hcju/isabelleps huggingface dataset:
simps_23: fixes e\<^sub>1 :: "('f, 'g, 'h) exp"   and C :: "String.literal"   and V :: "'f"   and e\<^sub>2 :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (try e\<^sub>1 catch(C V) e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<lfloor>{V}\<rfloor>))"
cong_solve_unique: fixes a :: "nat"   and n :: "nat"   and b :: "nat" assumes "coprime a n"   and "n \<noteq> 0" shows "\<exists>!x. x < n \<and> [a * x = b] (mod n)"
hyps: assumes "model valuation (filter (\<lambda>c. \<not> clauseTautology c) F')"   and "vars F' \<subseteq> vars valuation" shows "model valuation F'"
liveSimple: fixes n :: "'node"   and g :: "'g"   and val :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "val \<in> uses g n" shows "liveVal g val"
add_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and b :: "'b"   and b' :: "'b" assumes "bounded_cbilinear prod" shows "prod a (b + b') = prod a b + prod a b'"
rphil_state_update_def: fixes i :: "nat"   and s :: "nat"   and e :: "dining_event" shows "Up\<^sub>r\<^sub>p i s e = (if e = picks i i then 1 else if e = picks i (i - 1) then 2 else if e = putsdown i (i - 1) then 3 else 0)"
a1: shows "low_equal s1 s2 \<and> load_word_mem s1 addra 8 = load_word_mem s2 addra 8"
children_eq_h: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr' \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
adjoint_mult: fixes A :: "'c mat"   and n :: "nat"   and m :: "nat"   and B :: "'c mat"   and l :: "nat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m l" shows "adjoint (A * B) = adjoint B * adjoint A"
is_inf_binary: fixes x :: "'b"   and y :: "'b" shows "is_inf x y (\<Sqinter>{x, y})"
poly_deg_monom_mult_le: fixes c :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 'b"   and p :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd" shows "poly_deg (MPoly_Type_Class.punit.monom_mult c t p) \<le> deg_pm t + poly_deg p"
inverse_p_integer_def: fixes p :: "integer"   and x :: "integer" shows "inverse_p_integer p x = (if x = 0 then 0 else power_p_integer p x (p - 2))"
simps: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat"   and c :: "'c list"   and b :: "bool list"   and uu :: "'c list" assumes "BIT_Off acts qs init" shows "BIT_Off.phi acts qs init n (c, b, uu) = (\<Sum>(x, y)\<in>Inv c (BIT_Off.s_A acts qs init n). if b ! index init y then 2 else 1)"
eval_fps_add: fixes z :: "'a"   and f :: "'a fps"   and g :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f"   and "ereal (norm z) < fps_conv_radius g" shows "eval_fps (f + g) z = eval_fps f z + eval_fps g z"
node_relator_def: fixes x :: "'a"   and y :: "'a set" shows "node_relator x y = (x \<in> y)"
assms: shows "[a\<noteq>a]P \<longmapsto> Rs"
transitive: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>[Rel] Q"   and "Q \<leadsto>[Rel'] R"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''" shows "P \<leadsto>[Rel''] R"
that: fixes f :: "'b"   and T :: "('b, 'a) Term.term list" assumes "s x = Fun f T"   and "Fun f T \<in> subst_range s"   and "fv (Fun f T) = {}" shows "thesis"
fresh_PairD_1: fixes a :: "atom"   and x :: "'a"   and y :: "'b" assumes "a \<sharp> (x, y)" shows "a \<sharp> x"
real_cond_exp_cong: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "AE x in M. f x = g x"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AE x in M. real_cond_exp M F f x = real_cond_exp M F g x"
