hcju/isabelleps huggingface dataset:
crel_vs_elim: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and v :: "'d"   and v\<^sub>T :: "('mem, 'e) state"   and M :: "'mem" assumes "crel_vs R v v\<^sub>T"   and "cmem M"   and "P M" obtains v' :: "'e" and M' :: "'mem" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "cmem M'" and "P M'"
pr_auth_data_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and b :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Auth_Data (priAK n) b \<in> parts (A \<union> spies evs)" shows "\<exists>m run. IntMapK (S (Card m, m, run)) = Some b"
assms_4: shows "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
module_vec_simps_1: fixes ty :: "'a itself"   and n :: "nat" shows "(\<oplus>\<^bsub>module_vec ty n\<^esub>) = (+)"
index_mult_mat_1: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and B :: "'a mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = row A i \<bullet> col B j"
C: shows "OpposedThinChamberComplexFoldings X f g C"
select_convs_4: fixes Functions :: "'b \<Rightarrow> (real, 'a) vec \<Rightarrow> real"   and Predicates :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> bool"   and Contexts :: "'c \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set \<Rightarrow> ((real, 'a) vec \<times> (real, 'a) vec) set"   and Programs :: "'a \<Rightarrow> (((real, 'a) vec \<times> (real, 'a) vec) \<times> (real, 'a) vec \<times> (real, 'a) vec) set"   and ODEs :: "'a \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and ODEBV :: "'a \<Rightarrow> 'a set"   and more :: "'d" shows "Programs \<lparr>Functions = Functions, Predicates = Predicates, Contexts = Contexts, Programs = Programs, ODEs = ODEs, ODEBV = ODEBV, \<dots> = more\<rparr> = Programs"
IndP_cong: fixes H :: "fm set"   and x :: "tm"   and x' :: "tm" assumes "H \<turnstile> x EQ x'" shows "H \<turnstile> IndP x IFF IndP x'"
evenperm_inv: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "evenperm (inv p) = evenperm p"
is_word_Derives1: fixes a :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "is_word a"   and "Derives1 a i r b" shows "False"
simps: fixes \<C> :: "t_context"   and es :: "b_e list"   and tn :: "t list"   and tm :: "t list" shows "b_e_type_checker \<C> es (tn _> tm) = c_types_agree (check \<C> es (Type tn)) tm"
Bernstein_def: fixes n :: "nat"   and k :: "nat"   and x :: "real" shows "Bernstein n k x \<equiv> real (n choose k) * x ^ k * (1 - x) ^ (n - k)"
assms: shows "(\<lambda>x. real (nth_prime x)) \<sim>[sequentially] (\<lambda>x. real x * ln (real x))"
states_of_se_assign2: fixes c :: "('c, 'd) conf"   and v :: "'c"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and c' :: "('c, 'd) conf"   and \<sigma>' :: "'c \<Rightarrow> 'd" assumes "se c (Assign v e) c'"   and "\<exists>\<sigma>\<in>states c. \<sigma>' = \<sigma>(v := e \<sigma>)" shows "\<sigma>' \<in> states c'"
simps_1: fixes x :: "'b" shows "insert x \<langle>\<rangle> = \<langle>\<langle>\<rangle>, (x, 1), \<langle>\<rangle>\<rangle>"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "rel_rose_tree (Grp A f) = Grp {x. set_rose_tree x \<subseteq> A} (map_rose_tree f)"
zimage_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c zet" shows "zimage f A \<equiv> Abs_zet (f ` Rep_zet A)"
mu_le: fixes n :: "nat" shows "mu u n \<le> f n"
C\<^sub>1sat: shows "eval\<^sub>c F G C\<^sub>1"
source: fixes S1 :: "'procS"   and S2 :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "(S1, S2) \<in> SRel" shows "SourceTerm S1 \<R>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> SourceTerm S2"
