hcju/isabelleps huggingface dataset:
loc_reachp_loc_reach_eq: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "loc_reachp lc = (\<lambda>x xa xb. \<langle>x, xa, xb\<rangle> \<in> loc_reach lc)"
norm_PX1: fixes P :: "pol"   and i :: "nat"   and Q :: "pol" assumes "isnorm (PX P i Q)" shows "isnorm P"
\<beta>_nz: fixes p :: "int poly" assumes "p \<in> P" shows "\<beta> p \<noteq> 0"
positive_y_axis_def: shows "positive_y_axis = {z \<in> circline_set y_axis. z \<noteq> \<infinity>\<^sub>h \<and> 0 < Im (to_complex z)}"
ends_notin_segment: fixes a :: "'a"   and b :: "'a" shows "a \<notin> segment a b \<and> b \<notin> segment a b"
bubble_sort: fixes y :: "'a"   and x :: "'a" assumes "y ; x \<le> x ; y" shows "(x + y)\<^sup>\<star> = x\<^sup>\<star> ; y\<^sup>\<star>"
assms_4: shows "preds (slice_kinds S (a # as)) s"
delete_edges_wf: fixes G :: "'c graph"   and E :: "('c \<times> 'c) set" assumes "wf_graph G" shows "wf_graph (delete_edges G E)"
invertible_proportionality_equiv: shows "equiv (Collect invertible) invertible_proportionality"
simplices_conv_special_cosets: shows "\<Sigma> = dual_order.poset_simplex_map \<P> ` \<P>"
Contrapos2: fixes H :: "fm set"   and B :: "fm"   and A :: "fm" assumes "H \<turnstile> Neg B IMP Neg A" shows "H \<turnstile> A IMP B"
do_log_alt_def: fixes xa :: "lop"   and x1ca :: "v"   and x2ca :: "exp0" shows "do_log xa x1ca x2ca = (case xa of And \<Rightarrow> case x1ca of Litv x \<Rightarrow> Map.empty x | Conv None l2 \<Rightarrow> None | Conv (Some (s1, t1)) l2 \<Rightarrow> if s1 = ''true'' then case t1 of TypeId (Short s2) \<Rightarrow> if s2 = ''bool'' then case l2 of [] \<Rightarrow> Some (Exp x2ca) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else if s1 = ''false'' then case t1 of TypeId (Short s4) \<Rightarrow> if s4 = ''bool'' then case l2 of [] \<Rightarrow> Some (Val x1ca) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else None | Closure x xa xb \<Rightarrow> Map.empty xb | Recclosure x xa xb \<Rightarrow> Map.empty xb | Loc x \<Rightarrow> Map.empty x | Vectorv x \<Rightarrow> Map.empty x | Or \<Rightarrow> case x1ca of Litv x \<Rightarrow> Map.empty x | Conv None l6 \<Rightarrow> None | Conv (Some (s8, t0)) l6 \<Rightarrow> if s8 = ''false'' then case t0 of TypeId (Short s9) \<Rightarrow> if s9 = ''bool'' then case l6 of [] \<Rightarrow> Some (Exp x2ca) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else if s8 = ''true'' then case t0 of TypeId (Short s11) \<Rightarrow> if s11 = ''bool'' then case l6 of [] \<Rightarrow> Some (Val x1ca) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else None | Closure x xa xb \<Rightarrow> Map.empty xb | Recclosure x xa xb \<Rightarrow> Map.empty xb | Loc x \<Rightarrow> Map.empty x | Vectorv x \<Rightarrow> Map.empty x)"
padic_int_simps_4: fixes p :: "int" shows "(\<otimes>\<^bsub>padic_int p\<^esub>) = padic_mult p"
in_idem: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<in>| A" shows "max x (fMax A) = fMax A"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> r"
"1_1": shows "(xs, zs) \<in> \<langle>\<langle>A\<rangle>list_set_rel\<rangle>list_rel"
inv_comp_left_2: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.iso (VV.comp g f)"   and "VV.iso g" shows "VV.iso f"
sympartpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "r x y"   and "r y x" shows "sympartp r x y"
equator_def: shows "equator \<equiv> proj2_line_abs (vector [0, 1, 0])"
to_nat_from_nat_suc_k: shows "mod_type_class.to_nat (mod_type_class.from_nat (Suc k)) = Suc k"
stable_pair_on_CH: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "match XD_XH = CH (snd XD_XH)"
