hcju/isabelleps huggingface dataset:
fps_subdegree_mult_fps_X_power_2: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0" shows "subdegree (f * fps_X ^ n) = subdegree f + n"
dgrad_set_leI: fixes S :: "'e set"   and T :: "'e set"   and d :: "'e \<Rightarrow> nat" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. d s \<le> d t" shows "dgrad_set_le d S T"
reindex: fixes h :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and g :: "'d \<Rightarrow> 'e" assumes "inj_on h A" shows "prod g (h ` A) = prod (g \<circ> h) A"
gen_ramanujan_sum_fourier_expansion: fixes k :: "nat"   and f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex"   and n :: "nat" assumes "0 < k" shows "s f g k n = (\<Sum>m\<le>k - 1. 1 / of_nat k * (\<Sum>d | d dvd gcd m k. g d * f (k div d) * of_nat d) * unity_root k (int m * int n))"
currentLevelPrefix: fixes a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" assumes "isPrefix a b" shows "currentLevel a \<le> currentLevel b"
widest_spread_axis_Un_1: fixes k :: "'a"   and K :: "'a set"   and P :: "(real, 'a) vec set"   and k' :: "'a" assumes "widest_spread_axis k K P"   and "spread k' P \<le> spread k P" shows "widest_spread_axis k (K \<union> {k'}) P"
lem: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and t :: "lam"   and n :: "nat"   and z :: "'a" assumes "\<And>n ts z. (\<And>z. \<forall>t\<in>set ts. P z t) \<Longrightarrow> P z (Var n \<degree>\<degree> ts)"   and "\<And>x u ts z. \<lbrakk>x \<sharp> z; \<And>z. P z u; \<And>z. \<forall>t\<in>set ts. P z t\<rbrakk> \<Longrightarrow> P z ((Lam [x].u) \<degree>\<degree> ts)"   and "size t = n" shows "P z t"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Assumption x1 = Assumption y1) = (x1 = y1)"
le_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and o\<^sub>1 :: "'a option"   and o\<^sub>2 :: "'a option" shows "Opt.le r o\<^sub>1 o\<^sub>2 = (case o\<^sub>2 of None \<Rightarrow> o\<^sub>1 = None | \<lfloor>y\<rfloor> \<Rightarrow> case o\<^sub>1 of None \<Rightarrow> True | \<lfloor>x\<rfloor> \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
connected_Times_eq: fixes S :: "'c set"   and T :: "'d set" shows "connected (S \<times> T) = (S = {} \<or> T = {} \<or> connected S \<and> connected T)"
linear_sum: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd"   and S :: "'f set" assumes "clinear f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
Y_def: shows "(Y::'a::refinement_lattice set) = {(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) y |y::'a::refinement_lattice. (x::'a::refinement_lattice) \<sqsubseteq> F y}"
max3_def: fixes a :: "'b ivl"   and m :: "'b"   and n :: "'b" shows "max3 a m n = max (high a) (max m n)"
subgraph_awalk_imp_awalk: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "pre_digraph.awalk H u p v"   and "subgraph H G" shows "awalk u p v"
bind_return_val: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> val"   and t' :: "pdf_type" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M stock_measure t'" shows "M \<bind> (\<lambda>x. return_val (f x)) = distr M (stock_measure t') f"
countable_Int2: fixes B :: "'c set"   and A :: "'c set" assumes "countable B" shows "countable (A \<inter> B)"
partition_filter_conv: fixes f :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "List.partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
A: shows "Maps.Map (\<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<odot> \<lbrakk>\<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>) = Maps.Comp \<lbrakk>t\<^sub>0\<rbrakk> \<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk>"