l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs get_ancestors_si Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs set_disconnected_nodes set_disconnected_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs Shadow_DOM.get_owner_document Shadow_DOM.insert_before Shadow_DOM.insert_before_locs ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs"
orientable_irreflexive: fixes x :: "'a" assumes "orientable x" shows "irreflexive x"
double_ccos_square: fixes a :: "real" shows "2 * complex_of_real (cos a) * complex_of_real (cos a) = complex_of_real (cos (2 * a)) + 1"
nAct_active_suc: fixes t :: "cnf llist"   and n' :: "enat"   and n :: "nat"   and i :: "nat"   and c :: "'id" assumes "\<not> lfinite t \<or> n' - 1 < llength t"   and "n \<le> i"   and "enat i < n'"   and "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"   and "\<forall>i'. n \<le> i' \<and> enat i' < n' \<and> enat i' < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t i'\<^esub> \<longrightarrow> i' = i" shows "\<langle>c #\<^bsub>n'\<^esub>t\<rangle> = eSuc \<langle>c #\<^bsub>enat n\<^esub>t\<rangle>"
h3: shows "h2 \<turnstile> set_disconnected_nodes owner_document (remove1 node disconnected_nodes_h2) \<rightarrow>\<^sub>h h3"
swapEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "rho &[x \<and> y]_zs &[x' \<and> y']_zs' = rho &[x' \<and> y']_zs' &[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']_zs"
id_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(A ===> A) id id"
abs_eq: fixes xa :: "'a"   and x :: "'a" shows "(dual xa < dual x) = (x < xa)"
starfun2_star_of: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'c" shows "(*f2* f) (star_of x) = *f* f x"
ftf_syn21: fixes a :: "'c event"   and u :: "'c event list"   and t :: "'c event list"   and A :: "'c event set" shows "(a \<in> set u \<and> a \<notin> set t \<or> a \<in> set t \<and> a \<notin> set u) \<and> a \<in> A \<longrightarrow> setinterleaving (u, A, t) = {}"
projection_in_image: fixes M :: "'b set"   and h :: "'b" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "projection M h \<in> M"
eta_reducible_Abs1: fixes T :: "typ"   and t :: "Core.term" assumes "\<not> eta_reducible (Abs T (t $ Bv 0))" shows "\<not> eta_reducible t"
eval_var: fixes x :: "'a" assumes "x \<in> carrier R" shows "local.eval X x = x"
f': shows "\<forall>(x, K)\<in>p. f (x, K) \<notin> s \<and> f (x, K) \<in> K"
diameter_ball: fixes a :: "'b"   and r :: "real" shows "diameter (ball a r) = (if r < 0 then 0 else 2 * r)"
\<iota>_inf: shows "\<iota> \<in> Inf_FL"
cong: fixes f :: "bool list \<Rightarrow> bool list \<Rightarrow> bool list" shows "otp.decrypt f \<equiv> otp.decrypt f"
L': shows "((\<lambda>x. EM_remainder' n (\<lambda>x. cnj (f x)) (real_of_int a) (real_of_int x)) \<longlongrightarrow> cnj L) at_top"
VDM_Sound_n: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> c : A" shows "\<forall>n. G \<Turnstile>\<^sub>n c : A"
match_tcp_flags_conjunct: fixes f1 :: "ipt_tcp_flags"   and f2 :: "ipt_tcp_flags"   and pkt :: "tcp_flag set" shows "match_tcp_flags (match_tcp_flags_conjunct f1 f2) pkt = (match_tcp_flags f1 pkt \<and> match_tcp_flags f2 pkt)"
assms_2: fixes n :: "nat" assumes "N \<le> n" shows "norm (f (Suc n)) \<le> c * norm (f n)"
Diff: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a - b \<in> sets M"
connected_imp_perfect: fixes S :: "'b set"   and a :: "'b" assumes "connected S"   and "a \<in> S"   and "\<And>x. S \<noteq> {x}" shows "a islimpt S"
0: shows "(\<lambda>(f, y). f(n := y)) \<circ> (\<lambda>xa. (xa, x)) = (\<lambda>f. f(n := x))"
pre_subdivFace_pre_subdivFace': fixes v :: "nat"   and f :: "face"   and g :: "graph"   and vo :: "nat option"   and vol :: "nat option list" assumes "v \<in> \<V> f"   and "pre_subdivFace g f v (vo # vol)" shows "pre_subdivFace' g f v v 0 vol"