fold_mset_add_mset: fixes s :: "'b"   and x :: "'a"   and M :: "'a multiset" shows "fold_mset f s (add_mset x M) = f x (fold_mset f s M)"
roots_bound: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier (UP R)"   and "f \<noteq> \<zero>\<^bsub>UP R\<^esub>"   and "finite (carrier R)" shows "finite {a \<in> carrier R. eval R R id a f = \<zero>} \<and> card {a \<in> carrier R. eval R R id a f = \<zero>} \<le> deg R f"
BNTr5: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and W1 :: "'a set"   and W2 :: "'a set" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x"   and "W1 \<in> WWa D f a"   and "W2 \<in> WWa D f a"   and "ord_equiv (Iod D W1) (Iod D W2)" shows "W1 \<subseteq> W2"
down_def: shows "down = Grid.lift (\<lambda>d l p. down' d l p 0 0)"
qdfst_setsum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> QDelta" assumes "finite A" shows "qdfst (sum f A) = (\<Sum>x\<in>A. qdfst (f x))"
step_augmenter_eval: fixes step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and heuristic :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> nat"   and amount :: "nat"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "\<And>xs var L F \<Gamma>. length xs = var \<Longrightarrow> (\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))"   and "\<And>n var L F. heuristic n L F = var \<Longrightarrow> var \<le> n"   and "amount \<le> var + 1" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step_augment step heuristic amount var L F) (xs @ \<Gamma>))"
set_mset_union: fixes M :: "'c multiset"   and N :: "'c multiset" shows "set_mset (M + N) = set_mset M \<union> set_mset N"
assms_2: shows "r \<noteq> Rerr (Rabort Rtimeout_error)"
sup_def: fixes x :: "'b"   and y :: "'b" shows "sup x y = (if y \<le> x then x else y)"
comp_assoc_assoc'_1: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "\<a>\<^sub>D[f, g, h] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = f \<star>\<^sub>D g \<star>\<^sub>D h"
disj_of_clause_def: fixes c :: "'b literal list" shows "disj_of_clause c \<equiv> \<^bold>\<Or>map form_of_lit c"
finite_rabs_real_of_nat_le_real: fixes u :: "real" shows "finite {n. \<bar>real n\<bar> \<le> u}"
wfp_on_imp_minimal: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wfp_on P A" shows "\<forall>Q x. x \<in> Q \<and> Q \<subseteq> A \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. P y z \<longrightarrow> y \<notin> Q)"
le_neq_trans: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b"   and "a \<noteq> b" shows "a < b"
Suc_in_imp_not_subset_iMOD: fixes n :: "nat"   and S :: "nat set"   and m :: "nat"   and r :: "nat" assumes "n \<in> S"   and "Suc n \<in> S"   and "m \<noteq> Suc 0" shows "\<not> S \<subseteq> [ r, mod m ]"
bi_unique_rel_set: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "bi_unique (rel_set A)"
c: fixes z :: "complex" assumes "s0 < ereal (Re z)" shows "eval_fds f z * exp (- eval_fds (fds_ln l f) z) = c"
Sup_nres_def: fixes X :: "'b nres set" shows "Sup X \<equiv> if FAILi \<in> X then FAILi else RES (\<Union> {x. RES x \<in> X})"
ex_comm4: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s' :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" shows "(\<exists>c1 s a b. (a, b) \<in> fset (possible_steps e s' r l i) \<and> obtains s c1 e a (evaluate_updates b i r) t) = (\<exists>a b s c1. (a, b) \<in> fset (possible_steps e s' r l i) \<and> obtains s c1 e a (evaluate_updates b i r) t)"
