hcju/isabelleps huggingface dataset:
hyps: assumes "\<not> c \<le> d"   and "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))"   and "cone_decomp T (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))" shows "cone_decomp T (shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}))))"
map_index'_is_ConsD: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and y :: "'b"   and ys :: "'b list" assumes "map_index' n f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f n z = y \<and> map_index' (n + 1) f zs = ys"
comp_def: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "f \<circ> g = (\<lambda>x. f (g x))"
impl_correct: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "correct_subseqs_foldr_impl f (my_subseqs.impl f) (my_subseqs.invariant f)"
entails_trans: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real" assumes "P \<tturnstile> Q"   and "Q \<tturnstile> R" shows "P \<tturnstile> R"
sim: shows "weak_reduction_simulation ((indRelRTPO TRel)\<inverse>) (STCal Source Target)"
UN_I: fixes a :: "'c"   and A :: "'c set"   and b :: "'d"   and B :: "'c \<Rightarrow> 'd set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
Collect_case_prodD: fixes x :: "'a \<times> 'c"   and A :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
eq_MkI_MkI: fixes m :: "int"   and n :: "int" shows "eq\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n) = (if m = n then TT else FF)"
Fvars_fm_simps_9: fixes a :: "name"   and A :: "fm" shows "Fvars (SyntaxN.All a A) = Fvars A - {a}"
take_bit_signed_take_bit: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "m \<le> Suc n" shows "take_bit m (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a) = take_bit m a"
reflexivity_rule_def: fixes t :: "'b" shows "reflexivity_rule t = (LG {} {0}, LG {(t, 0, 0)} {0})"
match_def: fixes s :: "'s"   and w :: "'a list" shows "match s w = accept (fold delta w s)"
set_cases: fixes e :: "'b"   and a :: "'b ref" assumes "e \<in> set_ref a" obtains "a = Ref e"
rel_spmf_OO_trans: fixes R :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and p :: "'h spmf"   and q :: "'i spmf"   and S :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and r :: "'j spmf" assumes "rel_spmf R p q"   and "rel_spmf S q r" shows "rel_spmf (R OO S) p r"
Ord_lt_succ_iff_le: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" shows "(k < succ l) = (k \<le> l)"
suminf_nonneg: fixes f :: "nat \<Rightarrow> 'a" assumes "summable f"   and "\<And>n. (0::'a) \<le> f n" shows "(0::'a) \<le> suminf f"
trans: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b c" shows "less_eq a c"
dgrad_p_set_closed_except: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and S :: "'c set" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<in> gd_term.dgrad_p_set pair_of_term d m" shows "except p S \<in> gd_term.dgrad_p_set pair_of_term d m"
fgh_comp: fixes x :: "'a::refinement_lattice" shows "((F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)) (x::'a::refinement_lattice) = ((H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> F) x"
sin_less_zero_iff: fixes x :: "real" assumes "0 \<le> x"   and "x < 2 * pi" shows "(sin x < 0) = (pi < x)"
Lcm_fin_dvd_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "finite A" shows "dvd.dvd times (semiring_gcd.Lcm_fin one zero lcm A) b = (\<forall>a\<in>A. dvd.dvd times a b)"
intros_13: fixes P :: "'e set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set"   and c :: "('e, 'f, 'g) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
A_def: shows "A = (\<lambda>n. f ((1 / 2) ^ n))"
