hcju/isabelleps huggingface dataset:
set_integrable_subset: fixes M :: "'d measure"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d set" assumes "set_integrable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_integrable M B f"
ntrs_is_trss_s: fixes s :: "'a list"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> ntrs fg" shows "((s, c), w, s', c') \<in> trcl (trss fg)"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<nexists>x1 x2. prod = (x1, x2) \<and> \<not> P (f x1 x2))"
snd_fst_lens_indep: shows "snd\<^sub>L \<bowtie> fst\<^sub>L"
compute_list_of_pdevs: fixes xs :: "(nat, 'b) slist" shows "list_of_pdevs (Pdevs xs) = list_of_slist xs"
equiv_trans_comm: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "equiv_trans t u = equiv_trans u t"
pgalleryD_chamber: fixes xs :: "'a set list"   and x :: "'a set" assumes "pgallery xs"   and "x \<in> set xs" shows "chamber x"
axioms_1: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and basis1 :: "'d set"   and basis2 :: "'e set" assumes "finite_dimensional_vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1 basis2" shows "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1"
terminates_sequence_flatten_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>sequence Seq (flatten c) \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
FSbImorph_def: fixes h :: "'c \<Rightarrow> 'd"   and hA :: "'e \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'c, 'e, 'h) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'g, 'd, 'f, 'i) model_scheme" shows "FSbImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
fl: shows "round_num_denom (dmu_ij_state state i (j - 1)) (d_state state j) = fc"
incident_polar: fixes p :: "proj2"   and q :: "proj2" shows "proj2_incident p (polar q) = (proj2_rep p \<bullet> (M *v proj2_rep q) = 0)"
Liminf_add_ereal_right: fixes F :: "'a filter"   and c :: "ereal"   and g :: "'a \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "Liminf F (\<lambda>n. g n + c) = Liminf F g + c"
bmod_assoc_aux2: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'c"   and R :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" shows "\<Squnion>{\<Squnion>{f u \<cdot> g v \<cdot> h w |v w. R y v w} |u y. R x u y} = \<Squnion>{uu_. \<exists>u v w y. uu_ = f u \<cdot> g v \<cdot> h w \<and> R y v w \<and> R x u y}"
isometry_on_def: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "isometry_on X f = (\<forall>x\<in>X. \<forall>y\<in>X. dist (f x) (f y) = dist x y)"
zero_ext_fg: fixes y :: "'a" assumes "y \<in> FG - supp fg" shows "fg y * h (- y + x) = (0::'b)"
trunc_zero: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "deg R f = 0" shows "Cring_Poly.truncate R f = \<zero>\<^bsub>P\<^esub>"
residue_times_one_r: fixes k :: "nat"   and y :: "int" shows "1 \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> y = y mod p ^ k"
bindI: fixes P :: "'a Predicate.pred"   and x :: "'a"   and Q :: "'a \<Rightarrow> 'b Predicate.pred"   and y :: "'b" assumes "pred.eval P x"   and "pred.eval (Q x) y" shows "pred.eval (Predicate.bind P Q) y"
left_unique_rel_converter: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "left_total A"   and "left_unique B"   and "left_unique C"   and "left_total R" shows "left_unique (rel_converter A B C R)"
integral_Re: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> complex" assumes "complex_integrable M f" shows "LINT x|M. Re (f x) = Re (integral\<^sup>C M f)"
that: shows "real_of_float a \<in>\<^sub>r I"
homeomorphism_apply2: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "homeomorphism S T f g"   and "x \<in> T" shows "f (g x) = x"
sccs_verts_subsets: fixes S :: "'a set" assumes "S \<in> sccs_verts" shows "S \<subseteq> verts G"
