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theory Orderings(* Title: HOL/Orderings.thy ID: $Id: Orderings.thy,v 1.87 2007/11/10 17:36:06 wenzelm Exp $ Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson *) header {* Syntactic and abstract orders *} theory Orderings imports Set Fun uses "~~/src/Provers/order.ML" begin subsection {* Partial orders *} class order = ord + assumes less_le: "x < y <-> x ≤ y ∧ x ≠ y" and order_refl [iff]: "x ≤ x" and order_trans: "x ≤ y ==> y ≤ z ==> x ≤ z" assumes antisym: "x ≤ y ==> y ≤ x ==> x = y" begin text {* Reflexivity. *} lemma eq_refl: "x = y ==> x ≤ y" -- {* This form is useful with the classical reasoner. *} by (erule ssubst) (rule order_refl) lemma less_irrefl [iff]: "¬ x < x" by (simp add: less_le) lemma le_less: "x ≤ y <-> x < y ∨ x = y" -- {* NOT suitable for iff, since it can cause PROOF FAILED. *} by (simp add: less_le) blast lemma le_imp_less_or_eq: "x ≤ y ==> x < y ∨ x = y" unfolding less_le by blast lemma less_imp_le: "x < y ==> x ≤ y" unfolding less_le by blast lemma less_imp_neq: "x < y ==> x ≠ y" by (erule contrapos_pn, erule subst, rule less_irrefl) text {* Useful for simplification, but too risky to include by default. *} lemma less_imp_not_eq: "x < y ==> (x = y) <-> False" by auto lemma less_imp_not_eq2: "x < y ==> (y = x) <-> False" by auto text {* Transitivity rules for calculational reasoning *} lemma neq_le_trans: "a ≠ b ==> a ≤ b ==> a < b" by (simp add: less_le) lemma le_neq_trans: "a ≤ b ==> a ≠ b ==> a < b" by (simp add: less_le) text {* Asymmetry. *} lemma less_not_sym: "x < y ==> ¬ (y < x)" by (simp add: less_le antisym) lemma less_asym: "x < y ==> (¬ P ==> y < x) ==> P" by (drule less_not_sym, erule contrapos_np) simp lemma eq_iff: "x = y <-> x ≤ y ∧ y ≤ x" by (blast intro: antisym) lemma antisym_conv: "y ≤ x ==> x ≤ y <-> x = y" by (blast intro: antisym) lemma less_imp_neq: "x < y ==> x ≠ y" by (erule contrapos_pn, erule subst, rule less_irrefl) text {* Transitivity. *} lemma less_trans: "x < y ==> y < z ==> x < z" by (simp add: less_le) (blast intro: order_trans antisym) lemma le_less_trans: "x ≤ y ==> y < z ==> x < z" by (simp add: less_le) (blast intro: order_trans antisym) lemma less_le_trans: "x < y ==> y ≤ z ==> x < z" by (simp add: less_le) (blast intro: order_trans antisym) text {* Useful for simplification, but too risky to include by default. *} lemma less_imp_not_less: "x < y ==> (¬ y < x) <-> True" by (blast elim: less_asym) lemma less_imp_triv: "x < y ==> (y < x --> P) <-> True" by (blast elim: less_asym) text {* Transitivity rules for calculational reasoning *} lemma less_asym': "a < b ==> b < a ==> P" by (rule less_asym) text {* Reverse order *} lemma order_reverse: "order (op ≥) (op >)" by unfold_locales (simp add: less_le, auto intro: antisym order_trans) end subsection {* Linear (total) orders *} class linorder = order + assumes linear: "x ≤ y ∨ y ≤ x" begin lemma less_linear: "x < y ∨ x = y ∨ y < x" unfolding less_le using less_le linear by blast lemma le_less_linear: "x ≤ y ∨ y < x" by (simp add: le_less less_linear) lemma le_cases [case_names le ge]: "(x ≤ y ==> P) ==> (y ≤ x ==> P) ==> P" using linear by blast lemma linorder_cases [case_names less equal greater]: "(x < y ==> P) ==> (x = y ==> P) ==> (y < x ==> P) ==> P" using less_linear by blast lemma not_less: "¬ x < y <-> y ≤ x" apply (simp add: less_le) using linear apply (blast intro: antisym) done lemma not_less_iff_gr_or_eq: "¬(x < y) <-> (x > y | x = y)" apply(simp add:not_less le_less) apply blast done lemma not_le: "¬ x ≤ y <-> y < x" apply (simp add: less_le) using linear apply (blast intro: antisym) done lemma neq_iff: "x ≠ y <-> x < y ∨ y < x" by (cut_tac x = x and y = y in less_linear, auto) lemma neqE: "x ≠ y ==> (x < y ==> R) ==> (y < x ==> R) ==> R" by (simp add: neq_iff) blast lemma antisym_conv1: "¬ x < y ==> x ≤ y <-> x = y" by (blast intro: antisym dest: not_less [THEN iffD1]) lemma antisym_conv2: "x ≤ y ==> ¬ x < y <-> x = y" by (blast intro: antisym dest: not_less [THEN iffD1]) lemma antisym_conv3: "¬ y < x ==> ¬ x < y <-> x = y" by (blast intro: antisym dest: not_less [THEN iffD1]) text{*Replacing the old Nat.leI*} lemma leI: "¬ x < y ==> y ≤ x" unfolding not_less . lemma leD: "y ≤ x ==> ¬ x < y" unfolding not_less . (*FIXME inappropriate name (or delete altogether)*) lemma not_leE: "¬ y ≤ x ==> x < y" unfolding not_le . text {* Reverse order *} lemma linorder_reverse: "linorder (op ≥) (op >)" by unfold_locales (simp add: less_le, auto intro: antisym order_trans simp add: linear) text {* min/max *} text {* for historic reasons, definitions are done in context ord *} definition (in ord) min :: "'a => 'a => 'a" where [code unfold, code inline del]: "min a b = (if a ≤ b then a else b)" definition (in ord) max :: "'a => 'a => 'a" where [code unfold, code inline del]: "max a b = (if a ≤ b then b else a)" lemma min_le_iff_disj: "min x y ≤ z <-> x ≤ z ∨ y ≤ z" unfolding min_def using linear by (auto intro: order_trans) lemma le_max_iff_disj: "z ≤ max x y <-> z ≤ x ∨ z ≤ y" unfolding max_def using linear by (auto intro: order_trans) lemma min_less_iff_disj: "min x y < z <-> x < z ∨ y < z" unfolding min_def le_less using less_linear by (auto intro: less_trans) lemma less_max_iff_disj: "z < max x y <-> z < x ∨ z < y" unfolding max_def le_less using less_linear by (auto intro: less_trans) lemma min_less_iff_conj [simp]: "z < min x y <-> z < x ∧ z < y" unfolding min_def le_less using less_linear by (auto intro: less_trans) lemma max_less_iff_conj [simp]: "max x y < z <-> x < z ∧ y < z" unfolding max_def le_less using less_linear by (auto intro: less_trans) lemma split_min [noatp]: "P (min i j) <-> (i ≤ j --> P i) ∧ (¬ i ≤ j --> P j)" by (simp add: min_def) lemma split_max [noatp]: "P (max i j) <-> (i ≤ j --> P j) ∧ (¬ i ≤ j --> P i)" by (simp add: max_def) end subsection {* Reasoning tools setup *} ML {* signature ORDERS = sig val print_structures: Proof.context -> unit val setup: theory -> theory val order_tac: thm list -> Proof.context -> int -> tactic end; structure Orders: ORDERS = struct (** Theory and context data **) fun struct_eq ((s1: string, ts1), (s2, ts2)) = (s1 = s2) andalso eq_list (op aconv) (ts1, ts2); structure Data = GenericDataFun ( type T = ((string * term list) * Order_Tac.less_arith) list; (* Order structures: identifier of the structure, list of operations and record of theorems needed to set up the transitivity reasoner, identifier and operations identify the structure uniquely. *) val empty = []; val extend = I; fun merge _ = AList.join struct_eq (K fst); ); fun print_structures ctxt = let val structs = Data.get (Context.Proof ctxt); fun pretty_term t = Pretty.block [Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1, Pretty.str "::", Pretty.brk 1, Pretty.quote (Syntax.pretty_typ ctxt (type_of t))]; fun pretty_struct ((s, ts), _) = Pretty.block [Pretty.str s, Pretty.str ":", Pretty.brk 1, Pretty.enclose "(" ")" (Pretty.breaks (map pretty_term ts))]; in Pretty.writeln (Pretty.big_list "Order structures:" (map pretty_struct structs)) end; (** Method **) fun struct_tac ((s, [eq, le, less]), thms) prems = let fun decomp thy (Trueprop $ t) = let fun excluded t = (* exclude numeric types: linear arithmetic subsumes transitivity *) let val T = type_of t in T = HOLogic.natT orelse T = HOLogic.intT orelse T = HOLogic.realT end; fun rel (bin_op $ t1 $ t2) = if excluded t1 then NONE else if Pattern.matches thy (eq, bin_op) then SOME (t1, "=", t2) else if Pattern.matches thy (le, bin_op) then SOME (t1, "<=", t2) else if Pattern.matches thy (less, bin_op) then SOME (t1, "<", t2) else NONE | rel _ = NONE; fun dec (Const (@{const_name Not}, _) $ t) = (case rel t of NONE => NONE | SOME (t1, rel, t2) => SOME (t1, "~" ^ rel, t2)) | dec x = rel x; in dec t end; in case s of "order" => Order_Tac.partial_tac decomp thms prems | "linorder" => Order_Tac.linear_tac decomp thms prems | _ => error ("Unknown kind of order `" ^ s ^ "' encountered in transitivity reasoner.") end fun order_tac prems ctxt = FIRST' (map (fn s => CHANGED o struct_tac s prems) (Data.get (Context.Proof ctxt))); (** Attribute **) fun add_struct_thm s tag = Thm.declaration_attribute (fn thm => Data.map (AList.map_default struct_eq (s, Order_Tac.empty TrueI) (Order_Tac.update tag thm))); fun del_struct s = Thm.declaration_attribute (fn _ => Data.map (AList.delete struct_eq s)); val attribute = Attrib.syntax (Scan.lift ((Args.add -- Args.name >> (fn (_, s) => SOME s) || Args.del >> K NONE) --| Args.colon (* FIXME || Scan.succeed true *) ) -- Scan.lift Args.name -- Scan.repeat Args.term >> (fn ((SOME tag, n), ts) => add_struct_thm (n, ts) tag | ((NONE, n), ts) => del_struct (n, ts))); (** Diagnostic command **) val print = Toplevel.unknown_context o Toplevel.keep (Toplevel.node_case (Context.cases (print_structures o ProofContext.init) print_structures) (print_structures o Proof.context_of)); val _ = OuterSyntax.improper_command "print_orders" "print order structures available to transitivity reasoner" OuterKeyword.diag (Scan.succeed (Toplevel.no_timing o print)); (** Setup **) val setup = Method.add_methods [("order", Method.ctxt_args (Method.SIMPLE_METHOD' o order_tac []), "transitivity reasoner")] #> Attrib.add_attributes [("order", attribute, "theorems controlling transitivity reasoner")]; end; *} setup Orders.setup text {* Declarations to set up transitivity reasoner of partial and linear orders. *} context order begin (* The type constraint on @{term op =} below is necessary since the operation is not a parameter of the locale. *) lemmas [order add less_reflE: order "op = :: 'a => 'a => bool" "op <=" "op <"] = less_irrefl [THEN notE] lemmas [order add le_refl: order "op = :: 'a => 'a => bool" "op <=" "op <"] = order_refl lemmas [order add less_imp_le: order "op = :: 'a => 'a => bool" "op <=" "op <"] = less_imp_le lemmas [order add eqI: order "op = :: 'a => 'a => bool" "op <=" "op <"] = antisym lemmas [order add eqD1: order "op = :: 'a => 'a => bool" "op <=" "op <"] = eq_refl lemmas [order add eqD2: order "op = :: 'a => 'a => bool" "op <=" "op <"] = sym [THEN eq_refl] lemmas [order add less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = less_trans lemmas [order add less_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = less_le_trans lemmas [order add le_less_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = le_less_trans lemmas [order add le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = order_trans lemmas [order add le_neq_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = le_neq_trans lemmas [order add neq_le_trans: order "op = :: 'a => 'a => bool" "op <=" "op <"] = neq_le_trans lemmas [order add less_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] = less_imp_neq lemmas [order add eq_neq_eq_imp_neq: order "op = :: 'a => 'a => bool" "op <=" "op <"] = eq_neq_eq_imp_neq lemmas [order add not_sym: order "op = :: 'a => 'a => bool" "op <=" "op <"] = not_sym end context linorder begin lemmas [order del: order "op = :: 'a => 'a => bool" "op <=" "op <"] = _ lemmas [order add less_reflE: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = less_irrefl [THEN notE] lemmas [order add le_refl: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = order_refl lemmas [order add less_imp_le: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = less_imp_le lemmas [order add not_lessI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = not_less [THEN iffD2] lemmas [order add not_leI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = not_le [THEN iffD2] lemmas [order add not_lessD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = not_less [THEN iffD1] lemmas [order add not_leD: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = not_le [THEN iffD1] lemmas [order add eqI: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = antisym lemmas [order add eqD1: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = eq_refl lemmas [order add eqD2: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = sym [THEN eq_refl] lemmas [order add less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = less_trans lemmas [order add less_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = less_le_trans lemmas [order add le_less_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = le_less_trans lemmas [order add le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = order_trans lemmas [order add le_neq_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = le_neq_trans lemmas [order add neq_le_trans: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = neq_le_trans lemmas [order add less_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = less_imp_neq lemmas [order add eq_neq_eq_imp_neq: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = eq_neq_eq_imp_neq lemmas [order add not_sym: linorder "op = :: 'a => 'a => bool" "op <=" "op <"] = not_sym end setup {* let fun prp t thm = (#prop (rep_thm thm) = t); fun prove_antisym_le sg ss ((le as Const(_,T)) $ r $ s) = let val prems = prems_of_ss ss; val less = Const (@{const_name less}, T); val t = HOLogic.mk_Trueprop(le $ s $ r); in case find_first (prp t) prems of NONE => let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in case find_first (prp t) prems of NONE => NONE | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv1})) end | SOME thm => SOME(mk_meta_eq(thm RS @{thm order_class.antisym_conv})) end handle THM _ => NONE; fun prove_antisym_less sg ss (NotC $ ((less as Const(_,T)) $ r $ s)) = let val prems = prems_of_ss ss; val le = Const (@{const_name less_eq}, T); val t = HOLogic.mk_Trueprop(le $ r $ s); in case find_first (prp t) prems of NONE => let val t = HOLogic.mk_Trueprop(NotC $ (less $ s $ r)) in case find_first (prp t) prems of NONE => NONE | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})) end | SOME thm => SOME(mk_meta_eq(thm RS @{thm linorder_class.antisym_conv2})) end handle THM _ => NONE; fun add_simprocs procs thy = (Simplifier.change_simpset_of thy (fn ss => ss addsimprocs (map (fn (name, raw_ts, proc) => Simplifier.simproc thy name raw_ts proc)) procs); thy); fun add_solver name tac thy = (Simplifier.change_simpset_of thy (fn ss => ss addSolver (mk_solver' name (fn ss => tac (MetaSimplifier.prems_of_ss ss) (MetaSimplifier.the_context ss)))); thy); in add_simprocs [ ("antisym le", ["(x::'a::order) <= y"], prove_antisym_le), ("antisym less", ["~ (x::'a::linorder) < y"], prove_antisym_less) ] #> add_solver "Transitivity" Orders.order_tac (* Adding the transitivity reasoners also as safe solvers showed a slight speed up, but the reasoning strength appears to be not higher (at least no breaking of additional proofs in the entire HOL distribution, as of 5 March 2004, was observed). *) end *} subsection {* Dense orders *} class dense_linear_order = linorder + assumes gt_ex: "∃y. x < y" and lt_ex: "∃y. y < x" and dense: "x < y ==> (∃z. x < z ∧ z < y)" (*see further theory Dense_Linear_Order*) begin lemma interval_empty_iff: "{y. x < y ∧ y < z} = {} <-> ¬ x < z" by (auto dest: dense) end subsection {* Name duplicates *} lemmas order_less_le = less_le lemmas order_eq_refl = order_class.eq_refl lemmas order_less_irrefl = order_class.less_irrefl lemmas order_le_less = order_class.le_less lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq lemmas order_less_imp_le = order_class.less_imp_le lemmas order_less_imp_not_eq = order_class.less_imp_not_eq lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2 lemmas order_neq_le_trans = order_class.neq_le_trans lemmas order_le_neq_trans = order_class.le_neq_trans lemmas order_antisym = antisym lemmas order_less_not_sym = order_class.less_not_sym lemmas order_less_asym = order_class.less_asym lemmas order_eq_iff = order_class.eq_iff lemmas order_antisym_conv = order_class.antisym_conv lemmas order_less_trans = order_class.less_trans lemmas order_le_less_trans = order_class.le_less_trans lemmas order_less_le_trans = order_class.less_le_trans lemmas order_less_imp_not_less = order_class.less_imp_not_less