subdegree_lr_inverse_2: fixes g :: "'b fps"   and y :: "'b" shows "subdegree (fps_right_inverse g y) = 0"
sortedP_Cons_iff: fixes xa :: "real \<times> real"   and x :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "ccw'.sortedP xa (x # xs) = (Ball (set xs) (ccw' xa x) \<and> ccw'.sortedP xa xs)"
rep_eq: fixes x :: "'a mpoly" shows "mapping_of (- x) = - mapping_of x"
dom_eqI: fixes a :: "'b"   and f :: "'b" assumes "D.ide a"   and "D.seq f a" shows "D.dom f = a"
irrefl: fixes M :: "'a multiset" shows "(M, M) \<notin> mult {(x, y). x < y}"
witness_matricization: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row Aw'"   and "j < dim_col Aw'" shows "Aw' $$ (i, j) = (if i = j \<and> (\<forall>i0\<in>set (digit_encode (nths (dims Aw) (Collect even)) i). i0 < last (butlast rs)) then 1 else 0)"
complex_eqI: fixes x :: "complex"   and y :: "complex" assumes "Re x = Re y"   and "Im x = Im y" shows "x = y"
bind_def: fixes s :: "('e, 'g) state"   and k :: "'g \<Rightarrow> ('e, 'f) state" shows "s \<bind> k = State (\<lambda>M. let (a, M') = runState s M in runState (k a) M')"
reversepath_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "reversepath (subpath u v g) = subpath v u g"
HMA_diagonal_to_Smith_i_PQ_aux: fixes i :: "nat"   and xs :: "nat list"   and P :: "'d mat"   and A :: "'d mat"   and Q :: "'d mat"   and P' :: "(('d, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and A' :: "(('d, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and Q' :: "(('d, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "i < min CARD('b) CARD('c)"   and "\<forall>j\<in>set xs. j < min CARD('b) CARD('c)"   and "HMA_M3 (P, A, Q) (P', A', Q')" shows "HMA_M3 (diagonal_to_Smith_i_PQ_JNF xs i bezout (P, A, Q)) (diagonal_to_Smith_i_PQ xs i bezout (P', A', Q'))"
48: fixes x :: "'a"   and y :: "'a" shows "- - x \<squnion> (y \<squnion> - x) = y \<squnion> top"
\<psi>: shows "\<guillemotleft>\<psi> : f \<star> g \<Rightarrow> trg f\<guillemotright> \<and> local.iso \<psi>"
simps_19: fixes vb :: "'a"   and vc :: "value list"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_gexp_aux (In vb vc) (Eq v va) = False"
list_case_def: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'c SList.list \<Rightarrow> 'b"   and xs :: "'c SList.list" shows "list_case a f xs = list_rec xs a (\<lambda>x xs r. f x xs)"
stability: fixes j :: "nat" assumes "t j \<le> t0 + e1'" shows "dist (grid_function (discrete_evolution (euler_increment' e g)) x0' t j) (grid_function (discrete_evolution (euler_increment f)) (solution (t 0)) t j) \<le> euler_C TYPE('a) / B' * (exp (B' * (t0 + e1' - t 0) + 1) - 1) * max_stepsize j ^ 1"
eq_refl: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x = y" shows "ord (PP x) (PP y)"
memb_equiv_1: fixes f :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "\<F>C.arr f"   and "t \<in> f"   and "u \<in> f" shows "Par t u"
boundedE: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf" assumes "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf) \<sqinter> (c::'b::semilattice_inf)" obtains "(a::'b::semilattice_inf) \<le> (b::'b::semilattice_inf)" and "(a::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)"
monad_altc_envT: assumes "monad_altc return bind altc" shows "monad_altc local.return_env local.bind_env local.altc_env"
scaleR_sq_mtx_def: shows "(*\<^sub>R) \<equiv> map_fun id (map_fun to_vec to_mtx) (*\<^sub>R)"