input2state_def: fixes n :: "value list" shows "input2state n = fold (\<lambda>(k, v) f. f(k $:= Some v)) (enumerate 0 n) (K$ None)"
P_in: shows "P \<in> carrier_mat (dim_row A) (dim_row A)"
vsv_axioms: fixes A :: "V" shows "vsv (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
\<alpha>0: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale F n \<delta> f k \<beta> G1 G2"   and "1 < i" shows "(subresultant_prs_locale.\<alpha> \<delta> f i = 0) = (k < i - 1)"
k': shows "k' = (GREATEST k'. k' \<le> n - k \<and> (\<pi> \<guillemotleft> k) k' = x)"
slice_shiftr: fixes n :: "nat"   and w :: "'c word" shows "slice n w = ucast (w >> n)"
Pg_zero_def: shows "0 = Abs_Pg (eq_game_rel `` {zero_game})"
non_neg_vec_def: fixes A :: "('a, 'b) Finite_Cartesian_Product.vec" shows "non_neg_vec A \<equiv> \<forall>i. (0::'a) \<le> A $h i"
DIAndSubst12_def: shows "DIAndSubst12 = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some (Pc (Inl pid2)) else if C = pid2 then Some (Pc (Inl pid1) && Pc (Inl pid2)) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
leq_antisym: fixes x :: "'c llist"   and y :: "'c llist" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x" shows "x = y"
invar_uniqify: fixes q :: "('c, 'd) SkewBinomialTree list" assumes "queue_invar q" shows "queue_invar (uniqify q)"
Ds_1: fixes x :: "'a" shows "(s has_derivative blinfun_apply (Ds x)) (at x)"
ideal_prod_la1: fixes I :: "'a set"   and J :: "'a set" assumes "ideal R I"   and "ideal R J" shows "I \<diamondsuit>\<^sub>r J \<subseteq> I"
semiring_normalization_rules_13: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lx :: "'a"   and ly :: "'a"   and rx :: "'a"   and ry :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times (times lx ly) (times rx ry) = times (times lx rx) (times ly ry)"
elims: fixes x :: "('c \<times> nat) tree"   and y :: "nat" assumes "lvl x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | uu_ :: "('c \<times> nat) tree" and uv_ :: "'c" and lv :: "nat" and uw_ :: "('c \<times> nat) tree" where "x = \<langle>uu_, (uv_, lv), uw_\<rangle>" and "y = lv"
conv_abscissa_PInf_iff: fixes f :: "'c fds" shows "(conv_abscissa f = \<infinity>) = (\<forall>s. \<not> fds_converges f s)"
NF: shows "nofail (m0 \<bind> WHILE cond step)"
remove_from_disconnected_nodes_removes: fixes h :: "(_) heap"   and ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr"   and h' :: "(_) heap"   and disc_nodes' :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> set_disconnected_nodes ptr (remove1 node_ptr disc_nodes) \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> get_disconnected_nodes ptr \<rightarrow>\<^sub>r disc_nodes'" shows "node_ptr \<notin> set disc_nodes'"
interleaves_equal_nil_all: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and zs :: "'b list" assumes "xs \<simeq> {[], zs, P}" shows "xs = zs"
prob_compl: fixes A :: "'a set"   and M :: "'a pmf" assumes "A \<in> measure_pmf.events M" shows "measure_pmf.prob M (space (measure_pmf M) - A) = 1 - measure_pmf.prob M A"
seq_lexord_total: fixes R :: "('b \<times> 'b) set" assumes "total R" shows "total (seq_lexord R)"
ln_Gamma_stirling_complex: fixes s :: "complex"   and m :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "0 < m" shows "ln_Gamma s = (s - 1 / 2) * Ln s - s + complex_of_real (ln (2 * pi) / 2) + (\<Sum>k = 1..<m. complex_of_real (bernoulli (Suc k)) / (of_nat k * of_nat (Suc k) * s ^ k)) - stirling_integral m s / of_nat m"
dickson_leD2: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_le ord d m s t" shows "d t \<le> m"
bounded_antilinear_to_conjugate_space: shows "bounded_antilinear to_conjugate_space"
