hcju/isabelleps huggingface dataset:
fsps_chain_bound2: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and C :: "('a set \<times> ('a \<Rightarrow> 'g)) set" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "C \<subseteq> fsps R M N f H"   and "C \<noteq> {}"   and "\<forall>a\<in>C. \<forall>b\<in>C. fst a \<subseteq> fst b \<or> fst b \<subseteq> fst a" shows "\<forall>y\<in>\<Union> {a. \<exists>b. (a, b) \<in> C}. (\<lambda>x\<in>fgs R M (\<Union> {a. \<exists>b. (a, b) \<in> C}). THE y. y \<in> carrier N \<and> (\<exists>a b. (a, b) \<in> C \<and> x \<in> fgs R M a \<and> y = b x)) y = f y"
narrow_PrimT_strong: fixes G :: "prog"   and pt :: "prim_ty"   and T :: "ty" assumes "G\<turnstile>PrimT pt\<succ>T" shows "T = PrimT pt"
zipWithA_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and a :: "'a array"   and b :: "'b array" shows "zipWithA f a b = Array.len a \<bind> (\<lambda>n. Heap_Monad.fold_map (\<lambda>n. Array.nth a n \<bind> (\<lambda>x. Array.nth b n \<bind> (\<lambda>y. Array.upd n (f x y) a))) [0..<n] \<bind> (\<lambda>_. return a))"
exec: shows "(ts\<^sub>h, m, \<S>\<^sub>h) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>h'', m'', \<S>\<^sub>h'')"
intro: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset" assumes "selection S" shows "ground_resolution_with_selection S"
preserves_trg: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "trg\<^sub>C (F \<mu>) = \<tau>.F.map\<^sub>0 (trg\<^sub>B \<mu>)"
induct: fixes x :: "hmultiset"   and P :: "hmultiset \<Rightarrow> bool" assumes "local.well_base\<^sub>h x"   and "\<And>M. \<lbrakk>\<forall>N\<in>#hmsetmset M. local.well_base\<^sub>h N \<and> P N; local.well_base (hmsetmset M)\<rbrakk> \<Longrightarrow> P M" shows "P x"
encL: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> SourceTerm S"
steps_z_V: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "Z \<subseteq> V" shows "Z' \<subseteq> V"
assms: shows "\<rho> f|` (- domA h) \<sqsubseteq> \<rho>' f|` (- domA h)"
in_arcs_app_iso_eq: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'a" assumes "digraph_isomorphism hom"   and "u \<in> verts G" shows "in_arcs (app_iso hom G) (iso_verts hom u) = iso_arcs hom ` in_arcs G u"
B_def: shows "B \<equiv> (\<Sum>n. cmod (fds_nth f (n + 2)) / real (n + 2) powr c) * 2 powr c"
simps_7: fixes c :: "int"   and e :: "Ferrack.num" shows "isrlfm (Gt (CN 0 c e)) = (0 < c \<and> numbound0 e)"
differentiable_transform_eventually: fixes g :: "'c \<Rightarrow> 'd"   and x :: "'c"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "g differentiable at x within X"   and "f x = g x"   and "\<forall>\<^sub>F x in at x within X. f x = g x" shows "f differentiable at x within X"
constr: fixes ts :: "value list"   and us :: "value list"   and name :: "name" assumes "list_all2 erelated ts us" shows "Vconstr name ts \<approx>\<^sub>e Vconstr name us"
elims_3: fixes x :: "real_alg_2" assumes "\<not> invariant_2 x" obtains n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai" and "\<not> (invariant_1_2 rai \<and> n = card {y. y \<le> real_of_1 rai \<and> poly (real_of_int_poly (poly_real_alg_1 rai)) y = 0})"
\<nu>_stable_def: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<nu>_stable \<phi> w = (\<F>\<G> \<phi> w = \<G> \<phi> w)"
parts_mpair_pwd: fixes s :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "\<lbrace>X, Y\<rbrace> \<in> parts (used s)"   and "X = Pwd n \<or> Y = Pwd n" shows "{X, Y} \<subseteq> spied s"
in_RSpan_Cons_obtain_same_length_coeffs: fixes n :: "'m"   and m :: "'m"   and ms :: "'m list" assumes "n \<in> RSpan (m # ms)" shows "\<exists>r rs. set (r # rs) \<subseteq> R \<and> length rs = length ms \<and> n = r \<cdot> m + rs \<bullet>\<cdot> ms"
