hcju/isabelleps huggingface dataset:
non_persistent_clauses_are_redundant: fixes D :: "'at Literal set"   and R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set"   and N :: "nat" assumes "D \<in> inferred_clause_sets R S N"   and "D \<notin> saturation R S"   and "all_fulfill finite S"   and "derived_clauses_are_finite R" shows "\<exists>M. strictly_redundant D (inferred_clause_sets R S M)"
sees_method_mono: fixes P :: "'a prog"   and C' :: "String.literal"   and C :: "String.literal"   and wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option"   and D :: "String.literal" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
"case": fixes f :: "nat \<times> nat \<times> nat \<times> nat \<Rightarrow> 'a"   and x :: "nat \<times> nat \<times> nat \<times> nat" shows "(case Quad x of Quad x \<Rightarrow> f x) = f x"
succ_no_inclusion: shows "succ AB w (p1, p2) \<noteq> succ A w p1 \<times> succ B w p2"
lookup_single_eq: fixes k :: "'d"   and v :: "'c" shows "lookup (monomial v k) k = v"
simps_1: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and v :: "'e"   and p :: "('b, 'a) antecedent" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst" shows "tree.root (Tasked_Proof_Graph.edge_tree anyP antecedent nodeOf edges v p) = (Tasked_Proof_Graph.adjacentTo edges v p, v, p)"
simps_2: fixes n :: "nat" shows "bv (Suc n) = bv n \<bind> (\<lambda>xs. bernoulli_pmf (5 / 10) \<bind> (\<lambda>x. return_pmf (x # xs)))"
assms_2: shows "free_vars e \<subseteq> V \<union> V'"
has_derivative_on_open3: fixes s :: "'a" assumes "s \<in> S" obtains B :: "real set" where "0 \<in> B" and "open B" and "B \<subseteq> T" and "D (\<lambda>\<tau>. \<phi> \<tau> s) \<mapsto> (\<lambda>\<tau>. \<tau> *\<^sub>R f (\<phi> 0 s)) at 0 within B"
accept_def: fixes w :: "nat \<Rightarrow> 'l" shows "local.igba.accept w \<equiv> \<exists>r. igba.is_acc_run r \<and> (\<forall>i. local.igba.L (r i) (w i))"
mod_mod_trivial: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (modulo a b) b = modulo a b"
inf_demorgan: fixes x :: "'b"   and y :: "'b" shows "- (- x \<sqinter> - y) = - - x \<squnion> - - y"
univ_poly_is_monoid: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "domain R"   and "subring K R" shows "Group.monoid (K [X]\<^bsub>R\<^esub>)"
comp_reduce: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and k :: "'c \<times> 'c"   and h :: "'c \<times> 'c" assumes "f \<odot> g = k"   and "CopxC.seq f g"   and "CopxC.seq g h" shows "f \<odot> g \<odot> h = k \<odot> h"
subset_chain_def: fixes \<A> :: "'b set set"   and \<C> :: "'b set set" shows "subset.chain \<A> \<C> = (\<C> \<subseteq> \<A> \<and> (\<forall>X\<in>\<C>. \<forall>Y\<in>\<C>. X \<subseteq> Y \<or> Y \<subseteq> X))"
mat_inverse_2: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "mat_inverse A = Some B" shows "A * B = 1\<^sub>m n \<and> B * A = 1\<^sub>m n \<and> B \<in> carrier_mat n n"
anno_def: fixes C :: "'a acom"   and p :: "nat" shows "anno C p = annos C ! p"
INF_INF_seq_distrib: fixes Y :: "'d::type set"   and c :: "'e::type \<Rightarrow> 'a::refinement_lattice"   and X :: "'e::type set"   and d :: "'d::type \<Rightarrow> 'a::refinement_lattice" assumes "(Y::'d::type set) \<noteq> {}" shows "\<Sqinter> ((c::'e::type \<Rightarrow> 'a::refinement_lattice) ` (X::'e::type set)) ; \<Sqinter> ((d::'d::type \<Rightarrow> 'a::refinement_lattice) ` (Y::'d::type set)) = (\<Sqinter>x::'e::type\<in>X. \<Sqinter>y::'d::type\<in>Y. c x ; d y)"
minGraphProps8a: fixes g :: "graph"   and v :: "nat" assumes "minGraphProps g"   and "v \<in> \<V> g" shows "distinct (normFaces (faceListAt g ! v))"
natural_of_nat_of_nat: shows "natural_of_nat = of_nat"
C'_def: shows "C' = (\<Prod>x\<in>Roots'. \<Sum>q\<in>P. real_of_int \<bar>\<beta> q\<bar> * (\<Sum>\<alpha>\<in>Roots q. cmod \<alpha> * exp (cmod \<alpha>)))"