exhaust_sel: fixes sum :: "'d + 'e" obtains "sum = Inl (projl sum)"   | "sum = Inr (projr sum)"
listset_closed_map2: fixes ys1 :: "'b list"   and xs :: "'b set list"   and ys2 :: "'b list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "ys1 \<in> listset xs"   and "ys2 \<in> listset xs"   and "\<And>x y1 y2. \<lbrakk>x \<in> set xs; y1 \<in> x; y2 \<in> x\<rbrakk> \<Longrightarrow> f y1 y2 \<in> x" shows "map2 f ys1 ys2 \<in> listset xs"
face_cycle_succ_no_arc: fixes a :: "'b" assumes "a \<notin> arcs G" shows "G.face_cycle_succ a = a"
timpls_transformable_to_pred_if_timpls_transformable_to: fixes TI :: "('sets set \<times> 'sets set) list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and A :: "'sets set set" assumes "timpls_transformable_to TI t s"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<forall>f\<in>funs_term t. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s"
recurrent_subset_infty_inf_returns: fixes x :: "'a"   and A :: "'a set" shows "(x \<in> recurrent_subset_infty A) = (x \<in> A \<and> infinite {n. (T ^^ n) x \<in> A})"
insert_def: shows "Challenge1.insert \<equiv> \<lambda>uu uua. rec_textbuffer (\<lambda>a b x. BUF (a @ [x]) b) uua uu"
map_concat: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list list" shows "map f (concat xs) = concat (map (map f) xs)"
pelims_3: fixes x :: "'a val" assumes "\<not> is_Intg x"   and "is_Intg_dom x" obtains "x = Unit" and "is_Intg_dom Unit"   | "x = Null" and "is_Intg_dom Null"   | v :: "bool" where "x = Bool v__" and "is_Intg_dom (Bool v__)"   | v :: "'a" where "x = Addr v" and "is_Intg_dom (Addr v)"
arcsin_pi: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "- (pi / 2) \<le> arcsin y \<and> arcsin y \<le> pi \<and> sin (arcsin y) = y"
basis_subset: fixes a :: "'c" shows "directional_derivative \<infinity> a ` Basis \<subseteq> manifold_eucl.tangent_space \<infinity> a"
hyps: shows "bound (a * of_nat n) \<le> bound a * bound (of_nat n)"
1: shows "Sigma_Algebra.measure (measure_spmf p) {x. A x \<and> \<not> bad1 x} = Sigma_Algebra.measure (measure_spmf q) {y. B y \<and> \<not> bad2 y}"
neqs1_5: shows "(\<lambda>x. (diamond_x (x / 2), - diamond_x (x / 2) - d / 2)) \<noteq> (\<lambda>x. (diamond_x (x / 2 + 1 / 2), diamond_x (x / 2 + 1 / 2) - d / 2))"
b: shows "Cinv g ;; Cinv f maps dom Cinv g to cod Cinv f"
Sup_sup_dual: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_dual f" shows "sup_dual f"
HPhase1or2ReadElse_HInv4b: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc"   and r :: "Proc" assumes "HPhase1or2ReadElse s s' p d q"   and "HInv4b s r"   and "Inv2a s"   and "HInv4a s p" shows "HInv4b s' r"
unsigned_less: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'b \<times> 'b"   and divmod_step :: "num \<Rightarrow> 'b \<times> 'b \<Rightarrow> 'b \<times> 'b"   and w :: "'c word" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "less (semiring_1.unsigned one plus zero w) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) LENGTH('c))"
cases: fixes a :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "a \<in> K st" obtains l :: "'s" and u :: "'c \<Rightarrow> real" and t :: "real" where "a = return_pmf (la__, u__ \<oplus> t)" and "st \<in> S" and "st = (la__, u__)" and "0 \<le> t" and "u__ \<oplus> t \<turnstile> PTA.inv_of A la__"   | l :: "'s" and u :: "'c \<Rightarrow> real" and g :: "('c, real) cconstraint" and \<mu> :: "('c set \<times> 's) pmf" where "a = map_pmf (\<lambda>(X, l). (l, [(SOME r. set r = X)\<rightarrow>0]u)) \<mu>" and "st \<in> S" and "st = (l, u)" and "(l, g, \<mu>) \<in> PTA.trans_of A" and "u \<turnstile> g"   | "a = return_pmf st"