closed_segment_subset_domainI: fixes t0 :: "real"   and t :: "real"   and s :: "real" assumes "t0 \<in> cball t_max et"   and "t \<in> cball t_max et"   and "s \<in> {t0--t}" shows "s \<in> cball t_max et"
A: fixes v :: "nat list" assumes "v \<in> A" shows "(\<Prod>j = 0..m. f $ v ! j) = f $ k * (f $ 0) ^ m"
i_Exec_Stream_Acc_Output_append: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c \<frown> i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input (f_Exec_Comp trans_fun (xs \<odot> k) c)"
x: shows "x \<in> carrier (reduced_homology_group (int p) (nsphere p))"
num_params_polyadd: fixes p1 :: "'a poly"   and p2 :: "'a poly" shows "num_params (p1 +\<^sub>p p2) \<le> max (num_params p1) (num_params p2)"
Gcd_period_set_invariant: fixes x :: "'s"   and y :: "'s" assumes "(x, y) \<in> communicating" shows "Gcd (period_set x) = Gcd (period_set y)"
double_complement_above: fixes x :: "'a" shows "- - x \<sqinter> x = x"
negligible_imp_sets: fixes S :: "'a set" assumes "negligible S" shows "S \<in> sets lebesgue"
combine_linorder: fixes D1 :: "'a set"   and cmp1 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D2 :: "'a set"   and cmp2 :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and D :: "'a set" assumes "linorder_on D1 cmp1"   and "linorder_on D2 cmp2"   and "D = D1 \<union> D2" shows "linorder_on D (cmp_combine D1 cmp1 D2 cmp2)"
map_denote_positive_trace_dim: fixes x1 :: "nat"   and x2a :: "nat \<Rightarrow> complex mat"   and x3a :: "com list"   and x4 :: "complex mat" assumes "well_com (Measure x1 x2a x3a)"   and "x4 \<in> carrier_mat d d"   and "partial_density_operator x4"   and "\<And>x3aa \<rho>. \<lbrakk>x3aa \<in> set x3a; well_com x3aa; \<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (denote x3aa \<rho>) \<and> trace (denote x3aa \<rho>) \<le> trace \<rho> \<and> denote x3aa \<rho> \<in> carrier_mat d d" shows "\<forall>k<x1. positive ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<and> (map denote x3a ! k) (x2a k * x4 * adjoint (x2a k)) \<in> carrier_mat d d \<and> trace ((map denote x3a ! k) (x2a k * x4 * adjoint (x2a k))) \<le> trace (x2a k * x4 * adjoint (x2a k))"
abs_diff_le_iff: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and x :: "'e"   and a :: "'e"   and r :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs (minus x a)) r = (less_eq (minus a r) x \<and> less_eq x (plus a r))"
root_in_redexes: fixes c :: "('d, 'e, 'f) com" shows "c \<in> redexes c"
c: shows "\<^bold>g [^] d' \<otimes> (\<^bold>g [^] y) [^] m' = \<^bold>g [^] d \<otimes> (\<^bold>g [^] y) [^] m"
xor_one_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl x one = compl x"
first_summand_overlap: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "O z x" shows "O z (x \<oplus> y)"
lub_eq: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'c \<Rightarrow> 'b"   and Y :: "'c \<Rightarrow> 'b" assumes "class.cpo below"   and "\<And>i. X i = Y i" shows "po.lub below (range X) = po.lub below (range Y)"
less_deg_add_nonzero: fixes p :: "'a"   and q :: "'a" assumes "p \<in> carrier R"   and "p \<noteq> \<zero>"   and "q \<in> carrier R"   and "q \<noteq> \<zero>"   and "deg_n R S X p < deg_n R S X q" shows "p \<plusminus> q \<noteq> \<zero>"
