hcju/isabelleps huggingface dataset:
mertens_third_theorem: shows "(\<lambda>x. (\<Prod>p\<in>{p. prime p \<and> real p \<le> x}. 1 - 1 / real p) - third_mertens_const / ln x) \<in> O(\<lambda>x. 1 / (ln x)\<^sup>2)"
nested_prop_atoms\<^sub>\<nu>_finite: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "finite (nested_prop_atoms\<^sub>\<nu> \<phi> X)"
isPath_bwd_cases: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "cf.isPath s p t" obtains "p = []" and "t = s"   | p' :: "(nat \<times> nat) list" and u :: "nat" where "p = p' @ [(u, t)]" and "cf.isPath s p' u" and "(u, t) \<in> cf.E"
has_bochner_integral_lebesgue_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'c"   and i :: "'c"   and t :: "real" assumes "c \<noteq> 0" shows "has_bochner_integral lebesgue f i = has_bochner_integral lebesgue (\<lambda>x. f (t + c * x)) (i /\<^sub>R \<bar>c\<bar>)"
assms: shows "\<not> (f \<longlongrightarrow> l) F"
is_regular_spairD3_1: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "punit.lt (rep_list q) \<oplus> lt p \<noteq> punit.lt (rep_list p) \<oplus> lt q"
cupcake_nsBind_preserve: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and ns :: "(char list, char list, v) namespace"   and v0 :: "v"   and k :: "char list" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_ns static_cenv ns"   and "cakeml_static_env.is_cupcake_value static_cenv v0" shows "cakeml_static_env.is_cupcake_ns static_cenv (nsBind k v0 ns)"
basis_reduction_step_def: fixes \<alpha> :: "rat"   and n :: "nat"   and m :: "nat"   and upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "LLL_Impl.basis_reduction_step \<alpha> n m upw i state = (if i = 0 then (True, Suc i, inc_state state) else let state' = LLL_Impl.basis_reduction_add_rows n upw i state; di = d_state state' i; dsi = d_state state' (Suc i); dim1 = d_state state' (i - 1); (num, denom) = quotient_of \<alpha> in if di * di * denom \<le> num * dim1 * dsi then (True, Suc i, inc_state state') else LLL_Impl.basis_reduction_swap m i state')"
in_Basis_prod_iff: fixes i :: "'c \<times> 'd" shows "(i \<in> Basis) = (fst i = (0::'c) \<and> snd i \<in> Basis \<or> snd i = (0::'d) \<and> fst i \<in> Basis)"
sup_bot_left: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and bot :: "'d"   and x :: "'d" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup bot x = x"
\<tau>exec_moveE: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>exec_move ci P t e h (stk, loc, pc, xcp) s'" obtains "exec_move ci P t e h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'" and "\<tau>move2 P h stk e pc xcp"
sort_is_sorted: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "all_in_list p l"   and "distinct p"   and "singleCombinators p" shows "FWNormalisationCore.sorted (FWNormalisationCore.sort p l) l"
dfs_induct: fixes S :: "'b"   and xs :: "'a list"   and P :: "'b \<Rightarrow> 'a list \<Rightarrow> bool" assumes "invariant S"   and "list_all isNode xs"   and "\<And>S. invariant S \<Longrightarrow> P S []"   and "\<And>S x xs. \<lbrakk>invariant S; isNode x; list_all isNode xs; memb x S \<Longrightarrow> P S xs; \<not> memb x S \<Longrightarrow> P (ins x S) (succs x @ xs)\<rbrakk> \<Longrightarrow> P S (x # xs)" shows "P S xs"
IH: shows "keys (sum f A) = (\<Union>i\<in>A. keys (f i))"
Un_Int_eq_2: fixes S :: "'d set"   and T :: "'d set" shows "(S \<union> T) \<inter> T = T"
rcm_D: fixes a :: "'a" shows "a r\<rightarrow> a = (1::'a)"