trms\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_eq_ineq: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"   and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)"   and "\<A>' = \<A> @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t]" shows "\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>) = \<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>')"
sum_mset_0_iff: fixes M :: "'a multiset multiset" shows "(comm_monoid_add.sum_mset (+) {#} M = {#}) = (\<forall>x\<in>#M. x = {#})"
elims_2: fixes x :: "'d" assumes "terminal_vertex x" obtains v :: "'d" where "x = v" and "v |\<in>| vertices \<and> terminal_node (nodeOf v)"
zip_eq_ConsE: fixes xs :: "'b list"   and ys :: "'c list"   and xy :: "'b \<times> 'c"   and xys :: "('b \<times> 'c) list" assumes "zip xs ys = xy # xys" obtains x :: "'b" and xs' :: "'b list" and y :: "'c" and ys' :: "'c list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'"
is_OK_returns_result_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and y :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y" shows "h \<turnstile> ok f"
lfinite_ltl: fixes xs :: "'b llist" shows "lfinite (ltl xs) = lfinite xs"
v_gr: shows "\<forall>a b. ((S_Idt, a, b) \<in> edges G) = (a \<in> vertices G \<and> b = a)"
select_convs_4: fixes iso_verts :: "'e \<Rightarrow> 'd"   and iso_arcs :: "'f \<Rightarrow> 'c"   and iso_head :: "'c \<Rightarrow> 'd"   and iso_tail :: "'c \<Rightarrow> 'd"   and more :: "'g" shows "iso_tail \<lparr>iso_verts = iso_verts, iso_arcs = iso_arcs, iso_head = iso_head, iso_tail = iso_tail, \<dots> = more\<rparr> = iso_tail"
moebius_ocircline_comp_inv_right: fixes M :: "moebius"   and H :: "ocircline" shows "moebius_ocircline M (moebius_ocircline (moebius_inv M) H) = H"
e_type_const_list_snoc: fixes vs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and t :: "t" assumes "const_list vs"   and "\<S>\<bullet>\<C> \<turnstile> vs : [] _> ts @ [t]" shows "\<exists>vs1 v2. \<S>\<bullet>\<C> \<turnstile> vs1 : [] _> ts \<and> \<S>\<bullet>\<C> \<turnstile> [v2] : ts _> ts @ [t] \<and> vs = vs1 @ [v2] \<and> const_list vs1 \<and> is_const v2"
cnt_remove_all_mono: fixes y :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "cnt y (remove_all_cycles xs ys) \<le> max 1 (cnt y ys)"
inf_assoc: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type"   and z :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (inf (x::'b::type) (y::'b::type)) (z::'b::type) = inf x (inf y z)"
starfun_star_n: fixes f :: "'c \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'c" shows "(*f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
p'_def: shows "p' = (\<Sum>g\<in>G'. q (dehomogenize x g) * g)"
map_add_empty: fixes m :: "'d \<Rightarrow> 'e option" shows "m ++ Map.empty = m"
f: shows "f \<in>\<^sub>\<circ> Vset \<alpha>"
simps: fixes a1 :: "char list list"   and a2 :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "P,C \<turnstile> a1 \<sqsubset>\<^sup>1 a2 = ((\<exists>Cs Ds. a1 = Cs \<and> a2 = Ds \<and> Subobjs P C Cs \<and> Subobjs P C Ds \<and> Cs = butlast Ds) \<or> (\<exists>Cs D. a1 = Cs \<and> a2 = [D] \<and> Subobjs P C Cs \<and> P \<turnstile> last Cs \<prec>\<^sub>S D))"
antisym_conv3: fixes y :: "'t"   and x :: "'t" assumes "\<not> lt_of_nat_term_order cmp_term y x" shows "(\<not> lt_of_nat_term_order cmp_term x y) = (x = y)"
exec_instr_Goto: fixes i :: "int"   and P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr (Goto i) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, nat (int pc + i), ics) # frs, sh)"