lemmas order_less_imp_triv = order_class.less_imp_triv lemmas order_less_asym' = order_class.less_asym' lemmas linorder_linear = linear lemmas linorder_less_linear = linorder_class.less_linear lemmas linorder_le_less_linear = linorder_class.le_less_linear lemmas linorder_le_cases = linorder_class.le_cases lemmas linorder_not_less = linorder_class.not_less lemmas linorder_not_le = linorder_class.not_le lemmas linorder_neq_iff = linorder_class.neq_iff lemmas linorder_neqE = linorder_class.neqE lemmas linorder_antisym_conv1 = linorder_class.antisym_conv1 lemmas linorder_antisym_conv2 = linorder_class.antisym_conv2 lemmas linorder_antisym_conv3 = linorder_class.antisym_conv3 subsection {* Bounded quantifiers *} syntax "_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) "_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) "_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) "_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) "_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10) "_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10) "_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10) "_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10) syntax (xsymbols) "_All_less" :: "[idt, 'a, bool] => bool" ("(3∀_<_./ _)" [0, 0, 10] 10) "_Ex_less" :: "[idt, 'a, bool] => bool" ("(3∃_<_./ _)" [0, 0, 10] 10) "_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3∀_≤_./ _)" [0, 0, 10] 10) "_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3∃_≤_./ _)" [0, 0, 10] 10) "_All_greater" :: "[idt, 'a, bool] => bool" ("(3∀_>_./ _)" [0, 0, 10] 10) "_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3∃_>_./ _)" [0, 0, 10] 10) "_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3∀_≥_./ _)" [0, 0, 10] 10) "_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3∃_≥_./ _)" [0, 0, 10] 10) syntax (HOL) "_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) "_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) "_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) "_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) syntax (HTML output) "_All_less" :: "[idt, 'a, bool] => bool" ("(3∀_<_./ _)" [0, 0, 10] 10) "_Ex_less" :: "[idt, 'a, bool] => bool" ("(3∃_<_./ _)" [0, 0, 10] 10) "_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3∀_≤_./ _)" [0, 0, 10] 10) "_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3∃_≤_./ _)" [0, 0, 10] 10) "_All_greater" :: "[idt, 'a, bool] => bool" ("(3∀_>_./ _)" [0, 0, 10] 10) "_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3∃_>_./ _)" [0, 0, 10] 10) "_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3∀_≥_./ _)" [0, 0, 10] 10) "_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3∃_≥_./ _)" [0, 0, 10] 10) translations "ALL x<y. P" => "ALL x. x < y --> P" "EX x<y. P" => "EX x. x < y ∧ P" "ALL x<=y. P" => "ALL x. x <= y --> P" "EX x<=y. P" => "EX x. x <= y ∧ P" "ALL x>y. P" => "ALL x. x > y --> P" "EX x>y. P" => "EX x. x > y ∧ P" "ALL x>=y. P" => "ALL x. x >= y --> P" "EX x>=y. P" => "EX x. x >= y ∧ P" print_translation {* let val All_binder = Syntax.binder_name @{const_syntax All}; val Ex_binder = Syntax.binder_name @{const_syntax Ex}; val impl = @{const_syntax "op -->"}; val conj = @{const_syntax "op &"}; val less = @{const_syntax less}; val less_eq = @{const_syntax less_eq}; val trans = [((All_binder, impl, less), ("_All_less", "_All_greater")), ((All_binder, impl, less_eq), ("_All_less_eq", "_All_greater_eq")), ((Ex_binder, conj, less), ("_Ex_less", "_Ex_greater")), ((Ex_binder, conj, less_eq), ("_Ex_less_eq", "_Ex_greater_eq"))]; fun matches_bound v t = case t of (Const ("_bound", _) $ Free (v', _)) => (v = v') | _ => false fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false) fun mk v c n P = Syntax.const c $ Syntax.mark_bound v $ n $ P fun tr' q = (q, fn [Const ("_bound", _) $ Free (v, _), Const (c, _) $ (Const (d, _) $ t $ u) $ P] => (case AList.lookup (op =) trans (q, c, d) of NONE => raise Match | SOME (l, g) => if matches_bound v t andalso not (contains_var v u) then mk v l u P else if matches_bound v u andalso not (contains_var v t) then mk v g t P else raise Match) | _ => raise Match); in [tr' All_binder, tr' Ex_binder] end *} subsection {* Transitivity reasoning *} context ord begin lemma ord_le_eq_trans: "a ≤ b ==> b = c ==> a ≤ c" by (rule subst) lemma ord_eq_le_trans: "a = b ==> b ≤ c ==> a ≤ c" by (rule ssubst) lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c" by (rule subst) lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c" by (rule ssubst) end lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b < c" finally (order_less_trans) show ?thesis . qed lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_less_trans) show ?thesis . qed lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a < c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b < c" finally (order_le_less_trans) show ?thesis . qed lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a <= f b" also assume "b < c" hence "f b < f c" by (rule r) finally (order_le_less_trans) show ?thesis . qed lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b <= c" finally (order_less_le_trans) show ?thesis . qed lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a < f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a < f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_less_le_trans) show ?thesis . qed lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (order_trans) show ?thesis . qed lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b <= c" finally (order_trans) show ?thesis . qed lemma ord_le_eq_subst: "a <= b ==> f b = c ==> (!!x y. x <= y ==> f x <= f y) ==> f a <= c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a <= b" hence "f a <= f b" by (rule r) also assume "f b = c" finally (ord_le_eq_trans) show ?thesis . qed lemma ord_eq_le_subst: "a = f b ==> b <= c ==> (!!x y. x <= y ==> f x <= f y) ==> a <= f c" proof - assume r: "!!x y. x <= y ==> f x <= f y" assume "a = f b" also assume "b <= c" hence "f b <= f c" by (rule r) finally (ord_eq_le_trans) show ?thesis . qed lemma ord_less_eq_subst: "a < b ==> f b = c ==> (!!x y. x < y ==> f x < f y) ==> f a < c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a < b" hence "f a < f b" by (rule r) also assume "f b = c" finally (ord_less_eq_trans) show ?thesis . qed lemma ord_eq_less_subst: "a = f b ==> b < c ==> (!!x y. x < y ==> f x < f y) ==> a < f c" proof - assume r: "!!x y. x < y ==> f x < f y" assume "a = f b" also assume "b < c" hence "f b < f c" by (rule r) finally (ord_eq_less_trans) show ?thesis . qed text {* Note that this list of rules is in reverse order of priorities. *} lemmas order_trans_rules [trans] = order_less_subst2 order_less_subst1 order_le_less_subst2 order_le_less_subst1 order_less_le_subst2 order_less_le_subst1 order_subst2 order_subst1 ord_le_eq_subst ord_eq_le_subst ord_less_eq_subst ord_eq_less_subst forw_subst back_subst rev_mp mp order_neq_le_trans order_le_neq_trans order_less_trans order_less_asym' order_le_less_trans order_less_le_trans order_trans order_antisym ord_le_eq_trans ord_eq_le_trans ord_less_eq_trans ord_eq_less_trans trans (* FIXME cleanup *) text {* These support proving chains of decreasing inequalities a >= b >= c ... in Isar proofs. *} lemma xt1: "a = b ==> b > c ==> a > c" "a > b ==> b = c ==> a > c" "a = b ==> b >= c ==> a >= c" "a >= b ==> b = c ==> a >= c" "(x::'a::order) >= y ==> y >= x ==> x = y" "(x::'a::order) >= y ==> y >= z ==> x >= z" "(x::'a::order) > y ==> y >= z ==> x > z" "(x::'a::order) >= y ==> y > z ==> x > z" "(a::'a::order) > b ==> b > a ==> P" "(x::'a::order) > y ==> y > z ==> x > z" "(a::'a::order) >= b ==> a ~= b ==> a > b" "(a::'a::order) ~= b ==> a >= b ==> a > b" "a = f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" "a > b ==> f b = c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" "a = f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" "a >= b ==> f b = c ==> (!! x y. x >= y ==> f x >= f y) ==> f a >= c" by auto lemma xt2: "(a::'a::order) >= f b ==> b >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a >= f c" by (subgoal_tac "f b >= f c", force, force) lemma xt3: "(a::'a::order) >= b ==> (f b::'b::order) >= c ==> (!!x y. x >= y ==> f x >= f y) ==> f a >= c" by (subgoal_tac "f a >= f b", force, force) lemma xt4: "(a::'a::order) > f b ==> (b::'b::order) >= c ==> (!!x y. x >= y ==> f x >= f y) ==> a > f c" by (subgoal_tac "f b >= f c", force, force) lemma xt5: "(a::'a::order) > b ==> (f b::'b::order) >= c==> (!!x y. x > y ==> f x > f y) ==> f a > c" by (subgoal_tac "f a > f b", force, force) lemma xt6: "(a::'a::order) >= f b ==> b > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" by (subgoal_tac "f b > f c", force, force) lemma xt7: "(a::'a::order) >= b ==> (f b::'b::order) > c ==> (!!x y. x >= y ==> f x >= f y) ==> f a > c" by (subgoal_tac "f a >= f b", force, force) lemma xt8: "(a::'a::order) > f b ==> (b::'b::order) > c ==> (!!x y. x > y ==> f x > f y) ==> a > f c" by (subgoal_tac "f b > f c", force, force) lemma xt9: "(a::'a::order) > b ==> (f b::'b::order) > c ==> (!!x y. x > y ==> f x > f y) ==> f a > c" by (subgoal_tac "f a > f b", force, force) lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9 (* Since "a >= b" abbreviates "b <= a", the abbreviation "..." stands for the wrong thing in an Isar proof. The extra transitivity rules can be used as follows: lemma "(a::'a::order) > z" proof - have "a >= b" (is "_ >= ?rhs") sorry also have "?rhs >= c" (is "_ >= ?rhs") sorry also (xtrans) have "?rhs = d" (is "_ = ?rhs") sorry also (xtrans) have "?rhs >= e" (is "_ >= ?rhs") sorry also (xtrans) have "?rhs > f" (is "_ > ?rhs") sorry also (xtrans) have "?rhs > z" sorry finally (xtrans) show ?thesis . qed Alternatively, one can use "declare xtrans [trans]" and then leave out the "(xtrans)" above. *) subsection {* Order on bool *} instance bool :: order le_bool_def: "P ≤ Q ≡ P --> Q" less_bool_def: "P < Q ≡ P ≤ Q ∧ P ≠ Q" by intro_classes (auto simp add: le_bool_def less_bool_def) lemmas [code func del] = le_bool_def less_bool_def lemma le_boolI: "(P ==> Q) ==> P ≤ Q" by (simp add: le_bool_def) lemma le_boolI': "P --> Q ==> P ≤ Q" by (simp add: le_bool_def) lemma le_boolE: "P ≤ Q ==> P ==> (Q ==> R) ==> R" by (simp add: le_bool_def) lemma le_boolD: "P ≤ Q ==> P --> Q" by (simp add: le_bool_def) lemma [code func]: "False ≤ b <-> True" "True ≤ b <-> b" "False < b <-> b" "True < b <-> False" unfolding le_bool_def less_bool_def by simp_all subsection {* Order on sets *} instance set :: (type) order by (intro_classes, (assumption | rule subset_refl subset_trans subset_antisym psubset_eq)+) lemmas basic_trans_rules [trans] = order_trans_rules