hcju/isabelleps huggingface dataset:
FvarsT_VarD: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "Generic_Syntax var trm fmla Var FvarsT substT Fvars subst"   and "x \<in> FvarsT (Var y)"   and "y \<in> var" shows "x = y"
subgroup_inverse_equality: fixes u :: "'a" assumes "u \<in> Units" shows "inverse u = units.sub.inverse u"
summable_sums: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f" shows "f sums suminf f"
cd_cs_swap: fixes l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'n"   and l' :: "nat"   and k' :: "nat" assumes "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"   and "cs\<^bsup>\<pi>\<^esup> k = cs\<^bsup>\<pi>'\<^esup> k'" shows "l' cd\<^bsup>\<pi>'\<^esup>\<rightarrow> k'"
prove'_def: fixes s :: "(nat \<times> form) list list" shows "prove' s = (\<exists>m. iter (\<lambda>x. flatten (map subs x)) s m = [])"
atLeast_def: fixes l :: "'a" shows "{l..} \<equiv> \<up>l"
zero_smallest: fixes j :: "'b \<Rightarrow> 'a" assumes "j \<in> Field oexp" shows "(o.zero, j) \<in> oexp"
dom_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "local.dom (cod f) = cod f"
asInp_pickInp: fixes inp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option" assumes "goodInp inp" shows "asInp (pickInp inp) = inp"
assms: shows "\<langle>a, a\<rangle> \<in>\<^sub>\<circ> vid_on A"
degree_monic_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
simps_2: fixes a :: "'f"   and S :: "'f list"   and iostep :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e (a # S) iostep \<sigma> = (case iostep a \<sigma> of None \<Rightarrow> Some ([], \<sigma>) | Some (out, \<sigma>') \<Rightarrow> case mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e S iostep \<sigma>' of None \<Rightarrow> Some ([out], \<sigma>') | Some (outs, \<sigma>'') \<Rightarrow> Some (out # outs, \<sigma>''))"
trans_fm_perm: fixes c :: "name"   and i :: "name"   and j :: "name"   and A :: "fm"   and B :: "fm" assumes "atom c \<sharp> (i, j, A, B)"   and "trans_fm [i] A = trans_fm [j] B" shows "(i \<leftrightarrow> c) \<bullet> A = (j \<leftrightarrow> c) \<bullet> B"
nodes_target: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set"   and r :: "(nat \<times> 'c) list" assumes "p \<in> gnodes A w V"   and "gpath A w V r p" shows "gtarget r p \<in> gnodes A w V"
conforms_to_another_strategy: fixes \<sigma>' :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy p P \<sigma>'" shows "vmc_path G P v0 p \<sigma>'"
pf_\<alpha>_def: fixes n :: "nat" shows "pf_\<alpha> n = probGn p n (\<lambda>es. ereal (1 / 2 * real n / real k) \<le> ereal_of_enat (\<alpha> (edge_space.edge_ugraph n es)))"
l2_dy_fake_msg_def: fixes m :: "msg" shows "l2_dy_fake_msg m \<equiv> {(s, s'). m \<in> dy_fake_msg (bad s) (ik s) (chan s) \<and> s' = s\<lparr>ik := {m} \<union> ik s\<rparr>}"
simps_4: fixes tst :: "'test"   and c :: "('test, 'atom) com" shows "SC_siso (while tst do c) = (compatTst tst \<and> SC_siso c)"
L_zero_space_1: fixes M :: "'b measure" shows "space\<^sub>N (\<LL> 0 M) = borel_measurable M"
subgroup_conjugation_is_surj2: fixes g :: "'a"   and H :: "'a set" assumes "g \<in> carrier G"   and "subgroup H G" shows "subgroup (g <# H #> inv g) G"
alpha': fixes a :: "name"   and x :: "'d"   and b :: "name"   and y :: "'d" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
monoAux: fixes A :: "(pi \<times> pi) set"   and B :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi" assumes "A \<subseteq> B" shows "P \<leadsto>\<^sup>^<A> Q \<longrightarrow> P \<leadsto>\<^sup>^<B> Q"
rev_swap_def: fixes a :: "'b"   and b :: "'b" shows "rev_swap a b = \<lparr>edge_rev = perm_swap a b (edge_rev M), edge_succ = perm_swap a b (edge_succ M)\<rparr>"
emeasure_eq_AE: fixes A :: "'d set"   and B :: "'d set"   and M :: "'d measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A = emeasure M B"