singletons_except_one_def: shows "singletons_except_one = (\<lambda>a'. (\<lambda>a. if a = a' then {a, x} else {a}) ` (A - {x}))"
simps_2: fixes x :: "'e"   and xs :: "'e list"   and ys :: "'e list" shows "(x # xs) @ ys = x # xs @ ys"
bigo_plus_self_subset: fixes f :: "'c \<Rightarrow> 'd" shows "O(f) + O(f) \<subseteq> O(f)"
HInv5_inner_R_def: fixes s :: "state"   and p :: "Proc" shows "HInv5_inner_R s p = (maxBalInp s (bal (dblock s p)) (inp (dblock s p)) \<or> (\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s p) < mbal (disk s d q) \<and> \<not> hasRead s p d q))"
polypoly_polypoly': fixes p :: "'b poly"   and n0 :: "nat"   and x :: "'b"   and bs :: "'b list" assumes "isnpolyh p n0" shows "polypoly (x # bs) p = polypoly' bs p"
fin: shows "finite {k. \<exists>v. map_of xs k = Some v}"
countably_subadditive_subadditive: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "countably_subadditive M f" shows "subadditive M f"
prime_factors_Prod: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite S"   and "\<And>a. a \<in> S \<Longrightarrow> f a \<noteq> (0::'d)" shows "prime_factors (prod f S) = \<Union> (prime_factors ` f ` S)"
min_eq_arg_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'b"   and n :: "'b" assumes "class.linorder less_eq less" shows "(ord.min less_eq m n = m) = less_eq m n"
poly_add_rat_irreducible: fixes p :: "int poly"   and r :: "rat" assumes "irreducible p"   and "degree p \<noteq> 0" shows "irreducible (cf_pos_poly (poly_add_rat r p))"
Un_iff: fixes c :: "'l"   and A :: "'l set"   and B :: "'l set" shows "(c \<in> A \<union> B) = (c \<in> A \<or> c \<in> B)"
Confidentiality_Kas_lemma: fixes authK :: "nat"   and A :: "agent"   and evs :: "event list"   and Ta :: "nat" assumes "authK \<in> symKeys"   and "A \<notin> bad"   and "evs \<in> kerbV"   and "Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace>, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace> \<in> set evs"   and "Key authK \<in> analz (knows Spy evs)" shows "expiredAK Ta evs"
primitive_part_prim: fixes p :: "'a poly" assumes "content p = (1::'a)" shows "primitive_part p = p"
intros_7: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<equiv>\<^sub>s Q \<parallel> P"
cSUP_upper: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "x \<in> A"   and "preorder.bdd_above less_eq (f ` A)" shows "less_eq (f x) (Sup (f ` A))"
finite_measure_compl: fixes S :: "'a set" assumes "S \<in> sets M" shows "Sigma_Algebra.measure M (space M - S) = Sigma_Algebra.measure M (space M) - Sigma_Algebra.measure M S"
invertible_functor_axioms: shows "invertible_functor (\<cdot>) (\<cdot>\<acute>) \<Phi>"
independent_extend_basis: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B :: "'e set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "\<not> module.dependent scale (vector_space.extend_basis scale B)"
C1: fixes Q :: "('a, 'b, 'c) psi" shows "(\<Psi>, Q, Q) \<in> Rel"
prems: shows "\<Psi> \<simeq> \<Psi>'"
leg1_simps_3: shows "src rs.p\<^sub>1 = src rs.p\<^sub>0"
E_TApp: fixes t :: "trm"   and t' :: "trm"   and T :: "POPLmarkRecord.type" assumes "t \<longmapsto> t'" shows "t \<bullet>\<^sub>\<tau> T \<longmapsto> t' \<bullet>\<^sub>\<tau> T"
ivl_disj_int_two_6: fixes l :: "'a"   and m :: "'a"   and u :: "'a" shows "{l<..m} \<inter> {m<..u} = {}"
less_eq_prod_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "less_eq_prod x y \<equiv> fst x \<le> fst y \<and> snd x \<le> snd y"
