imports Pure

(* Title: FOL/ex/First_Order_Logic.thy Author: Markus Wenzel, TU Munich *) section {* A simple formulation of First-Order Logic *} theory First_Order_Logic imports Pure begin text {* The subsequent theory development illustrates single-sorted intuitionistic first-order logic with equality, formulated within the Pure framework. Actually this is not an example of Isabelle/FOL, but of Isabelle/Pure. *} subsection {* Syntax *} typedecl i typedecl o judgment Trueprop :: "o => prop" ("_" 5) subsection {* Propositional logic *} axiomatization false :: o ("⊥") and imp :: "o => o => o" (infixr "-->" 25) and conj :: "o => o => o" (infixr "∧" 35) and disj :: "o => o => o" (infixr "∨" 30) where falseE [elim]: "⊥ ==> A" and impI [intro]: "(A ==> B) ==> A --> B" and mp [dest]: "A --> B ==> A ==> B" and conjI [intro]: "A ==> B ==> A ∧ B" and conjD1: "A ∧ B ==> A" and conjD2: "A ∧ B ==> B" and disjE [elim]: "A ∨ B ==> (A ==> C) ==> (B ==> C) ==> C" and disjI1 [intro]: "A ==> A ∨ B" and disjI2 [intro]: "B ==> A ∨ B" theorem conjE [elim]: assumes "A ∧ B" obtains A and B proof from `A ∧ B` show A by (rule conjD1) from `A ∧ B` show B by (rule conjD2) qed definition true :: o ("\<top>") where "\<top> ≡ ⊥ --> ⊥" definition not :: "o => o" ("¬ _" [40] 40) where "¬ A ≡ A --> ⊥" definition iff :: "o => o => o" (infixr "<->" 25) where "A <-> B ≡ (A --> B) ∧ (B --> A)" theorem trueI [intro]: \<top> proof (unfold true_def) show "⊥ --> ⊥" .. qed theorem notI [intro]: "(A ==> ⊥) ==> ¬ A" proof (unfold not_def) assume "A ==> ⊥" then show "A --> ⊥" .. qed theorem notE [elim]: "¬ A ==> A ==> B" proof (unfold not_def) assume "A --> ⊥" and A then have ⊥ .. then show B .. qed theorem iffI [intro]: "(A ==> B) ==> (B ==> A) ==> A <-> B" proof (unfold iff_def) assume "A ==> B" then have "A --> B" .. moreover assume "B ==> A" then have "B --> A" .. ultimately show "(A --> B) ∧ (B --> A)" .. qed theorem iff1 [elim]: "A <-> B ==> A ==> B" proof (unfold iff_def) assume "(A --> B) ∧ (B --> A)" then have "A --> B" .. then show "A ==> B" .. qed theorem iff2 [elim]: "A <-> B ==> B ==> A" proof (unfold iff_def) assume "(A --> B) ∧ (B --> A)" then have "B --> A" .. then show "B ==> A" .. qed subsection {* Equality *} axiomatization equal :: "i => i => o" (infixl "=" 50) where refl [intro]: "x = x" and subst: "x = y ==> P x ==> P y" theorem trans [trans]: "x = y ==> y = z ==> x = z" by (rule subst) theorem sym [sym]: "x = y ==> y = x" proof - assume "x = y" from this and refl show "y = x" by (rule subst) qed subsection {* Quantifiers *} axiomatization All :: "(i => o) => o" (binder "∀" 10) and Ex :: "(i => o) => o" (binder "∃" 10) where allI [intro]: "(!!x. P x) ==> ∀x. P x" and allD [dest]: "∀x. P x ==> P a" and exI [intro]: "P a ==> ∃x. P x" and exE [elim]: "∃x. P x ==> (!!x. P x ==> C) ==> C" lemma "(∃x. P (f x)) --> (∃y. P y)" proof assume "∃x. P (f x)" then show "∃y. P y" proof fix x assume "P (f x)" then show ?thesis .. qed qed lemma "(∃x. ∀y. R x y) --> (∀y. ∃x. R x y)" proof assume "∃x. ∀y. R x y" then show "∀y. ∃x. R x y" proof fix x assume a: "∀y. R x y" show ?thesis proof fix y from a have "R x y" .. then show "∃x. R x y" .. qed qed qed end