lincomb_as_lincomb_list: fixes ws :: "'b vec list"   and n :: "nat"   and f :: "'b vec \<Rightarrow> 'b" assumes "set ws \<subseteq> carrier_vec n" shows "module.lincomb (module_vec TYPE('b) n) f (set ws) = module.lincomb_list (module_vec TYPE('b) n) (\<lambda>i. if \<exists>j<i. ws ! i = ws ! j then 0::'b else f (ws ! i)) ws"
wgraph_specs_6: fixes g :: "'g"   and u :: "'v"   and v :: "'v"   and d :: "nat" assumes "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "\<alpha>g (add_edge (u, v) d g) = ins_edge (u, v) (\<alpha>g g)"
\<sigma>_def: shows "\<sigma> = (\<lambda>\<Sigma> n. concat (map (\<lambda>bs. map (\<lambda>a. (a, bs)) \<Sigma>) (List.n_lists n [True, False])))"
list_gcd_simps_1: shows "gcd_list [] = (0::'a)"
INV1: fixes I :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and N :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "|~ I \<and> _actrans N f \<longrightarrow> \<circle>I" shows "\<turnstile> I \<and> \<box>[N]_f \<longrightarrow> \<box>I"
simps_1: fixes c :: "'b"   and c' :: "'b" shows "(c)\<^sub>p *\<^sub>p (c')\<^sub>p = (c * c')\<^sub>p"
acomplete_subtreeR: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "acomplete \<langle>l, x, r\<rangle>" shows "acomplete r"
mult_coprime: fixes a :: "nat"   and b :: "nat" assumes "coprime a b" shows "dirichlet_prod f g (a * b) = dirichlet_prod f g a * dirichlet_prod f g b"
Gcd_dvd_nat: fixes a :: "nat"   and A :: "nat set" assumes "a \<in> A" shows "Gcd A dvd a"
iFROM_pred: fixes n :: "nat"   and x :: "nat" assumes "n < x" shows "x - Suc 0 \<in> [n\<dots>]"
ek_analysis_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "ek_analysis c \<equiv> Finite_Graph c"
lvl_skew: fixes t :: "('c \<times> nat) tree" shows "lvl (skew t) = lvl t"
cring_simprules_12: fixes x :: "int" shows "1 * x = x"
get_shadow_root_ptr_simp4: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and element_ptr :: "(_) element_ptr"   and f :: "('m, 'n, 'o, 'q, 't option) RElement_scheme"   and h :: "(_) heap" shows "get shadow_root_ptr (put element_ptr f h) = get shadow_root_ptr h"
fits_conf: fixes G :: "prog"   and s :: "st"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "G,s\<turnstile>v\<Colon>\<preceq>T"   and "G\<turnstile>T\<preceq>? T'"   and "G,s\<turnstile>v fits T'"   and "ws_prog G" shows "G,s\<turnstile>v\<Colon>\<preceq>T'"
mat_of_rows_def: fixes n :: "nat"   and rs :: "'b vec list" shows "mat_of_rows n rs = mat (length rs) n (\<lambda>(i, j). rs ! i $ j)"
pref_in_at_least_as: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "rational_preference carrier relation"   and "x \<succeq>[relation] y" shows "x \<in> at_least_as_good y carrier relation"
nt_match_list_matches: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and a :: "action"   and p :: "'b"   and l :: "'a negation_type list" shows "nt_match_list \<gamma> a p l = matches \<gamma> (alist_and l) a p"
map_removeAll_inj_on: fixes f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and xs :: "'g list" assumes "inj_on f (insert x (set xs))" shows "map f (removeAll x xs) = removeAll (f x) (map f xs)"
primes_omega_pos: fixes n :: "nat" assumes "1 < n" shows "0 < primes_omega n"
D_def: shows "D \<equiv> {0<..<1} \<times> {0<..<1}"
obligation1: fixes s :: "(nat list \<times> (bool list \<times> nat list + nat list)) pmf"   and Da :: "(nat list \<times> bool list \<times> nat list) pmf"   and Db :: "(nat list \<times> nat list) pmf"   and qs :: "nat list" assumes "map_pmf con s = Sum_pmf (8 / 10) Da Db" shows "map_pmf con (Partial_Cost_Model.config'_rand (COMB []) s qs) = Sum_pmf (8 / 10) (Partial_Cost_Model.config'_rand BIT Da qs) (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS [])) Db qs)"