ring_mat_simps_1: fixes ty :: "'c itself"   and n :: "nat"   and b :: "'d" shows "(\<otimes>\<^bsub>ring_mat ty n b\<^esub>) = (*)"
simps_2: fixes a :: "ereal" shows "\<infinity> + a = \<infinity>"
w_nonneg: shows "\<forall>i. 0 \<le> w i"
normalized_gauss_int_1: fixes z :: "gauss_int" assumes "normalize z = z" shows "0 \<le> ReZ z"
sup_aforms_refine: fixes xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and d :: "nat"   and optns :: "nat" assumes "(xi, x) \<in> aforms_rel"   and "length xi = d" shows "(RETURN (sup_aforms optns xi), Sup_specs d x) \<in> \<langle>rl_rel\<rangle>nres_rel"
pinf_4: fixes t :: "'b" shows "\<exists>z. \<forall>x>z. (x \<noteq> t) = True"
lowerPowers_higherPowers: fixes n :: "nat"   and i :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 'b" assumes "\<And>j. \<lbrakk>n \<le> j; j < n + i\<rbrakk> \<Longrightarrow> lookup x j = (0::'b)" shows "lowerPowers n i (higherPowers n i x) = x"
rep_eq: fixes x :: "nat"   and xa :: "integer" shows "int_of_integer (drop_bit x xa) = drop_bit x (int_of_integer xa)"
mod_ring_normalize: shows "(mod_ring_rel ===> mod_ring_rel) (\<lambda>x. if x = 0 then 0 else 1) normalize"
freshResidualChain4: fixes xvec :: "name list"   and \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi" assumes "xvec \<sharp>* (\<alpha> \<prec> P)" shows "xvec \<sharp>* subject \<alpha>"
maxR_def: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "MaxR r" shows "MaxR.maxR r x y = (if (x, y) \<in> r then y else x)"
natUnion_elem: fixes x :: "'b"   and f :: "nat \<Rightarrow> 'b set"   and n :: "nat" assumes "x \<in> f n" shows "x \<in> natUnion f"
unat_power_lower: fixes n :: "nat" assumes "n < LENGTH('a)" shows "unat (2 ^ n) = 2 ^ n"
assms: shows "to_gc_conf r c = (\<Gamma>, e, S)"
analz_auth_data_in: fixes x :: "rat"   and H :: "msg set"   and y :: "rat" assumes "Pri_AgrK x \<in> analz H \<or> Pri_AgrK y \<in> analz H" shows "analz (insert (Auth_Data x y) H) = insert (Auth_Data x y) (analz ({Pri_AgrK x, Pri_AgrK y} \<union> H))"
set_plus_rearrange2: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" shows "a +o (b +o C) = (a + b) +o C"
f: fixes a :: "'a \<Rightarrow> 'b" assumes "a \<in> A" shows "poly_eval a f = (0::'b)"
assms_1: shows "prv (dsj \<phi>1 \<phi>2)"
omit_branch_mem_diff_formula: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set"   and q :: "('c, 'd) fm" assumes "all_is_branch p i branch xs"   and "q at i in branch"   and "p \<noteq> q" shows "q at i in omit_branch xs branch"
infinity_unat_def: shows "\<infinity> \<equiv> Abs_unat None"
T_monotonic: shows "MonotProblem TE_wtFsym wtPsym TE_arOf TE_resOf parOf tPB"
set_mset_add_mset_insert: fixes a :: "'b"   and A :: "'b multiset" shows "set_mset (add_mset a A) = insert a (set_mset A)"
jumpNestingOk_evalE: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and jmps :: "jump set"   and Env :: "env"   and T :: "tys" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "jumpNestingOk jmps t"   and "Env\<turnstile>t\<Colon>T"   and "wf_prog G"   and "prg Env = G"   and "\<And>j. fst s0 = Some (Jump j) \<Longrightarrow> j \<in> jmps" obtains "\<And>j. fst s1 = Some (Jump j) \<Longrightarrow> j \<in> jmps" and "\<And>w upd s j val. \<lbrakk>normal s1; v = In2 (w, upd); fst s \<noteq> Some (Jump j)\<rbrakk> \<Longrightarrow> fst (upd val s) \<noteq> Some (Jump j)"