shift_prefix: fixes xl :: "'a list"   and xs :: "'a stream"   and yl :: "'a list"   and ys :: "'a stream" assumes "xl @- xs = yl @- ys"   and "length xl \<le> length yl" shows "prefix xl yl"
full_quicksort_poly_def: shows "full_quicksort_poly = full_quicksort_ref (\<lambda>x y. x = y \<or> (x, y) \<in> term_order_rel) fst"
simps: fixes vs :: "'b list"   and as :: "('b \<times> 'b) list" shows "grouped_out_arcs (vs, as) = grouped_by_fst as"
blues_seen_def: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and w :: "'b \<Rightarrow> 'a"   and p :: "'b" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "Blue_Eyes.blues_seen blue w p = {p'. w p' = blue} - {p}"
poly_eval_zero: fixes a :: "'b \<Rightarrow> 'a" shows "poly_eval a 0 = (0::'a)"
a_known_ptrs_def: fixes h :: "(_) heap" shows "ShadowRootClass.known_ptrs h = (\<forall>ptr\<in>fset (object_ptr_kinds h). ShadowRootClass.known_ptr ptr)"
simps_7: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm"   and q :: "fol fm" shows "semantics e f g (Or p q) = (semantics e f g p \<or> semantics e f g q)"
distproj_def: fixes x :: "'b"   and A :: "'b set" shows "distproj x A = (if proj_set x A \<noteq> {} then SOME y. y \<in> proj_set x A else x)"
Abs_BetasE: fixes x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" assumes "Abs x s \<rightarrow>\<^sub>\<beta>\<^sup>* t" obtains u :: "Lambda_Z.term" where "t = Abs x u" and "s \<rightarrow>\<^sub>\<beta>\<^sup>* u"
outer_IH: shows "\<not> \<bar>f p - f r\<bar> \<le> \<Delta>"
r_neg: fixes x :: "'e" assumes "x \<in> carrier N" shows "x \<oplus>\<^bsub>N\<^esub> \<ominus>\<^bsub>N\<^esub> x = \<zero>\<^bsub>N\<^esub>"
M_not_dvd_24: fixes p :: "nat" assumes "mersenne_prime p" shows "\<not> 2 ^ p - 1 dvd 24"
less_eq_pos: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (pos x \<le> pos y)"
symmetric_mpoly_symmetric_prod': fixes A :: "nat set"   and g :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "\<And>\<pi>. \<pi> permutes A \<Longrightarrow> g \<pi> permutes X"   and "\<And>x \<pi>. \<lbrakk>x \<in> X; \<pi> permutes A\<rbrakk> \<Longrightarrow> mpoly_map_vars \<pi> (f x) = f (g \<pi> x)" shows "symmetric_mpoly A (prod f X)"
simps: fixes \<Sigma> :: "'a set set"   and \<phi> :: "'a ltl" shows "\<A> \<Sigma> \<phi> = (delta \<Sigma>, initial \<phi>, rabin_pairs \<Sigma> \<phi>)"
foldr_disj_map: fixes bs :: "real list"   and f :: "'a \<Rightarrow> fm"   and xs :: "'a list" shows "Ifm bs (foldr MIR.disj (map f xs) F) = (\<exists>x\<in>set xs. Ifm bs (f x))"
ex_reg: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<forall>x. P x \<longrightarrow> Q x"   and "Ex P" shows "Ex Q"
measure_lborel_cbox_eq: fixes l :: "'c"   and u :: "'c" shows "content (cbox l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
sum_abs_triangle_ineq: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S" shows "\<bar>sum f S\<bar> \<le> (\<Sum>l\<in>S. \<bar>f l\<bar>)"
finter_lower1: fixes A :: "'a fset"   and B :: "'a fset" shows "A |\<inter>| B |\<subseteq>| A"
suminf: fixes x :: "'v"   and y :: "'v" shows "(\<Sum>i. (k_i' i (x, y) - k_i' i (y, x)) * indicat_real \<^bold>E (x, y)) = f' (x, y)"
coeff_X: fixes i :: "nat" shows "coeff X i = (if i = 1 then 1::'b else (0::'b))"
rev_bexI: fixes x :: "'d"   and A :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
monic_prod_list_pow: fixes xis :: "('b \<times> nat) list" shows "monic (\<Prod>(x, i)\<leftarrow>xis. [:- x, 1::'b:] ^ Suc i)"
simps_2: fixes l1 :: "value"   and v :: "'a" shows "local.less_aexp_aux (L l1) (V v) = True"
state_succ_pair: fixes s :: "('a, 'b) fmap"   and p :: "('a, 'b) fmap"   and e :: "('a, 'b) fmap" shows "state_succ s (p, e) = (if p \<subseteq>\<^sub>f s then e ++ s else s)"