min_int_poly_real_code_unfold: shows "min_int_poly = min_int_poly_real"
distinct_permutations_of_set_aux_list: fixes xs :: "'a list"   and acc :: "'a list" assumes "distinct xs" shows "distinct (permutations_of_set_aux_list acc xs)"
distinct_84: fixes pi' :: "pi"   and pi1 :: "pi"   and pi2 :: "pi" shows "!pi' \<noteq> pi1 \<oplus> pi2"
TID_sym: fixes x :: "'a"   and y :: "'a" shows "turn_into_distance f x y = turn_into_distance f y x"
case_v_leq_u: assumes "v \<le> u" shows "\<exists>u. p @ [a] \<in> \<P> (charslength (p @ [a])) u"
sub_add_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "sub_add t = (\<forall>P Q s. sound P \<and> sound Q \<longrightarrow> t P s + t Q s \<le> t (\<lambda>s'. P s' + Q s') s)"
UNION_empty_conv_1: fixes B :: "'h \<Rightarrow> 'g set"   and A :: "'h set" shows "({} = \<Union> (B ` A)) = (\<forall>x\<in>A. B x = {})"
a_def: shows "a = [:lead_coeff q ^ (Suc (degree p) - degree q):]"
full_quicksort_vars_def: shows "full_quicksort_vars = full_quicksort_ref (\<lambda>x y. x = y \<or> (x, y) \<in> var_order_rel) id"
affine_independent_subset: fixes t :: "'a set"   and s :: "'a set" assumes "\<not> affine_dependent t"   and "s \<subseteq> t" shows "\<not> affine_dependent s"
atms_simps_5: fixes r :: "Formula.formula Regex.regex" shows "atms (regex.Star r) = atms r"
double_of_man_exp_def: fixes m :: "integer"   and e :: "integer" shows "double_of_man_exp m e = (let f = Float (int_of_integer m) (int_of_integer e) in if is_finite_Float TYPE((11, 52) IEEE.float) f then Some (Double.of_finite_Float f) else None)"
map_vector_def: fixes f :: "'c \<Rightarrow> 'a"   and v :: "('c, 'b) Finite_Cartesian_Product.vec" shows "map_vector f v \<equiv> \<chi>i. f (v $h i)"
sum_list_list_of_pdevs: fixes x :: "'a pdevs" shows "sum_list (map snd (list_of_pdevs x)) = sum_list (dense_list_of_pdevs x)"
ASSIGN: fixes H :: "Var \<Rightarrow> L"   and G :: "Var \<Rightarrow> L"   and x :: "Var"   and p :: "L"   and t :: "L"   and e :: "Expr"   and X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and q :: "L" assumes "H = upd G x (LUB p t)"   and "\<forall>s ss. EQ G t s ss \<longrightarrow> evalE e s = evalE e ss" shows "X \<rhd> Assign x e : Sec (Q p H) (EQ G q) (EQ H q) (\<lambda>(s, t). \<exists>r. s = update r x (evalE e r) \<and> EQ G q r t)"
g: shows "g \<in> M \<rightarrow>\<^sub>M subprob_algebra L"
seqr_inf_distr: fixes P :: "('b, 'd) urel"   and Q :: "('d, 'c) urel"   and R :: "('d, 'c) urel" shows "P ;; Q \<sqinter> R = (P ;; Q) \<sqinter> (P ;; R)"
length_of_bin_rep_aux: fixes m :: "nat"   and n :: "nat" assumes "m < 2 ^ n" shows "length (bin_rep_aux n m) = n + 1"
length_enumerate: fixes n :: "nat"   and xs :: "'a list" shows "length (enumerate n xs) = length xs"
1: shows "\<guillemotleft>\<Psi> f b b' : G f \<cong>\<^sub>D G (F (\<eta>'.map\<^sub>0 b' \<star>\<^sub>D G f \<star>\<^sub>D \<eta>\<^sub>0 b))\<guillemotright>"
chine_eq_apex: shows "\<nu>.chine = \<nu>.apex"
crename_fresh_interesting1: fixes c :: "coname"   and M :: "trm"   and a :: "coname"   and b :: "coname" assumes "c \<sharp> M[a\<turnstile>c>b]"   and "c \<sharp> (a, b)" shows "c \<sharp> M"
span_scale: fixes x :: "'a"   and S :: "'a set"   and c :: "'a" assumes "x \<in> ideal S" shows "c * x \<in> ideal S"
abs_eq: shows "id_blinfun = Blinfun (\<lambda>x. x)"