GFP_upperbound: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "x \<in> carrier L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> f x" shows "x \<sqsubseteq>\<^bsub>L\<^esub> GFP\<^bsub>L\<^esub> f"
ordertype_eq_Ord: fixes \<alpha> :: "V" assumes "Ord \<alpha>" shows "tp (elts \<alpha>) = \<alpha>"
fin_digraphI_app_iso: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "fin_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "fin_digraph (app_iso hom G)"
Cauchy_Schwarz_ineq2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
coeff_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly"   and n :: "nat" assumes "f (0::'c) = (0::'b)" shows "coeff (map_poly f p) n = f (coeff p n)"
bits_of_1_int: shows "bits_of 1 = [True, False]"
solve_depressed_quartic_real: fixes x :: "real"   and p :: "real"   and q :: "real"   and r :: "real" shows "(x \<in> set (solve_depressed_quartic_real p q r)) = (x ^ 4 + p * x\<^sup>2 + q * x + r = 0)"
lem: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and P :: "'a \<Rightarrow> bool"   and b :: "'a" assumes "\<forall>y. r\<^sup>+\<^sup>+ a y \<longrightarrow> P a \<longrightarrow> P y" shows "((b, a) \<in> {(y, x). P x \<and> r x y}\<^sup>+) = ((b, a) \<in> {(y, x). P x \<and> r\<^sup>+\<^sup>+ x y})"
numeral_mult: fixes m :: "num"   and n :: "num" shows "numeral (m * n) = numeral m * numeral n"
coeff_lp_monom: fixes c :: "rat"   and v :: "nat"   and v' :: "nat" shows "coeff (lp_monom c v) v' = (if v = v' then c else 0)"
asm4: shows "\<forall>d\<in>D. b \<le> bal (disk s d p) \<and> (\<forall>q. phase s q = 1 \<and> b \<le> mbal (dblock s q) \<and> hasRead s q d p \<longrightarrow> (\<exists>br\<in>blocksRead s q d. b \<le> bal (block br)))"
asm: shows "infinite {t. linepath a b t \<bullet> c = d \<and> 0 \<le> t \<and> t \<le> 1}"
WTrtSeq: fixes 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2" shows "P,E,h \<turnstile> e\<^sub>1;; e\<^sub>2 : T\<^sub>2"
group_ow_axioms_def: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a" shows "group_ow_axioms U f z inverse \<equiv> (z \<in> U \<and> (\<forall>a. a \<in> U \<longrightarrow> inverse a \<in> U)) \<and> (\<forall>a. a \<in> U \<longrightarrow> f z a = a) \<and> (\<forall>a. a \<in> U \<longrightarrow> f (inverse a) a = z)"
even_odd_mirror_path_injective_2: fixes path :: "dir list"   and path' :: "dir list" shows "(odd_mirror_path path = odd_mirror_path path') = (path = path')"
prems_3: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
intro: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "H \<subseteq> carrier G"   and "\<And>x y. \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H"   and "\<one>\<^bsub>G\<^esub> \<in> H"   and "\<And>x. x \<in> H \<Longrightarrow> inv\<^bsub>G\<^esub> x \<in> H" shows "subgroup H G"
midpoint_not_midpoint: fixes A :: "'p"   and B :: "'p"   and I :: "'p" assumes "A \<noteq> B"   and "I Midpoint A B" shows "\<not> B Midpoint A I"
FW_canonical_id: fixes i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c" assumes "canonical M i" shows "FW M i = M"
ensures_simple: fixes P :: "state \<Rightarrow> bool"   and N :: "state \<times> state \<Rightarrow> bool"   and Q :: "state \<Rightarrow> bool"   and A :: "state \<times> state \<Rightarrow> bool" assumes "\<turnstile> $P \<and> N \<longrightarrow> P$ \<or> Q$"   and "\<turnstile> ($P \<and> N) \<and> A \<longrightarrow> Q$" shows "\<turnstile> \<box>N \<and> \<box>\<diamond>A \<longrightarrow> (P \<leadsto> Q)"
is_rbt_impl_of: fixes t :: "('b, 'c) RBT.rbt" shows "is_rbt (rbt.impl_of t)"
from_bool_to_bool_iff: fixes w :: "'b word"   and b :: "bool" shows "(w = from_bool b) = (to_bool w = b \<and> (w = 0 \<or> w = 1))"