lens_indep_comm: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'d"   and u :: "'b" assumes "x \<bowtie> y" shows "put\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) u = put\<^bsub>y\<^esub> (put\<^bsub>x\<^esub> \<sigma> u) v"
set_map_2: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'a"   and f3 :: "'f \<Rightarrow> 'c"   and v :: "('d, 'e, 'f) expr" shows "LID\<^sub>E (map_expr f1 f2 f3 v) = f2 ` LID\<^sub>E v"
higher_part_zero: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k < fst c"   and "\<forall>j\<in>nset (Suc k) (fst c). snd c j = \<zero>\<^bsub>S\<^esub>" shows "\<Sigma>\<^sub>f R (\<lambda>j. snd c j \<cdot>\<^sub>r X^\<^bsup>R j\<^esup>) Suc k fst c = \<zero>"
Nat2nat_nat2Nat: fixes n :: "nat" shows "Nat2nat (nat2Nat n) = n"
simps_3: fixes r :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp list" shows "seq (r # v # va) = Times r (seq (v # va))"
inject: fixes x2 :: "nat"   and y2 :: "nat" shows "(Friend x2 = Friend y2) = (x2 = y2)"
replicating_fair_price: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and der :: "'a \<Rightarrow> real"   and matur :: "nat" assumes "viable_market Mkt"   and "replicating_portfolio pf der matur"   and "support_adapt Mkt pf" shows "fair_price Mkt (initial_value pf) der matur"
extend_Join: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and G :: "'c program" assumes "Extend h" shows "extend h (F \<squnion> G) = extend h F \<squnion> extend h G"
that: fixes x :: "'x" assumes "h \<turnstile> f x \<rightarrow>\<^sub>r Some y"   and "x \<in> set xs" shows "thesis"
prems_1: shows "atom ` domA \<Delta> \<sharp>* \<Gamma>"
outputFreshDerivative_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "xvec \<sharp>* M"   and "distinct xvec"   and "x \<sharp> P"   and "x \<sharp> xvec" shows "x \<sharp> N"
divide_by_norm_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "divide_by_norm A = (\<chi>a b. Projections.normalize (column b A) $ a)"
lunit_def: fixes a :: "'a" shows "\<l>[a] \<equiv> THE f. \<guillemotleft>f : \<I> \<otimes> a \<rightarrow> a\<guillemotright> \<and> \<I> \<otimes> f = (\<iota> \<otimes> a) \<cdot> local.inv \<a>[\<I>, \<I>, a]"
many_reducible_to_def: shows "many_reducible_to = (\<lambda>A B. \<exists>f. many_reducible_to_via A B f)"
less_pratt2_def: fixes x :: "pratt2"   and y :: "pratt2" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
intros_2: fixes c :: "('fsym, 'psym) lit list"   and \<sigma> :: "'tp" assumes "\<And>l x. \<lbrakk>l \<in> set c; x \<in> nv2L l; tpOfV x = \<sigma>\<rbrakk> \<Longrightarrow> isGuard x (grdOf c l x)" shows "\<sigma> \<turnstile>2 c"
hereditarily_fs_implies_finite_supp: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "finite (supp t\<^sub>\<alpha>)"
vec_hom_Ints: fixes i :: "nat"   and n :: "nat"   and xs :: "int vec" assumes "i < n"   and "xs \<in> carrier_vec n" shows "of_int_hom.vec_hom xs $ i \<in> \<int>"
round_def: fixes x :: "'b" shows "round x = \<lfloor>x + (1::'b) / (2::'b)\<rfloor>"
sum_list_map_closed: fixes f :: "'a \<Rightarrow> 'g"   and as :: "'a list" assumes "order.greater_eq G (set (map f as))" shows "sum_list (map f as) \<in> G"
hyps: assumes "distinct (map fst xs)" shows "lookup_pair (sort_oalist xs) = lookup_dflt xs"
simps_6: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brn (Par cl) = lsum brn cl"
timpl_closure'_inv': fixes s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) set" assumes "(s, t) \<in> timpl_closure' TI" shows "(\<exists>x. s = Var x \<and> t = Var x) \<or> (\<exists>f g S T. s = Fun f S \<and> t = Fun g T \<and> length S = length T \<and> (\<forall>i<length T. (S ! i, T ! i) \<in> timpl_closure' TI) \<and> (f \<noteq> g \<longrightarrow> is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> TI\<^sup>+))"
stream_enc_welldef: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "stream_enc (w, I) = stream_enc (w', I')"   and "wf_formula (length I) \<phi>"   and "wf_interp_for_formula (w, I) \<phi>"   and "wf_interp_for_formula (w', I') \<phi>" shows "((w, I) \<Turnstile> \<phi>) = ((w', I') \<Turnstile> \<phi>)"