sig_red_subset: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q"   and "sing_reg = (\<preceq>\<^sub>t) \<or> sing_reg = (\<prec>\<^sub>t)" shows "sig_red sing_reg top_tail {f \<in> F. sing_reg (lt f) (lt p)} p q"
f: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set xs" shows "f x \<circ> f y = f y \<circ> f x"
src_src: fixes \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>" shows "S.src (S.src \<mu>) = S.src \<mu>"
pred_prod_inject: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" shows "pred_prod P1 P2 (a, b) = (P1 a \<and> P2 b)"
trg_hcomp: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.hseq \<mu> \<nu>" shows "trg\<^sub>C (\<mu> \<star>\<^sub>C \<nu>) = trg\<^sub>C \<mu>"
oalist_inv_raw_map_val_pair: fixes x :: "'o"   and xs :: "('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e" assumes "oalist_inv_raw x xs" shows "oalist_inv_raw x (map_val_pair f xs)"
cadjoint_univ_prop': fixes G :: "'b \<Rightarrow> 'c" assumes "bounded_clinear G" shows "\<forall>x y. \<langle>x, (G\<^sup>\<dagger>) y\<rangle> = \<langle>G x, y\<rangle>"
simps: fixes it :: "('a, 'b, 'c, 'd) itnode rose_tree" shows "iAnnot it = iAnnot' (root it)"
subst3: fixes xvec :: "name list"   and Tvec :: "'b list"   and T :: "'a"   and x :: "name" assumes "length xvec = length Tvec"   and "distinct xvec"   and "set xvec \<subseteq> supp T"   and "x \<sharp> T[xvec::=Tvec]" shows "x \<sharp> Tvec"
suf_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
if_true_symeval: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and b :: "'a upred"   and P :: "'a hrel"   and Q :: "'a hrel" assumes "\<Gamma> \<dagger> b = true" shows "\<Gamma> \<Turnstile> (P \<triangleleft> b \<triangleright>\<^sub>r Q) = \<Gamma> \<Turnstile> P"
sup_compl: fixes x :: "'a" shows "x \<squnion> - x = \<top>"
prv_implies_scnj_insert: fixes F :: "'fmla set"   and \<phi> :: "'fmla" assumes "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "prv (scnj (insert \<phi> F))" shows "prv (cnj \<phi> (scnj F))"
closed: shows "rel D \<subseteq> carrier D \<times> carrier D"
fps_XD'_prod: fixes a :: "'a fps"   and b :: "'a fps"   and c :: "'a fps" shows "fps_XD' a (b * c) = fps_XD' a b * c + b * fps_XD' a c"
assms_1: shows "factorize_polys qs = (fs, data)"
same_values_def: fixes f :: "'a trm \<Rightarrow> 'a trm" shows "same_values f = (\<lambda>x y. f x = f y)"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b" assumes "bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B" shows "op_bicategory V H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
numeral_le_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus m) (numeral.numeral one plus n) = (m \<le> n)"
l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_root_node_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_host get_host_locs get_ancestors_si get_ancestors_si_locs get_root_node_si get_root_node_si_locs"
surj_xor: fixes x :: "'a" assumes "x \<in> carrier L" shows "(\<oplus>) x ` carrier L = carrier L"
scf_list_def: fixes scf :: "nat \<Rightarrow> nat"   and xs :: "'a list" shows "scf_list scf xs = concat (map2 (\<lambda>x i. replicate (scf i) x) xs [0..<length xs])"
deg_belowI: fixes R :: "('a, 'b) ring_scheme"   and n :: "nat"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "n \<noteq> 0 \<Longrightarrow> coeff (UP R) p n \<noteq> \<zero>\<^bsub>R\<^esub>"   and "p \<in> carrier (UP R)" shows "n \<le> deg R p"
