hcju/isabelleps huggingface dataset:
FG_advice_monotone: fixes X :: "'b ltln set"   and Y :: "'b ltln set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" assumes "X \<subseteq> Y"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<mu>" shows "w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>"
B: shows "U \<subseteq> unaffected_domains I D (D ` set ys) (zs @ [z])"
Cons_replicate_eq: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat"   and y :: "'a" shows "(x # xs = y \<up> n) = (x = y \<and> 0 < n \<and> xs = x \<up> (n - 1))"
diff_inv_eq_inv_set: fixes I :: "'b \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and G :: "'b \<Rightarrow> bool" shows "diff_invariant I f U S t\<^sub>0 G = (\<forall>s. I s \<longrightarrow> g_orbital f G U S t\<^sub>0 s \<subseteq> {s. I s})"
to_tree_order_pure: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs to_tree_order" shows "pure (to_tree_order ptr) h"
is_unit_pCons_iff: fixes a :: "'a"   and p :: "'a poly" shows "is_unit (pCons a p) = (p = 0 \<and> a \<noteq> (0::'a))"
down'_\<beta>: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and l :: "nat"   and lm :: "nat"   and dm :: "nat"   and p :: "(nat \<times> int) list"   and fl :: "real"   and fr :: "real"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" assumes "d < length b"   and "l + level b = lm"   and "b \<in> sparsegrid' dm"   and "p \<in> sparsegrid' dm" shows "down' d l b fl fr \<alpha> p = (if p \<in> lgrid b {d} lm then (fl + (fr - fl) / 2 * (real_of_int (ix p d) / 2 ^ (lv p d - lv b d) - real_of_int (ix b d) + 1)) / 2 ^ (lv p d + 1) + (\<Sum>p'\<in>parents d b p. \<alpha> p' * l2_\<phi> (p ! d) (p' ! d)) else \<alpha> p)"
psubst_LLq: fixes t1 :: "'trm"   and t2 :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "t1 \<in> atrm"   and "t2 \<in> atrm"   and "fst ` set txs \<subseteq> atrm"   and "snd ` set txs \<subseteq> var"   and "distinct (map snd txs)" shows "psubst (LLq t1 t2) txs = LLq (psubstT t1 txs) (psubstT t2 txs)"
alpha_trans_twice: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X'' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X''' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X \<or> qGood X' \<or> qGood X'' \<or> qGood X'''"   and "X #= X'"   and "X' #= X''"   and "X'' #= X'''" shows "X #= X'''"
subsumption_impl_axioms: shows "subsumption_impl r subsumes set (\<lambda>x. case x of h # t \<Rightarrow> (h, t)) (@) length"
lincomb_list_append: fixes Ws :: "'a vec list"   and Vs :: "'a vec list"   and f :: "nat \<Rightarrow> 'a" assumes "set Ws \<subseteq> carrier_vec n"   and "set Vs \<subseteq> carrier_vec n" shows "lincomb_list f (Vs @ Ws) = lincomb_list f Vs + lincomb_list (\<lambda>i. f (i + length Vs)) Ws"
monom_mult_eq_zero_iff: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "(local.punit.monom_mult c t p = 0) = (c = (0::'b) \<or> p = 0)"
mult_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<oplus> b \<oplus> c = a \<oplus> (b \<oplus> c)"
tendsto_sup: fixes X :: "'a \<Rightarrow> 'b"   and x :: "'b"   and net :: "'a filter"   and Y :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "(X \<longlongrightarrow> x) net"   and "(Y \<longlongrightarrow> y) net" shows "((\<lambda>i. sup (X i) (Y i)) \<longlongrightarrow> sup x y) net"
\<tau>\<^sub>2: shows "\<tau>' = \<lfloor>\<tau>\<^sub>2\<rfloor>"
set_insert: fixes x :: "'d"   and xs :: "'d list" shows "set (List.insert x xs) = insert x (set xs)"
CNOT_inv: shows "CNOT * CNOT = 1\<^sub>m 4"
fresh_atm_4: fixes a :: "vrs"   and b :: "tyvrs" shows "a \<sharp> b"
