hcju/isabelleps huggingface dataset:
greedy_conforming_path_properties_3: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a" assumes "v0 \<in> V"   and "strategy p \<sigma>"   and "strategy p** \<sigma>'" shows "valid_path (greedy_conforming_path p \<sigma> \<sigma>' v0)"
digraphD_6: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
SDG_Def_parent_Def: 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 V :: "'c"   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.SDG_Def sourcenode targetnode kind valid_edge get_proc procs Def ParamDefs V n" shows "V \<in> Def (parent_node n)"
degree_aform_mult_aform': fixes x :: "(real \<times> real pdevs) \<times> real"   and n :: "nat"   and y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "degree_aform_err x \<le> n"   and "degree_aform_err y \<le> n" shows "degree_aform_err (mult_aform' p x y) \<le> n"
rel_with_target_impl_transC_TRel_is_weak_reduction_bisimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_bisimulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "weak_reduction_bisimulation (TRel\<^sup>+) Target"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'g, 'h, 'var, 'c, 'd, 'i) Iteration.model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
seqI': fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and c :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] c\<guillemotright>" shows "A_B.seq g f"
dom_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "ar ar x \<le> ar ar y"
open_UNIV: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "open UNIV"
defs_finite: fixes g :: "'g"   and n :: "'node" shows "finite (defs g n)"
map_upd_Some_unfold: fixes m :: "'c \<Rightarrow> 'b option"   and a :: "'c"   and b :: "'b"   and x :: "'c"   and y :: "'b" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
wf_max: fixes R :: "('c, 'd) ordered_semiring_scheme"   and x :: "'c"   and y :: "'c" assumes "ordered_semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "Max\<^bsub>R\<^esub> x y \<in> carrier R"
weak_group_morphism_range: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set"   and a :: "'a" assumes "weak_group_morphism f H (add_monoid R)"   and "a \<in> carrier R" shows "f ` (H #>\<^bsub>add_monoid R\<^esub> a) = {f a}"
fmlookup_fmrestrict_set_dom: fixes vs :: "'a set"   and s :: "('a, 'b) fmap" shows "dom (fmlookup (fmrestrict_set vs s)) = vs \<inter> fmdom' s"
f_snth: fixes n :: "nat" shows "(\<lambda>x. f x !! n) \<in> N \<rightarrow>\<^sub>M M"
list_update_same_conv: fixes i :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "i < length xs" shows "(xs[i := x] = xs) = (xs ! i = x)"
sorted_descending_split: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list" assumes "sorted_descending (map f l)" shows "\<exists>m n. l = m @ n \<and> (\<forall>e\<in>set m. f (hd l) = f e) \<and> (\<forall>e\<in>set n. f e < f (hd l))"
homotopic_loops_imp_loop: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_loops s p q" shows "pathfinish p = pathstart p \<and> pathfinish q = pathstart q"
hom_closed: fixes x :: "'a" assumes "x \<in> carrier A" shows "i x \<in> carrier G"
eventually_at_bot_linorder: fixes P :: "'b \<Rightarrow> bool" shows "eventually P at_bot = (\<exists>N. \<forall>n\<le>N. P n)"
A_def: shows "A = (\<Sum>i\<le>n. 2 * (poly.coeff p i)\<^sup>2)"
topo_open: fixes t :: "'a set" shows "(t open) = (t \<in> topo A)"
aux: fixes m :: "nat \<Rightarrow> nat set option" shows "fold (\<lambda>(u, v) g. case g u of None \<Rightarrow> g(u \<mapsto> {v}) | Some s \<Rightarrow> g(u \<mapsto> insert v s)) l m = (\<lambda>u. let s = set l `` {u} in if s = {} then m u else Some (the_default {} (m u) \<union> s))"
solve_system_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and b :: "('d, 'f) vec" shows "solve_system A b = (let A' = Gauss_Jordan_PA A in (snd A', fst A' *v b))"