has_subprob_density_dens_AE: fixes f :: "'c \<Rightarrow> ennreal"   and f' :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure" assumes "AE y in N. f y = f' y"   and "f' \<in> borel_measurable N"   and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f' x"   and "has_subprob_density M N f" shows "has_subprob_density M N f'"
RETURN_to_SPEC_rule: fixes m :: "'b nres"   and v :: "'b" assumes "m \<le> SPEC ((=) v)" shows "m \<le> RETURN v"
card_less_if_surj_not_inj: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "finite A"   and "f ` A = B"   and "\<not> inj_on f A" shows "card B < card A"
pair_spmf_return_spmf2: fixes p :: "'c spmf"   and y :: "'d" shows "pair_spmf p (return_spmf y) = map_spmf (\<lambda>x. (x, y)) p"
S_Suc: fixes n :: "nat" shows "S (Suc n) = x * S n /\<^sub>R real (Suc n)"
indaut_adj_halfchsys_im_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "C \<sim> D" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g ` D = C"
preserves_reflects_arr: fixes f :: "'a" shows "arr (\<I> \<otimes> f) = arr f"
append_eq_Cons_conv: fixes ys :: "'c list"   and zs :: "'c list"   and x :: "'c"   and xs :: "'c list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
smallest_dist_is_ortho: fixes M :: "'a set"   and h :: "'a"   and k :: "'a" assumes "closed_csubspace M" shows "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k = (h - k \<in> orthogonal_complement M \<and> k \<in> M)"
np2: shows "num_params (tm_poly t2) \<le> length I"
to_list_correct: fixes e :: "'f" shows "map_of (Fubx_to_list e) = Fubx_get e"
succ_inv_5: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and s' :: "('a, 'b) state"   and v :: "nat \<Rightarrow> 'b" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> PivotUpdateMinVars.gt_state eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "v \<Turnstile>\<^sub>t \<T> s = v \<Turnstile>\<^sub>t \<T> s'"
IH_1: fixes V :: "nat set" assumes "randomfree e1"   and "free_vars e1 \<subseteq> V" shows "(\<lambda>\<sigma>. expr_sem_rf \<sigma> e1) \<in> state_measure V \<Gamma> \<rightarrow>\<^sub>M stock_measure t1"
subst_Var_simp1: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and xs :: "'varSort"   and y :: "'var"   and x :: "'var" assumes "good Y"   and "ys \<noteq> xs \<or> y \<noteq> x" shows "Var xs x #[Y / y]_ys = Var xs x"
cl_type_def: fixes cl :: "cl" shows "cl_type cl = (case cl of Func_native x tf xa xb \<Rightarrow> tf | Func_host tf x \<Rightarrow> tf)"
size_union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "size (M + N) = size M + size N"
stepE: fixes i :: "nat" obtains l :: "nat" where "alstep l (s i) (s (Suc i))"
estep_Mn_def: fixes e :: "nat" shows "estep_Mn e \<equiv> if e2ls e = 0 then prod_encode (e_cons (triple_encode (pdec22 (e2i e)) (e_cons 0 (e2xs e)) 0) (e_cons (triple_encode (e2i e) (e2xs e) (singleton_encode 0)) (e2tail e)), 0) else if e2rv e = 1 then prod_encode (e2tail e, Suc (e_hd (e2ls e))) else prod_encode (e_cons (triple_encode (pdec22 (e2i e)) (e_cons (Suc (e_hd (e2ls e))) (e2xs e)) 0) (e_cons (triple_encode (e2i e) (e2xs e) (singleton_encode (Suc (e_hd (e2ls e))))) (e2tail e)), 0)"
val_Zp_not_equal_ord_plus_minus: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp x \<noteq> val_Zp y" shows "val_Zp (x \<ominus> y) = val_Zp (x \<oplus> y)"
subst_w2rexp: fixes xs :: "nat list"   and ys :: "nat list"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "lang (subst (w2rexp (xs @ ys)) \<sigma>) = lang (subst (w2rexp xs) \<sigma>) @@ lang (subst (w2rexp ys) \<sigma>)"
ord_strict_pI: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lookup p v = (0::'c)"   and "lookup q v \<noteq> (0::'c)"   and "\<And>u. lt_of_nat_term_order cmp_term v u \<Longrightarrow> lookup p u = lookup q u" shows "ord_strict_p p q"
