1. Type checking and inference
Using the type constraints approach
Question 3 (cont’d): Typing the application (lambda (x) (x x))
STAGE-II: Construct type equations.
The rules:
1. Atomic expressions / primitive-procedures: Construct equations using their types.
2. Lambda expressions:
For (lambda (v vn) e em), construct:
3. Application expressions:
For (f e en), construct:
Expression
Equation
(lambda(x) (x x))
T0 = [Tx → T1 ]
(x x)
Tx = [Tx → T1 ]
Expression
Var
(lambda(x) (x x)) T0
(x x) T1 x Tx

2. Type checking and inference
Using the type constraints approach
Question 3 (cont’d): Typing the application (lambda (x) (x x))
STAGE-III: Solving the equations.
For each equation:
Replace vars by their substituting expressions.
Both sides of the eq. are atomic?
If equal, ignore equation.
Else, output FAIL.
Both sides are different vars?
Apply the equation to the current substitution.
Add the equation to the substitution.
A circular substitution occurred? Output FAIL.
Both side are composite with the same type constructor? Split into equations between corresponding components.
Equation
Substitution
1. T0 = [Tx → T1 ]
2. Tx = [Tx → T1 ]
Equation 1:
Initially, the substitution is empty. Step 1 is ignored.
Eq1 is moved to the substitution.

3. Type checking and inference
Using the type constraints approach
Question 3 (cont’d): Typing the application (lambda (x) (x x))
STAGE-III: Solving the equations.
For each equation:
Replace vars by their substituting expressions.
Both sides of the eq. are atomic?
If equal, ignore equation.
Else, output FAIL.
Both sides are different vars?
Apply the equation to the current substitution.
Add the equation to the substitution.
A circular substitution occurred? Output FAIL.
Both side are composite with the same type constructor? Split into equations between corresponding components.
Equation
Substitution
2. Tx = [Tx → T1 ]
T0 := [Tx → T1 ]
Equation 1:
Initially, the substitution is empty. Step 1 is ignored.
Eq1 is moved to the substitution.

4. Type checking and inference
Using the type constraints approach
Question 3 (cont’d): Typing the application (lambda (x) (x x))
STAGE-III: Solving the equations.
For each equation:
Replace vars by their substituting expressions.
Both sides of the eq. are atomic?
If equal, ignore equation.
Else, output FAIL.
Both sides are different vars?
Apply the equation to the current substitution.
Add the equation to the substitution.
A circular substitution occurred? Output FAIL.
Both side are composite with the same type constructor? Split into equations between corresponding components.
Equation
Substitution
2. Tx = [Tx → T1 ]
T0 := [Tx → T1 ]
Equation 2:
Apply step 1: No change.
Eq2 is moved to the substitution.

5. Type checking and inference
Using the type constraints approach
Question 3 (cont’d): Typing the application (lambda (x) (x x))
STAGE-III: Solving the equations.
For each equation:
Replace vars by their substituting expressions.
Both sides of the eq. are atomic?
If equal, ignore eq.
Else, output FAIL.
Both side are different vars?
Apply the equation to the current substitution.
Add the equation to the substitution.
A circular substitution occurred? Output FAIL.
Both side are composite with the same type constructor? Split into equations between corresponding components.
Equation
Substitution
T0 := [[Tx → T1 ] → T1 ]
Tx := [Tx → T1 ]
Equation 2:
We got a circular substitution: Tx := [Tx → T1 ]