hP: fixes k :: "'a" shows "degree (P k) \<le> p"
real_root_le_1_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x \<le> 1) = (x \<le> 1)"
aff_dim_subspace: fixes S :: "'b set" assumes "subspace S" shows "aff_dim S = int (dim S)"
assms_1: shows "atom ` domA \<Delta> \<sharp>* S"
symD: fixes r :: "('e \<times> 'e) set"   and b :: "'e"   and a :: "'e" assumes "sym r"   and "(b, a) \<in> r" shows "(a, b) \<in> r"
prems_2: shows "t \<notin> ik\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t (A @ [Decomp t'])) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
imgTr9: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and H :: "'e set"   and u :: "'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "F \<guillemotright> H "   and "u \<in> f ` H" shows "\<rho>\<^bsub>G\<^esub> u \<in> f ` H"
order_reflect_poly: fixes c :: "'a"   and p :: "'a poly" assumes "c \<noteq> (0::'a)"   and "constant_term p \<noteq> (0::'a)" shows "order c (reflect_poly p) = order ((1::'a) / c) p"
is_partial_magma_2: shows "partial_magma.null (\<star>\<^sub>D) = D.null"
subsetD: fixes x :: "'a" assumes "x \<in> H" shows "x \<in> E"
flrestrictionD_1: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> A"
replicate_append_same: fixes i :: "nat"   and x :: "'a" shows "x \<up> i @ [x] = x # x \<up> i"
hyps_1: fixes t' :: "'a tree" assumes "delete x l = Some t'" shows "set_of t' \<subseteq> set_of l"
scale_shift_ms_aux_MSLNil_iff: fixes x :: "'a \<times> real"   and F :: "('a \<times> real) msllist" shows "(scale_shift_ms_aux x F = MSLNil) = (F = MSLNil)"
copyp1: fixes x :: "'a"   and y :: "'a"   and k :: "nat" assumes "- (- (x \<squnion> - y) \<squnion> - y) = x" shows "- (y \<squnion> k \<otimes> (x \<squnion> - (x \<squnion> - y))) = - y"
the_inv_into_f_f: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c" assumes "inj_on f A"   and "s2p A x" shows "the_inv_into A f (f x) = x"
B: shows "S (Inf (S' ` X)) = Inf (S ` S' ` X)"
hoare_tripleD_3: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'a" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "relH {a. a < lim h \<and> a \<notin> as} h (the_state \<sigma>)"
Leaf: fixes a :: "'d" shows "Countable.finite_item (Old_Datatype.Leaf a)"
qp_ineq: fixes p :: "nat"   and q :: "nat"   and k :: "int" assumes "QR p q" shows "(int p * k \<le> (int p * int q - 1) div 2) = (k \<le> (int q - 1) div 2)"
dom_fun_upd: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else x \<triangleright> dom f)"
take_drop_CycleI: fixes x :: "'d list"   and L :: "'d list set"   and i :: "nat" assumes "x \<in> L" shows "drop i x @ take i x \<in> Cycle L"
walk_decomp_2: fixes xs :: "'a list"   and ys :: "'a list" assumes "Q.walk (xs @ ys)" shows "Q.walk ys"
Sigma_fm_All2_fresh: fixes A :: "fm"   and i :: "name"   and j :: "name" assumes "Sigma_fm A"   and "i \<noteq> j" shows "Sigma_fm (All2 i (Var j) A)"
gcd_proj2_if_dvd: 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 b :: "'a"   and a :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b a" shows "gcd a b = normalize b"
inducts: fixes x :: "exp"   and P :: "exp \<Rightarrow> bool" assumes "isval x"   and "\<And>n. P (ENat n)"   and "\<And>x e. P (ELam x e)" shows "P x"
0: shows "one_chain_line_integral F basis subdiv = one_chain_line_integral F basis (\<Union> (f ` one_chain1))"
nonneg_Reals_of_nat_I: fixes n :: "nat" shows "of_nat n \<in> \<real>\<^sub>\<ge>\<^sub>0"
le_Sup_iff: fixes A :: "V set"   and x :: "V" assumes "A \<subseteq> ON"   and "Ord x"   and "small A" shows "(x \<le> \<Squnion> A) = (\<forall>y\<in>ON. y < x \<longrightarrow> (\<exists>a\<in>A. y < a))"
remainder_theorem_expression: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "q \<in> carrier P"   and "r \<in> carrier P"   and "carrier R \<noteq> {\<zero>}"   and "f = (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r"   and "deg R r = 0" shows "r = monom P (eval R R id a f) 0"