assms: shows "RETURN c \<le> \<Down> R (SPEC \<Phi>)"
distrib_imp2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<And>x y z. x + y \<cdot> z = (x + y) \<cdot> (x + z)" shows "x \<cdot> (y + z) = x \<cdot> y + x \<cdot> z"
le1_2: shows "sum (wt (ParT dl) t) (BrnFT dl cl) < 1"
rbt_lookup_rbt_map_entry: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and f :: "'d \<Rightarrow> 'd"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (ord.rbt_map_entry less k f t) = (ord.rbt_lookup less t)(k := map_option f (ord.rbt_lookup less t k))"
select_convs_1: fixes ok :: "bool"   and more :: "'b" shows "ok \<lparr>ok = ok, \<dots> = more\<rparr> = ok"
add_2_1: fixes x :: "real_alg_2"   and y :: "real_alg_2" assumes "invariant_2 x"   and "invariant_2 y" shows "invariant_2 (add_2 x y)"
3: shows "continuous_on {w. - pi < Im w \<and> Im w < pi} (\<lambda>x. Blinfun ((*) (exp x)))"
aless_nat_less: fixes n :: "nat"   and m :: "nat" shows "(an n < an m) = (n < m)"
assms: shows "\<forall>n'. enat n' \<le> llength t \<longrightarrow> n \<le> n' \<longrightarrow> \<not> \<parallel>c\<parallel>\<^bsub>lnth t n'\<^esub>"
antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
eq_infI: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1" shows "w\<^sub>1 =\<^sub>I w\<^sub>2"
infnorm_def: fixes x :: "'a" shows "infnorm x = Sup {\<bar>x \<bullet> b\<bar> |b. b \<in> Basis}"
ldropn_lnull: fixes xs :: "'c llist"   and n :: "nat" assumes "lnull xs" shows "ldropn n xs = LNil"
max_extp_max_ext_eq: fixes R :: "('a \<times> 'a) set" shows "max_extp (\<lambda>x xa. (x, xa) \<in> R) = (\<lambda>x xa. (x, xa) \<in> max_ext R)"
arr_char: fixes f :: "'a" shows "R.arr f = right f f"
x_in: shows "x \<in> sum_list ` listset (map cone qs)"
nthtriple_chainmap: fixes i :: "nat"   and \<Pi> :: "('a, 'b) chain"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" assumes "i < chainlen \<Pi>" shows "nthtriple (chainmap f g \<Pi>) i = (f (fst3 (nthtriple \<Pi> i)), g (snd3 (nthtriple \<Pi> i)), f (thd3 (nthtriple \<Pi> i)))"
obind_def: fixes f :: "'c \<Rightarrow> 'e option"   and g :: "'e \<Rightarrow> 'c \<Rightarrow> 'd option" shows "f |>> g \<equiv> \<lambda>s. case f s of None \<Rightarrow> None | Some x \<Rightarrow> g x s"
simulation1_aux: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and tlsim :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and s1 :: "'g"   and s2 :: "'i"   and tl1 :: "'h"   and s1' :: "'g" assumes "bisimulation_into_delay trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"   and "bisim s1 s2"   and "trsys1 s1 tl1 s1'"   and "\<not> \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> trsys2 s2' tl2 s2'' \<and> \<not> \<tau>move2 s2' tl2 s2'' \<and> bisim s1' s2'' \<and> tlsim tl1 tl2"
typing_fnE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and \<theta> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> Fn x T A : \<theta>" shows "\<exists>\<sigma>. \<theta> = TArr T \<sigma> \<and> \<Gamma>(x \<mapsto> T) \<turnstile> A : \<sigma>"
smult_distrib_left: fixes r :: "'f"   and m :: "'v"   and n :: "'v" assumes "r \<in> UNIV"   and "m \<in> V"   and "n \<in> V" shows "r \<cdot> (m + n) = r \<cdot> m + r \<cdot> n"
ws_def: shows "ws \<equiv> [countVertices g'..<countVertices g' + n]"
sel_4: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "key (Node x21 x22 x23 x24 x25) = x23"
vrat_mult_def: shows "vrat_mult = (\<lambda>x\<in>\<^sub>\<circ>\<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (rat_of_vrat (x\<lparr>0\<^sub>\<nat>\<rparr>) * rat_of_vrat (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<rat>)"