cos_45: shows "cos (pi / 4) = sqrt 2 / 2"
euler_maclaurin_strong_nat': fixes b :: "nat" assumes "1 \<le> b" shows "(\<Sum>k = 1..b. 1 / real k) = ln (real b) + euler_mascheroni + (1 / 2) *\<^sub>R (1 / real b) + (\<Sum>i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\<^sub>R fs (2 * i - 1) (real b)) - EM_remainder (2 * N + 1) (fs (2 * N + 1)) (int b)"
simps: fixes a1 :: "freeExp"   and a2 :: "freeExp" shows "(a1 \<sim> a2) = ((\<exists>X Y Z. a1 = PLUS X (PLUS Y Z) \<and> a2 = PLUS (PLUS X Y) Z) \<or> (\<exists>N. a1 = VAR N \<and> a2 = VAR N) \<or> (\<exists>X X' Y Y'. a1 = PLUS X Y \<and> a2 = PLUS X' Y' \<and> X \<sim> X' \<and> Y \<sim> Y') \<or> (\<exists>Xs Xs' F. a1 = FNCALL F Xs \<and> a2 = FNCALL F Xs' \<and> (Xs, Xs') \<in> listrel exprel) \<or> (\<exists>X Y. a1 = Y \<and> a2 = X \<and> X \<sim> Y) \<or> (\<exists>X Y Z. a1 = X \<and> a2 = Z \<and> X \<sim> Y \<and> Y \<sim> Z))"
ta_hb_consistent_into_non_speculative: fixes P :: "'a prog"   and obs0 :: "('b \<times> ('c, 'b) obs_event action) list"   and obs :: "('b \<times> ('c, 'b) obs_event action) llist" assumes "ta_hb_consistent P obs0 obs" shows "non_speculative P (w_values P (\<lambda>_. {}) (map snd obs0)) (lmap snd obs)"
poly_const_pow: fixes a :: "'b"   and b :: "nat" shows "[:a:] ^ b = [:a ^ b:]"
poly_minus: fixes p :: "'a list"   and x :: "'a" shows "poly (-- p) x = - poly p x"
char_poly_factorized: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "\<exists>as. char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1:]) \<and> length as = n"
mset_set_insert: fixes x :: "'c"   and E :: "'c set" assumes "x \<notin> E"   and "finite E" shows "mset_set (E \<union> {x}) = mset_set E + {#x#}"
ordLess_imp_ordLeq: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "ordLess2 r r'" shows "ordLeq3 r r'"
notindom: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "a \<in> set p"   and "x \<notin> dom (Cp (list2FWpolicy p))" shows "x \<notin> dom (Cp a)"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl t0 x0" shows "x0 \<in> X"
countable_basis: fixes p :: "'c set \<Rightarrow> bool"   and B :: "'c set set" assumes "countable_basis p B" shows "countable B"
idiff_self: fixes n :: "eint" assumes "n \<noteq> \<infinity>" shows "n - n = 0"
induct: fixes P :: "natural \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b) \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "natural"   and a1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"   and a2 :: "'a" assumes "\<And>k f x. (\<And>xa. k \<noteq> 0 \<Longrightarrow> P (k - 1) f xa) \<Longrightarrow> P k f x" shows "P a0 a1 a2"
sum_le_1_prod: fixes d :: "real"   and c :: "real" assumes "1 \<le> d"   and "1 \<le> c" shows "c + d \<le> 1 + c * d"
comp_ide_self: fixes a :: "'a2" assumes "C2.ide a" shows "a \<cdot>\<^sub>2 a = a"
diag_mat_zero: fixes n :: "nat" shows "diag_mat (0\<^sub>m n n) = replicate n (0::'a)"
1: shows "prv (imp (PP \<langle>\<phi>\<rangle>) (imp (PP \<langle>neg \<phi>\<rangle>) (PP \<langle>fls\<rangle>)))"
eval_in_hom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'a \<Rightarrow> 'd"   and t :: "'a monoidal_language.term" assumes "MonoidalCategory.evaluation_map C D T \<alpha> \<iota> V"   and "monoidal_language.Arr C t" shows "partial_magma.in_hom D (MonoidalCategory.evaluation_map.eval D T \<alpha> \<iota> V t) (MonoidalCategory.evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Dom C t)) (MonoidalCategory.evaluation_map.eval D T \<alpha> \<iota> V (monoidal_language.Cod C t))"