perm_closed: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list" assumes "Group.monoid G"   and "as <~~> bs"   and "set as \<subseteq> carrier G" shows "set bs \<subseteq> carrier G"
aencrypt_def: fixes \<alpha> :: "'a"   and msg :: "bool list" shows "aencrypt \<alpha> msg = lift_spmf (sample_uniform (order \<G>)) \<bind> (\<lambda>y. hash (\<alpha> [^] y) \<bind> (\<lambda>h. Generative_Probabilistic_Value.Done (\<^bold>g [^] y, h [\<oplus>] msg)))"
simps: fixes k :: "nat"   and i :: "nat"   and j :: "nat" shows "h1 k i j = (if i = j then One else if j \<le> k then Zero else Two)"
isCont_indicator: fixes x :: "'a"   and A :: "'a set" shows "isCont (indicat_real A) x = (x \<notin> frontier A)"
InvariantReasonClausesAfterDecide: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list"   and M' :: "(Literal \<times> bool) list"   and literal :: "Literal" assumes "InvariantReasonClauses F M"   and "InvariantUniq M"   and "M' = M @ [(literal, True)]" shows "InvariantReasonClauses F M'"
Lcm_factorial_def: fixes A :: "'a set" shows "Lcm_factorial A = (if A = {} then 1::'a else if (0::'a) \<notin> A \<and> subset_mset.bdd_above (prime_factorization ` (A - {0::'a})) then normalize (\<Prod>\<^sub># (Sup (prime_factorization ` A))) else (0::'a))"
lookupDef_fst: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)" shows "fst (lookupDef g n v) = v"
sup_Sup: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = \<Squnion> insert x (\<eta> y)"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "Inl x1 \<noteq> Inr x2"
cball_eq_ball_iff: fixes x :: "'b"   and d :: "real"   and y :: "'b"   and e :: "real" shows "(cball x d = ball y e) = (d < 0 \<and> e \<le> 0)"
min_le_iff_disj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.linorder less_eq less" shows "less_eq (ord.min less_eq x y) z = (less_eq x z \<or> less_eq y z)"
LFP_lowerbound: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> carrier L"   and "f x \<sqsubseteq> x" shows "LFP f \<sqsubseteq> x"
rev_image_eqI: fixes x :: "'f"   and A :: "'f set"   and b :: "'g"   and f :: "'f \<Rightarrow> 'g" assumes "x \<in> A"   and "b = f x" shows "b \<in> f ` A"
numsub_def: fixes s :: "Ferrack.num"   and t :: "Ferrack.num" shows "numsub s t = (if s = t then C 0 else numadd s (numneg t))"
length_coeffs_degree: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "length (coeffs p) = Suc (degree p)"
subset_subterms_Union: fixes M :: "('c, 'd) Term.term set" shows "M \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t M"
ntrs_c_cases_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" obtains csp :: "'a list multiset" where "c' = csp + c" and "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"
ID3: fixes f :: "'c \<rightarrow> 'd" shows "ID oo f = f"
laz_swap_ex: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'a list"   and B :: "'b list" assumes "list_all_zip (\<lambda>a b. \<exists>c. P a b c) A B" obtains C :: "'c list" where "list_all_zip (\<lambda>a c. \<exists>b. P a b c) A C" and "list_all_zip (\<lambda>b c. \<exists>a. P a b c) B C"
abs_eq_iff: fixes x :: "'d list"   and y :: "'d list" shows "(abs_dlist x = abs_dlist y) = dlist_eq x y"
invc_baldL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invc2 l"   and "invc r"   and "color r = Black" shows "invc (baldL l a r)"
assms_1: shows "\<not> satisfiable (F @ val2form (oppositeLiteralList c))"
indexed_pmult_index_free: fixes P :: "'c multiset \<Rightarrow> 'a"   and j :: "'c"   and i :: "'c" assumes "index_free P j"   and "i \<noteq> j" shows "index_free (P \<Otimes> i) j"
simps_1: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list" shows "l_sort_inv_1 (x # xs, y # ys, z # zs) = (l_sorted (y # ys) \<and> l_sorted (z # zs) \<and> last (y # ys) \<le> x \<and> last (y # ys) \<le> z)"