sums_minus: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f sums a" shows "(\<lambda>n. - f n) sums - a"
stc_Infinitesimal_add_SComplex2: fixes x :: "complex star"   and e :: "complex star" assumes "x \<in> SComplex"   and "e \<in> Infinitesimal" shows "stc (e + x) = x"
wellformed_items_\<J>: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and u :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.\<J> \<TT> \<RR> \<SS> Lex Sel Doc k u)"
inverse_formula_renaming: fixes A :: "'at set"   and L :: "'at Literal set set" shows "rename_formula A (rename_formula A L) = L"
monotone_max_enat1: fixes y :: "enat" shows "monotone (\<le>) (\<le>) (\<lambda>x. max x y)"
backflows_wf: fixes N :: "'b set"   and E :: "('b \<times> 'b) set" assumes "wf_graph \<lparr>nodes = N, edges = E\<rparr>" shows "wf_graph \<lparr>nodes = N, edges = backflows E\<rparr>"
simps: fixes T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "arrV T a.[i] = arrLoc T a i"
induces_elementary_monoidal_category: shows "elementary_monoidal_category (\<cdot>) tensor unity lunit runit assoc"
flow_initial_time: fixes t0 :: "real"   and x0 :: "real" assumes "t0 \<in> T"   and "x0 \<in> X" shows "flow t0 x0 t0 = x0"
inverse_negative_imp_negative: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less (inverse a) zero"   and "a \<noteq> zero" shows "less a zero"
measurable_convex: fixes S :: "'c set" assumes "convex S"   and "bounded S" shows "S \<in> lmeasurable"
progress_time_conv: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool"   and progress :: "'c prefix \<Rightarrow> nat"   and \<pi> :: "'c prefix"   and \<pi>' :: "'c prefix" assumes "timed_progress nfv fv sat progress"   and "pts \<pi> = pts \<pi>'" shows "progress \<pi> = progress \<pi>'"
resComm: fixes Rel :: "(pi \<times> pi) set"   and a :: "name"   and b :: "name"   and P :: "pi" assumes "\<And>c d Q. (<\<nu>c><\<nu>d>Q, <\<nu>d><\<nu>c>Q) \<in> Rel"   and "Id \<subseteq> Rel"   and "eqvt Rel" shows "<\<nu>a><\<nu>b>P \<leadsto>[Rel] <\<nu>b><\<nu>a>P"
cinner_real_hermiteanI: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "\<And>\<psi>. \<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<in> \<real>" shows "A = A*"
hta_ensure_idx_f_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_ensure_idx_f H \<equiv> case hta_idx_f H of None \<Rightarrow> H\<lparr>hta_idx_f := Some (build_rule_index_f (hta_\<delta> H))\<rparr> | Some x \<Rightarrow> H"
T_off_neq0: fixes alg :: "'b list \<Rightarrow> 'b list \<Rightarrow> (nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" assumes "\<And>rs s0. length (alg s0 rs) = length rs"   and "rs \<noteq> []" shows "T_off alg s0 rs \<noteq> 0"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "K.A.seq g f" shows "K (B g f) = C (K g) (K f)"
s_prod_test: fixes P :: "('a \<times> 'a set) set"   and a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" assumes "P \<subseteq> 1\<^sub>\<sigma>" shows "((a, A) \<in> R \<cdot> P) = ((a, A) \<in> R \<and> (\<forall>a\<in>A. (a, {a}) \<in> P))"
prj_simps_8: shows "C.cod \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 = \<nu>\<pi>.apex"
case_prod_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and a :: "'c"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
lprefix_lappend_sameI: fixes xs :: "'c llist"   and ys :: "'c llist"   and zs :: "'c llist" assumes "xs \<sqsubseteq> ys" shows "lappend zs xs \<sqsubseteq> lappend zs ys"
mut_initial_state_def: fixes s :: "('e, 'f, 'g, 'h) Model.local_state" shows "mut_initial_state s = (ghost_hs_phase s = hp_IdleMarkSweep \<and> ghost_honorary_grey s = {} \<and> ghost_honorary_root s = {} \<and> W s = {})"
lstrict_prefix_lfinite1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lstrict_prefix xs ys" shows "lfinite xs"