gbinomial_int_mono: fixes x :: "int"   and y :: "int"   and k :: "nat" assumes "0 \<le> x"   and "x \<le> y" shows "x gchoose k \<le> y gchoose k"
alpha_res_alpha_set: fixes bs :: "atom set"   and x :: "'b"   and p :: "perm"   and cs :: "atom set"   and y :: "'b" shows "(bs, x) \<approx>res (=) supp p (cs, y) = (bs \<inter> supp x, x) \<approx>set (=) supp p (cs \<inter> supp y, y)"
star_inductl_star: fixes x :: "'a"   and y :: "'a" assumes "x ; y\<^sup>\<star> \<le> y\<^sup>\<star>" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
valid_strategy_updates_set: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "strategy p \<sigma>"   and "\<And>v. \<lbrakk>v \<in> A; v \<in> VV p; \<not> deadend v\<rbrakk> \<Longrightarrow> v \<rightarrow> \<sigma>' v" shows "strategy p (override_on \<sigma> \<sigma>' A)"
sum_right: fixes x :: "'b \<Rightarrow>\<^sub>L 'a"   and g :: "'c \<Rightarrow> 'b"   and S :: "'c set" shows "blinfun_apply x (sum g S) = (\<Sum>i\<in>S. blinfun_apply x (g i))"
enc_def: fixes \<eta> :: "nat" shows "CIPHER.enc \<eta> \<equiv> CNV (stateless_callee (\<lambda>m. Pause (Inl ()) Done \<bind> (\<lambda>k. lift_spmf (enc \<eta> (projl k) m) \<bind> (\<lambda>c. Pause (Inr c) Done \<bind> (\<lambda>_. Done ()))))) ()"
map_snd_half_segments_aux_eq: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "map snd (half_segments_of_aform X) = tl (map fst (half_segments_of_aform X)) @ [snd (last (half_segments_of_aform X))]"
f2: shows "length (get_subsets (take_rows_from_matrix (mat_of_cols (dim_row (M_mat signs subsets)) (map ((!) (cols (M_mat signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))))) (rows_to_keep (mat_of_cols (dim_row (M_mat signs subsets)) (map ((!) (cols (M_mat signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))))), map ((!) subsets) (rows_to_keep (mat_of_cols (dim_row (M_mat signs subsets)) (map ((!) (cols (M_mat signs subsets))) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))))), map ((!) signs) (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))))) = length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
z_def: shows "z \<equiv> getFr [xx, yy, x] [t1, t2, s] []"
derived_set_of_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" shows "X derived_set_of S \<subseteq> topspace X"
map\<^sub>0_simp: fixes a :: "'b" assumes "B.obj a" shows "map\<^sub>0 a = G.map\<^sub>0 (F.map\<^sub>0 a)"
divide_poly_main_list: fixes lc :: "'b"   and d :: "'b list"   and n :: "nat"   and r :: "'b list"   and q :: "'b list" assumes "lc \<noteq> (0::'b)"   and "last d = lc"   and "d \<noteq> []"   and "n = 1 + length r - length d" shows "Poly (divide_poly_main_list lc q (rev r) (rev d) n) = divide_poly_main lc (monom (1::'b) n * Poly q) (Poly r) (Poly d) (length r - 1) n"
Class_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and T :: "ty" assumes "P \<turnstile> Class C \<le> T" shows "\<exists>D. T = Class D"
IH_1: fixes T1 :: "tree"   and T2 :: "tree" assumes "T1' = Branching T1 T2" shows "\<exists>B. branch (B @ [True]) T1' \<and> branch (B @ [False]) T1'"
assms_2: shows "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
has_field_derivative_pbernpoly_Suc': fixes f :: "real \<Rightarrow> real"   and D :: "real"   and x :: "real"   and s :: "real set"   and n :: "nat" assumes "(f has_real_derivative D) (at x within s)"   and "2 \<le> n \<or> f x \<notin> \<int>" shows "((\<lambda>x. pbernpoly (Suc n) (f x)) has_real_derivative real (Suc n) * pbernpoly n (f x) * D) (at x within s)"
rprodl_simps: fixes a :: "'a"   and b :: "'b"   and c :: "'c" shows "rprodl ((a, b), c) = (a, b, c)"