successors_predecessors: fixes n :: "'node"   and g :: "'g"   and m :: "'node" assumes "n \<in> set (\<alpha>n g)" shows "(n \<in> set (successors g m)) = (m \<in> set (predecessors g n))"
right_lesseq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a r\<rightarrow> b = (1::'a))"
measure_lborel_singleton: fixes x :: "'a" shows "content {x} = 0"
const_on2: shows "const_on (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (carrier (Fstack as S)) TTree.empty"
intro: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" assumes "carrier R \<noteq> I"   and "\<And>a b. \<lbrakk>a \<in> carrier R; b \<in> carrier R; a \<otimes>\<^bsub>R\<^esub> b \<in> I\<rbrakk> \<Longrightarrow> a \<in> I \<or> b \<in> I" shows "primeideal_axioms I R"
reachable_adjI: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>H\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
left_commute: fixes x :: "'b"   and y :: "'b"   and z :: "'a" assumes "x \<in> B"   and "y \<in> B"   and "z \<in> D" shows "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
cgenideal_minimal: fixes J :: "'a list set"   and a :: "'a list" assumes "ideal J (K [X])"   and "a \<in> J" shows "PIdl\<^bsub>K [X]\<^esub> a \<subseteq> J"
simps_3: fixes x1 :: "'b"   and l1 :: "'b list"   and x2 :: "'b"   and l2 :: "'b list" shows "inter_sorted (x1 # l1) (x2 # l2) = (if x1 < x2 then inter_sorted l1 (x2 # l2) else if x1 = x2 then x1 # inter_sorted l1 l2 else inter_sorted (x1 # l1) l2)"
foldr_map: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and a :: "'b" shows "foldr g (map f xs) a = foldr (g \<circ> f) xs a"
R1_def: fixes input\<^sub>1 :: "bool \<times> bool"   and \<sigma> :: "bool" shows "R1 input\<^sub>1 \<sigma> = (let (b\<^sub>0, b\<^sub>1) = input\<^sub>1 in I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>\<sigma>'. let y\<^sub>\<sigma> = F \<alpha> x\<^sub>\<sigma> in return_spmf ((b\<^sub>0, b\<^sub>1), if \<sigma> then y\<^sub>\<sigma>' else y\<^sub>\<sigma>, if \<sigma> then y\<^sub>\<sigma> else y\<^sub>\<sigma>')))))"
fa_refl: fixes q0 :: "'a"   and Q0 :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" assumes "q0 \<in> Q0" shows "q0 \<in> f_accessible_alt \<delta> Q0"
exhaust: fixes y :: "'b nres" obtains "y = FAILi"   | x2 :: "'b set" where "y = RES x2"
rev_swap: fixes xs :: "'c list"   and ys :: "'c list" shows "(rev xs = ys) = (xs = rev ys)"
inverse_nonnegative_iff_nonnegative: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (inverse a) = less_eq zero a"
store_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and rd :: "word5"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> (fst instr = load_store_type STB \<or> fst instr = load_store_type STH \<or> fst instr = load_store_type ST \<or> fst instr = load_store_type STD) \<and> t1 = snd (fst (store_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (store_sub1 instr rd 0 s2))" shows "low_equal t1 t2"
rtranclp_conjD: fixes r1 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and r2 :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'d"   and x2 :: "'d" assumes "(\<lambda>x1 x2. r1 x1 x2 \<and> r2 x1 x2)\<^sup>*\<^sup>* x1 x2" shows "r1\<^sup>*\<^sup>* x1 x2 \<and> r2\<^sup>*\<^sup>* x1 x2"
dim_span_eq_card_independent: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'c set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "vector_space.dim scale (module.span scale B) = card B"
scale_minus_right: fixes a :: "complex"   and x :: "'b" shows "a *\<^sub>C - x = - (a *\<^sub>C x)"
size_bubble_min: fixes xs :: "'a list" shows "length (bubble_min xs) = length xs"
exhaust: fixes y :: "'b list" obtains "y = \<epsilon>"   | x21 :: "'b" and x22 :: "'b list" where "y = x21 # x22"
unit_disc_fix_moebius_comp: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "unit_disc_fix M1"   and "unit_disc_fix M2" shows "unit_disc_fix (moebius_comp M1 M2)"