assms: fixes \<eta> :: "nat" shows "\<I> \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res res \<eta> \<surd>"
nz1: fixes n :: "int" shows "real_of_int n + 1 / 2 \<noteq> 0"
ffun_plus_minus: fixes f :: "('a, 'b) ffun"   and g :: "('a, 'b) ffun" assumes "f \<subseteq>\<^sub>f g" shows "g - f + f = g"
execute'_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "execute' problem name s \<equiv> case lookup_operator' problem name of Some \<pi> \<Rightarrow> execute_opr' \<pi> s"
stateT_eqI: fixes a :: "'b\<cdot>('c, 'a) stateT"   and b :: "'b\<cdot>('c, 'a) stateT" assumes "\<And>s. runStateT\<cdot>a\<cdot>s = runStateT\<cdot>b\<cdot>s" shows "a = b"
simps_4: fixes E_ES :: "'b set"   and I_ES :: "'b set"   and O_ES :: "'b set"   and Tr_ES :: "'b list set"   and more :: "'c" shows "Tr_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = Tr_ES"
accfield_declC_is_class: fixes G :: "prog"   and C :: "qtname"   and S :: "qtname"   and en :: "vname"   and fd :: "qtname"   and f :: "Decl.field" assumes "is_class G C"   and "accfield G S C en = Some (fd, f)"   and "ws_prog G" shows "is_class G fd"
maddux_21: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq x (composition top x)"
prime_factorization_cong: fixes x :: "'a"   and y :: "'a" assumes "normalize x = normalize y" shows "prime_factorization x = prime_factorization y"
right_neutral: fixes a :: "'c" shows "a \<squnion> \<bottom> = a"
upd_locks_append: fixes l :: "('c \<times> nat) option"   and t :: "'c"   and Ls :: "lock_action list"   and Ls' :: "lock_action list" shows "upd_locks l t (Ls @ Ls') = upd_locks (upd_locks l t Ls) t Ls'"
rep_eq: fixes x :: "'a prefix"   and xa :: "'k" shows "Rep_prefix (splitter x xa) = map (\<lambda>(D, t). ({e \<in> D. xa \<in> event_splitter e}, t)) (Rep_prefix x)"
aux: fixes gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "set_spmf (inline1 callee gpv s) \<subseteq> {Inr (out, callee', rpv') |out callee' rpv'. \<exists>call\<in>outs_\<I> \<I>. \<exists>s. I s \<and> (\<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (callee s call))} \<union> {Inl (x, s') |x s'. x \<in> results_gpv \<I> gpv \<and> I s'}"
coprime_r_zero1: fixes qs :: "real poly list" shows "poly (coprime_r qs) (real_of_int (crb (prod_list qs))) = 0"
Large: fixes n :: "nat" shows "\<not> \<bar>set_lebesgue_integral M (A n) h\<bar> < delta"
leadsTo_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<longmapsto> B \<equiv> {F. (A, B) \<in> leads F}"
vertical_line_affine: fixes x :: "'b" shows "affine {(x, m) |m. m \<in> UNIV}"
shead_bottom_iff: fixes xs :: "[:'a:]" shows "(shead\<cdot>xs = \<bottom>) = (xs = \<bottom> \<or> xs = [::])"
Union_mono: fixes A :: "'d set set"   and B :: "'d set set" assumes "A \<subseteq> B" shows "\<Union> A \<subseteq> \<Union> B"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "Bossy'.stable_on ds X" shows "Bossy''.CD_on ds X = X"
closed_map_imp_subset: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "closed_map X1 X2 f"   and "S \<subseteq> topspace X1" shows "f ` S \<subseteq> topspace X2"
intro: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat" assumes "wf_nfa A n" shows "subset_DFS A n"
l10_2_existence_spec: fixes P :: "'p"   and A :: "'p"   and B :: "'p" shows "\<exists>P'. P' P ReflectL A B"
dvd_diffD1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k dvd m - n"   and "k dvd m"   and "n \<le> m" shows "k dvd n"
dsum_iOp_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_mOp I A \<in> carr_dsumag I A \<rightarrow> carr_dsumag I A"
Complete_Lattice_axioms: shows "Complete_Lattice L Glb"
degree_zero_imp_not_is_root: fixes p :: "'a list list"   and x :: "'a list" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "degree p = 0" shows "\<not> UP.is_root p x"