nop_v11_n: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_v11 A n f" shows "n \<in>\<^sub>\<circ> \<omega>"
scaleR_half_double: fixes a :: "'c" shows "(1 / 2) *\<^sub>R (a + a) = a"
b_e_type_value_list_1: fixes \<C> :: "t_context"   and es :: "b_e list"   and v :: "v"   and ts :: "t list"   and ts' :: "t list"   and t :: "t" assumes "\<C> \<turnstile> es @ [C v] : ts _> ts' @ [t]" shows "\<C> \<turnstile> es : ts _> ts'"
testop: fixes t :: "t"   and \<C> :: "t_context"   and ux :: "testop" assumes "is_int_t t" shows "\<C> \<turnstile> [Testop t ux] : [t] _> [T_i32]"
integrable_const_bound: fixes f :: "'c \<Rightarrow> 'd"   and B :: "real"   and p :: "'c spmf" assumes "AE x in measure_spmf p. norm (f x) \<le> B"   and "f \<in> borel_measurable (measure_spmf p)" shows "integrable (measure_spmf p) f"
relpowp_relpow_eq: fixes n :: "nat"   and R :: "('b \<times> 'b) set" shows "(\<lambda>x y. (x, y) \<in> R) ^^ n = (\<lambda>x y. (x, y) \<in> R ^^ n)"
contlub_cfun_arg: fixes Y :: "nat \<Rightarrow> 'd"   and f :: "'d \<rightarrow> 'e" assumes "chain Y" shows "f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
PO_m1_refines_init_a0n: shows "init a0i \<subseteq> R0n0i `` init a0n"
getEnv_vsubstEnv1: fixes ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and y1 :: "'var" assumes "ys \<noteq> xs \<or> y \<noteq> x"   and "rho xs x = None" shows "(rho &[y1 // y]_ys) xs x = None"
chamber_pconnect: fixes x :: "'a set"   and y :: "'a set" assumes "x \<noteq> y"   and "domain.chamber x"   and "domain.chamber y" shows "\<exists>xs. domain.pgallery (x # xs @ [y])"
three_lists_cases_snoc_mid_Cons: fixes as :: "'d list"   and bs :: "'b list"   and cs :: "'c list" obtains ys :: "'b list" and zs :: "'c list" where "as = []" and "bs = ys__" and "cs = zs__"   | xs :: "'d list" and zs :: "'c list" where "as = xs__" and "bs = []" and "cs = zsa__"   | xs :: "'d list" and ys :: "'b list" where "as = xsa__" and "bs = ysa__" and "cs = []"   | xs :: "'d list" and x :: "'d" and y :: "'b" and z :: "'c" and zs :: "'c list" where "as = xsb__ @ [x__]" and "bs = [y__]" and "cs = z__ # zsb__"   | xs :: "'d list" and x :: "'d" and w :: "'b" and ys :: "'b list" and y :: "'b" and z :: "'c" and zs :: "'c list" where "as = xs @ [x]" and "bs = w # ys @ [y]" and "cs = z # zs"
strict_trans: fixes b :: "'c word"   and a :: "'c word"   and c :: "'c word" assumes "b <s a"   and "c <s b" shows "c <s a"
inject: fixes x1 :: "'a"   and y1 :: "'a" shows "(prot_atom.Atom x1 = prot_atom.Atom y1) = (x1 = y1)"
f: shows "(\<Sum>x\<in>UNIV. f x *s X $ x) = v"
restrict_bind_spmf: fixes p :: "'b spmf"   and g :: "'b \<Rightarrow> 'a spmf"   and A :: "'a set" shows "bind_spmf p g \<upharpoonleft> A = bind_spmf p (\<lambda>x. g x \<upharpoonleft> A)"
simps: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'d"   and a2 :: "'d" shows "r\<^sup>+\<^sup>+ a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> r a b) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>+\<^sup>+ a b \<and> r b c))"
boundOutputChainAlpha'': fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and M :: "'e"   and P :: "('e, 'f, 'g) psi"   and yvec :: "name list" assumes "(p \<bullet> xvec) \<sharp>* M"   and "(p \<bullet> xvec) \<sharp>* P"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)"   and "set xvec \<subseteq> set yvec" shows "\<lparr>\<nu>*yvec\<rparr>M \<prec>' P = \<lparr>\<nu>*p \<bullet> yvec\<rparr>(p \<bullet> M) \<prec>' (p \<bullet> P)"
