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+Calculational Proofs
+====================
+
+A calculational proof is just a chain of intermediate results that are
+meant to be composed by basic principles such as the transitivity of
+`=`. In Lean, a calculation proof starts with the keyword `calc`, and has
+the following syntax
+
+        calc  <expr>_0  'op_1'  <expr>_1  ':'  <proof>_1
+                '...'   'op_2'  <expr>_2  ':'  <proof>_2
+          ...
+                '...'   'op_n'  <expr>_n  ':'  <proof>_n
+
+Each `<proof>_i` is a proof for `<expr>_{i-1} op_i <expr>_i`.
+Recall that proofs are also expressions in Lean. The `<proof>_i`
+may also be of the form `{ <pr> }`, where `<pr>` is a proof
+for some equality `a = b`. The form `{ <pr> }` is just syntax sugar
+for
+
+          Subst (Refl <expr>_{i-1}) <pr>
+
+That is, we are claiming we can obtain `<expr>_i` by replacing `a` with `b`
+in `<expr>_{i-1}`.
+
+Here is an example
+
+```lean
+        Variables a b c d e : Nat.
+        Axiom Ax1 : a = b.
+        Axiom Ax2 : b = c + 1.
+        Axiom Ax3 : c = d.
+        Axiom Ax4 : e = 1 + d.
+
+        Theorem T : a = e
+        := calc a    =  b     : Ax1
+                ...  =  c + 1 : Ax2
+                ...  =  d + 1 : { Ax3 }
+                ...  =  1 + d : Nat::PlusComm d 1
+                ...  =  e     : Symm Ax4.
+```
+
+The proof expressions `<proof>_i` do not need to be explicitly provided.
+We can use `by <tactic>` or `by <solver>` to (try to) automatically construct the
+proof expression using the given tactic or solver.
+
+Even when tactics and solvers are not used, we can still use the elaboration engine to fill
+gaps in our calculational proofs. In the previous examples, we can use `_` as arguments for the
+`Nat::PlusComm` theorem. The Lean elaboration engine will infer `d` and `1` for us.
+Here is the same example using placeholders.
+
+```lean
+        Theorem T' : a = e
+        := calc a    =  b     : Ax1
+                ...  =  c + 1 : Ax2
+                ...  =  d + 1 : { Ax3 }
+                ...  =  1 + d : Nat::PlusComm _ _
+                ...  =  e     : Symm Ax4.
+```
+
+We can also use the operators `=>`, `⇒`, `<=>`, `⇔` and `≠` in calculational proofs.
+Here is an example.
+
+```lean
+       Theorem T2 (a b c : Nat) (H1 : a = b) (H2 : b = c + 1) : a ≠ 0
+       := calc  a = b      : H1
+              ... = c + 1  : H2
+              ... ≠ 0      : Nat::SuccNeZero _.
+```
+
+The Lean `let` construct can also be used to build calculational-like proofs.
+
+```lean
+      Variable P : Nat → Nat → Bool.
+      Variable f : Nat → Nat.
+      Axiom Axf (a : Nat) : f (f a) = a.
+
+      Theorem T3 (a b : Nat) (H : P (f (f (f (f a)))) (f (f b))) : P a b
+      := let s1 : P (f (f a)) (f (f b))   :=   Subst H  (Axf a),
+             s2 : P a (f (f b))           :=   Subst s1 (Axf a),
+             s3 : P a b                   :=   Subst s2 (Axf b)
+         in s3.
+```
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