coboundedI: fixes A :: "'c DBMEntry set"   and a :: "'c DBMEntry" assumes "finite A"   and "a \<in> A" shows "a \<preceq> linordered_monoid.Max A"
asymptotics8: shows "\<forall>\<^sub>F x in at_top. 1 < x * (1 - b - hb / ln x powr (1 + e))"
ASSERT_leI: fixes \<Phi> :: "bool"   and M :: "'c nres"   and M' :: "'c nres" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> M \<le> M'" shows "ASSERT \<Phi> \<bind> (\<lambda>_. M) \<le> M'"
assoc: fixes a :: "'g"   and b :: "'g"   and c :: "'g" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
intros: fixes p :: "'a \<times> 'b" shows "sndsp p (snd p)"
mult_unit_dvd_iff': fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a dvd (1::'b)" shows "(a * b dvd c) = (b dvd c)"
vimage_eq: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" shows "(a \<in> f -` B) = (f a \<in> B)"
is_top_sorted_def: fixes R :: "('b \<times> 'b) set"   and l :: "'b list" shows "is_top_sorted R l \<equiv> list_before_rel l \<inter> (R\<^sup>*)\<inverse> = {}"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "ring_hom hom"   and "zero_hom_0 hom" shows "inj_ring_hom hom"
B_def: shows "B = (\<lambda>x. (- fst x, snd x)) -` A"
f_Exec_Stream_Acc_Output_take: fixes n :: "nat"   and k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c \<down> n = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (xs \<down> n) c"
Subobjs_Rep: 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 C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "Subobjs P C Cs"
approximating_semantics_iff_fun_good_ruleset: fixes rs :: "'a rule list"   and \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and p :: "'b"   and s :: "state"   and t :: "state" assumes "good_ruleset rs" shows "\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t = (approximating_bigstep_fun \<gamma> p rs s = t)"
assms: shows "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
fdr_step_unfold: fixes v :: "nat"   and c :: "nat" shows "local.fdr_step v c = (if v = 0 then return_pmf None else if n \<le> v then return_spmf (v, c) else coin_spmf \<bind> (\<lambda>b. local.fdr_step (2 * v) (2 * c + (if b then 1 else 0))))"
sorted_job_lower_bound_machine: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j"   and "m < j"   and "sorted j" shows "\<exists>x\<in>{1..m}. 2 * t j \<le> T x"
assms_2: assumes "A \<subseteq>\<^sub>\<circ> B"   and "B \<subseteq>\<^sub>\<circ> A" shows "P"
NewInit: fixes sh :: "sheap"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and l :: "sfields"   and v' :: "val"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and a :: "nat"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h'' :: "heap" assumes "\<nexists>sfs. sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C ([C],False) \<leftarrow> unit,(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v',(h', l', sh')\<rangle>"   and "new_Addr h' = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h'' = h'(a \<mapsto> blank P C)" shows "P \<turnstile> \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>addr a,(h'', l', sh')\<rangle>"
elims_2: fixes x :: "'a \<times> nat"   and xa :: "'a \<times> nat" assumes "happened_immediately_before x xa" obtains i :: "'a \<times> nat" and j :: "'a \<times> nat" where "x = i" and "xa = j" and "is_valid_event_id i \<and> is_valid_event_id j \<and> (fst i = fst j \<and> Suc (snd i) = snd j \<or> is_reception i j)"
elims: fixes x :: "'a extended"   and y :: "'a extended" assumes "- x = y" obtains xa :: "'a" where "x = Fin xa" and "y = Fin (- xa)"   | "x = \<infinity>" and "y = -\<infinity>"   | "x = -\<infinity>" and "y = \<infinity>"