fresh_string_if: fixes Xs :: "char list set"   and Y :: "char list" shows "fresh_string Xs Y = (if Y \<in> Xs \<and> finite Xs then fresh_string (Xs - {Y}) (upChar Y) else Y)"
range_dehomogenize: fixes x :: "'b" shows "range (dehomogenize x) = P[- {x}]"
abs_e: fixes i :: "nat" shows "\<bar>e i\<bar> \<le> 1"
SC_sublist: fixes xs :: "('b, 'c) Combinators list"   and P :: "('b, 'c) Combinators \<Rightarrow> bool"   and l :: "'b set list" assumes "singleCombinators xs" shows "singleCombinators (qsort (filter P xs) l)"
cases: fixes x :: "tm \<times> tm" obtains u :: "name" and p :: "tm" and q :: "tm" and v :: "name" where "atom u \<sharp> (p, q, v)" and "atom v \<sharp> (p, q)" and "x = (p, q)"
hd_take: fixes j :: "nat"   and xs :: "'b list" assumes "0 < j" shows "hd (xs \<down> j) = hd xs"
\<iota>\<^sub>0_is': shows "\<iota>\<^sub>0 = Infer ((CAs0 @ [DA0]) \<cdot>\<cdot>cl (\<eta>s @ [\<eta>])) (E0 \<cdot> \<eta>2)"
sup_upper: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> A" shows "x \<sqsubseteq> \<Squnion>A"
has_field_derivative_Arcsin: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "(Arcsin has_field_derivative inverse (cos (Arcsin z))) (at z)"
in_dgrad_p_set_iff: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "(p \<in> gd_term.dgrad_p_set pair_of_term d m) = (\<forall>v\<in>keys p. d (term_powerprod.pp_of_term pair_of_term v) \<le> m)"
strict_mono_to_nat: shows "strict_mono mod_type_class.to_nat"
reachable_snapshot_inv_dequeue_Mutate: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and m' :: "'h"   and r :: "'g"   and f :: "'e"   and opt_r' :: "'g option"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers (mutator m') s = mw_Mutate r f opt_r' # ws"   and "mut_m.marked_insertions m' s"   and "mut_m.marked_deletions m' s"   and "mut_m.reachable_snapshot_inv m s"   and "strong_tricolour_inv s"   and "valid_refs_inv s" shows "mut_m.reachable_snapshot_inv m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := opt_r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(mutator m' := ws)\<rparr>))"
WTrtBinOpAdd: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P,E,h,sh \<turnstile> e\<^sub>1 : Integer"   and "P,E,h,sh \<turnstile> e\<^sub>2 : Integer" shows "P,E,h,sh \<turnstile> e\<^sub>1 \<guillemotleft>Add\<guillemotright> e\<^sub>2 : Integer"
H_iso_cond: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and p :: "'c"   and x :: "'c"   and y :: "'b"   and q :: "'c" assumes "mono f"   and "mono g"   and "p \<sqinter> x \<le> f y"   and "q \<sqinter> x \<le> g y" shows "x \<le> ((\<lfloor>p\<rfloor> \<circ> f) \<sqinter> (\<lfloor>q\<rfloor> \<circ> g)) y"
Join_welldef_D1: fixes F :: "'b program"   and G :: "'b program" assumes "F \<squnion> G \<in> welldef" shows "F \<in> welldef"
incseq_imp_monoseq: fixes X :: "nat \<Rightarrow> 'a" assumes "incseq X" shows "monoseq X"
ltake_iterates_Suc: fixes n :: "nat"   and m :: "nat" shows "ltake (enat n) (iterates Suc m) = llist_of [m..<n + m]"
delete_point_wf: fixes pa :: "'a"   and p :: "'a" shows "incidence_system (add_point_sys.del_point pa p) (add_point_sys.del_point_blocks p)"
enabledPrefixSingleFinalStep: fixes SES :: "('c, 'd) SES_rec"   and s :: "'c"   and t :: "'d list"   and e :: "'d" assumes "enabled SES s (t @ [e])" shows "\<exists>t' t''. t' e\<longrightarrow>\<^bsub>SES\<^esub> t''"
subtype_refl: fixes t :: "'BExp set"   and P :: "'BExp set" shows "t \<le>:\<^sub>P t"
complex_mod_minus_le_complex_mod: fixes x :: "complex" shows "- cmod x \<le> cmod x"