LeftDerivationLadder_propagate: fixes \<alpha> :: "symbol list"   and \<delta> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and n :: "nat"   and index :: "nat" assumes "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"   and "ladder_i L 0 < length \<alpha>"   and "n = ladder_n L index"   and "index < length L" shows "if index + 1 < length L then \<exists>\<beta>. LeftDerivation \<alpha> (take n D) \<beta> \<and> ladder_\<gamma> (\<alpha> @ \<delta>) D L index = \<beta> @ \<delta> \<and> ladder_j L index < length \<beta> else \<exists>n' \<beta> \<delta>'. (index = 0 \<or> ladder_prev_n L index < n') \<and> n' \<le> n \<and> LeftDerivation \<alpha> (take n' D) \<beta> \<and> LeftDerivation (\<alpha> @ \<delta>) (take n' D) (\<beta> @ \<delta>) \<and> derivation_ge (drop n' D) (length \<beta>) \<and> LeftDerivation \<delta> (derivation_shift (drop n' D) (length \<beta>) 0) \<delta>' \<and> ladder_\<gamma> (\<alpha> @ \<delta>) D L index = \<beta> @ \<delta>' \<and> ladder_j L index < length \<beta>"
AxiomC: fixes t :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term"   and M :: "(('fun, 'atom, 'sets) prot_fun, 'c) Term.term set" assumes "t \<in> M" shows "M \<turnstile>\<^sub>c t"
neqprop: shows "\<exists>y'>k. \<forall>(d, e, f)\<in>set d. \<forall>x\<in>{k<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
interleave_set: fixes w :: "'d list"   and w1 :: "'d list"   and w2 :: "'d list" assumes "w \<in> w1 \<otimes> w2" shows "set w = set w1 \<union> set w2"
numbers_of_marks_Suc: fixes n :: "nat"   and bs :: "bool list" shows "numbers_of_marks (Suc n) bs = Suc ` numbers_of_marks n bs"
sink_nondet_envcomm: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "A \<rightarrow>\<^sub>\<E> B" shows "A \<rightarrow>\<^sub>\<E> B + C"
tick_count_latest: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n\<^sub>p :: "nat"   and n :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "f n\<^sub>p < n \<and> (\<forall>k. f n\<^sub>p < k \<and> k \<le> n \<longrightarrow> (\<nexists>k\<^sub>0. f k\<^sub>0 = k))" shows "tick_count r c n = tick_count r c (f n\<^sub>p)"
order_iff_strict: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term" shows "a \<unrhd> b = (a \<rhd> b \<or> a = b)"
Union_disjoint: fixes C :: "'b set set"   and A :: "'b set" shows "(\<Union> C \<inter> A = {}) = (\<forall>B\<in>C. B \<inter> A = {})"
mem_simps_1: fixes a :: "'e"   and b :: "'e"   and A :: "'e set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
AE_not_in: fixes N :: "'c set"   and M :: "'c measure" assumes "N \<in> null_sets M" shows "AE x in M. x \<notin> N"
singletonI: fixes a :: "'d" shows "a \<in> {a}"
finite_total_preorder_on_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "finite_total_preorder_on carrier le \<equiv> total_preorder_on carrier le \<and> finite_total_preorder_on_axioms carrier"
uniqAppendIff: fixes l1 :: "'b list"   and l2 :: "'b list" shows "uniq (l1 @ l2) = (uniq l1 \<and> uniq l2 \<and> set l1 \<inter> set l2 = {})"
elims: fixes x :: "(char list \<Rightarrow> 'av) option"   and xa :: "char list set"   and y :: "nat" assumes "m\<^sub>o x xa = y" obtains S :: "char list \<Rightarrow> 'av" and X :: "char list set" where "x = Some S" and "xa = Xa__" and "y = m\<^sub>s S Xa__"   | X :: "char list set" where "x = None" and "xa = X" and "y = h * card X + 1"
not_in_Gromov_boundary: fixes x :: "'b Gromov_completion" assumes "x \<notin> Gromov_boundary" shows "\<exists>a. x = to_Gromov_completion a"
fmfilter_false: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> \<not> P x" shows "fmfilter P m = fmempty"
fBexE: fixes A :: "'b fset"   and P :: "'b \<Rightarrow> bool" assumes "fBex A P" obtains x :: "'b" where "x |\<in>| A" and "P x"
Unit_eq_dividesone: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c" assumes "Group.comm_monoid G"   and "u \<in> carrier G" shows "(u \<in> Units G) = u divides\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub>"
on_stack_less_def: fixes I :: "'c \<Rightarrow> node_state option"   and u :: "nat"   and v :: "'c" shows "on_stack_less I u v \<equiv> case I v of None \<Rightarrow> False | Some (STACK j) \<Rightarrow> j < u | Some DONE \<Rightarrow> False"
