This document summarizes and discusses type checking algorithms for programming languages. It introduces constraint-based type checking, which separates type checking into constraint generation and constraint solving. This provides a more declarative way to specify type checkers. The document discusses using variables and constraints to represent types during type checking. It introduces NaBL2, a domain-specific language for writing constraint generators to specify name and type constraints for programming language static semantics. NaBL2 uses scope graphs to represent name binding structures and supports features like type equality, subtyping, and type-dependent name resolution through constraint rules. An example scope graph and constraint rule for let-bindings is provided.

1. Hendrik van Antwerpen
IN4303 Compiler Construction
TU Delft
September 2017
Declare Your Language
Chapter 6: Name & Type Constraints

2. Reading Material
2

3. 3
Type checkers are algorithms that
check names and types in programs.
This paper introduces the NaBL Name
Binding Language, which supports the
declarative definition of the name
binding and scope rules of
programming languages through the
definition of scopes, declarations,
references, and imports associated.
http://dx.doi.org/10.1007/978-3-642-36089-3_18
SLE 2012
Background

4. 4
This paper introduces scope graphs as a
language-independent representation for
the binding information in programs.
http://dx.doi.org/10.1007/978-3-662-46669-8_9
ESOP 2015

5. 5
Separating type checking into
constraint generation and
constraint solving provides more
declarative definition of type
checkers. This paper introduces a
constraint language integrating
name resolution into constraint
resolution through scope graph
constraints.
This is the basis for the design of
the NaBL2 static semantics
specification language.
https://doi.org/10.1145/2847538.2847543
PEPM 2016

6. 6
Documentation for NaBL2 at
the metaborg.org website.
http://www.metaborg.org/en/latest/source/langdev/meta/lang/nabl2/index.html

7. Types and Type Checking
7

8. 8
Source
Code
Editor
Parse
Abstract
Syntax
Tree
Type Check
Check that names are used correctly and that expressions are well-typed
Errors

9. Goal of type checking
- Prove the absence of certain (wrong) runtime behavior
‣ “Well-typed programs cannot go wrong.” [Reynolds1985]
- Check correct usage of names, and types of expressions
- Beyond catching runtime errors
‣ How many resources are use?
‣ Do calculations respect units?
‣ Is any sensitive information leaked through this function?
Types influence language design
- Types abstract over implementation
‣ Any value with the correct type is accepted
- Types enable separate or incremental compilation
‣ As long as the public interface is implemented, using modules do not change
Why types?
9

10. Many different type system features
- Simple types
- Sub-typing
- Nominal versus structural types
- Ad-hoc polymorphism (overloading)
- Parametric polymorphism (generics)
- Dependent types (vector with length)
- Units of numbers (meters vs seconds)
Feature interaction is not trivial!
Why types?
10

11. Type system specification
- (Formal) description of the typing rules
- For human comprehension
- To prove properties
Type checking algorithm
- Executable version of the specification
- Often contains details that are omitted from the specification
Developing both is a lot of work, often only an algorithm
Type system of a language
11

12. What should a type checker do?
- Resolve names, and check or compute types
- Report useful error messages
- Provide a representation of name and type information
‣ Type annotated AST
‣ Scope graphs for name binding structure
This information is used for
- Next compiler steps (optimization, code generation, …)
- IDE (error reporting, code completion, refactoring, …)
- Other tools (API documentation, …)
How are type checkers implemented?
- Algorithms for specific languages / feature combinations
- Common elements (unification, constraints)
How to check types?
12

13. Type Checking
with Constraints
13

14. Two phase approach
- First record constraints (type equality, name resolution)
- Then solve constraints
Separation of concern
- Language specific specification in terms of constraints
- Language independent algorithm to solve constraints
- Write specification, get an executable checker
Advantages
- Order of solving independent of order of collection
- Natural support for inference
- Many constraint-based formulations of typing features exist
- Constraint variables act as interface between different kinds of constraints
Combining different kinds of constraints is still non-trivial!
Constraint-based type checking
14