irreducible\<^sub>dI: fixes p :: "'d poly" assumes "0 < degree p"   and "\<And>q r. \<lbrakk>0 < degree q; degree q < degree p; 0 < degree r; degree r < degree p; p = q * r\<rbrakk> \<Longrightarrow> False" shows "irreducible\<^sub>d p"
pr_weak'_def: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b \<times> nat"   and g :: "'b \<times> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "admissible_kbo.pr_weak' pr_weak least f g = (f \<notin> admissible_kbo.lConsts least \<and> pr_weak f g \<or> g \<in> admissible_kbo.lConsts least)"
finite_params_fm: fixes p :: "('a, 'b) fm" shows "finite (params_fm p)"
supp_Abs_formula: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "supp (Abs_formula t\<^sub>\<alpha>) = supp t\<^sub>\<alpha>"
nn_integral_snd_count_space: fixes f :: "'c \<times> 'b \<Rightarrow> ennreal" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>count_space UNIV \<partial>count_space UNIV = integral\<^sup>N (count_space UNIV) f"
that: fixes T :: "('a, 'b, 'c) psi" assumes "P = Q \<parallel> T"   and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' T = \<lparr>\<nu>*yvec\<rparr>N \<prec>' R" shows "thesis"
e': shows "ball (apply_chart c x) e \<subseteq> codomain c"
alllsts_empty: shows "{}\<^sup>\<infinity> = {LNil}"
concat_eq_Nil_conv: fixes xss :: "'b list list" shows "(concat xss = \<epsilon>) = (\<forall>xs\<in>set xss. xs = \<epsilon>)"
fmap_induct: fixes P :: "('g \<Rightarrow> 'h option) \<Rightarrow> bool"   and F' :: "'g \<Rightarrow> 'h option" assumes "P Map.empty"   and "\<And>F x z. \<lbrakk>x \<notin> dom F; P F\<rbrakk> \<Longrightarrow> P (F(x \<mapsto> z))" shows "P F'"
part: shows "List.partition (\<lambda>gi. gi dvdm f1) gs = (gs1, gs2)"
ppoint_mult_correct: fixes a :: "int"   and b :: "int"   and p :: "int \<times> int \<times> int"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curvep a b p" shows "make_affine (ppoint_mult a n p) = point_mult a n (make_affine p)"
tendsto_sum: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and a :: "'c \<Rightarrow> 'e"   and F :: "'d filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F" shows "((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> sum a I) F"
rsp: shows "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) (0::'b, 0::'b) (0::'b, 0::'b)"
regular_def: fixes L :: "'b list set" shows "regular L \<equiv> \<exists>M. dfa M \<and> dfa.language M = L"
simps_2: fixes xs :: "'c llist"   and n :: "nat" shows "lnth xs (Suc n) = (case xs of LNil \<Rightarrow> undefined (Suc n) | LCons x xs' \<Rightarrow> lnth xs' n)"
intro: assumes "OFCLASS('a, euclidean_semiring_gcd_class)"   and "OFCLASS('a, idom_class)" shows "OFCLASS('a, euclidean_ring_gcd_class)"
extremum_unique: fixes a :: "'d" shows "(a \<le> bot) = (a = bot)"
rule2: shows "verticesFrom f v ! (|vertices f| - Suc 0) = f\<^bsup>|vertices f| - Suc 0\<^esup> \<bullet> v"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P \<alpha> P' A\<^sub>P \<Psi>\<^sub>P"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P"   and "\<Psi>\<^sub>P \<simeq> \<one>"   and "supp \<Psi>\<^sub>P = {}"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* C" shows "Prop C \<Psi> (Cases Cs) \<alpha> P' [] (\<one>)"
m_def: shows "m = \<lfloor>r\<rfloor> mod n"
psubst_fresh_fact: fixes x :: "name"   and t :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "x \<sharp> t"   and "x \<sharp> \<theta>" shows "x \<sharp> \<theta><t>"