spr_abs_def: fixes t :: "'s Trace" shows "spr_abs t \<equiv> tLast ` rel_ext (\<lambda>t'. t' \<in> SPR.jkbpC \<and> spr_jview agent t' = spr_jview agent t)"
cs_implies_srtc: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "relation.cs r a b" shows "relation.srtc r a b"
isNullV_def: fixes v :: "Value" shows "isNullV v = (case v of nullV \<Rightarrow> True | _ \<Rightarrow> False)"
simps_3: fixes \<phi> :: "formula"   and \<psi> :: "formula" shows "convert_multiway (And \<phi> \<psi>) = (if safe_assignment (fv \<phi>) \<psi> then And (convert_multiway \<phi>) \<psi> else if safe_formula \<psi> then Ands (get_and_list (convert_multiway \<phi>) @ get_and_list (convert_multiway \<psi>)) else if is_constraint \<psi> then And (convert_multiway \<phi>) \<psi> else Ands (convert_multiway \<psi> # get_and_list (convert_multiway \<phi>)))"
prems: shows "\<forall>j<Suc m. \<exists>no. \<forall>n\<ge>no. P n j"
find_left_rl: fixes i\<^sub>0 :: "nat"   and i :: "nat" assumes "i\<^sub>0 < length xs"   and "i < i\<^sub>0"   and "local.left_spec i\<^sub>0 \<le> Some i" shows "if xs ! i < xs ! i\<^sub>0 then local.left_spec i\<^sub>0 = Some i else local.left_spec i\<^sub>0 \<le> local.left_spec i"
nrename_AndL2_aux: fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL2 v.N u" shows "x = u \<and> x = y \<or> x \<noteq> u"
prefix_le_fin_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item list" assumes "w\<^sub>1 \<le> w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>3"
int1: shows "integrable (measure_spmf (pair_spmf coin_spmf coin_spmf)) (\<lambda>x. spmf (case x of (m0, m1) \<Rightarrow> R1_OT12 (m0, m1) c0 \<bind> (\<lambda>view. A1 view (m0, m1))) True)"
ereal_SUP_uminus: fixes f :: "'a \<Rightarrow> ereal"   and R :: "'a set" shows "(SUP i\<in>R. - f i) = - Inf (f ` R)"
seq'_def: fixes x :: "'a"   and i :: "nat" shows "seq' x i \<equiv> (f ^^ i) x"
minf_le_any: fixes x :: "ant" shows "- \<infinity> \<le> x"
discI: fixes tree :: "('c, 'd) tree"   and x21 :: "('c, 'd) tree"   and x22 :: "'c"   and x23 :: "'d"   and x24 :: "('c, 'd) tree" assumes "tree = Node x21 x22 x23 x24" shows "tree \<noteq> Tip"
is_generic_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c point"   and q :: "'c point" assumes "ell_field R" shows "ell_field.is_generic R p q = (p \<noteq> Infinity \<and> q \<noteq> Infinity \<and> p \<noteq> q \<and> p \<noteq> ell_field.opp R q)"
oexp_add: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>"   and "Ord \<gamma>" shows "\<alpha> \<up> (\<beta> + \<gamma>) = \<alpha> \<up> \<beta> * \<alpha> \<up> \<gamma>"
assms_1: shows "to_nat a \<in> {i..<min (nrows A) (ncols A)}"
fail_reachable_by_sequence_to_failure: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list"   and FAIL :: "'f \<times> 'f"   and PM :: "('d, 'e, 'f \<times> 'f) FSM" assumes "sequence_to_failure M1 M2 io"   and "well_formed M1"   and "well_formed M2"   and "productF M2 M1 FAIL PM" obtains p :: "('f \<times> 'f) list" where "path PM (io || p) (initial PM) \<and> length p = length io \<and> target (io || p) (initial PM) = FAIL"
that: fixes k :: "'a" assumes "k \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys)"   and "aux = f k (OAlist_tc_lookup xs k) (OAlist_tc_lookup ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (list_of_oalist_tc xs) \<union> fst ` set (list_of_oalist_tc ys); k' < k\<rbrakk> \<Longrightarrow> f k' (OAlist_tc_lookup xs k') (OAlist_tc_lookup ys k') = Some Eq" shows "thesis"
set_vs_Un: shows "set vs - A \<union> A = set vs"