map\<^sub>1_simps_1: fixes f :: "'a" assumes "C.ide f" shows "D.arr (\<tau>\<^sub>1 f)"
real_transfer_operator_pos: fixes f :: "'a \<Rightarrow> real" assumes "AE x in M. 0 \<le> f x"   and "f \<in> borel_measurable M" shows "AE x in M. 0 \<le> real_transfer_operator f x"
Max_wrt_among_of_weak_ranking_Cons: fixes A :: "'alt set" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = (if x \<inter> A = {} then Max_wrt_among (of_weak_ranking xs) A else x \<inter> A)"
star_of_less_HNatInfinite: fixes N :: "nat star"   and n :: "nat" assumes "N \<in> HNatInfinite" shows "hypnat_of_nat n < N"
mplus_def: shows "mplus = COERCE(udom\<cdot>'d \<rightarrow> udom\<cdot>'d \<rightarrow> udom\<cdot>'d, 'c\<cdot>'d \<rightarrow> 'c\<cdot>'d \<rightarrow> 'c\<cdot>'d)\<cdot>plusU"
assms_1: shows "is_pole (moebius_similarity a b) z"
blinding_respects_hashes_F: fixes rha :: "'e \<Rightarrow> 'f"   and boa :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and rhb :: "'g \<Rightarrow> 'h"   and bob :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "blinding_respects_hashes rha boa"   and "blinding_respects_hashes rhb bob" shows "blinding_respects_hashes (root_hash_F rha rhb) (blinding_of_F rha boa rhb bob)"
card_SigmaI: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A"   and "\<forall>a\<in>A. finite (B a)" shows "card (Sigma A B) = (\<Sum>a\<in>A. card (B a))"
induced_induce: fixes vs :: "'a set" assumes "vs \<subseteq> verts (G \<restriction> verts H)" shows "induced_subgraph ((G \<restriction> verts H) \<restriction> vs) (G \<restriction> verts H)"
Cauchy_def: fixes X :: "nat \<Rightarrow> 'b" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
zero_hom_0_axioms: shows "zero_hom_0 (inv (inv (inv hom)))"
n_circ_7: fixes p' :: "complex"   and k :: "nat" assumes "p' \<notin> s - pts" shows "winding_number (n_circ k) p' = 0 \<and> p' \<notin> path_image (n_circ k)"
assms_4: shows "\<not> is_zero_row (i + (1::'c)) A"
subset_imp: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "Min B \<le> Min A"
step_phi: fixes v :: "'val" assumes "v \<in> allVars g"   and "v \<noteq> chooseNext g" shows "step.phi g v = map_option (map (substNext g)) (phi g v)"
assert_demonic_comp: fixes p :: "'b \<Rightarrow> bool"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p' :: "'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'a \<Rightarrow> bool" shows "{. p .} \<circ> [: r :] \<circ> {. p' .} \<circ> [: r' :] = {.x. p x \<and> (\<forall>y. r x y \<longrightarrow> p' y).} \<circ> [: r OO r' :]"
hom_mult_eq_zero: fixes x :: "'b"   and y :: "'b" assumes "x * y = (0::'b)" shows "to_fract x * to_fract y = 0"
real_sqrt_sum_squares_less: fixes x :: "real"   and u :: "real"   and y :: "real" assumes "\<bar>x\<bar> < u / sqrt 2"   and "\<bar>y\<bar> < u / sqrt 2" shows "sqrt (x\<^sup>2 + y\<^sup>2) < u"
sum_product: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'c"   and B :: "'e set" shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
\<phi>: shows "\<guillemotleft>\<phi> : g \<star> f\<^sup>* \<Rightarrow> r\<guillemotright> \<and> local.iso \<phi>"
sorted_wrt1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" shows "sorted_wrt P [x] = True"
poly_roots_mult: fixes p :: "'b poly"   and q :: "'b poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "poly_roots (p * q) = poly_roots p + poly_roots q"
r_prod_encode_def: shows "r_prod_encode \<equiv> Cn 2 r_add [Cn 2 r_triangle [r_add], recf.Id 2 0]"
cltn2_inverse_compose: fixes x :: "cltn2"   and y :: "cltn2" shows "cltn2_inverse (cltn2_compose x y) = cltn2_compose (cltn2_inverse y) (cltn2_inverse x)"