can_select_set_list_ex1: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b list" shows "can_select P (set A) = list_ex1 P A"
simps_4: fixes dims :: "nat list"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list"   and \<rho> :: "complex mat" shows "state_sig.denote dims (Measure n M S) \<rho> = state_sig.denote_measure dims n M (map (state_sig.denote dims) S) \<rho>"
midpoint_in_closed_segment: fixes a :: "'b"   and b :: "'b" shows "midpoint a b \<in> closed_segment a b"
insert_dom: fixes f :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a" assumes "f x = Some y" shows "insert x (dom f) = dom f"
assms_6: shows "(A, B) \<in> s_mul_ext ns s"
th: shows "- k *b f a \<in> vs2.span (f ` b)"
subtype_weaken': fixes \<Gamma> :: "binding list"   and P :: "POPLmark.type"   and Q :: "POPLmark.type"   and \<Delta> :: "binding list" assumes "\<Gamma> \<turnstile> P <: Q"   and "\<Delta> @ \<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "\<Delta> @ \<Gamma> \<turnstile> \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 P <: \<up>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 Q"
LxxI: fixes qs :: "nat list"   and x :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"   and y :: "nat" assumes "qs \<in> lang (seq [Atom x, Atom x]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x]) \<Longrightarrow> P x y qs"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y]) \<Longrightarrow> P x y qs"   and "qs \<in> Lxx x y" shows "P x y qs"
monom_Suc_2: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a (Suc n) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
dis_rews_6: fixes discr :: "bool discr" shows "is_CFn\<cdot>(CB\<cdot>discr) = FF"
hyps_1: shows "cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> inj_on Xd (CH fp) \<and> cop_F_rejected_inv ds fp"
distinct_3: shows "ou_witness.a \<noteq> ou_witness.c"
iAll_imp_iEx: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "I \<noteq> {}"   and "\<box> t I. P t" shows "\<diamond> t I. P t"
lincomb_closed: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and rs :: "'a list"   and ms :: "'b list" assumes "VectorSpace smult V"   and "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "scalar_mult.lincomb smult rs ms \<in> V"
root_info_condD_2: fixes ri :: "root_info"   and p :: "int poly"   and a :: "rat" assumes "root_info_cond ri p" shows "root_info.number_root ri a = card {x. x \<le> real_of_rat a \<and> poly (real_of_int_poly p) x = 0}"
Mapping_keys_dest: fixes x :: "'a"   and f :: "('a, 'b) mapping" assumes "x \<in> Mapping.keys f" shows "\<exists>y. Mapping.lookup f x = Some y"
scan_upto_length: fixes "end" :: "char list"   and s :: "char list"   and y :: "char list"   and r :: "char list" assumes "scan_upto end s = Inr (y, r)" shows "length end + length r \<le> length s"
unfold: shows "lrev_body = (\<Lambda> r v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>lnil) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(r\<cdot>xs :++ x :@ lnil))))"
tensor_in_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d"   and d :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow>\<^sub>D d\<guillemotright>" shows "\<guillemotleft>f \<otimes>\<^sub>D g : a \<otimes>\<^sub>D c \<rightarrow>\<^sub>D b \<otimes>\<^sub>D d\<guillemotright>"
diff_order_le: fixes l :: "enat" assumes "l \<le> k" shows "diff l charts charts_eucl f"
dist_Vf_\<mu>: fixes v :: "'a" assumes "v \<in> verts G"   and "\<exists>r. dist v = ereal r" shows "dist v = \<mu> c s v"
assms_3: shows "Q * P = 1\<^sub>m n"
retract_of_space_def: fixes S :: "'c set"   and X :: "'c topology" shows "S retract_of_space X \<equiv> S \<subseteq> topspace X \<and> (\<exists>r. continuous_map X (subtopology X S) r \<and> (\<forall>x\<in>S. r x = x))"
Int_Diff: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "finite A" shows "prod g A = prod g (A \<inter> B) * prod g (A - B)"