domsMT_notND_DD4: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent"   and a :: "(int \<times> int) set set"   and b :: "(int \<times> int) set set"   and c :: "(int \<times> int) set set"   and d :: "(int \<times> int) set set" assumes "x \<in> dom (C (DenyAllFromTo a b))"   and "x \<in> dom (C (DenyAllFromTo c d))" shows "\<not> netsDistinct b d"
augmented_set_minimum: fixes X :: "'a set"   and a :: "'a"   and Y :: "'a set"   and y :: "'a set"   and x :: "'a" assumes "X \<subseteq> carrier D"   and "a \<in> carrier D"   and "Y - {a} \<subseteq> X"   and "y - {a} \<noteq> {}"   and "minimum_elem (Iod D X) (Y - {a}) x"   and "\<forall>x\<in>X. x \<preceq> a" shows "minimum_elem (Iod D (insert a X)) Y x"
prod_rel_def: fixes R1 :: "('a \<times> 'c) set"   and R2 :: "('b \<times> 'd) set" shows "R1 \<times>\<^sub>r R2 \<equiv> {((a, b), a', b'). (a, a') \<in> R1 \<and> (b, b') \<in> R2}"
c: shows "card (set (Gram_Schmidt V)) = vec.dim (set V)"
assms_2: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "P \<leadsto>[X] Q"
independent_bound: fixes S :: "'a set" assumes "independent S" shows "finite S \<and> card S \<le> DIM('a)"
dim_insert: fixes x :: "'a"   and S :: "'a set" shows "dim (insert x S) = (if x \<in> span S then dim S else dim S + 1)"
elims_2: fixes x :: "'a ifex" assumes "ifex_minimal x" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "t \<noteq> e \<and> ifex_minimal t \<and> ifex_minimal e"   | "x = Trueif"   | "x = Falseif"
filtered_prop: fixes X :: "'a set" assumes "X \<noteq> {}"   and "\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y" shows "filtered X"
subgroup_is_submonoid: shows "submonoid (derived G (carrier G)) G"
not_lex_eq: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "(\<not> Counterclockwise_2D_Arbitrary.lex a b) = (Counterclockwise_2D_Arbitrary.lex b a \<and> a \<noteq> b)"
smallest_accepting_rank_properties_3: fixes i :: "nat" assumes "smallest_accepting_rank = Some i" shows "finite (local.merge i)"
exhaust: fixes y :: "B" obtains "y = \<bottom>"   | A :: "A" where "y = B\<cdot>A"
rely_refinement: fixes i :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)" shows "((c::'a::refinement_lattice) // (i::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)) = (c \<sqsubseteq> d \<parallel> i)"
r_i: fixes a :: "'a" assumes "a \<in> carrier G" shows "a \<cdot> \<rho> a = \<one>"
and_mask_bl_take: fixes n :: "nat"   and x :: "bool list" assumes "n \<le> length x" shows "of_bl x && mask n = of_bl (rev (take n (rev x)))"
ord_\<Gamma>_def: shows "gd.ord_\<Gamma> = {uu_. \<exists>CAs DA AAs As E. uu_ = Infer (mset CAs) DA E \<and> gd.ord_resolve CAs DA AAs As E}"
basic_just_sp_axioms_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat" shows "basic_just_sp_axioms G dist c s num \<equiv> \<forall>v. v \<in> verts G \<longrightarrow> v \<noteq> s \<longrightarrow> num v \<noteq> \<infinity> \<longrightarrow> (\<exists>e\<in>arcs G. v = head G e \<and> dist v = dist (tail G e) + ereal (c e) \<and> num v = num (tail G e) + enat 1)"
size_index_conv: fixes xs :: "'c list"   and n :: "nat"   and x :: "'c" assumes "length xs = n" shows "(n = index xs x) = (x \<notin> set xs)"
sim_first: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b" assumes "s \<approx> t" shows "first s = first t"
that: fixes I :: "'a \<Rightarrow> 'c"   and J :: "'c" assumes "\<And>n. (f n has_integral I n) (cbox a b)"   and "(g has_integral J) (cbox a b)"   and "(I \<longlongrightarrow> J) F" shows "thesis"
less_eq_insert2_iff: fixes z :: "hf"   and x :: "hf"   and y :: "hf" shows "(z \<le> y \<triangleleft> x) = (z \<le> y \<or> (\<exists>u. u \<triangleleft> x = z \<and> x \<^bold>\<notin> u \<and> u \<le> y))"
split_list_last_prop_iff: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" shows "(\<exists>x\<in>set xs. P x) = (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z))"