hyps: shows "concat (map2 (\<lambda>x i. replicate (scf i) x) (bef @ i # aft) [0..<length (bef @ i # aft)]) = concat (map2 (\<lambda>x i. replicate (scf i) x) bef [0..<length bef]) @ replicate (scf (length bef)) i @ concat (map2 (\<lambda>x i. replicate (scf (Suc (length bef + i))) x) aft [0..<length aft])"
ordinal_minus_monoR: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" assumes "x \<le> y" shows "z - y \<le> z - x"
orbit_inverse: fixes f :: "'a perm" shows "orbit (inverse f) = orbit f"
UP_ring_subring_car_subset: fixes S :: "'a set" assumes "subring S R" shows "carrier (UP (R\<lparr>carrier := S\<rparr>)) \<subseteq> carrier (UP R)"
freshAbs_vsubstAbs_ident: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x :: "'var"   and x1 :: "'var" assumes "goodAbs A"   and "freshAbs xs x A" shows "A $[x1 // x]_xs = A"
frame: fixes p :: "assertion"   and c :: "command"   and q :: "assertion"   and r :: "assertion" assumes "prov_triple (p, c, q)"   and "wr_com c \<inter> rd_ass r = {}" shows "prov_triple (p \<star> r, c, q \<star> r)"
finite_elements_mat: fixes A :: "'b mat" shows "finite (elements_mat A)"
multiplicity_normalize_left: fixes p :: "'a"   and x :: "'a" shows "multiplicity (normalize p) x = multiplicity p x"
STA_potential: fixes A :: "'c set"   and n :: "nat"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and m :: "nat" assumes "A, n \<turnstile> branch" shows "A, Suc m \<turnstile> branch"
R: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "A = B" shows "finite B"
iprev_predicate_change_exists2: fixes a :: "nat"   and b :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "a \<le> b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P b"   and "P a" shows "\<exists>n\<in>I. a < n \<and> n \<le> b \<and> \<not> P n \<and> (\<forall>k\<in>I. a \<le> k \<and> k < n \<longrightarrow> P k)"
is_prefix_def: fixes u :: "'b list"   and v :: "'b list" shows "is_prefix u v = (\<exists>w. u @ w = v)"
finite_dimensional_vector_space_pair_axioms: shows "finite_dimensional_vector_space_pair (*a) B1 (*b) B2"
MC': fixes n :: "nat"   and \<omega>' :: "nat \<Rightarrow> 's" assumes "prob {\<omega> \<in> space (T' I). \<forall>t\<le>n. \<omega> !! t = \<omega>' t} \<noteq> 0" shows "prob {\<omega> \<in> space (T' I). \<omega> !! n = \<omega>' n} \<noteq> 0"
inner_same_Basis: fixes u :: "'a" assumes "u \<in> Basis" shows "inner u u = 1"
simps_1: fixes k :: "udom \<rightarrow> udom\<cdot>llist" shows "bindU\<cdot>LNil\<cdot>k = LNil"
open_Gromov_completion_def: fixes U :: "'a Gromov_completion set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> U)"
lec_is_cbox: fixes L :: "'a set" assumes "content (L \<inter> {x. x \<bullet> i \<le> c}) \<noteq> 0"   and "L \<in> \<D>" shows "\<exists>a b. L \<inter> {x. x \<bullet> i \<le> c} = cbox a b"
noDAsortQ: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "noDenyAll1 p" shows "noDenyAll1 (qsort p l)"
functor_axioms: shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a)"
split_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b"   and f5 :: "char list \<Rightarrow> 'b"   and f6 :: "'b"   and f7 :: "char list \<Rightarrow> 'b"   and f8 :: "'b"   and f9 :: "'b"   and action :: "action" shows "P (case action of Accept \<Rightarrow> f1 | Drop \<Rightarrow> f2 | Log \<Rightarrow> f3 | Reject \<Rightarrow> f4 | Call x \<Rightarrow> f5 x | Return \<Rightarrow> f6 | Goto x \<Rightarrow> f7 x | Empty \<Rightarrow> f8 | Unknown \<Rightarrow> f9) = (\<not> (action = Accept \<and> \<not> P f1 \<or> action = Drop \<and> \<not> P f2 \<or> action = Log \<and> \<not> P f3 \<or> action = Reject \<and> \<not> P f4 \<or> (\<exists>x5. action = Call x5 \<and> \<not> P (f5 x5)) \<or> action = Return \<and> \<not> P f6 \<or> (\<exists>x7. action = Goto x7 \<and> \<not> P (f7 x7)) \<or> action = Empty \<and> \<not> P f8 \<or> action = Unknown \<and> \<not> P f9))"