cInf_atLeastAtMost: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "less_eq y x" shows "Inf (ord.atLeastAtMost less_eq y x) = y"
poly_deg_poly_subst_le: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and f :: "'c \<Rightarrow> ('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>x. x \<in> indets p \<Longrightarrow> poly_deg (f x) \<le> 1" shows "poly_deg (poly_subst f p) \<le> poly_deg p"
diff_sign_vec: shows "sign_vec qs x1 \<noteq> sign_vec qs x2"
assms_3: assumes "p \<in> dterms \<Gamma> p1" shows "P"
map: fixes f :: "'b \<Rightarrow> 'a"   and x1 :: "'b match_expr"   and x2 :: "action" shows "map_rule f (Rule x1 x2) = Rule (map_match_expr f x1) x2"
boundedI: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> inf b c"
compatible_def: fixes x :: "'b"   and y :: "'b" shows "compatible x y = (\<exists>z. {x, y} <<| z)"
sim_actions_action_tidD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "E [\<approx>] E'" shows "action_tid E a = action_tid E' a"
matrix_vector_mult_diff_rdistrib: fixes A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and x :: "('b, 'd) Finite_Cartesian_Product.vec" shows "(A - B) *v x = A *v x - B *v x"
uint_m2p_not_non_neg: fixes x :: "'b word" shows "\<not> 0 \<le> uint x - 2 ^ LENGTH('b)"
wtE_def: fixes \<xi> :: "'a \<Rightarrow> univ" shows "U.wtE \<xi> \<equiv> \<forall>x. D (\<xi> x)"
invh_paint: fixes t :: "('c \<times> color) tree"   and c :: "color" assumes "invh t" shows "invh (paint c t)"
handy_if_lemma: fixes P :: "bool"   and A :: "'a"   and B :: "'a" shows "((if P then Some A else None) = Some B) = (P \<and> A = B)"
realrel_def: shows "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
to_nat_le: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and y :: "'b"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less y (mod_type.from_nat Abs k)" shows "mod_type.to_nat Rep y < k"
ide_is_iso: fixes a :: "'a" assumes "\<phi>.A.ide a" shows "\<phi>.A.iso a"
add_mset_commute: fixes x :: "'f"   and y :: "'f"   and M :: "'f multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
is_parallel_solution_for_problem_plan_operator_set: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list list" assumes "is_parallel_solution_for_problem \<Psi> \<psi>" shows "\<forall>ops\<in>set \<psi>. \<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"
invertible_iff_is_unit_JNF: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "invertible_mat A = (Determinant.det A dvd (1::'a))"
carrier_direct_lim_def: shows "stfx.carrier_direct_lim \<equiv> eqY.Partition"
elims_3: fixes x :: "'a kdt" assumes "\<not> invar x" obtains k :: "'a" and v :: "real" and l :: "'a kdt" and r :: "'a kdt" where "x = Node k v l r" and "\<not> ((\<forall>p\<in>set_kdt l. p $ k \<le> v) \<and> (\<forall>p\<in>set_kdt r. v < p $ k) \<and> widest_spread_axis k UNIV (set_kdt l \<union> set_kdt r) \<and> invar l \<and> invar r)"
ik_snd_empty': fixes t :: "('a, 'b) Term.term" shows "ik\<^sub>s\<^sub>t [send\<langle>t\<rangle>\<^sub>s\<^sub>t] = {}"
\<phi>: shows "\<guillemotleft>\<phi> : f' \<Rightarrow>\<^sub>D F f\<guillemotright> \<and> D.iso \<phi>"
pderiv_smult: fixes a :: "'c"   and p :: "'c poly" shows "pderiv (smult a p) = smult a (pderiv p)"
FAssReds2: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>F{D} := e,s\<rangle> \<rightarrow>* \<langle>Val v\<bullet>F{D} := e',s'\<rangle>"
g_def: shows "g = (\<lambda>p a. signof p * (\<Prod>i\<in>{0..<n} - {a}. Sum $$ (i, p i)))"
simps_1: fixes x :: "'sz"   and y :: "'sz"   and \<theta>1 :: "('sf, 'sz) trm"   and \<theta>2 :: "('sf, 'sz) trm" shows "FUrename x y (Geq \<theta>1 \<theta>2) = Geq (TUrename x y \<theta>1) (TUrename x y \<theta>2)"