ext_inject: fixes Chn :: "'b"   and Dom :: "'b span_data"   and Cod :: "'b span_data"   and more :: "'c"   and Chn' :: "'b"   and Dom' :: "'b span_data"   and Cod' :: "'b span_data"   and more' :: "'c" shows "(\<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr> = \<lparr>Chn = Chn', Dom = Dom', Cod = Cod', \<dots> = more'\<rparr>) = (Chn = Chn' \<and> Dom = Dom' \<and> Cod = Cod' \<and> more = more')"
dominoes_tile_matrix: fixes m :: "nat"   and n :: "nat" shows "below m \<times> below (2 * n) \<in> tiling domino"
composable: shows "src (Maps.REP R\<^sub>0)\<^sup>* = trg (Maps.REP S\<^sub>1)"
ordinal_less_times_div_plus: fixes y :: "ordinal"   and x :: "ordinal" assumes "0 < y" shows "x < y * (x div y) + y"
simps: fixes stop :: "'a \<Rightarrow> bool"   and "next" :: "'a \<Rightarrow> 'a"   and a :: "'a" shows "enat_unfold stop next a = (if stop a then 0 else eSuc (enat_unfold stop next (next a)))"
ring_mat: fixes n :: "nat"   and b :: "'c" shows "ring (ring_mat TYPE('b) n b)"
lcm_altdef_int: fixes a :: "int"   and b :: "int" shows "lcm a b = \<bar>a\<bar> * \<bar>b\<bar> div gcd a b"
psi_pos: fixes n :: "nat" shows "0 \<le> psi n"
ivl_subset: fixes i :: "'d"   and j :: "'d"   and m :: "'d"   and n :: "'d" shows "({i..<j} \<subseteq> {m..<n}) = (j \<le> i \<or> m \<le> i \<and> j \<le> n)"
\<G>_subset: fixes N1 :: "'f set"   and N2 :: "'f set" assumes "N1 \<subseteq> N2" shows "\<G>_Fset N1 \<subseteq> \<G>_Fset N2"
set_mp: fixes A :: "'c set"   and B :: "'c set"   and c :: "'c" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
varset_action_pair: fixes p :: "('c, bool) fmap"   and e :: "('c, bool) fmap"   and vs :: "'c set" shows "varset_action (p, e) vs = (fmdom' e \<subseteq> vs)"
small_insert_iff: fixes a :: "'a"   and X :: "'a set" shows "small (insert a X) = small X"
arr_MkArr: fixes A :: "'a \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "A \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))"   and "B \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))"   and "f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) A B)" shows "arr (MkArr A B f)"
Axiom_of_Infinity: fixes \<alpha> :: "V" assumes "\<Z> \<alpha>" shows "\<omega> \<in>\<^sub>\<circ> Vset \<alpha>"
assms: shows "Q \<in> sets (N \<Otimes>\<^sub>M M)"
a_notin_segment: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c" assumes "Order D" shows "a \<notin> segment D a"
extremum: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and a :: "'a" assumes "ordering_top less_eq less top" shows "less_eq a top"
FStep: shows "\<forall>cfgList msgList. \<exists>cfgList' msgList'. fStepCfg cfgList msgList = cfgList' \<and> fStepMsg cfgList msgList = msgList' \<and> (initial (hd cfgList) \<and> nonUniform (last cfgList) \<and> execution trans sends start cfgList msgList \<longrightarrow> nonUniform (last (fStepCfg cfgList msgList)) \<and> prefixList cfgList (fStepCfg cfgList msgList) \<and> prefixList msgList (fStepMsg cfgList msgList) \<and> execution trans sends start (fStepCfg cfgList msgList) (fStepMsg cfgList msgList) \<and> initial (hd (fStepCfg cfgList msgList)) \<and> (\<exists>msg'. execution.minimalEnabled cfgList msgList msg' \<and> msg' \<in> set (drop (length msgList) (fStepMsg cfgList msgList))))"