simps_3: fixes C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (C\<^sub>1;; C\<^sub>2) Q = pre C\<^sub>1 (pre C\<^sub>2 Q)"
sd_distr: fixes X :: "('a \<times> 'b) set"   and y :: "'a \<times> 'b" shows "sd_prod (Sup_prod X) y = Sup_prod {sd_prod x y |x. x \<in> X}"
a_d_closed: fixes x :: "('b \<times> 'b) set" shows "rel_antidomain_kleene_algebra.apd_d (rel_ad x) = rel_ad x"
inf_sup_ord_2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
simps_17: fixes v :: "nat" shows "fold_const_fa (Var v) = Var v"
only_new_introduces_lids: fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s s'"   and "\<And>\<sigma> \<tau> \<E> v. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])" shows "LID\<^sub>G s' \<subseteq> LID\<^sub>G s"
cl10_d_var3: fixes x :: "'a" shows "d (x \<sqinter> nc) = 1\<^sub>\<sigma> \<sqinter> x \<cdot> U"
normalize_right_idem: fixes a :: "'b"   and b :: "'b" shows "lcm a (normalize b) = lcm a b"
\<sigma>_6: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<sigma>)"
simps_6: fixes f1 :: "CTypeId \<Rightarrow> 'a"   and f2 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "Arraytype"   and x22 :: "nat" shows "(case new_array x21 x22 of new_instance x \<Rightarrow> f1 x | new_array x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
le_word_or1: fixes c :: "'b word"   and y :: "'b word" shows "c \<le> y OR c"
termFSbMorph_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) Recursion.model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.termFSbMorph Delta h hA MOD \<equiv> FixSyn.presWlsAll Delta h hA MOD \<and> FixSyn.presCons Delta h hA MOD \<and> FixSyn.presFreshAll Delta h hA MOD \<and> FixSyn.presSubstAll Delta h hA MOD"
component_path: fixes u :: "'b"   and g :: "'b ugraph"   and r :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "u \<in> nodes (component_of g r)" shows "path (component_of g r) u p v = path g u p v"
infinite_nonempty: fixes A :: "'a set" assumes "infinite A" shows "\<exists>x. x \<in> A"
B: shows "f_\<upsilon> (f_lam a1 \<tau>) \<tau> = f_\<upsilon> a1 \<tau>"
comp_proj_i_def: fixes X :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat"   and y :: "'b" shows "comp_proj_i X n i y = {z \<in> range (proj_stoch_proc X n). z !! i = y}"
Suc: fixes x :: "real_alg_3" shows "real_of_3 x \<in>\<^sub>i get_itvl_3 ((tighten_bounds_3 ^^ n) x)"
rotate_Suc: fixes n :: "nat"   and xs :: "'c list" shows "rotate (Suc n) xs = rotate1 (rotate n xs)"
le_numeral_extra_3: fixes one :: "'r"   and times :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and zero :: "'r" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero zero"
preserves_reflects_arr: fixes f :: "'a" shows "D.arr (G f) = C.arr f"
ball_eq_ball_iff: fixes x :: "'a"   and d :: "real"   and y :: "'a"   and e :: "real" shows "(ball x d = ball y e) = (d \<le> 0 \<and> e \<le> 0 \<or> x = y \<and> d = e)"
read_tmps_distinct: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "distinct_read_tmps sb"
distinct_35: fixes "term" :: "'d"   and input :: "('d, 'e, 'f) input"   and psi' :: "('d, 'e, 'f) psi" shows "term\<lparr>input \<noteq> !psi'"
index_distinct_eqI: fixes xs :: "'b list"   and i :: "nat"   and x :: "'b" assumes "distinct xs"   and "i < length xs"   and "xs ! i = x" shows "index xs x = i"
NSLIM_const: fixes k :: "'d"   and x :: "'c" shows "(\<lambda>x. k) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S k"
order_refl: fixes x :: "'b word" shows "x \<le>s x"
mbufnt_take_induct: fixes f :: "event_data option list set list \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbufnt_take f z buf nts = (z', buf', nts')"   and "wf_mbufn i js Ps buf"   and "list_all2 R [i..<j] nts"   and "\<And>k. k \<in> set js \<Longrightarrow> k \<le> j"   and "U i z"   and "\<And>k xs t z. \<lbrakk>i \<le> k; Suc k \<le> Mini j js; list_all2 (\<lambda>P. P k) Ps xs; R k t; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f xs t z)" shows "U (Mini (i + length nts) js) z'"