m_o1: fixes X :: "char list set"   and o1 :: "(char list \<Rightarrow> 'av) option"   and o2 :: "(char list \<Rightarrow> 'av) option" assumes "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 \<le> o2" shows "m\<^sub>o o2 X \<le> m\<^sub>o o1 X"
Suc: shows "\<Sum>\<^sub># (mset (replicate n m1)) \<subseteq># \<Sum>\<^sub># (mset (replicate n m2))"
simps_20: fixes \<C> :: "t_context"   and "is" :: "nat list"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Br_table is i) ts = (case same_lab (is @ [i]) (label \<C>) of None \<Rightarrow> Bot | Some tls \<Rightarrow> type_update ts (to_ct_list (tls @ [T_i32])) (TopType []))"
measure_scale_measure: fixes r :: "real"   and M :: "'c measure"   and A :: "'c set" assumes "0 \<le> r" shows "Sigma_Algebra.measure (scale_measure (ennreal r) M) A = r * Sigma_Algebra.measure M A"
dsum_pOp_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_pOp I A \<in> carr_dsumag I A \<rightarrow> carr_dsumag I A \<rightarrow> carr_dsumag I A"
is_arg_min_size: fixes m :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "is_arg_min m P x" shows "m x = m (arg_min m P)"
funpow_simps_right_1: fixes f :: "'b \<Rightarrow> 'b" shows "f ^^ 0 = id"
char_to_digit_def: fixes c :: "char" shows "char_to_digit c = (if c = CHR ''0'' then 0 else if c = CHR ''1'' then 1 else if c = CHR ''2'' then 2 else if c = CHR ''3'' then 3 else if c = CHR ''4'' then 4 else if c = CHR ''5'' then 5 else if c = CHR ''6'' then 6 else if c = CHR ''7'' then 7 else if c = CHR ''8'' then 8 else if c = CHR ''9'' then 9 else 10)"
simps_1: fixes k :: "int" shows "del (Some k) Empty = None"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_discard_axioms bind" shows "monad_discard return bind"
\<o>_\<iota>: fixes x :: "'j \<Rightarrow> 'c" assumes "x \<in> UNIV" shows "\<o> (\<iota> x) = x"
lossless_bind_spmf: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "lossless_spmf (p \<bind> f) = (lossless_spmf p \<and> (\<forall>x\<in>set_spmf p. lossless_spmf (f x)))"
ruc_wfc_failures: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set"   and U :: "'b set"   and xs :: "'a list"   and xs' :: "'a list"   and X :: "'a set" assumes "ref_union_closed P"   and "weakly_future_consistent P I D R"   and "U \<subseteq> range D \<inter> (- I) `` range D"   and "U \<noteq> {}"   and "\<forall>u\<in>U. (xs, xs') \<in> R u"   and "(xs, X) \<in> failures P" shows "(xs', X \<inter> D -` U) \<in> failures P"
halfspace_eq_empty_lt: fixes a :: "'a"   and b :: "real" shows "({x. a \<bullet> x < b} = {}) = (a = (0::'a) \<and> b \<le> 0)"
V_bclim_def: fixes i :: "nat" shows "V_bclim i \<equiv> if case_two i then let k = gap i (LEAST t. \<forall>T. gap i T \<le> gap i t) in {\<tau> i(k \<mapsto> 0), \<tau> i(k \<mapsto> 1)} else {\<tau> i}"
step_2: shows "\<mu>_bound_row_inner (fs[i := fs ! i - round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j) \<cdot>\<^sub>v fs ! j]) i j"
prim_invar2_ctd_alt_aux1: fixes Q :: "'v \<Rightarrow> 'a"   and \<pi> :: "'v \<Rightarrow> 'v option"   and u :: "'v" assumes "prim_invar1 (A Q \<pi>)"   and "Q u \<noteq> \<infinity>"   and "u \<noteq> r" shows "u \<notin> S (A Q \<pi>)"