typedef_right_total: fixes Rep :: "'d \<Rightarrow> 'e"   and Abs :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and T :: "'e \<Rightarrow> 'd \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "T \<equiv> \<lambda>x y. x = Rep y" shows "right_total T"
rs_props_3: shows "defNode g r \<notin> set (tl rs)"
negligible_standard_hyperplane: fixes k :: "'c"   and c :: "real" assumes "k \<in> Basis" shows "negligible {x. x \<bullet> k = c}"
bits_div_by_1: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide a one = a"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x xs. \<lbrakk>x \<in> set xs \<Longrightarrow> P (dropWhile (\<lambda>y. y \<noteq> x) xs); x \<notin> set xs \<Longrightarrow> P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
nat_pow_Suc: fixes x :: "'d"   and n :: "nat" shows "x ^ Suc n = x ^ n * x"
above_aboveS_trans: fixes r :: "('b \<times> 'b) set"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "trans r"   and "antisym r"   and "b \<in> above r a"   and "c \<in> aboveS r b" shows "c \<in> aboveS r a"
open_box: fixes a :: "'b"   and b :: "'b" shows "open (box a b)"
Red_F_def: fixes N :: "('a literal multiset \<times> label) set" shows "FL.Red_F_\<G> N = (\<Inter>q. FL.Red_F_\<G>_q q N)"
gt_imp_vars_mset: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "kbo_std.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym t s" shows "vars_mset s \<subseteq># vars_mset t"
cring_def: fixes R :: "('c, 'd) ring_scheme" shows "cring R \<equiv> ring R \<and> Group.comm_monoid R"
dickson_leD3: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "dickson_le d m s t" shows "s \<preceq> t"
reciprocal_cvec_def: fixes z :: "complex \<times> complex" shows "reciprocal_cvec z = (let (z1, z2) = z in (z2, z1))"
reindex: fixes h :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'd" assumes "inj_on h A" shows "sum g (h ` A) = sum (g \<circ> h) A"
Dom_g: shows "Dom g = \<lparr>Leg0 = ra, Leg1 = r1\<rparr>"
observable_transition_stepI: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and P :: "'a"   and \<alpha> :: "'c"   and P' :: "'a" assumes "weak_nominal_ts satisfies transition \<tau>"   and "transition P \<langle>\<alpha>,P'\<rangle>" shows "weak_nominal_ts.observable_transition transition \<tau> P \<alpha> P'"
det_transpose: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Determinants.det (Finite_Cartesian_Product.transpose A) = Determinants.det A"
resComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<sim>\<^sup>s <\<nu>y><\<nu>x>P"
uminus_add_minus_mat: fixes l :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and r :: "'c mat" assumes "l \<in> carrier_mat nr nc"   and "r \<in> carrier_mat nr nc" shows "- (l + r) = - l - r"
scale_right_distrib: fixes x :: "'vs"   and y :: "'vs"   and a :: "'f" assumes "x \<in> U\<^sub>V\<^sub>S"   and "y \<in> U\<^sub>V\<^sub>S" shows "a *s\<^sub>o\<^sub>w (x +\<^sub>V\<^sub>S y) = a *s\<^sub>o\<^sub>w x +\<^sub>V\<^sub>S a *s\<^sub>o\<^sub>w y"
assms: shows "(c2, s) \<rightarrow>*c (c2', s')"
abs_eq: fixes x :: "char list"   and xa :: "char" assumes "eq_onp (\<lambda>s. length s \<le> MAX_STR) x x" shows "fin_string_cons xa (Abs_fin_string x) = Abs_fin_string (string_cons xa x)"
rep_eq: fixes x :: "'c"   and xa :: "'c set['d]" shows "set_bset (binsert x xa) = insert x (set_bset xa)"
access_mthd_simp: fixes m :: "'b \<times> 'a decl_scheme" shows "access (mthd m) = accmodi m"
new_tv_compatible_W: fixes n :: "nat"   and A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ"   and t :: "typ"   and m :: "nat"   and e :: "expr" assumes "new_tv n A"   and "Some (S, t, m) = W e A n" shows "new_tv m A"
connected_edgeRtc: fixes u :: "nat"   and v :: "nat" shows "connected u v = ((u, v) \<in> E\<^sup>*)"