set_rev_mp set_mp subsection {* Order on functions *} instance "fun" :: (type, ord) ord le_fun_def: "f ≤ g ≡ ∀x. f x ≤ g x" less_fun_def: "f < g ≡ f ≤ g ∧ f ≠ g" .. lemmas [code func del] = le_fun_def less_fun_def instance "fun" :: (type, order) order by default (auto simp add: le_fun_def less_fun_def expand_fun_eq intro: order_trans order_antisym) lemma le_funI: "(!!x. f x ≤ g x) ==> f ≤ g" unfolding le_fun_def by simp lemma le_funE: "f ≤ g ==> (f x ≤ g x ==> P) ==> P" unfolding le_fun_def by simp lemma le_funD: "f ≤ g ==> f x ≤ g x" unfolding le_fun_def by simp text {* Handy introduction and elimination rules for @{text "≤"} on unary and binary predicates *} lemma predicate1I [Pure.intro!, intro!]: assumes PQ: "!!x. P x ==> Q x" shows "P ≤ Q" apply (rule le_funI) apply (rule le_boolI) apply (rule PQ) apply assumption done lemma predicate1D [Pure.dest, dest]: "P ≤ Q ==> P x ==> Q x" apply (erule le_funE) apply (erule le_boolE) apply assumption+ done lemma predicate2I [Pure.intro!, intro!]: assumes PQ: "!!x y. P x y ==> Q x y" shows "P ≤ Q" apply (rule le_funI)+ apply (rule le_boolI) apply (rule PQ) apply assumption done lemma predicate2D [Pure.dest, dest]: "P ≤ Q ==> P x y ==> Q x y" apply (erule le_funE)+ apply (erule le_boolE) apply assumption+ done lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" by (rule predicate1D) lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" by (rule predicate2D) subsection {* Monotonicity, least value operator and min/max *} context order begin definition mono :: "('a => 'b::order) => bool" where "mono f <-> (∀x y. x ≤ y --> f x ≤ f y)" lemma monoI [intro?]: fixes f :: "'a => 'b::order" shows "(!!x y. x ≤ y ==> f x ≤ f y) ==> mono f" unfolding mono_def by iprover lemma monoD [dest?]: fixes f :: "'a => 'b::order" shows "mono f ==> x ≤ y ==> f x ≤ f y" unfolding mono_def by iprover end context linorder begin lemma min_of_mono: fixes f :: "'a => 'b::linorder" shows "mono f ==> min (f m) (f n) = f (min m n)" by (auto simp: mono_def Orderings.min_def min_def intro: Orderings.antisym) lemma max_of_mono: fixes f :: "'a => 'b::linorder" shows "mono f ==> max (f m) (f n) = f (max m n)" by (auto simp: mono_def Orderings.max_def max_def intro: Orderings.antisym) end lemma LeastI2_order: "[| P (x::'a::order); !!y. P y ==> x <= y; !!x. [| P x; ALL y. P y --> x ≤ y |] ==> Q x |] ==> Q (Least P)" apply (unfold Least_def) apply (rule theI2) apply (blast intro: order_antisym)+ done lemma Least_mono: "mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y ==> (LEAST y. y : f ` S) = f (LEAST x. x : S)" -- {* Courtesy of Stephan Merz *} apply clarify apply (erule_tac P = "%x. x : S" in LeastI2_order, fast) apply (rule LeastI2_order) apply (auto elim: monoD intro!: order_antisym) done lemma Least_equality: "[| P (k::'a::order); !!x. P x ==> k <= x |] ==> (LEAST x. P x) = k" apply (simp add: Least_def) apply (rule the_equality) apply (auto intro!: order_antisym) done lemma min_leastL: "(!!x. least <= x) ==> min least x = least" by (simp add: min_def) lemma max_leastL: "(!!x. least <= x) ==> max least x = x" by (simp add: max_def) lemma min_leastR: "(!!x::'a::order. least ≤ x) ==> min x least = least" apply (simp add: min_def) apply (blast intro: order_antisym) done lemma max_leastR: "(!!x::'a::order. least ≤ x) ==> max x least = x" apply (simp add: max_def) apply (blast intro: order_antisym) done end
lemma eq_refl:
x = y ==> x ≤ y
lemma less_irrefl:
¬ x < x
lemma le_less:
(x ≤ y) = (x < y ∨ x = y)
lemma le_imp_less_or_eq:
x ≤ y ==> x < y ∨ x = y
lemma less_imp_le:
x < y ==> x ≤ y
lemma less_imp_neq:
x < y ==> x ≠ y
lemma less_imp_not_eq:
x < y ==> (x = y) = False
lemma less_imp_not_eq2:
x < y ==> (y = x) = False
lemma neq_le_trans:
[| a ≠ b; a ≤ b |] ==> a < b
lemma le_neq_trans:
[| a ≤ b; a ≠ b |] ==> a < b
lemma less_not_sym:
x < y ==> ¬ y < x
lemma less_asym:
[| x < y; ¬ P ==> y < x |] ==> P
lemma eq_iff:
(x = y) = (x ≤ y ∧ y ≤ x)
lemma antisym_conv:
y ≤ x ==> (x ≤ y) = (x = y)
lemma less_imp_neq:
x < y ==> x ≠ y
lemma less_trans:
[| x < y; y < z |] ==> x < z
lemma le_less_trans:
[| x ≤ y; y < z |] ==> x < z
lemma less_le_trans:
[| x < y; y ≤ z |] ==> x < z
lemma less_imp_not_less:
x < y ==> (¬ y < x) = True
lemma less_imp_triv:
x < y ==> (y < x --> P) = True
lemma less_asym':
[| a < b; b < a |] ==> P
lemma order_reverse:
order greater_eq greater
lemma less_linear:
x < y ∨ x = y ∨ y < x
lemma le_less_linear:
x ≤ y ∨ y < x
lemma le_cases:
[| x ≤ y ==> P; y ≤ x ==> P |] ==> P
lemma linorder_cases:
[| x < y ==> P; x = y ==> P; y < x ==> P |] ==> P
lemma not_less:
(¬ x < y) = (y ≤ x)
lemma not_less_iff_gr_or_eq:
(¬ x < y) = (y < x ∨ x = y)
lemma not_le:
(¬ x ≤ y) = (y < x)
lemma neq_iff:
(x ≠ y) = (x < y ∨ y < x)
lemma neqE:
[| x ≠ y; x < y ==> R; y < x ==> R |] ==> R
lemma antisym_conv1:
¬ x < y ==> (x ≤ y) = (x = y)
lemma antisym_conv2:
x ≤ y ==> (¬ x < y) = (x = y)
lemma antisym_conv3:
¬ y < x ==> (¬ x < y) = (x = y)
lemma leI:
¬ x < y ==> y ≤ x
lemma leD:
y ≤ x ==> ¬ x < y
lemma not_leE:
¬ y ≤ x ==> x < y
lemma linorder_reverse:
linorder greater_eq greater
lemma min_le_iff_disj:
(min x y ≤ z) = (x ≤ z ∨ y ≤ z)
lemma le_max_iff_disj:
(z ≤ max x y) = (z ≤ x ∨ z ≤ y)
lemma min_less_iff_disj:
(min x y < z) = (x < z ∨ y < z)
lemma less_max_iff_disj:
(z < max x y) = (z < x ∨ z < y)
lemma min_less_iff_conj:
(z < min x y) = (z < x ∧ z < y)
lemma max_less_iff_conj:
(max x y < z) = (x < z ∧ y < z)
lemma split_min:
P (min i j) = ((i ≤ j --> P i) ∧ (¬ i ≤ j --> P j))
lemma split_max:
P (max i j) = ((i ≤ j --> P j) ∧ (¬ i ≤ j --> P i))
lemma
x1 < x1 ==> R
lemma
x ≤ x
lemma
x < y ==> x ≤ y
lemma
[| x ≤ y; y ≤ x |] ==> x = y
lemma
x = y ==> x ≤ y
lemma
y = x ==> x ≤ y
lemma
[| x < y; y < z |] ==> x < z
lemma
[| x < y; y ≤ z |] ==> x < z
lemma
[| x ≤ y; y < z |] ==> x < z
lemma
[| x ≤ y; y ≤ z |] ==> x ≤ z
lemma
[| a ≤ b; a ≠ b |] ==> a < b
lemma
[| a ≠ b; a ≤ b |] ==> a < b
lemma
x < y ==> x ≠ y
lemma
[| x = a; a ≠ b; b = y |] ==> x ≠ y
lemma
t ≠ s ==> s ≠ t
lemma
PROP psi ==> PROP psi
lemma
x1 < x1 ==> R
lemma
x ≤ x
lemma
x < y ==> x ≤ y
lemma
y1 ≤ x1 ==> ¬ x1 < y1
lemma
y1 < x1 ==> ¬ x1 ≤ y1
lemma
¬ x1 < y1 ==> y1 ≤ x1
lemma
¬ x1 ≤ y1 ==> y1 < x1
lemma
[| x ≤ y; y ≤ x |] ==> x = y
lemma
x = y ==> x ≤ y
lemma
y = x ==> x ≤ y
lemma
[| x < y; y < z |] ==> x < z
lemma
[| x < y; y ≤ z |] ==> x < z
lemma
[| x ≤ y; y < z |] ==> x < z
lemma
[| x ≤ y; y ≤ z |] ==> x ≤ z
lemma
[| a ≤ b; a ≠ b |] ==> a < b
lemma
[| a ≠ b; a ≤ b |] ==> a < b
lemma
x < y ==> x ≠ y
lemma
[| x = a; a ≠ b; b = y |] ==> x ≠ y
lemma
t ≠ s ==> s ≠ t
lemma interval_empty_iff:
({y. x < y ∧ y < z} = {}) = (¬ x < z)
lemma order_less_le:
(x < y) = (x ≤ y ∧ x ≠ y)
lemma order_eq_refl:
x = y ==> x ≤ y
lemma order_less_irrefl:
¬ x < x
lemma order_le_less:
(x ≤ y) = (x < y ∨ x = y)
lemma order_le_imp_less_or_eq:
x ≤ y ==> x < y ∨ x = y
lemma order_less_imp_le:
x < y ==> x ≤ y
lemma order_less_imp_not_eq:
x < y ==> (x = y) = False
lemma order_less_imp_not_eq2:
x < y ==> (y = x) = False
lemma order_neq_le_trans:
[| a ≠ b; a ≤ b |] ==> a < b
lemma order_le_neq_trans:
[| a ≤ b; a ≠ b |] ==> a < b
lemma order_antisym:
[| x ≤ y; y ≤ x |] ==> x = y
lemma order_less_not_sym:
x < y ==> ¬ y < x
lemma order_less_asym:
[| x < y; ¬ P ==> y < x |] ==> P
lemma order_eq_iff:
(x = y) = (x ≤ y ∧ y ≤ x)
lemma order_antisym_conv:
y ≤ x ==> (x ≤ y) = (x = y)
lemma order_less_trans:
[| x < y; y < z |] ==> x < z
lemma order_le_less_trans:
[| x ≤ y; y < z |] ==> x < z
lemma order_less_le_trans:
[| x < y; y ≤ z |] ==> x < z
lemma order_less_imp_not_less:
x < y ==> (¬ y < x) = True
lemma order_less_imp_triv:
x < y ==> (y < x --> P) = True
lemma order_less_asym':
[| a < b; b < a |] ==> P
lemma linorder_linear:
x ≤ y ∨ y ≤ x
lemma linorder_less_linear:
x < y ∨ x = y ∨ y < x
lemma linorder_le_less_linear:
x ≤ y ∨ y < x
lemma linorder_le_cases:
[| x ≤ y ==> P; y ≤ x ==> P |] ==> P
lemma linorder_not_less:
(¬ x < y) = (y ≤ x)
lemma linorder_not_le:
(¬ x ≤ y) = (y < x)
lemma linorder_neq_iff:
(x ≠ y) = (x < y ∨ y < x)
lemma linorder_neqE:
[| x ≠ y; x < y ==> R; y < x ==> R |] ==> R
lemma linorder_antisym_conv1:
¬ x < y ==> (x ≤ y) = (x = y)
lemma linorder_antisym_conv2:
x ≤ y ==> (¬ x < y) = (x = y)
lemma linorder_antisym_conv3:
¬ y < x ==> (¬ x < y) = (x = y)
lemma ord_le_eq_trans:
[| a ≤ b; b = c |] ==> a ≤ c
lemma ord_eq_le_trans:
[| a = b; b ≤ c |] ==> a ≤ c
lemma ord_less_eq_trans:
[| a < b; b = c |] ==> a < c
lemma ord_eq_less_trans:
[| a = b; b < c |] ==> a < c
lemma order_less_subst2:
[| a < b; f b < c; !!x y. x < y ==> f x < f y |] ==> f a < c
lemma order_less_subst1:
[| a < f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
lemma order_le_less_subst2:
[| a ≤ b; f b < c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a < c
lemma order_le_less_subst1:
[| a ≤ f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
lemma order_less_le_subst2:
[| a < b; f b ≤ c; !!x y. x < y ==> f x < f y |] ==> f a < c
lemma order_less_le_subst1:
[| a < f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a < f c
lemma order_subst1:
[| a ≤ f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
lemma order_subst2:
[| a ≤ b; f b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
lemma ord_le_eq_subst:
[| a ≤ b; f b = c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
lemma ord_eq_le_subst:
[| a = f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
lemma ord_less_eq_subst:
[| a < b; f b = c; !!x y. x < y ==> f x < f y |] ==> f a < c
lemma ord_eq_less_subst:
[| a = f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
lemma order_trans_rules:
[| a < b; f b < c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a < f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a ≤ b; f b < c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a < c
[| a ≤ f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a < b; f b ≤ c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a < f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a < f c
[| a ≤ b; f b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
[| a ≤ f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
[| a ≤ b; f b = c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
[| a = f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
[| a < b; f b = c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a = f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a = b; P b |] ==> P a
[| P a; a = b |] ==> P b
[| P; P --> Q |] ==> Q
[| P --> Q; P |] ==> Q
[| a ≠ b; a ≤ b |] ==> a < b
[| a ≤ b; a ≠ b |] ==> a < b
[| x < y; y < z |] ==> x < z
[| a < b; b < a |] ==> P
[| x ≤ y; y < z |] ==> x < z
[| x < y; y ≤ z |] ==> x < z
[| x ≤ y; y ≤ z |] ==> x ≤ z