assms: fixes i :: "nat" shows "(coeff p i = (0::'a)) = (coeff q i = (0::'b))"
smult_monom: fixes a :: "'b"   and b :: "'b"   and n :: "nat" shows "smult a (monom b n) = monom (a * b) n"
is_SDG_path_slice1: 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 Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n' \<in> SDG.sum_SDG_slice1 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n''" shows "n \<in> SDG.sum_SDG_slice1 sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n''"
execute_def: fixes instrs :: "('d, 'c) instr list"   and env :: "'d \<Rightarrow> 'c" shows "execute instrs env = hd (exec instrs [] env)"
unop_v11I: fixes A :: "V"   and f :: "V" assumes "nop_v11 A (1\<^sub>\<nat>) f" shows "unop_v11 A f"
l2_step3_def: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and K :: "msg" shows "l2_step3 Ra A B K \<equiv> {(s, s'). guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress s Ra = Some {xpkE, xskE} \<and> guessed_frame Ra xsk = Some K \<and> Auth B A (Aenc K (epubKF (Ra $ kE))) \<in> chan s \<and> s' = s\<lparr>progress := progress s(Ra \<mapsto> {xpkE, xskE, xsk}), l1_state.signals := if can_signal s A B then addSignal (l1_state.signals s) (Commit A B \<langle>epubKF (Ra $ kE), K\<rangle>) else l1_state.signals s, secret := {x. x = K \<and> Ra = test} \<union> secret s\<rparr>}"
mvt_simple_closed_segmentE: fixes f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes " D f = f' on {a--b}"   and "a \<noteq> b" obtains y :: "real" where "y \<in> {a<--<b}" and "f b - f a = (b - a) * f' y"
lex_fun_plus_monotone: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "lex_fun s t" shows "lex_fun (s + u) (t + u)"
app_invimageD1: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
cases: fixes x :: "'b list" obtains a :: "'b" where "x = [a__]"   | a :: "'b" and b :: "'b" where "x = [aa__, b__]"   | a :: "'b" and b :: "'b" and c :: "'b" where "x = [ab__, ba__, c__]"   | a :: "'b" and b :: "'b" and c :: "'b" and d :: "'b" where "x = [ac__, bb__, ca__, d__]"   | a :: "'b" and b :: "'b" and c :: "'b" and d :: "'b" and e :: "'b" where "x = [a, b, c, d, e]"   | "x = []"   | v :: "'b" and vb :: "'b" and va :: "'b" and vc :: "'b" and vd :: "'b" and ve :: "'b" and vg :: "'b list" where "x = v # vb # va # vc # vd # ve # vg"
sin_le_arg: fixes x :: "real" assumes "0 \<le> x" shows "sin x \<le> x"
length_remdups_card_conv: fixes xs :: "'a list" shows "length (remdups xs) = card (set xs)"
is_dyn_operand_eq_typeof: shows "is_dyn_operand = (\<lambda>x. typeof x = None)"
invertibleE: fixes u :: "'a"   and thesis :: "bool" assumes "invertible u"   and "\<And>v. \<lbrakk>u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>; v \<in> M\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> M" shows "thesis"
Neg_E: fixes H :: "fm set"   and A :: "fm" assumes "H \<turnstile> A" shows "insert (Neg A) H \<turnstile> Fls"
nodes_from_rel: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and q0 :: "'a" shows "nodes (from_rel rel q0) \<subseteq> insert q0 ((snd \<circ> snd) ` rel)"
path_image_subpath_commute: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b" shows "path_image (subpath u v g) = path_image (subpath v u g)"
Derives1_prefix: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and a :: "symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 \<delta> i r (a @ b)"   and "splits_at \<delta> i \<alpha> N \<beta>" shows "is_prefix \<alpha> a \<or> is_prefix a \<alpha>"
contlub_def: fixes f :: "'a \<Rightarrow> 'b" shows "contlub f = (\<forall>Y. chain Y \<longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i)))"