axioms_1: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "galois_connection G" shows "connection G"
atMost_Suc: fixes k :: "nat" shows "{..Suc k} = insert (Suc k) {..k}"
isPath_rtc: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "h.cf.isPath u p v" shows "(u, v) \<in> h.cf.E\<^sup>*"
prems: shows "eval_abv\<^sub>C \<langle>Stop, mds\<^sub>C_, mem\<^sub>C_\<rangle>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C"
th3: shows "s @ t setinterleaves ((ta, u), A) \<longrightarrow> (\<forall>t1. t1 \<in> set u \<longrightarrow> t1 \<notin> A)"
rep_eq: fixes x :: "'b ttree"   and xa :: "'b" shows "Rep_ttree (nxt x xa) = insert [] {xs |xs. xa # xs \<in> Rep_ttree x}"
asn_simps_2: fixes P :: "interface"   and Q :: "interface" shows "asn (P \<otimes> Q) = asn P \<star> asn Q"
complete_if_size1_height: fixes t :: "'b tree" assumes "size1 t = 2 ^ height t" shows "Tree.complete t"
simps_5: fixes \<sigma> :: "nat \<Rightarrow> val"   and e1 :: "expr"   and e2 :: "expr" shows "expr_sem_rf \<sigma> (LET e1 IN e2) = expr_sem_rf (case_nat (expr_sem_rf \<sigma> e1) \<sigma>) e2"
complete_ins: fixes t :: "'b tree23"   and a :: "'b" assumes "complete t" shows "complete (treeI (ins a t)) \<and> hI (ins a t) = height t"
simps_5: fixes r :: "'c rexp"   and s :: "'c rexp" shows "final (Plus r s) = (final r \<or> final s)"
loxodromic_repelling_fixed_point_1: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "repelling_fixed_point f \<in> Gromov_boundary"
simps_10: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'a exp" shows "max_vars (C\<bullet>\<^sub>sF{D} := e\<^sub>2) = max_vars e\<^sub>2"
disj_to_dimacs_sound: fixes n_vars :: "nat"   and f :: "nat formula"   and M :: "nat \<Rightarrow> bool" assumes "1 < n_vars"   and "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_disj f"   and "M \<Turnstile> f" shows "\<exists>l\<in>set (model_to_dimacs_model M [1..<n_vars]). l \<in> set (disj_to_dimacs f)"
a: shows "(\<lambda>n. f (u n)) \<longlonglongrightarrow> f x"
linear: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less_eq y x"
u_def: shows "u = (\<lambda>n. SOME z. z \<in> F n \<and> z \<in> A)"
fun_def: fixes tm :: "('a, 'b) tm" shows "fun tm = (case tm of Hd \<zeta> \<Rightarrow> Hd \<zeta> | App x21 x22 \<Rightarrow> x21)"
le_cases: fixes x :: "'a Word.word"   and y :: "'a Word.word" obtains "x \<le>s y"   | "y \<le>s x"
sec_id_pubk_less_def: fixes n :: "nat" shows "sec_id_pubk_less n \<equiv> insert (Spy, \<langle>n, PubKey {Tok_PriK n, Rev_PriK}\<rangle>) (sec_pubk_less n)"
mono_Retr_2: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "Bisim.mono (PL_Indis.ZOretr tval aval indis)"
assms_2: shows "DIM('a) + 2 \<le> card s"
dgrad_p_setD_lp: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "p \<in> dgrad_p_set d m"   and "p \<noteq> 0" shows "d (lp p) \<le> m"
space_embed_measure: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "space (embed_measure M f) = f ` space M"
tangent_space_restrict: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and k :: "enat"   and p :: "'c" assumes "X \<in> manifold_eucl.tangent_space k p" shows "extensional0 (manifold_eucl.diff_fun_space k) X"
weakBisimScopeExtSym: fixes x :: "name"   and \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> \<Psi>"   and "x \<sharp> Q" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(P \<parallel> Q) \<approx> \<lparr>\<nu>x\<rparr>P \<parallel> Q"
exhaust: fixes y :: "'b extended" obtains x1 :: "'b" where "y = Fin x1"   | "y = \<infinity>"   | "y = -\<infinity>"