15. Computing Type of Expression (recap)
15
function (a : int) = a + 1
rules
type-check(|env):
Fun(x, t, e) -> FUN(t, t')
where <type-check(|[(x, t) | env])> e => t'
type-check(|env):
Var(x) -> t
where <lookup> (x, env) => t
type-check(|env):
Int(_) -> INT()
type-check(|env):
Plus(e1, e2) -> INT()
where <type-check(|env)> e1 => INT()
where <type-check(|env)> e2 => INT()
Fun("i", INT(),
Plus(Var("i"), Int(1)))
FUN(INT(), INT())

16. rules
type-check(|env):
Fun(x, e) -> FUN(??, t')
where <type-check(|[(x, ??) | env])> e => t'
type-check(|env):
Var(x) -> t
where <lookup> (x, env) => t
type-check(|env):
Int(_) -> INT()
type-check(|env):
Plus(e1, e2) -> INT()
where <type-check(|env)> e1 => INT()
where <type-check(|env)> e2 => INT()
Inferring Type of Parameter?
16
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(??, INT())

17. rules
type-check(|env):
Fun(x, e) -> FUN(??, t')
where <type-check(|[(x, ??) | env])> e => t'
type-check(|env):
Var(x) -> t
where <lookup> (x, env) => t
type-check(|env):
Int(_) -> INT()
type-check(|env):
Plus(e1, e2) -> INT()
where <type-check(|env)> e1 => INT()
where <type-check(|env)> e2 => INT()
Inferring Type of Parameter?
16
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(??, INT())

18. rules
type-check(|env):
Fun(x, e) -> FUN(??, t')
where <type-check(|[(x, ??) | env])> e => t'
type-check(|env):
Var(x) -> t
where <lookup> (x, env) => t
type-check(|env):
Int(_) -> INT()
type-check(|env):
Plus(e1, e2) -> INT()
where <type-check(|env)> e1 => INT()
where <type-check(|env)> e2 => INT()
Inferring Type of Parameter?
16
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(??, INT())

19. rules
type-check(|env):
Fun(x, e) -> FUN(??, t')
where <type-check(|[(x, ??) | env])> e => t'
type-check(|env):
Var(x) -> t
where <lookup> (x, env) => t
type-check(|env):
Int(_) -> INT()
type-check(|env):
Plus(e1, e2) -> INT()
where <type-check(|env)> e1 => INT()
where <type-check(|env)> e2 => INT()
Inferring Type of Parameter?
16
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(??, INT())

20. Use Variables and Constraints
17
Eq(VAR("a"), INT())
Eq(INT(), INT())
+
VAR("a") => INT()
rules
type-con(|env):
Fun(x, e) -> (FUN(t, t'), C)
where !VAR(<fresh>) => t;
<type-con(|[(x, t) | env])> e => (t', C)
type-con(|env):
Var(x) -> (t, [])
where <lookup> (x, env) => t
type-con(|env):
Int(_) -> (INT(), [])
type-con(|env):
Plus(e1, e2) -> (INT(), [ C1, C2,
Eq(t1, INT()),
Eq(t2, INT()) ])
where <type-con(|env)> e1 => (t1, C2)
where <type-con(|env)> e2 => (t2, C1)
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(VAR("a"), INT())

21. Use Variables and Constraints
17
Eq(VAR("a"), INT())
Eq(INT(), INT())
+
VAR("a") => INT()
rules
type-con(|env):
Fun(x, e) -> (FUN(t, t'), C)
where !VAR(<fresh>) => t;
<type-con(|[(x, t) | env])> e => (t', C)
type-con(|env):
Var(x) -> (t, [])
where <lookup> (x, env) => t
type-con(|env):
Int(_) -> (INT(), [])
type-con(|env):
Plus(e1, e2) -> (INT(), [ C1, C2,
Eq(t1, INT()),
Eq(t2, INT()) ])
where <type-con(|env)> e1 => (t1, C2)
where <type-con(|env)> e2 => (t2, C1)
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(VAR("a"), INT())

22. Use Variables and Constraints
17
Eq(VAR("a"), INT())
Eq(INT(), INT())
+
VAR("a") => INT()
rules
type-con(|env):
Fun(x, e) -> (FUN(t, t'), C)
where !VAR(<fresh>) => t;
<type-con(|[(x, t) | env])> e => (t', C)
type-con(|env):
Var(x) -> (t, [])
where <lookup> (x, env) => t
type-con(|env):
Int(_) -> (INT(), [])
type-con(|env):
Plus(e1, e2) -> (INT(), [ C1, C2,
Eq(t1, INT()),
Eq(t2, INT()) ])
where <type-con(|env)> e1 => (t1, C2)
where <type-con(|env)> e2 => (t2, C1)
function (a) = a + 1
Fun("i", Plus(Var("i"), Int(1)))
FUN(VAR("a"), INT())