max_y_le_take: fixes x :: "nat list"   and j :: "nat"   and k :: "nat" assumes "length x \<le> m" shows "max_y x j \<le> max_y (take k x) j"
step: fixes y :: "'w"   and ss :: "'w list"   and ts :: "'w list" assumes "order.greater (S_length w) (S_length y)"   and "y \<noteq> (0::'w)"   and "S_reduced_for y ss"   and "S_reduced_for y ts" shows "\<exists>xss. flip_altsublist_chain (ss # xss @ [ts])"
node_deliver_messages_empty: shows "trivial_network.node_deliver_messages [] = []"
assms: fixes \<xi> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and a :: "'f"   and \<xi>' :: "'a"   and p' :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e"   and l' :: "'e" assumes "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p), a, \<xi>', p') \<in> automaton.trans A"   and "I a"   and "l \<in> labels \<Gamma> p"   and "l' \<in> labels \<Gamma> p'" shows "P ((\<xi>, l), a, \<xi>', l')"
le_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
finite_set_mset: fixes M :: "'a multiset" shows "finite (set_mset M)"
assms_1: shows "dom (Cp (AllowPortFromTo a b po)) \<noteq> {}"
process_ret_add_match_dist1: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), s\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m (process_ret rs), s\<rangle> \<Rightarrow> t"
hyps_2: fixes bb :: "(name \<times> ty) list"   and b :: "name"   and ba :: "trm" assumes "\<Delta> @ [(x, U)] @ \<Gamma> = bb @ [(b, U)] @ \<Gamma>"   and "\<Gamma> \<turnstile> ba : U" shows "bb @ \<Gamma> \<turnstile> t\<^sub>1[b\<mapsto>ba] : T \<rightarrow> S"
vector_smult_rneg: fixes c :: "'c"   and x :: "('c, 'd) vec" shows "c *s - x = - (c *s x)"
IH_1: fixes N :: "nat"   and k :: "nat" assumes "n \<le> N" shows "n choose k \<le> N choose k"
valid_null_ref_def: fixes r :: "'h option" shows "valid_null_ref r \<equiv> case r of None \<Rightarrow> \<lambda>s. True | Some r' \<Rightarrow> obj_at (\<lambda>s. True) r'"
divisor_sigma_conv_dirichlet_prod: fixes x :: "real"   and n :: "nat" shows "divisor_sigma x n = dirichlet_prod (\<lambda>n. real n powr x) (\<lambda>_. 1) n"
dflt_cmp_def: fixes le :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and lt :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" shows "dflt_cmp le lt a b \<equiv> if lt a b then LESS else if le a b then EQUAL else GREATER"
intro: fixes type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool" assumes "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"   and "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf" shows "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs"
derivation_shift_right_left_cancel: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and r :: "nat" shows "derivation_shift (derivation_shift D 0 r) r 0 = D"
"2_2": shows "exec_plan s as_2 = last slist_2"
s2: shows "0 \<le> root p (real_of_int x)"
op_nres_ASSUME_bnd_def: fixes \<Phi> :: "bool"   and m :: "'b nres" shows "op_nres_ASSUME_bnd \<Phi> m \<equiv> ASSUME \<Phi> \<bind> (\<lambda>_. m)"
diff_iff: fixes f :: "'a \<Rightarrow> 'a" shows "diff k charts charts f = (\<forall>x\<in>carrier. \<exists>c1\<in>atlas. \<exists>c2\<in>atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
coms_skip_inv: fixes p :: "assertion"   and c :: "command" assumes "coms_ass (Rib p) c" obtains "c = Skip"
i: shows "(cms\<^sub>2 ! k, mem\<^sub>2) \<leadsto> \<langle>c\<^sub>2', snd (cms\<^sub>1' ! k), mem\<^sub>2'\<rangle>"
oldUses_correct: fixes n :: "'a"   and g :: "('a, 'b, 'c) gen_cfg_wf" assumes "n \<in> set (gen_wf_\<alpha>n g)" shows "gen_wf_uses g n = gen_wf_var g ` gen_wf.uses' g n"
left_diff_distrib': 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 b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times (minus b c) a = minus (times b a) (times c a)"
