(* Title: HOL/Isar_Examples/Basic_Logic.thy Author: Makarius Basic propositional and quantifier reasoning. *) section ‹Basic logical reasoning› theory Basic_Logic imports Main begin subsection ‹Pure backward reasoning› text ‹ In order to get a first idea of how Isabelle/Isar proof documents may look like, we consider the propositions ‹I›, ‹K›, and ‹S›. The following (rather explicit) proofs should require little extra explanations. › lemma I: "A ⟶ A" proof assume A show A by fact qed lemma K: "A ⟶ B ⟶ A" proof assume A show "B ⟶ A" proof show A by fact qed qed lemma S: "(A ⟶ B ⟶ C) ⟶ (A ⟶ B) ⟶ A ⟶ C" proof assume "A ⟶ B ⟶ C" show "(A ⟶ B) ⟶ A ⟶ C" proof assume "A ⟶ B" show "A ⟶ C" proof assume A show C proof (rule mp) show "B ⟶ C" by (rule mp) fact+ show B by (rule mp) fact+ qed qed qed qed text ‹ Isar provides several ways to fine-tune the reasoning, avoiding excessive detail. Several abbreviated language elements are available, enabling the writer to express proofs in a more concise way, even without referring to any automated proof tools yet. Concluding any (sub-)proof already involves solving any remaining goals by assumption⁋‹This is not a completely trivial operation, as proof by assumption may involve full higher-order unification.›. Thus we may skip the rather vacuous body of the above proof. › lemma "A ⟶ A" proof qed text ‹ Note that the ⬚‹proof› command refers to the ‹rule› method (without arguments) by default. Thus it implicitly applies a single rule, as determined from the syntactic form of the statements involved. The ⬚‹by› command abbreviates any proof with empty body, so the proof may be further pruned. › lemma "A ⟶ A" by rule text ‹ Proof by a single rule may be abbreviated as double-dot. › lemma "A ⟶ A" .. text ‹ Thus we have arrived at an adequate representation of the proof of a tautology that holds by a single standard rule.⁋‹Apparently, the rule here is implication introduction.› ┉ Let us also reconsider ‹K›. Its statement is composed of iterated connectives. Basic decomposition is by a single rule at a time, which is why our first version above was by nesting two proofs. The ‹intro› proof method repeatedly decomposes a goal's conclusion.⁋‹The dual method is ‹elim›, acting on a goal's premises.› › lemma "A ⟶ B ⟶ A" proof (intro impI) assume A show A by fact qed text ‹Again, the body may be collapsed.› lemma "A ⟶ B ⟶ A" by (intro impI) text ‹ Just like ‹rule›, the ‹intro› and ‹elim› proof methods pick standard structural rules, in case no explicit arguments are given. While implicit rules are usually just fine for single rule application, this may go too far with iteration. Thus in practice, ‹intro› and ‹elim› would be typically restricted to certain structures by giving a few rules only, e.g.\ ⬚‹proof (intro impI allI)› to strip implications and universal quantifiers. Such well-tuned iterated decomposition of certain structures is the prime application of ‹intro› and ‹elim›. In contrast, terminal steps that solve a goal completely are usually performed by actual automated proof methods (such as ⬚‹by blast›. › subsection ‹Variations of backward vs.\ forward reasoning› text ‹ Certainly, any proof may be performed in backward-style only. On the other hand, small steps of reasoning are often more naturally expressed in forward-style. Isar supports both backward and forward reasoning as a first-class concept. In order to demonstrate the difference, we consider several proofs of ‹A ∧ B ⟶ B ∧ A›. The first version is purely backward. › lemma "A ∧ B ⟶ B ∧ A" proof assume "A ∧ B" show "B ∧ A" proof show B by (rule conjunct2) fact show A by (rule conjunct1) fact qed qed text ‹ Above, the projection rules ‹conjunct1› / ‹conjunct2› had to be named explicitly, since the goals ‹B› and ‹A› did not provide any structural clue. This may be avoided using ⬚‹from› to focus on the ‹A ∧ B› assumption as the current facts, enabling the use of double-dot proofs. Note that ⬚‹from› already does forward-chaining, involving the ‹conjE› rule here. › lemma "A ∧ B ⟶ B ∧ A" proof assume "A ∧ B" show "B ∧ A" proof from ‹A ∧ B› show B .. from ‹A ∧ B› show A .. qed qed text ‹ In the next version, we move the forward step one level upwards. Forward-chaining from the most recent facts is indicated by the ⬚‹then› command. Thus the proof of ‹B ∧ A› from ‹A ∧ B› actually becomes an elimination, rather than an introduction. The resulting proof structure directly corresponds to that of the ‹conjE› rule, including the repeated goal proposition that is abbreviated as ‹?thesis› below. › lemma "A ∧ B ⟶ B ∧ A" proof assume "A ∧ B" then show "B ∧ A" proof ― ‹rule ‹conjE› of ‹A ∧ B›› assume B A then show ?thesis .. ― ‹rule ‹conjI› of ‹B ∧ A›› qed qed text ‹ In the subsequent version we flatten the structure of the main body by doing forward reasoning all the time. Only the outermost decomposition step is left as backward. › lemma "A ∧ B ⟶ B ∧ A" proof assume "A ∧ B" from ‹A ∧ B› have A .. from ‹A ∧ B› have B .. from ‹B› ‹A› show "B ∧ A" .. qed text ‹ We can still push forward-reasoning a bit further, even at the risk of getting ridiculous. Note that we force the initial proof step to do nothing here, by referring to the ‹-› proof method. › lemma "A ∧ B ⟶ B ∧ A" proof - { assume "A ∧ B" from ‹A ∧ B› have A .. from ‹A ∧ B› have B .. from ‹B› ‹A› have "B ∧ A" .. } then show ?thesis .. ― ‹rule ‹impI›› qed text ‹ ┉ With these examples we have shifted through a whole range from purely backward to purely forward reasoning. Apparently, in the extreme ends we get slightly ill-structured proofs, which also require much explicit naming of either rules (backward) or local facts (forward). The general lesson learned here is that good proof style would achieve just the ∗‹right› balance of top-down backward decomposition, and bottom-up forward composition. In general, there is no single best way to arrange some pieces of formal reasoning, of course. Depending on the actual applications, the intended audience etc., rules (and methods) on the one hand vs.