higher_deriv_Polygamma: fixes z :: "'a"   and n :: "nat"   and m :: "nat" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(deriv ^^ n) (Polygamma m) z = Polygamma (m + n) z"
Rep_FingerTree_invar: fixes t :: "('a, 'b) FingerTree" shows "FingerTreeStruc.ft_invar (Rep_FingerTree t)"
plus_inf: fixes f :: "atom fm"   and xs :: "'a list" assumes "nqfree f" shows "\<exists>x. \<forall>y\<ge>x. DLO.I (inf\<^sub>+ f) xs = DLO.I f (y # xs)"
subst_subst_back: fixes x :: "Vars.var"   and e :: "exp"   and y :: "Vars.var" assumes "atom x \<sharp> e" shows "e[y::=x][x::=y] = e"
reciprocal_monom: fixes n :: "nat"   and p :: "nat"   and a :: "'a" assumes "n \<le> p" shows "reciprocal_poly p (monom a n) = monom a (p - n)"
ran_map_of_zip: fixes xs :: "'b list"   and ys :: "'c list" assumes "length xs = length ys"   and "distinct xs" shows "ran (map_of (zip xs ys)) = set ys"
div_free_is_lifelock_free_v2: fixes P :: "'b process" shows "lifelock_free_v2 P = (D P = {})"
Reach_def: fixes E :: "(nat \<times> nat) list" shows "Reach E \<equiv> {x. (\<exists>path. 1 < length path \<and> path ! (length path - 1) \<in> Roots \<and> x = path ! 0 \<and> (\<forall>i<length path - 1. \<exists>j<length E. E ! j = (path ! (i + 1), path ! i))) \<or> x \<in> Roots}"
trms_of_deg_leq_0: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "trms_of_deg_leq 0 f = monom P (f 0) 0"
denote_while_n_sum_dim: fixes \<rho> :: "complex mat"   and M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat" assumes "\<rho> \<in> carrier_mat d d"   and "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<And>\<rho>'. \<lbrakk>\<rho>' \<in> carrier_mat d d; partial_density_operator \<rho>'\<rbrakk> \<Longrightarrow> positive (DS \<rho>') \<and> trace (DS \<rho>') \<le> trace \<rho>' \<and> DS \<rho>' \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<in> carrier_mat d d"
open_segment_commute: fixes a :: "'b"   and b :: "'b" shows "open_segment a b = open_segment b a"
fp_cop_F_def: fixes ds :: "'d set" shows "fp_cop_F ds = COP.fixp_above ds {}"
float_val_ge_largest: fixes a :: "('a, 'b) IEEE.float" assumes "1 < LENGTH('a)"   and "is_finite a" shows "- largest TYPE(('a, 'b) IEEE.float) \<le> valof a"
jointfun_hom0: fixes f :: "nat \<Rightarrow> 'e"   and n :: "nat"   and A :: "'e set"   and g :: "nat \<Rightarrow> 'e"   and m :: "nat"   and B :: "'e set" assumes "f \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {k. k \<le> m} \<rightarrow> B" shows "jointfun n f m g \<in> {l. l \<le> Suc (n + m)} \<rightarrow> A \<union> B"
N: shows "partn_lst {..<N} [m, n] 2"
1: shows "w1 = basic (make_vert_block (nat (domain_block B))) \<circ> basic B \<and> w2 = basic B \<circ> basic (make_vert_block (nat (codomain_block B))) \<and> domain_block B \<noteq> 0"
minus_zero_does_nothing: fixes x :: "'b list"   and y :: "'b list" shows "minus_poly_rev_list x (map ((*) (0::'b)) y) = x"
assms_2: shows "splits_at \<alpha> i a1 A a2"
snd_divmod_abs: fixes k :: "integer"   and l :: "integer" shows "snd (Code_Numeral.divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
p_subid_glb: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> (0::'a) \<le> x \<cdot> (0::'a) \<and> z \<cdot> (0::'a) \<le> y \<cdot> (0::'a)" shows "z \<cdot> (0::'a) \<le> x \<cdot> (0::'a) \<parallel> (y \<cdot> (0::'a))"
infer_fstE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and P :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (Fst P) = Some \<tau>" shows "\<exists>T S. infer \<Gamma> P = Some (TPair T S) \<and> \<tau> = T"
