88 lines
2.7 KiB
Org Mode
88 lines
2.7 KiB
Org Mode
* Calculational Proofs
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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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#+BEGIN_SRC
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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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#+END_SRC
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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 =subst <pr> (refl <expr>_{i-1})=
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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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#+BEGIN_SRC lean
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import nat
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using nat
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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 : add_comm d 1
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... = e : symm Ax4.
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#+END_SRC
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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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=add_comm= 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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#+BEGIN_SRC lean
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import nat
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using nat
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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 : add_comm _ _
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... = e : symm Ax4.
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#+END_SRC
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The =calc= command can be configured for any relation that supports
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some form of transitivity. It can even combine different relations.
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#+BEGIN_SRC lean
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import nat
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using nat
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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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... = succ c : add_one _
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... ≠ 0 : succ_ne_zero _
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#+END_SRC
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The Lean simplifier can be used to reduce the size of calculational proofs.
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In the following example, we create a rewrite rule set with basic theorems from the Natural number library, and some of the axioms
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declared above.
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