23. 18
Source
Code
Editor
Parse
Abstract
Syntax
Tree
Type Check
Check that names are used correctly and that expressions are well-typed
Errors

24. 19
Source
Code
Editor
Abstract
Syntax
Tree
ErrorsCollect Constraints Solve
Type checking proceeds in two steps
Parse

25. 19
Source
Code
Editor
Abstract
Syntax
Tree
ErrorsCollect Constraints Solve
Type checking proceeds in two steps
language
specific
Parse

26. 19
Source
Code
Editor
Abstract
Syntax
Tree
ErrorsCollect Constraints Solve
Type checking proceeds in two steps
language
specific
language
independent
Parse

27. What is NaBL2?
- Domain-specific specification language…
- … to write constraint generators
- Comes with a generic solver
What features does it support?
- Rich binding structures using scope graphs
- Type equality and nominal sub-typing
- Type-dependent name resolution
Limitations
- Restricted to the domain-specific (= restricted) model
‣ Not all name binding patterns in the wild can be expressed
- Hypothesis is that all sensible patterns are expressible
NaBL2
20

28. [[ Let(x, e, e') ^ (s) : t' ]] :=
new s', s' -P-> s,
Var{x} <- s', Var{x} : v,
[[ e ^ (s') : t ]],
v == t,
[[ e' ^ (s') : t' ]].
Constraint Rules
21
match pattern scope parameters type (optional)
subterm recursion
constraints

29. Scope Graph Constraints
22
new s // create a new scope
s1 -L-> s2 // edge labeled with L from scope s1 to scope s2
N{x} <- s // x is a declaration in scope s for namespace N
N{x} -> s // x is a reference in scope s for namespace N
N{x} =L=> s // declaration x is associated with scope s
N{x} <=L= s // scope s imports via reference x
N{x} |-> d // reference x resolves to declaration d
[[ e ^ (s) ]] // subterm e is scoped by scoped s
s
s s
N xis
N xi s
N xi s
N xis
N xi N xi

30. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end

31. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)

32. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)

33. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=

34. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_body
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope

35. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_body
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
P

36. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body P

37. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
P

38. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
P

39. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
P

40. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
P

41. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
x
P

42. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
x
P

43. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
x
?
P

44. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
x
?
P

45. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
x
?
P

46. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
?
P

47. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
?
P

48. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
x
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
?
P

49. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
x
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
?
P

50. Scope Graph Example
23
let
var x : int := x + 1
in
x + 1
end
s
s_bodyx
x
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
[[ e ^ (s) ]], // init expression
[[ e_body ^ (s_body) ]]. // body expression
[[ Var(x) ^ (s') ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
?
P

51. Name Set Constraints & Expressions
24
distinct S distinct/name S // elements/names in S are distinct
S1 subseteq S2 S1 subseteq/name S2 // elements/names in S1 are a subset of S2
S1 seteq S2 S1 seteq/name S2 // elements/names in S1 are equal to S2
0 // empty set
D(s) D(s)/N // all/namespace N declarations in s
R(s) R(s)/N // all/namespace N references in s
V(s) V(s)/N // visible declarations in s (w/ shadowing)
W(s) W(s)/N // reachable declarations in s (no shadowing)
(S1 union S2) // union of S1 and S2
(S1 isect S2) (S1 isect/name S2) // intersection of elements/names S1 and S2
(S1 lsect S2) (S1 lsect/name S2) // elements/names in S1 that are in S2
(S1 minus S2) (S1 minus/name S2) // elements/names in S1 that are not in S2

52. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4

53. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
D(s0)

54. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
D(s1)

55. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
R(s2)

56. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
V(s1)/Var

57. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
W(s1)/Var

58. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
(W(s1) minus D(s1)/Var)

59. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
(R(s1) isect/name D(s1))

60. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4
(R(s1) lsect/name D(s1))

61. Name Set Examples
25
s0
s1
s2
Var
x1
Var
x2
Type
y5
Type
y3
Var
x4

62. Type Constraints
26
ty1 == ty2 // types ty1 and ty2 are equal
ty1 <R! ty2 // declare ty1 to be related to ty2 in R
ty1 <R? ty2 // require ty1 to be related to ty2 in R
.... // … extensions …
d : ty // declaration d has type ty
[[ e ^ (s) : ty ]] // subterm e has type ty
N xi ty

63. Type Example
27
s
s_bodyx
xs’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)

64. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)

65. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)

66. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

67. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

68. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
[[ e ^ (s) : ty ]], // type of expression
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

69. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
[[ e ^ (s) : ty ]], // type of expression
[[ e_body ^ (s_body) : ty' ]]. // constraints for body
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

70. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
[[ e ^ (s) : ty ]], // type of expression
[[ e_body ^ (s_body) : ty' ]]. // constraints for body
[[ Var(x) ^ (s') : ty ]] :=
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

71. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
[[ e ^ (s) : ty ]], // type of expression
[[ e_body ^ (s_body) : ty' ]]. // constraints for body
[[ Var(x) ^ (s') : ty ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

72. Type Example
27
s
s_bodyx
x
[[ Let([VarDec(x, t, e)], [e_body]) ^ (s) : ty' ]] :=
new s_body, // new scope
s_body -P-> s, // parent edge to enclosing scope
Var{x} <- s_body, // x is a declaration in s_body
Var{x} : ty, // associate type
[[ t ^ (s) : ty ]], // type of type
[[ e ^ (s) : ty ]], // type of expression
[[ e_body ^ (s_body) : ty' ]]. // constraints for body
[[ Var(x) ^ (s') : ty ]] :=
Var{x} -> s', // x is a reference in s'
Var{x} |-> d, // check that x resolves to a declaration
d : ty. // type of declaration is type of reference
s’
x
let
var x : int := x + 1
in
x + 1
end
Let(
[VarDec(
"x"
, Tid("int")
, Plus(Var("x"), Int("1"))
)]
, [Plus(Var("x"), Int("1"))]
)
INT

73. NaBL2 Configuration
28
signature
namespaces
Var
name resolution
labels P I
order D < P, D < I, I < P
well-formedness P* I*
relations
reflexive, transitive, anti-symmetric sub : Type * Type

74. Tiger in NaBL2
29

75. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

76. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

77. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

78. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

79. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

80. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

81. Variable Declarations and References
30
s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

82. Variable Declarations and References
30
x1s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

83. Variable Declarations and References
30
x1s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

84. Variable Declarations and References
30
x1s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

85. Variable Declarations and References
30
x1s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

86. Variable Declarations and References
30
x1s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

87. Variable Declarations and References
30
x1x2 s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

88. Variable Declarations and References
30
x1x2 s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

89. Variable Declarations and References
30
x1x2 s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

90. Variable Declarations and References
30
x1x2 s1
Dec[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
[[ Var(x) ^ (s) ]] :=
Var{x} -> s,
Var{x} |-> d.
s0
s1
s0 Let(
[VarDec("x", INT(), Int("1"))]
, [Plus(Var("x"), Int("1"))]
)
let
var x1 : int := 1
in
x2 + 1
end