i2_ips: shows "ipassmt i2 = Some i2_ips"
distinct_sort_on: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "distinct (sort_key_on r f xs) = distinct xs"
mem_Collect_eq: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
iptables_goto_bigstepD_9: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and chain :: "char list"   and rs :: "'c rule list" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m (Call chain)]"   and "s = Undecided"   and "matches \<gamma> m p"   and "\<Gamma> chain = Some rs" obtains "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m' :: "'c match_expr" where "rs = rs\<^sub>1 @ Rule m' Return # rs\<^sub>2" and "matches \<gamma> m' p" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> Undecided" and "no_matching_Goto \<gamma> p rs\<^sub>1" and "t = Undecided"
trivial_homomorphism_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and f :: "'c \<Rightarrow> 'e" shows "trivial_homomorphism G H f \<equiv> f \<in> hom G H \<and> (\<forall>x\<in>carrier G. f x = \<one>\<^bsub>H\<^esub>)"
strict_mono_imp_increasing: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "strict_mono f" shows "n \<le> f n"
face_cycle_succ_in_arcsI: fixes a :: "'b" assumes "a \<in> arcs G" shows "GM.face_cycle_succ a \<in> arcs G"
dist_complex_def: fixes x :: "complex"   and y :: "complex" shows "dist x y = cmod (x - y)"
iffs: fixes hostsL :: "'b list"   and flows_fixL :: "('b \<times> 'b) list"   and flows_stateL :: "('b \<times> 'b) list"   and more :: "'c"   and hostsL' :: "'b list"   and flows_fixL' :: "('b \<times> 'b) list"   and flows_stateL' :: "('b \<times> 'b) list"   and more' :: "'c" shows "(\<lparr>hostsL = hostsL, flows_fixL = flows_fixL, flows_stateL = flows_stateL, \<dots> = more\<rparr> = \<lparr>hostsL = hostsL', flows_fixL = flows_fixL', flows_stateL = flows_stateL', \<dots> = more'\<rparr>) = (hostsL = hostsL' \<and> flows_fixL = flows_fixL' \<and> flows_stateL = flows_stateL' \<and> more = more')"
psGeZero: fixes ts :: "traffic" shows "\<forall>c. 0 < physical_size ts c"
b_Lam: fixes x :: "name"   and e :: "trm" shows "Lam [x].e \<Down> Lam [x].e"
WHILE_nofail_imp_rwof_nofail: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "nofail (m0 \<bind> WHILE cond step)"   and "rwof m0 cond step s"   and "cond s" shows "nofail (step s)"
the_inv_f_f: fixes f :: "'b \<Rightarrow> 'd"   and x :: "'b" assumes "inj f" shows "the_inv f (f x) = x"
integrable_imp_measurable: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f integrable_on S" shows "f \<in> borel_measurable (lebesgue_on S)"
simps_2: fixes \<sigma> :: "'c"   and x :: "'d"   and \<Pi> :: "('c, 'd) chain"   and y :: "'d"   and \<tau> :: "'c" shows "cSnoc (\<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> \<Pi>) y \<tau> = \<lbrace> \<sigma> \<rbrace> \<cdot> x \<cdot> cSnoc \<Pi> y \<tau>"
simps_21: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Skip"
rows_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rows A = {row i A |i. i \<in> UNIV}"
\<Delta>_naturality: shows "(\<pi> \<star> s\<^sub>0) \<cdot> \<sigma> = (t \<star> \<pi>.the_\<theta>) \<cdot> \<a>[t, t\<^sub>0, \<pi>.chine] \<cdot> (\<tau> \<star> \<pi>.chine) \<cdot> \<pi>.the_\<nu>"
mHom_add: fixes M :: "('h, 'i, 'j) Module_scheme"   and R :: "('i, 'k) Ring_scheme"   and N :: "('l, 'i, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'l"   and m :: "'h"   and n :: "'h" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "f (m \<plusminus>\<^bsub>M\<^esub> n) = f m \<plusminus>\<^bsub>N\<^esub> f n"
notin: shows "n \<notin> set (sourcenodes (a # asx))"
map_graph_inv: fixes f :: "'c \<Rightarrow> 'd option" shows "graph_map (map_graph f) = f"
segment_degen_1: fixes u :: "real"   and a :: "'b"   and b :: "'b" shows "((1 - u) *\<^sub>R a + u *\<^sub>R b = b) = (a = b \<or> u = 1)"