simps_3: fixes u :: "'b sort_key extended"   and i :: "'b"   and l :: "'b sort_key extended" assumes "u \<le> \<lbrakk>Final i\<rbrakk> \<and> l < u" shows "\<Psi> (l, u) i = NonFinal (\<gamma> u sort_dir.Left) (\<Psi> (derive_right (l, u)) i)"
that: fixes tr :: "'c list" assumes "{t. path M (io || t) q \<and> length io = length t} = {tr}" shows "thesis"
distrib_lincomb: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "T (lincomb' rs ms) = lincomb' rs (map T ms)"
trg: fixes n :: "nat" shows "trace (g n) = (\<Sum>k = 0..<d. g n $$ (k, k))"
node_ptr_kinds_eq_h3: shows "node_ptr_kinds h' = node_ptr_kinds h3"
select_convs_6: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_sport \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr> = p_sport"
sum_add_nat_ivl_singleton: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m < n" shows "f m + sum f {m<..<n} = sum f {m..<n}"
add\<^sub>O: fixes n :: "Ord"   and m :: "Ord" shows "\<langle>addO n m\<rangle>\<^sub>O = (\<lambda>z s l. \<langle>m\<rangle>\<^sub>O (\<langle>n\<rangle>\<^sub>O z s l) s l)"
zmod_zminus_eq_conv_pos: fixes m :: "int"   and a :: "int"   and b :: "int" assumes "0 < m" shows "(a mod - m = b mod - m) = (a mod m = b mod m)"
disj_evens_odds: fixes A :: "nat set"   and B :: "nat set" shows "(*) 2 ` A \<inter> Suc ` (*) 2 ` B = {}"
le_multiset_empty_right: fixes M :: "'b multiset" shows "\<not> M < {#}"
m_lcomm: fixes x :: "int vec"   and y :: "int vec"   and z :: "int vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + (y + z) = y + (x + z)"
prod_simps_1: fixes f0 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f1 :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f0"   and "arr f1" shows "arr (HF'.prod f0 f1)"
Plus_upper: fixes r1 :: "'a language"   and s :: "'a language"   and r2 :: "'a language" assumes "r1 \<le> s"   and "r2 \<le> s" shows "Plus r1 r2 \<le> s"
assms_3: shows "cnf \<Phi>' \<subseteq> cnf \<Phi>"
valid_10: shows "valid_program_history ts\<^sub>s\<^sub>b'"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a \<squnion> b \<squnion> c = a \<squnion> (b \<squnion> c)"
f: shows "\<guillemotleft>f : src\<^sub>C f \<rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright> \<and> C.ide f"
T1_7': fixes init :: "'c list"   and qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and n :: "nat"   and x :: "'c"   and y :: "'c" assumes "T\<^sub>p init qs Strat = T\<^sub>p_opt init qs"   and "length Strat = length qs"   and "n \<le> length qs"   and "x \<noteq> y"   and "x \<in> set init"   and "y \<in> set init"   and "distinct init"   and "set qs \<subseteq> set init" shows "\<exists>Strat2 sws. length Strat2 = length (Lxy (take n qs) {x, y}) \<and> x < y in steps' init (take n qs) (take n Strat) n = x < y in swaps sws (steps' (Lxy init {x, y}) (Lxy (take n qs) {x, y}) Strat2 (length Strat2)) \<and> T\<^sub>p (Lxy init {x, y}) (Lxy (take n qs) {x, y}) Strat2 + length sws = ALGxy_det Strat (take n qs) init x y + ALG_Pxy Strat (take n qs) init x y"
PEE: fixes a :: "nat \<Rightarrow> ValD" assumes "a \<in> unlr PE.delta" obtains "a = \<bottom>"   | "a = (\<lambda>i. ValTT)"   | "a = (\<lambda>i. ValFF)"   | n :: "nat" where "a = (\<lambda>i. ValN\<cdot>n)"   | "a 1 = \<bottom>"   | "a 2 = \<bottom>"   | fs :: "nat \<Rightarrow> ValD \<rightarrow> ValD" where "a = (\<lambda>j. ValF\<cdot>(fs j))" and "\<And>xs. xs \<in> unlr PE.delta \<Longrightarrow> (\<lambda>j. fs j\<cdot>(xs j)) \<in> unlr PE.delta"
cong_0_iff: fixes b :: "'a"   and a :: "'a" shows "[b = 0::'a] (mod a) = (a dvd b)"
hyps: fixes p :: "nat"   and q :: "nat" assumes "p + p + q < pa__ + pa__ + qa__"   and "\<not> p dvd q"   and "\<not> q dvd p" shows "\<^bold>|FW_word p q\<^bold>| = p + q - gcd p q - 1 \<and> periodN (FW_word p q) p \<and> periodN (FW_word p q) q \<and> \<not> periodN (FW_word p q) (gcd p q)"
