Tutorial on Isabelle/HOL

!!!!2.6 Some Basic Types 23The symbols > and ≥ are merely syntax: x > y stands for y < x and similaryfor ≥ and ≤.Constant 1::nat is defined to equal Suc 0. This definition (see Sect. 2.7.2) isunfolded automatically by some tactics (like auto, simp and arith) but not byothers (especially the single step tactics in Chapter 5). If you need the full set ofnumerals, see Sect. 8.4.1. Novices are advised to stick to 0 and Suc.Both auto and simp (a method introduced below, Sect. 3.1) prove simplearithmetic goals automatically:lemma "[[ ¬ m < n; m < n + (1::nat) ]] =⇒ m = n"For efficiency’s sake, this built-in prover ignores quantified formulae, manylogical connectives, and all arithmetic operations apart from addition. Inconsequence, auto and simp cannot prove this slightly more complex goal:lemma "m ≠ (n::nat) =⇒ m < n ∨ n < m"The method arith is more general. It attempts to prove the first subgoalprovided it is a linear arithmetic formula. Such formulas may involve theusual logical connectives (¬, ∧, ∨, −→, =, ∀ , ∃ ), the relations =, ≤ and