f g h k \<equiv> C.cospan f g \<and> C.span h k \<and> C.dom f = C.cod h \<and> f \<cdot> h = g \<cdot> k"
quadratic_form_mat: fixes v :: "complex vec"   and m :: "nat"   and A :: "complex mat" assumes "dim_vec v = m"   and "A \<in> carrier_mat m m" shows "inner_prod v (A *\<^sub>v v) = (\<Sum>i = 0..<m. \<Sum>j = 0..<m. conjugate (v $ i) * A $$ (i, j) * v $ j)"
validVi: shows "isViewOn \<V>i E\<^bsub>ESi\<^esub>"
ipcidr_conjunct_valid: fixes p1 :: "'a word \<times> nat"   and p2 :: "'a word \<times> nat"   and p :: "'a word \<times> nat" assumes "valid_prefix_fw p1"   and "valid_prefix_fw p2"   and "ipcidr_conjunct p1 p2 = Some p" shows "valid_prefix_fw p"
redT_updW_preserve_wset_thread_ok: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'d"   and wa :: "('d, 'e) wait_set_action"   and ws' :: "'d \<Rightarrow> 'e wait_set_status option"   and xln :: "'f \<times> 'g \<Rightarrow>f nat" assumes "wset_thread_ok ws ts"   and "redT_updW t ws wa ws'"   and "ts t = \<lfloor>xln\<rfloor>" shows "wset_thread_ok ws' ts"