\ facts on the other hand have to be emphasized in an appropriate way. This requires the proof writer to develop good taste, and some practice, of course. ┉ For our example the most appropriate way of reasoning is probably the middle one, with conjunction introduction done after elimination. › lemma "A ∧ B ⟶ B ∧ A" proof assume "A ∧ B" then show "B ∧ A" proof assume B A then show ?thesis .. qed qed subsection ‹A few examples from ``Introduction to Isabelle''› text ‹ We rephrase some of the basic reasoning examples of \<^cite>‹"isabelle-intro"›, using HOL rather than FOL. › subsubsection ‹A propositional proof› text ‹ We consider the proposition ‹P ∨ P ⟶ P›. The proof below involves forward-chaining from ‹P ∨ P›, followed by an explicit case-analysis on the two ∗‹identical› cases. › lemma "P ∨ P ⟶ P" proof assume "P ∨ P" then show P proof ― ‹rule ‹disjE›: \smash{$\infer{‹C›}{‹A ∨ B› & \infer*{‹C›}{[‹A›]} & \infer*{‹C›}{[‹B›]}}$}› assume P show P by fact next assume P show P by fact qed qed text ‹ Case splits are ∗‹not› hardwired into the Isar language as a special feature. The ⬚‹next› command used to separate the cases above is just a short form of managing block structure. ┉ In general, applying proof methods may split up a goal into separate ``cases'', i.e.\ new subgoals with individual local assumptions. The corresponding proof text typically mimics this by establishing results in appropriate contexts, separated by blocks. In order to avoid too much explicit parentheses, the Isar system implicitly opens an additional block for any new goal, the ⬚‹next› statement then closes one block level, opening a new one. The resulting behaviour is what one would expect from separating cases, only that it is more flexible. E.g.\ an induction base case (which does not introduce local assumptions) would ∗‹not› require ⬚‹next› to separate the subsequent step case. ┉ In our example the situation is even simpler, since the two cases actually coincide. Consequently the proof may be rephrased as follows. › lemma "P ∨ P ⟶ P" proof assume "P ∨ P" then show P proof assume P show P by fact show P by fact qed qed text ‹Again, the rather vacuous body of the proof may be collapsed. Thus the case analysis degenerates into two assumption steps, which are implicitly performed when concluding the single rule step of the double-dot proof as follows.› lemma "P ∨ P ⟶ P" proof assume "P ∨ P" then show P .. qed subsubsection ‹A quantifier proof› text ‹ To illustrate quantifier reasoning, let us prove ‹(∃x. P (f x)) ⟶ (∃y. P y)›. Informally, this holds because any ‹a› with ‹P (f a)› may be taken as a witness for the second existential statement. The first proof is rather verbose, exhibiting quite a lot of (redundant) detail. It gives explicit rules, even with some instantiation. Furthermore, we encounter two new language elements: the ⬚‹fix› command augments the context by some new ``arbitrary, but fixed'' element; the ⬚‹is› annotation binds term abbreviations by higher-order pattern matching. › lemma "(∃x. P (f x)) ⟶ (∃y. P y)" proof assume "∃x. P (f x)" then show "∃y. P y" proof (rule exE) ― ‹rule ‹exE›: \smash{$\infer{‹B›}{‹∃x. A(x)› & \infer*{‹B›}{[‹A(x)›]_x}}$}› fix a assume "P (f a)" (is "P ?witness") then show ?thesis by (rule exI [of P ?witness]) qed qed text ‹ While explicit rule instantiation may occasionally improve readability of certain aspects of reasoning, it is usually quite redundant. Above, the basic proof outline gives already enough structural clues for the system to infer both the rules and their instances (by higher-order unification). Thus we may as well prune the text as follows. › lemma "(∃x. P (f x)) ⟶ (∃y. P y)" proof assume "∃x. P (f x)" then show "∃y. P y" proof fix a assume "P (f a)" then show ?thesis .. qed qed text ‹ Explicit ‹∃›-elimination as seen above can become quite cumbersome in practice. The derived Isar language element ``⬚‹obtain›'' provides a more handsome way to do generalized existence reasoning. › lemma "(∃x. P (f x)) ⟶ (∃y. P y)" proof assume "∃x. P (f x)" then obtain a where "P (f a)" .. then show "∃y. P y" .. qed text ‹ Technically, ⬚‹obtain› is similar to ⬚‹fix› and ⬚‹assume› together with a soundness proof of the elimination involved. Thus it behaves similar to any other forward proof element. Also note that due to the nature of general existence reasoning involved here, any result exported from the context of an ⬚‹obtain› statement may ∗‹not› refer to the parameters introduced there. › subsubsection ‹Deriving rules in Isabelle› text ‹ We derive the conjunction elimination rule from the corresponding projections. The proof is quite straight-forward, since Isabelle/Isar supports non-atomic goals and assumptions fully transparently. › theorem conjE: "A ∧ B ⟹ (A ⟹ B ⟹ C) ⟹ C" proof - assume "A ∧ B" assume r: "A ⟹ B ⟹ C" show C proof (rule r) show A by (rule conjunct1) fact show B by (rule conjunct2) fact qed qed end