82 lines
No EOL
2.8 KiB
Markdown
82 lines
No EOL
2.8 KiB
Markdown
Calculational Proofs
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====================
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A calculational proof is just a chain of intermediate results that are
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meant to be composed by basic principles such as the transitivity of
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`=`. In Lean, a calculation proof starts with the keyword `calc`, and has
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the following syntax
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calc <expr>_0 'op_1' <expr>_1 ':' <proof>_1
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'...' 'op_2' <expr>_2 ':' <proof>_2
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...
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'...' 'op_n' <expr>_n ':' <proof>_n
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Each `<proof>_i` is a proof for `<expr>_{i-1} op_i <expr>_i`.
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Recall that proofs are also expressions in Lean. The `<proof>_i`
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may also be of the form `{ <pr> }`, where `<pr>` is a proof
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for some equality `a = b`. The form `{ <pr> }` is just syntax sugar
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for
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Subst (Refl <expr>_{i-1}) <pr>
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That is, we are claiming we can obtain `<expr>_i` by replacing `a` with `b`
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in `<expr>_{i-1}`.
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Here is an example
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```lean
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Variables a b c d e : Nat.
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Axiom Ax1 : a = b.
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Axiom Ax2 : b = c + 1.
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Axiom Ax3 : c = d.
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Axiom Ax4 : e = 1 + d.
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Theorem T : a = e
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:= calc a = b : Ax1
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... = c + 1 : Ax2
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... = d + 1 : { Ax3 }
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... = 1 + d : Nat::PlusComm d 1
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... = e : Symm Ax4.
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```
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The proof expressions `<proof>_i` do not need to be explicitly provided.
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We can use `by <tactic>` or `by <solver>` to (try to) automatically construct the
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proof expression using the given tactic or solver.
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Even when tactics and solvers are not used, we can still use the elaboration engine to fill
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gaps in our calculational proofs. In the previous examples, we can use `_` as arguments for the
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`Nat::PlusComm` theorem. The Lean elaboration engine will infer `d` and `1` for us.
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Here is the same example using placeholders.
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```lean
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Theorem T' : a = e
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:= calc a = b : Ax1
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... = c + 1 : Ax2
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... = d + 1 : { Ax3 }
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... = 1 + d : Nat::PlusComm _ _
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... = e : Symm Ax4.
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```
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We can also use the operators `=>`, `⇒`, `<=>`, `⇔` and `≠` in calculational proofs.
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Here is an example.
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```lean
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Theorem T2 (a b c : Nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0
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:= calc a = b : H1
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... = c + 1 : H2
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... ≠ 0 : Nat::SuccNeZero _.
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```
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The Lean `let` construct can also be used to build calculational-like proofs.
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```lean
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Variable P : Nat → Nat → Bool.
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Variable f : Nat → Nat.
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Axiom Axf (a : Nat) : f (f a) = a.
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Theorem T3 (a b : Nat) (H : P (f (f (f (f a)))) (f (f b))) : P a b
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:= let s1 : P (f (f a)) (f (f b)) := Subst H (Axf a),
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s2 : P a (f (f b)) := Subst s1 (Axf a),
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s3 : P a b := Subst s2 (Axf b)
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in s3.
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``` |