FunctorId2: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and X :: "'d" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "F @@ X \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
bind_ask: fixes f :: "'r \<Rightarrow> 'm"   and g :: "'s \<Rightarrow> 'm" shows "bind (ask f) g = ask (\<lambda>r. bind (f r) g)"
q: shows "\<integral>\<^sup>+ x. ennreal (max 0 (- f x)) \<partial>lborel = ennreal q"
MaxR_f_def: fixes x :: "'a"   and acc :: "'a option" shows "MaxR_f x acc = (if x \<in> Field r then Some (case acc of None \<Rightarrow> x | Some y \<Rightarrow> maxR x y) else acc)"
decr_grading_term_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and d :: "'d \<Rightarrow> nat"   and n :: "nat"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.decr_grading_term pair_of_term term_of_pair d n v = term_of_pair (decr_grading d n (term_powerprod.pp_of_term pair_of_term v), term_powerprod.component_of_term pair_of_term v)"
simps_1: fixes p :: "(action \<times> nat) list"   and b :: "cell" shows "fetch p 0 b = (Nop, 0)"
suminf_le_pos: fixes f :: "nat \<Rightarrow> ereal"   and g :: "nat \<Rightarrow> ereal" assumes "\<And>N. f N \<le> g N"   and "\<And>N. 0 \<le> f N" shows "suminf f \<le> suminf g"
ProofStepInduct_axioms: shows "ProofStepInduct G v0 v1 paths' P_new' sep_size"
midpoint_def: fixes a :: "'b"   and b :: "'b" shows "midpoint a b = (a + b) /\<^sub>R 2"
cfun_cong: fixes f :: "'e \<rightarrow> 'f"   and g :: "'e \<rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
H_2: shows "(\<lambda>v. Cauchy v \<and> (\<lambda>n. dist (x n) (v n)) \<longlonglongrightarrow> 0) = (\<lambda>v. Cauchy v \<and> (\<lambda>n. dist (y n) (v n)) \<longlonglongrightarrow> 0)"
semicomplete_nonempty_imp_extremed: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semicomplete A r"   and "A \<noteq> {}" shows "extremed A r"
assms_3: shows "\<forall>B\<in>M. B \<inter> {u \<in> U. real c / 2 < w u} \<noteq> {}"
permutation_mat_dim_3: fixes n :: "nat"   and p :: "nat \<Rightarrow> nat" shows "dim_col (permutation_mat n p) = n"
ETP_base_axioms: shows "ETP_base I domain range F F\<^sub>i\<^sub>n\<^sub>v"
poly_power_n_odd: fixes n :: "nat"   and a :: "real"   and x :: "real" assumes "odd n" shows "(0 < poly ([:- a, 1:] ^ n) x) = (a < x)"
restrict_domain: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and W :: "'b set set" assumes "ChamberComplexMorphism X Y f"   and "ChamberComplex.ChamberSubcomplex X W" shows "ChamberComplexMorphism W Y f"
M\<^sub>R_8: shows "\<forall>i\<le>n. \<forall>j\<le>n. M\<^sub>R i j \<noteq> \<infinity> \<and> 0 < i \<and> 0 < j \<longrightarrow> (\<exists>d. (M\<^sub>R i j = Le (real_of_int d) \<or> M\<^sub>R i j = Lt (real_of_int d)) \<and> - int (k (THE c. c \<in> X \<and> v c = j)) \<le> d \<and> d \<le> int (k (THE c. c \<in> X \<and> v c = i)))"
semilat_le_err_OK2: fixes x :: "'c"   and A :: "'c set"   and y :: "'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c err \<Rightarrow> 'c err \<Rightarrow> 'c err"   and z :: "'c" assumes "x \<in> A"   and "y \<in> A"   and "semilat (err A, Err.le r, f)"   and "OK x \<squnion>\<^bsub>f\<^esub> OK y = OK z" shows "y \<sqsubseteq>\<^bsub>r\<^esub> z"
bin_nth_Bit1: fixes w :: "num"   and n :: "nat" shows "bit (numeral (num.Bit1 w)) n = (n = 0 \<or> (\<exists>m. n = Suc m \<and> bit (numeral w) m))"
inf_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
len_lexext_rev_singleton: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" shows "len_lexext_rev gt [y] [x] = gt y x"
main_connect: assumes "length fs \<le> n" shows "gram_schmidt n fs = map gso [0..<length fs]"
compare: fixes x :: "'b comparator" shows "compare x \<in> {cmp. comparator cmp}"