ext: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. f x = g x" shows "f = g"
subdivide_simps_2: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b"   and w :: "'b" shows "parcs (subdivide G (u, v) w) = parcs G - {(u, v), (v, u)} \<union> {(u, w), (w, u), (w, v), (v, w)}"
Outside_def: fixes R :: "('a \<times> 'b) set"   and X :: "'a set" shows "R outside X = R - X \<times> Range R"
cpx_vec_length_def: fixes v :: "complex Matrix.vec" shows "\<parallel>v\<parallel> \<equiv> sqrt (\<Sum>i<dim_vec v. (cmod (v $ i))\<^sup>2)"
total_preorder_on_axioms: shows "total_preorder_on carrier le"
listall_cons_eq: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" shows "listall P (x # xs) = (P x \<and> listall P xs)"
insert2_insert: fixes x :: "'a"   and t :: "'a tree"   and t' :: "'a tree"   and n' :: "nat" shows "(insert2 x (t, size t) = (t', n')) = (t' = local.insert x t \<and> n' = size t')"
invariant3p_def: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and I :: "'c set" shows "invariant3p r I = (\<forall>s tl s'. s \<in> I \<longrightarrow> r s tl s' \<longrightarrow> s' \<in> I)"
simps_173: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x81 :: "expr"   and x82 :: "char list"   and x83 :: "char list list" shows "Call x101 x102 x103 x104 \<noteq> x81\<bullet>x82{x83}"
d2: shows "(r|labels \<sigma> @ labels \<tau>'|, r|labels \<sigma>| + r|labels \<tau>|) \<in> mul_eq r"
set_eq_fold: fixes x :: "'a"   and xs :: "'a list" shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>n set (x # xs) = fold (\<squnion>) xs x"
enum_ex_bool_def: fixes P :: "bool \<Rightarrow> bool" shows "enum_ex P = (P False \<or> P True)"
b: fixes x :: "real" assumes "b \<le> x" shows "norm (f x) \<le> c' * norm (g' x)"
adj_mk_symmetric_eq: assumes "symmetric G" shows "parcs (mk_symmetric G) = arcs_ends G"
left_def: fixes tree :: "'b tree" shows "left tree = (case tree of Node x1 x2 x3 \<Rightarrow> x2)"
termi_call_steps_def: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" shows "termi_call_steps \<Gamma> = {((t, q), s, p). \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal s \<and> (\<exists>css. \<Gamma>\<turnstile> ([the (\<Gamma> p)], [], Normal s) \<rightarrow>\<^sup>+ ([the (\<Gamma> q)], css, Normal t))}"
C: fixes x :: "real" assumes "x \<in> {0..1}" shows "norm (P x) \<le> C"
simps_1: fixes x :: "'a" shows "map_aux (basepoint x) = zero wo_aux"
sba3_inf_left_unit: fixes x :: "'a" shows "top \<sqinter> x = x"
arr_iff_in_hom: fixes f :: "'d" shows "D.arr f = \<guillemotleft>f : D.dom f \<Rightarrow>\<^sub>D D.cod f\<guillemotright>"
induct: fixes x :: "'b formula"   and P :: "'b formula \<Rightarrow> bool" assumes "x \<in> AX10"   and "\<And>F. F \<in> AX0 \<Longrightarrow> P F"   and "\<And>F G. P (F \<^bold>\<rightarrow> F \<^bold>\<or> G)"   and "\<And>G F. P (G \<^bold>\<rightarrow> F \<^bold>\<or> G)"   and "\<And>F H G. P ((F \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<or> G) \<^bold>\<rightarrow> H)"   and "\<And>F G. P ((F \<^bold>\<and> G) \<^bold>\<rightarrow> F)"   and "\<And>F G. P ((F \<^bold>\<and> G) \<^bold>\<rightarrow> G)"   and "\<And>F G. P (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F \<^bold>\<and> G)"   and "\<And>F. P ((F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> F)"   and "\<And>F. P (\<^bold>\<not> F \<^bold>\<rightarrow> F \<^bold>\<rightarrow> \<bottom>)"   and "\<And>F. P ((\<^bold>\<not> F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F)" shows "P x"
fold_union_fold_product: fixes X :: "'b ltln fset set" assumes "finite X"   and "\<And>\<Psi> \<psi>. \<lbrakk>\<Psi> \<in> X; \<psi> \<in> fset \<Psi>\<rbrakk> \<Longrightarrow> dnf \<psi> = {{|\<psi>|}}" shows "Finite_Set.fold (\<lambda>x. (\<union>) (Finite_Set.fold (\<lambda>\<phi>. (\<otimes>) (dnf \<phi>)) {{||}} (fset x))) {} X = X"