[| x ≤ y; y ≤ x |] ==> x = y
[| a ≤ b; b = c |] ==> a ≤ c
[| a = b; b ≤ c |] ==> a ≤ c
[| a < b; b = c |] ==> a < c
[| a = b; b < c |] ==> a < c
[| r = s; s = t |] ==> r = t
lemma xt1:
[| a = b; c < b |] ==> c < a
[| b < a; b = c |] ==> c < a
[| a = b; c ≤ b |] ==> c ≤ a
[| b ≤ a; b = c |] ==> c ≤ a
[| y ≤ x; x ≤ y |] ==> x = y
[| y ≤ x; z ≤ y |] ==> z ≤ x
[| y < x; z ≤ y |] ==> z < x
[| y ≤ x; z < y |] ==> z < x
[| b < a; a < b |] ==> P
[| y < x; z < y |] ==> z < x
[| b ≤ a; a ≠ b |] ==> b < a
[| a ≠ b; b ≤ a |] ==> b < a
[| a = f b; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
[| b < a; f b = c; !!x y. y < x ==> f y < f x |] ==> c < f a
[| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a
[| b ≤ a; f b = c; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a
lemma xt2:
[| f b ≤ a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a
lemma xt3:
[| b ≤ a; c ≤ f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a
lemma xt4:
[| f b < a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c < a
lemma xt5:
[| b < a; c ≤ f b; !!x y. y < x ==> f y < f x |] ==> c < f a
lemma xt6:
[| f b ≤ a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
lemma xt7:
[| b ≤ a; c < f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c < f a
lemma xt8:
[| f b < a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
lemma xt9:
[| b < a; c < f b; !!x y. y < x ==> f y < f x |] ==> c < f a
lemma xtrans:
[| a = b; c < b |] ==> c < a
[| b < a; b = c |] ==> c < a
[| a = b; c ≤ b |] ==> c ≤ a
[| b ≤ a; b = c |] ==> c ≤ a
[| y ≤ x; x ≤ y |] ==> x = y
[| y ≤ x; z ≤ y |] ==> z ≤ x
[| y < x; z ≤ y |] ==> z < x
[| y ≤ x; z < y |] ==> z < x
[| b < a; a < b |] ==> P
[| y < x; z < y |] ==> z < x
[| b ≤ a; a ≠ b |] ==> b < a
[| a ≠ b; b ≤ a |] ==> b < a
[| a = f b; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
[| b < a; f b = c; !!x y. y < x ==> f y < f x |] ==> c < f a
[| a = f b; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a
[| b ≤ a; f b = c; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a
[| f b ≤ a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c ≤ a
[| b ≤ a; c ≤ f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c ≤ f a
[| f b < a; c ≤ b; !!x y. y ≤ x ==> f y ≤ f x |] ==> f c < a
[| b < a; c ≤ f b; !!x y. y < x ==> f y < f x |] ==> c < f a
[| f b ≤ a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
[| b ≤ a; c < f b; !!x y. y ≤ x ==> f y ≤ f x |] ==> c < f a
[| f b < a; c < b; !!x y. y < x ==> f y < f x |] ==> f c < a
[| b < a; c < f b; !!x y. y < x ==> f y < f x |] ==> c < f a
lemma
P ≤ Q == P --> Q
P < Q == P ≤ Q ∧ P ≠ Q
lemma le_boolI:
(P ==> Q) ==> P ≤ Q
lemma le_boolI':
P --> Q ==> P ≤ Q
lemma le_boolE:
[| P ≤ Q; P; Q ==> R |] ==> R
lemma le_boolD:
P ≤ Q ==> P --> Q
lemma
(False ≤ b) = True
(True ≤ b) = b
(False < b) = b
(True < b) = False
lemma basic_trans_rules:
[| a < b; f b < c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a < f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a ≤ b; f b < c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a < c
[| a ≤ f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a < b; f b ≤ c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a < f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a < f c
[| a ≤ b; f b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
[| a ≤ f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
[| a ≤ b; f b = c; !!x y. x ≤ y ==> f x ≤ f y |] ==> f a ≤ c
[| a = f b; b ≤ c; !!x y. x ≤ y ==> f x ≤ f y |] ==> a ≤ f c
[| a < b; f b = c; !!x y. x < y ==> f x < f y |] ==> f a < c
[| a = f b; b < c; !!x y. x < y ==> f x < f y |] ==> a < f c
[| a = b; P b |] ==> P a
[| P a; a = b |] ==> P b
[| P; P --> Q |] ==> Q
[| P --> Q; P |] ==> Q
[| a ≠ b; a ≤ b |] ==> a < b
[| a ≤ b; a ≠ b |] ==> a < b
[| x < y; y < z |] ==> x < z
[| a < b; b < a |] ==> P
[| x ≤ y; y < z |] ==> x < z
[| x < y; y ≤ z |] ==> x < z
[| x ≤ y; y ≤ z |] ==> x ≤ z
[| x ≤ y; y ≤ x |] ==> x = y
[| a ≤ b; b = c |] ==> a ≤ c
[| a = b; b ≤ c |] ==> a ≤ c
[| a < b; b = c |] ==> a < c
[| a = b; b < c |] ==> a < c
[| r = s; s = t |] ==> r = t
[| x ∈ A; A ⊆ B |] ==> x ∈ B
[| A ⊆ B; x ∈ A |] ==> x ∈ B
lemma
f ≤ g == ∀x. f x ≤ g x
f < g == f ≤ g ∧ f ≠ g
lemma le_funI:
(!!x. f x ≤ g x) ==> f ≤ g
lemma le_funE:
[| f ≤ g; f x ≤ g x ==> P |] ==> P
lemma le_funD:
f ≤ g ==> f x ≤ g x
lemma predicate1I:
(!!x. P x ==> Q x) ==> P ≤ Q
lemma predicate1D:
[| P ≤ Q; P x |] ==> Q x
lemma predicate2I:
(!!x y. P x y ==> Q x y) ==> P ≤ Q
lemma predicate2D:
[| P ≤ Q; P x y |] ==> Q x y
lemma rev_predicate1D:
[| P x; P ≤ Q |] ==> Q x
lemma rev_predicate2D:
[| P x y; P ≤ Q |] ==> Q x y
lemma monoI:
(!!x y. x ≤ y ==> f x ≤ f y) ==> mono f
lemma monoD:
[| mono f; x ≤ y |] ==> f x ≤ f y
lemma min_of_mono:
mono f ==> min (f m) (f n) = f (min m n)
lemma max_of_mono:
mono f ==> max (f m) (f n) = f (max m n)
lemma LeastI2_order:
[| P x; !!y. P y ==> x ≤ y; !!x. [| P x; ∀y. P y --> x ≤ y |] ==> Q x |]
==> Q (Least P)
lemma Least_mono:
[| mono f; ∃x∈S. ∀y∈S. x ≤ y |] ==> (LEAST y. y ∈ f ` S) = f (LEAST x. x ∈ S)
lemma Least_equality:
[| P k; !!x. P x ==> k ≤ x |] ==> (LEAST x. P x) = k
lemma min_leastL:
(!!x. least ≤ x) ==> min least x = least
lemma max_leastL:
(!!x. least ≤ x) ==> max least x = x
lemma min_leastR:
(!!x. least ≤ x) ==> min x least = least
lemma max_leastR:
(!!x. least ≤ x) ==> max x least = x