mult_left_le_imp_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq (times c a) (times c b)"   and "less zero c" shows "less_eq a b"
widen_st_def: fixes F1 :: "'a Abs_State.st"   and F2 :: "'a Abs_State.st" shows "F1 \<nabla> F2 = FunDom (\<lambda>x. fun F1 x \<nabla> fun F2 x) (inter_list (Abs_State.dom F1) (Abs_State.dom F2))"
is_word_def: fixes s :: "symbol list" shows "is_word s = list_all is_terminal s"
mpi_less_Im_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "- pi < Im (Ln z)"
dbl_dec_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one (numeral.numeral one plus k) = numeral.numeral one plus (Num.BitM k)"
eqI: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<sigma> :: "'c \<Rightarrow> 'd"   and \<sigma>' :: "'c \<Rightarrow> 'd" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation A B F G \<sigma>'"   and "\<And>a. partial_magma.ide A a \<Longrightarrow> \<sigma> a = \<sigma>' a" shows "\<sigma> = \<sigma>'"
view_trees_for_same_hash: fixes vt :: "view\<^sub>m"   and view :: "view" assumes "vt \<in> set (local.view_trees_for view)" shows "hash_view vt = hash_view (embed_view view)"
functor_HoVH: shows "functor VVV.comp (\<cdot>) HoVH"
simps: fixes x :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_loop6_loop x (l, r) = (\<exists>i j k t. i + j = Suc x \<and> 0 < i \<and> k + t + 1 = j \<and> (l, r) = (Bk \<up> k @ Oc \<up> i, Bk \<up> Suc t @ Oc \<up> j))"
concurrent_create_delete_independent: fixes i :: "'a"   and e1 :: "'b"   and ir :: "'a"   and "is" :: "'a set"   and e2 :: "'b"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Create i e1) (ir, Delete is e2)"   and "\<not> hb (ir, Delete is e2) (i, Create i e1)"   and "xs prefix of j"   and "(i, Create i e1) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e2) \<in> set (node_deliver_messages xs)" shows "i \<notin> is"
foo28a: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "x \<in> dom (Cp a)" shows "dom (Cp a) \<noteq> {}"
Upper_semi_cong: fixes A :: "'a set"   and A' :: "'a set" assumes "A \<subseteq> carrier L"   and "A {.=} A'" shows "Upper L A \<subseteq> Upper L A'"
SomeGcd_def: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'c set" shows "SomeGcd G A = \<Sqinter>\<^bsub>division_rel G\<^esub>A"
inv_inv: fixes f :: "'a" assumes "C.iso f" shows "C.inv (C.inv f) = f"
chain_subdiv_chain_pathimg_subset': fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "chain_subdiv_chain one_chain subdiv"   and "(k, \<gamma>) \<in> subdiv" shows "\<exists>k' \<gamma>'. (k', \<gamma>') \<in> one_chain \<and> path_image \<gamma> \<subseteq> path_image \<gamma>'"
is_span: shows "C.span \<nu>\<pi>.leg0 \<nu>\<pi>.leg1"
add: fixes b1 :: "'m_1"   and b2 :: "'m_1" shows "f (b1 +\<^sub>M\<^sub>_\<^sub>1 b2) = f b1 +\<^sub>M\<^sub>_\<^sub>2 f b2"
hd: shows "C = hd ((C # Cs') @\<^sub>p Ds)"
SK_bindS_bindK: fixes x :: "nat discr llist"   and g :: "nat discr \<rightarrow> nat discr llist" shows "SK\<cdot>(bindS\<cdot>x\<cdot>g) = bindK\<cdot>(SK\<cdot>x)\<cdot>(SK oo g)"
f_is_g_indicator: shows "f = (\<lambda>x. if x \<in> s then g x else (0::'c))"
disjointFrame_L2: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and rcv :: "nat \<Rightarrow> 'a Frame list"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list" assumes "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "\<forall>i<n. FlexRayController (nReturn i) rcv (nC i) (nStore i) (nSend i) (nGet i)" shows "inf_disj n nSend"