one_unique: fixes G :: "('c, 'd) monoid_scheme"   and u :: "'c" assumes "Group.monoid G"   and "u \<in> carrier G"   and "\<And>x. x \<in> carrier G \<Longrightarrow> u \<otimes>\<^bsub>G\<^esub> x = x" shows "u = \<one>\<^bsub>G\<^esub>"
abs_payload: fixes x :: "cmsg" shows "(Ab x \<in> payload) = (x \<in> cpayload)"
Arity_exhaust: fixes y :: "Arity" obtains "y = 0"   | x :: "Arity" where "y = inc\<cdot>x"
GoTo: fixes i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and A :: "'b set"   and n :: "nat" assumes "i \<in> branch_nominals branch"   and "A, n \<turnstile> ([], i) # branch" shows "A, Suc n \<turnstile> branch"
distinct_first: fixes w :: "'b list"   and z :: "'b list"   and w' :: "'b list"   and z' :: "'b list" assumes "w \<noteq> \<epsilon>"   and "z \<noteq> \<epsilon>"   and "w ! 0 \<noteq> z ! 0" shows "w \<cdot> w' \<noteq> z \<cdot> z'"
Minimal_Solutions_min: fixes x :: "nat list"   and y :: "nat list"   and u :: "nat list"   and v :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions"   and "u @ v <\<^sub>v x @ y"   and "a \<bullet> u = b \<bullet> v"   and "length u = m"   and "nonzero (u @ v)" shows "False"
res_in_all_clauses: fixes S :: "'b literal set set" shows "res S \<subseteq> all_clauses S"
iv1: shows "2 ^ r \<le> size (tree_to_multiset (Node e1 a1 r ts1))"
Methd: fixes G :: "prog"   and A :: "'b triple set"   and P :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and Q :: "qtname \<Rightarrow> sig \<Rightarrow> 'b assn"   and ms :: "(qtname \<times> sig) set" assumes "G,A \<union> {{P} Methd-\<succ> {Q} | ms}|\<turnstile>{{P} body G-\<succ> {Q} | ms}" shows "G,A|\<turnstile>{{P} Methd-\<succ> {Q} | ms}"
incidence_system_isomorphism_axioms_def: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" shows "incidence_system_isomorphism_axioms \<V> \<B> \<V>' \<B>' bij_map \<equiv> bij_betw bij_map \<V> \<V>' \<and> image_mset ((`) bij_map) \<B> = \<B>'"
diff_pCons: fixes a :: "'b"   and p :: "'b poly"   and b :: "'b"   and q :: "'b poly" shows "pCons a p - pCons b q = pCons (a - b) (p - q)"
assms_1: shows "B \<subseteq> A \<union> C"
acc_sim2: fixes G :: "('a, 'b) gb_graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and ecnv :: "('a, 'b) gb_graph_rec_scheme \<Rightarrow> 'd"   and r :: "nat \<Rightarrow> 'c" assumes "gb_rename_precond G f"   and "gb_graph.is_acc (gb_rename_ext ecnv f G) r" shows "gb_graph.is_acc G (g_rename_precond.fi G f \<circ> r)"
graph_plan_not_eq_last_diff_paths: fixes s :: "('d, 'e) fmap"   and PROB :: "(('d, 'f) fmap \<times> ('d, 'g) fmap) set"   and vs :: "'d set" assumes "s \<in> valid_states PROB" shows "fmrestrict_set vs s \<in> valid_states (prob_proj PROB vs)"
countable_subset: fixes A :: "'f set"   and B :: "'f set" assumes "A \<subseteq> B"   and "countable B" shows "countable A"
3: shows "t = [Suc k..<Suc k + length r]"
open_subset_interior: fixes S :: "'a set"   and T :: "'a set" assumes "open S" shows "(S \<subseteq> interior T) = (S \<subseteq> T)"
arg_space_complete_ps: fixes states :: "('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a + call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr set"   and p :: "lambda" assumes "states \<subseteq> arg_poss p" shows "\<^ps> abs_R\<cdot>states \<subseteq> arg_poss p"
single_valued_has_lubs: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and u :: "'b"   and y :: "'b" assumes "single_valued r"   and "(x, u) \<in> r\<^sup>*"   and "(y, u) \<in> r\<^sup>*" shows "Ex (is_lub (r\<^sup>*) x y)"