not_SN_onE: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "\<not> SN_on r A" obtains f :: "nat \<Rightarrow> 'c" where "f 0 \<in> A" and "\<forall>i. (f i, f (Suc i)) \<in> r"
is_natural_1: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "HoVH f \<cdot> \<a>[fst (VVV.dom f), fst (snd (VVV.dom f)), snd (snd (VVV.dom f))] = \<a>[fst f, fst (snd f), snd (snd f)]"
dg_in_Obj_in_Vset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<DD>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
prime_nat_iff': fixes p :: "nat" shows "prime p = (1 < p \<and> (\<forall>n\<in>{2..<p}. \<not> n dvd p))"
slp_vp_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n'' :: "'b"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.same_level_path' sourcenode targetnode kind valid_edge get_return_edges n as n''"   and "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n'' as' n'" shows "CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n (as @ as') n'"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b exp"   and b :: "'c exp" assumes "rel_exp R a b" obtains x :: "'b" and y :: "'c" where "a = Var x" and "b = Var y" and "R x y"   | xa :: "int" and ya :: "int" where "a = Const xa" and "b = Const ya" and "xa = ya"   | x1a :: "'b exp" and x2a :: "'b exp" and y1a :: "'c exp" and y2a :: "'c exp" where "a = Plus x1a x2a" and "b = Plus y1a y2a" and "rel_exp R x1a y1a" and "rel_exp R x2a y2a"   | x1b :: "'b exp" and x2b :: "'b exp" and y1b :: "'c exp" and y2b :: "'c exp" where "a = Div x1b x2b" and "b = Div y1b y2b" and "rel_exp R x1b y1b" and "rel_exp R x2b y2b"
rotate1_map: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "rotate1 (map f xs) = map f (rotate1 xs)"
commute: fixes a :: "'c"   and b :: "'c" shows "a * b = b * a"
differentiable_within_subset: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "f differentiable at x within s"   and "t \<subseteq> s" shows "f differentiable at x within t"
permute_flip_at: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso (D.\<rho> a)"
msed_rel_invR: fixes R :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'b multiset"   and b :: "'e"   and N :: "'e multiset" assumes "rel_mset R M (add_mset b N)" shows "\<exists>M1 a. M = add_mset a M1 \<and> R a b \<and> rel_mset R M1 N"
snoc_1: assumes "is_regular_sequence fs" shows "is_regular_sequence (removeAll 0 fs)"
rep_eq: fixes x :: "'a" shows "generator (replicate_generator x) = ((<) 0, \<lambda>n. (x, n - 1))"
pa_pref_frp: fixes H1 :: "('a, 'b) hashedTa"   and H2 :: "('c, 'b) hashedTa" assumes "hashedTa H1"   and "hashedTa H2"   and "hta_has_idx_s H1"   and "hta_has_idx_sf H2" shows "wa_precise_refine (det_wa_wa (pa_det_algo H1 H2)) (frp_algo (hta_\<alpha> H1) (hta_\<alpha> H2)) pa_\<alpha>"
mem_UnitsI: fixes u :: "'a" assumes "units.sub.invertible u"   and "u \<in> Units" shows "u \<in> units.sub.Units"
shiftl: fixes xs :: "bool list" shows "of_bl xs * 2 * y = of_bl xs * y << 1"
main: fixes i :: "nat"   and p :: "(nat, nat) fm" assumes "i \<notin> nominals p" shows "valid p = (nominals p, 1 \<turnstile> [([\<^bold>\<not> p], i)])"
fs: shows "y \<sharp> (x, t1, s1, t2, s2)"
distincts_distinct_set: fixes xss :: "'b list list" assumes "distincts xss" shows "distinct (map set xss)"
Lim_def: fixes A :: "'c filter"   and f :: "'c \<Rightarrow> 'b" shows "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
k_i': fixes i :: "nat"   and e :: "'v \<times> 'v" shows "k_i i e = ennreal (k_i' i e)"
increasing_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "increasing f \<equiv> \<forall>e b c d. e \<le> c \<and> b \<le> d \<longrightarrow> f e b \<le> f c d"