comp_FSwSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'h, 'i, 'j) model_scheme"   and h :: "'h \<Rightarrow> 'k"   and hA :: "'i \<Rightarrow> 'l"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'k, 'l, 'm) model_scheme"   and h' :: "'k \<Rightarrow> 'n"   and hA' :: "'l \<Rightarrow> 'o"   and MOD'' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'n, 'o, 'p) model_scheme" assumes "igWlsAbsIsInBar MOD"   and "FSwSbImorph h hA MOD MOD'"   and "FSwSbImorph h' hA' MOD' MOD''" shows "FSwSbImorph (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
simps_1: fixes a :: "'a"   and uu :: "'a code_status" shows "merge_cstatus (CFAILED a) uu = CFAILED a"
A_trusts_NS4_lemma: fixes evs :: "event list"   and K :: "nat"   and A :: "agent"   and NA :: "msg"   and B :: "agent"   and X :: "msg"   and NB :: "nat" assumes "evs \<in> ns_shared"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs"   and "Crypt K (Nonce NB) \<in> parts (knows Spy evs)" shows "Says B A (Crypt K (Nonce NB)) \<in> set evs"
pcompose_smult: fixes a :: "'a"   and p :: "'a poly"   and r :: "'a poly" shows "smult a p \<circ>\<^sub>p r = smult a (p \<circ>\<^sub>p r)"
assms_2: assumes "b"   and "b'" shows "ERETURN S1 \<le> \<Down>\<^sub>E E R S1'"
C1_differentiable_on_subset: fixes f :: "real \<Rightarrow> 'a"   and T :: "real set"   and S :: "real set" assumes "f C1_differentiable_on T"   and "S \<subseteq> T" shows "f C1_differentiable_on S"
assm: shows "len v ts c = ext v \<and> v=v1\<parallel>v2"
LLL_invariant_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and \<alpha> :: "rat"   and upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "LLL.LLL_invariant n m fs_init \<alpha> upw i fs = (cof_vec_space.lin_indpt_list n (map of_int_hom.vec_hom fs) \<and> vec_module.lattice_of n fs = LLL.L n fs_init \<and> gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i \<and> i \<le> m \<and> length fs = m \<and> (upw \<or> LLL.\<mu>_small n fs i))"
\<rr>_ide_simp: fixes f :: "'a" assumes "ide f" shows "\<rr> f = \<r>[f]"
Var: fixes A :: "'b set set"   and x :: "'e" shows "timpls_transformable_to_pred A (Var x) (Var x)"
gAbsCongS_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gAbsCongS MOD \<equiv> \<forall>xs x x2 y s X' X X2' X2. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<and> wls s X2' \<and> gWls MOD s X2 \<longrightarrow> fresh xs y X' \<and> gFresh MOD xs y X' X \<and> fresh xs y X2' \<and> gFresh MOD xs y X2' X2 \<and> X' #[y \<and> x]_xs = X2' #[y \<and> x2]_xs \<longrightarrow> gSwap MOD xs y x X' X = gSwap MOD xs y x2 X2' X2 \<longrightarrow> gAbs MOD xs x X' X = gAbs MOD xs x2 X2' X2"
intros_2: fixes h :: "'a"   and H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
to_bl_n1: shows "to_bl (- 1) = replicate LENGTH('b) True"
assms_1: shows "h \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
Queue_def: shows "Queue \<equiv> \<lambda>s. length (filter (\<lambda>i. \<not> Z i \<and> M s ! T i \<noteq> Black) (Muts s))"
PO_m1_inv1r_cache_trans: shows "{m1_inv1r_cache} TS.trans m1 {> m1_inv1r_cache}"
vec_plus_index: fixes nr :: "nat"   and v1 :: "'c list"   and v2 :: "'c list"   and i :: "nat"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "vec nr v1"   and "vec nr v2"   and "i < nr" shows "vec_plusI pl v1 v2 ! i = pl (v1 ! i) (v2 ! i)"
signed_take_bit_diff: fixes n :: "nat"   and k :: "int"   and l :: "int" shows "signed_take_bit n (signed_take_bit n k - signed_take_bit n l) = signed_take_bit n (k - l)"
prems: shows "a \<in> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b (x # xs)"
is_filter_Rep_filter: fixes F :: "'a filter" shows "is_filter (Rep_filter F)"