nlex_ccw_left: fixes x :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex x 0" shows "ccw 0 (0, 1) x"
conv: fixes as :: "nat list" assumes "k \<notin> set as" shows "concat (map (\<lambda>i. if i \<noteq> k then [unit_vec n i] else []) as) = map (unit_vec n) as"
Union_Pow_eq: fixes A :: "'c set" shows "\<Union> (Pow A) = A"
add_mult_distrib1: fixes x :: "'a"   and y :: "'a"   and a :: "real" assumes "x \<in> V"   and "y \<in> V" shows "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
partList4_complete0: fixes s :: "'b set"   and ts :: "'b set list" assumes "s \<subseteq> \<Union> (set ts)" shows "\<Union> (set (partList4 s ts)) = \<Union> (set ts)"
tendsto_prod': fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F" shows "((\<lambda>x. \<Prod>i\<in>I. f i x) \<longlongrightarrow> prod a I) F"
well_analyzed_split_left_single: fixes A :: "('fun, 'var) extstrand_step list"   and a :: "('fun, 'var) extstrand_step" assumes "well_analyzed (A @ [a])" shows "well_analyzed A"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta>, m) \<in> derivable (ext R R2 M N) \<longrightarrow> ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable (ext R R2 M N)"
inf_conj_product: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) ((a::'b::refinement_lattice) \<sqinter> (b::'b::refinement_lattice)) ((c::'b::refinement_lattice) \<sqinter> (d::'b::refinement_lattice)) = conj a c \<sqinter> conj a d \<sqinter> conj b c \<sqinter> conj b d"
space_as_set: fixes x :: "'a ccsubspace" shows "space_as_set x \<in> {S. closed_csubspace S}"
inducts: fixes x1 :: "('procS, 'procT) Proc"   and x2 :: "('procS, 'procT) Proc"   and P :: "('procS, 'procT) Proc \<Rightarrow> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "indRelLp x1 x2"   and "\<And>S. P (TargetTerm (\<lbrakk>S\<rbrakk>)) (SourceTerm S)" shows "P x1 x2"
numgcd_pos: fixes t :: "MIR.num" shows "0 \<le> numgcd t"
image_transfer: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "((A ===> B) ===> rel_set A ===> rel_set B) (`) (`)"
perm_mem_left: fixes x :: "name"   and pi :: "(name \<times> name) list"   and A :: "name set" assumes "x \<in> pi \<bullet> A" shows "rev pi \<bullet> x \<in> A"
compat_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.compat less_eq less \<equiv> compatible_ordering UNIV less_eq less"
rel_intros_2: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x21 :: "'e"   and y21 :: "'f"   and x22 :: "'e list"   and y22 :: "'f list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
nat_le_iff_add: fixes m :: "nat"   and n :: "nat" shows "(m \<le> n) = (\<exists>k. n = m + k)"
mpoly_map_vars_prod: fixes f :: "nat \<Rightarrow> nat"   and g :: "'c \<Rightarrow> 'b mpoly"   and A :: "'c set" shows "mpoly_map_vars f (prod g A) = (\<Prod>x\<in>A. mpoly_map_vars f (g x))"
asIMOD_igSwap: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b" shows "igSwap (asIMOD SEM) zs z1 z2 X = (\<lambda>val. if sWlsVal SEM val then X (val ^[z1 \<and> z2]_zs) else undefined)"
B: fixes x :: "'a" assumes "x \<in># B" shows "prime x"
map_matrix_def: fixes f :: "'d \<Rightarrow> 'a"   and x :: "(('d, 'b) vec, 'c) vec" shows "map_matrix f x = (\<chi>i j. f (x $ i $ j))"
positive_iff_normalized_vec: fixes A :: "complex mat" shows "positive A = (A \<in> carrier_mat (dim_col A) (dim_col A) \<and> (\<forall>v. dim_vec v = dim_col A \<and> vec_norm v = 1 \<longrightarrow> 0 \<le> inner_prod v (A *\<^sub>v v)))"