simps_1: fixes \<Psi> :: "'b"   and G :: "'b frame" shows "(\<langle>\<epsilon>, \<Psi>\<rangle>) \<otimes>\<^sub>F G = insertAssertion G \<Psi>"
norm_prod_minus1_le_prod_minus1: fixes f :: "nat \<Rightarrow> 'e"   and A :: "nat set" shows "norm ((\<Prod>n\<in>A. (1::'e) + f n) - (1::'e)) \<le> (\<Prod>n\<in>A. 1 + norm (f n)) - 1"
expand_and_propag: fixes \<mu> :: "'b ltlr"   and \<eta> :: "'b ltlr"   and n_ns :: "'b node \<times> 'b node set" assumes "\<mu> and\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old (fst n_ns) \<union> new (fst n_ns)"   and "\<forall>nd\<in>snd n_ns. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd" shows "expand n_ns \<le> SPEC (\<lambda>nm_nds. \<forall>nd\<in>snd nm_nds. \<mu> and\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<subseteq> old nd)"
ChiRel_HARoot: fixes x :: "'d"   and y :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(x, y) \<in> ChiRel A" shows "y \<notin> States (HARoot A)"
HBex_def: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool" shows "HBex A P = (\<exists>x. x \<^bold>\<in> A \<and> P x)"
dim_tangent_space_src_dest_eq: fixes p :: "'a" assumes "p \<in> a.sub.carrier"   and "0 < dim (a.dest.tangent_space (apply_chart c p))" shows "dim (a.sub.tangent_space p) = dim (a.dest.tangent_space (apply_chart c p))"
abs_eq: shows "(\<le>) = (\<lambda>X Y. Rep_downsets X \<subseteq> Rep_downsets Y)"
dim_le_card': fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b set" assumes "vector_space scale"   and "finite s" shows "vector_space.dim scale s \<le> card s"
lt_eqI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "lookup p v \<noteq> (0::'c)"   and "\<And>u. lookup p u \<noteq> (0::'c) \<Longrightarrow> u \<preceq>\<^sub>t v" shows "lt p = v"
UV: shows "(u, v) \<in> (edges (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2)))\<^sup>*"
retain_TCons_def: shows "local.retain_TCons \<equiv> map_fun ((`) (\<lambda>ys. (llist_of_tllist ys, terminal ys))) ((`) (\<lambda>(xs, a). tllist_of_llist a xs)) (\<lambda>A. A \<inter> {(xs, b). \<not> lnull xs})"
aodv_labels_not_empty: fixes p :: "(state, msg, pseqp, pseqp label) seqp" shows "labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p \<noteq> {}"
preorder_on_restrict: fixes A :: "'a set" shows "Order_Predicates.preorder_on (carrier \<inter> A) (restrict_relation A le)"
eq: shows "n div 2 + Suc n div 2 = n"
unfold_lesub_err: fixes e1 :: "'c err"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e2 :: "'c err" shows "(e1 \<sqsubseteq>\<^bsub>Err.le r\<^esub> e2) = Err.le r e1 e2"
alive_trivial_simps_4: fixes "by" :: "int"   and s :: "Store" shows "alive (byteV by) s"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> 'b"   and f2 :: "'b"   and x :: "enat" shows "P (case x of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = ((\<forall>nat. x = enat nat \<longrightarrow> P (f1 nat)) \<and> (x = \<infinity> \<longrightarrow> P f2))"
atLeastLessThan_singleton: fixes m :: "nat" shows "{m..<Suc m} = {m}"
abs_eq: fixes x :: "nat \<times> nat" shows "nat (Abs_Integ x) = (case x of (x, xa) \<Rightarrow> x - xa)"
max_0L: fixes n :: "nat" shows "max 0 n = n"
o: shows "ord n a = (LEAST d. 0 < d \<and> [a ^ d = 1] (mod n))"
cross_orthogonal_matrix: fixes A :: "((real, 3) vec, 3) vec"   and x :: "(real, 3) vec"   and y :: "(real, 3) vec" assumes "orthogonal_matrix A" shows "(A *v x) \<times> (A *v y) = det A *\<^sub>R (A *v x \<times> y)"
FAddSA_dom_insert: fixes S :: "'d"   and A :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> dom A"   and "S \<notin> States SA" shows "(A [f+] (S, SA)) S = Some (insert SA (the (A S)))"
norm_Pair2: fixes x :: "'b" shows "norm (x, 0::'c) = norm x"