fib_add: fixes n :: "nat"   and k :: "nat" shows "fib (n + k + 1) = fib (k + 1) * fib (n + 1) + fib k * fib n"
adjoint_pair_def: fixes f :: "'c"   and g :: "'c" shows "C.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
INTERP_def: fixes N :: "'a literal multiset set" shows "INTERP N = \<Union> (production N ` N)"
append_assoc: fixes xs :: "'k list"   and ys :: "'k list"   and zs :: "'k list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
rs_subset_def: fixes as :: "('c, unit) RBT.rbt"   and bs :: "('c, unit) RBT.rbt" shows "rs_subset as bs = rs.iteratei as (\<lambda>maybe. case maybe of None \<Rightarrow> True | Some x \<Rightarrow> False) (\<lambda>a _. if rs.memb a bs then None else Some a) None"
crename_aredu: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and M' :: "trm" assumes "M[a\<turnstile>c>b] \<longrightarrow>\<^sub>a M'"   and "a \<noteq> b" shows "\<exists>M0. M0[a\<turnstile>c>b] = M' \<and> M \<longrightarrow>\<^sub>a M0"
cases: fixes x :: "('a, 'b) BinomialHeapStruc.BinomialTree" obtains e :: "'a" and a :: "'b" and r :: "nat" and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" where "x = BinomialHeapStruc.Node e a r ts"
pos_prod_le: fixes c :: "'a"   and x :: "'a" assumes "(0::'a) < c" shows "c * x \<le> (0::'a) \<equiv> x \<le> (0::'a)"
fun_left_inverse: fixes f :: "'c \<Rightarrow> 'd" assumes "zero_free f" shows "inverse f * f = 1"
neg_le_iff_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus b) (uminus a) = less_eq a b"
solve_consistent_rref_code: fixes A :: "(('c, 'b) vec, 'd) vec"   and b :: "('c, 'd) vec" shows "($) (solve_consistent_rref A b) = (\<lambda>j. if \<exists>i. A $ i $ j = (1::'c) \<and> j = (LEAST n. A $ i $ n \<noteq> (0::'c)) then b $ (THE i. A $ i $ j = (1::'c)) else (0::'c))"
ide_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "partial_magma C" shows "partial_magma.ide C a = partial_magma.in_hom C a a a"
BF: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
\<rho>_in_terms_of_rep: shows "\<rho> = (T0.trnr\<^sub>\<epsilon> r \<rho> \<star> f) \<cdot> T0.trnr\<^sub>\<eta> g (g \<star> f\<^sup>*)"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> = b"
cSUP_upper: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset" assumes "x \<in> A"   and "subset_mset.bdd_above (f ` A)" shows "f x \<subseteq># Sup (f ` A)"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and f :: "'d"   and Tl :: "'d trm list" assumes "ProblemIk wtFsym wtPsym arOf resOf parOf \<Phi> infTp" shows "ProblemIk.nvT (Fn f Tl) = {}"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "Ik.SAT \<Phi> \<equiv> \<forall>\<xi>. TE.wtE \<xi> \<longrightarrow> Ik.satPB \<xi> \<Phi>"
pprt_neg: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.pprt zero sup (uminus x) = uminus (lattice_ab_group_add.nprt zero inf x)"
surjective: fixes r :: "('d, 'e) eq_object_scheme" shows "r = \<lparr>carrier = carrier r, eq = (.=\<^bsub>r\<^esub>), \<dots> = eq_object.more r\<rparr>"
compass_reflect_compass_3: shows "apply_cltn2 north meridian_reflect = north"
sparse_row_matrix_def: fixes arr :: "(nat \<times> (nat \<times> 'a) list) list" shows "sparse_row_matrix arr = foldl (\<lambda>m r. m + move_matrix (sparse_row_vector (snd r)) (int (fst r)) 0) 0 arr"
set_tree_diff: fixes t1 :: "('a \<times> 'b) tree"   and t2 :: "('a \<times> 'b) tree" assumes "Tree2.bst t1"   and "Tree2.bst t2" shows "Tree2.set_tree (diff t1 t2) = Tree2.set_tree t1 - Tree2.set_tree t2"
lfp_fixpoint: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "f (lfp f) = lfp f"