91. Scoping of Loop Variable
31
s1 x1
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

92. Scoping of Loop Variable
31
s1 x1
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

93. Scoping of Loop Variable
31
s1 x1
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

94. Scoping of Loop Variable
31
s1 x1
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

95. Scoping of Loop Variable
31
s1 x1
s2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

96. Scoping of Loop Variable
31
s1 x1
s2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

97. Scoping of Loop Variable
31
s1 x1
s2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

98. Scoping of Loop Variable
31
s1 x1
s2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

99. Scoping of Loop Variable
31
s1 x1
s2 i2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

100. Scoping of Loop Variable
31
s1 x1
s2 i2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

101. Scoping of Loop Variable
31
s1 x1
s2 i2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

102. Scoping of Loop Variable
31
s1 x1
s2 i2
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

103. Scoping of Loop Variable
31
s1 x1
s2 i2x3
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

104. Scoping of Loop Variable
31
s1 x1
s2 i2x3
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

105. Scoping of Loop Variable
31
s1 x1
s2 i2x3
x4
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

106. Scoping of Loop Variable
31
s1 x1
s2 i2x3
x4
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

107. Scoping of Loop Variable
31
s1 x1
s2 i2x3
i5x4
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

108. Scoping of Loop Variable
31
s1 x1
s2 i2x3
i5x4
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

109. Scoping of Loop Variable
31
s1 x1
s2 i2x3
i5x4
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

110. Scoping of Loop Variable
31
s1 x1
s2 i2x3
i5x4
x6
s1
[[ stm@For(Var(x), e1, e2, e3) ^ (s) ]] :=
new s_for,
s_for -P-> s,
Var{x} <- s_for,
Loop{Break()@stm} <- s_for,
[[ e1 ^ (s) ]],
[[ e2 ^ (s) ]],
[[ e3 ^ (s_for) ]].
s0
s0
s2
let
var x1 : int := 0
in
for i2 := 1 to 4 do
x3 := x4 + i5;
x6 * 7
end

111. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

112. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

113. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

114. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

115. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

116. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

117. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

118. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1 pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

119. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1 pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

120. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

121. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

122. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

123. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

124. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

125. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

126. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

127. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

128. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

129. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

130. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

131. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

132. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

133. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

134. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

135. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1pow9
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

136. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1pow9
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

137. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1pow9
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

138. Function Declarations and References
32
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
Var{f} <- s,
new s_fun,
s_fun -P-> s,
Map2[[ args ^ (s_fun, s_outer) ]],
distinct/name D(s_fun),
[[ e ^ (s_fun) ]].
[[ FArg(x, t) ^ (s_fun, s_outer) ]] :=
Var{x} <- s_fun,
[[ t ^ (s_outer) ]].
[[ Call(f, exps) ^ (s) ]] :=
Var{f} -> s,
Var{f} |-> d,
Map1[[ exps ^ (s) ]].
s0
s1
s2
s0
s1
s2
pow1pow9
b2
n3
n4
b5
n8
b7
pow6
let
function pow1(b2 : int, n3 : int) : int =
if n4 < 1
then 1
else (b5 * pow6(b7, n8 - 1))
in
pow9(2, 7)
end

139. Sequential Let
33
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

140. Sequential Let
33
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

141. Sequential Let
33
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

142. Sequential Let
33
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

143. Sequential Let
33
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

144. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

145. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

146. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

147. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

148. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

149. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

150. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

151. Sequential Let
33
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

152. Sequential Let
33
s2
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

153. Sequential Let
33
s2
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

154. Sequential Let
33
s2
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

155. Sequential Let
33
s2
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

156. Sequential Let
33
s2 x1
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

157. Sequential Let
33
s2 x1
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

158. Sequential Let
33
s2 x1
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

159. Sequential Let
33
s2 x1
s3
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

160. Sequential Let
33
s2 x1
s3
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

161. Sequential Let
33
s2 x1
s3
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

162. Sequential Let
33
s2 x1
s3
s1
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

163. Sequential Let
33
s2 x1
s3 y2
s1
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

164. Sequential Let
33
s2 x1
s3 y2
s1
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

165. Sequential Let
33
s2 x1
s3 y2
s1
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

166. Sequential Let
33
s2 x1
s3 y2
s1
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

167. Sequential Let
33
s2 x1
s3 y2
s1
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

168. Sequential Let
33
s2 x1
s3 y2
s1 z4x5 y6
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

169. Sequential Let
33
s2 x1
s3 y2
s1 z4x5 y6
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

170. Sequential Let
33
s2 x1
s3 y2
s1 z4x5
z7
y6
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

171. Sequential Let
33
s2 x1
s3 y2
s1 z4x5
z7
y6
z3
[[ Let(blocks, exps) ^ (s) ]] :=
new s_body,
Decs[[ blocks ^ (s, s_body) ]],
Seq[[ exps ^ (s_body) ]],
distinct D(s_body).
Decs[[ [] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer.
Decs[[ [block] ^ (s_outer, s_body) ]] :=
s_body -P-> s_outer,
Dec[[ block ^ (s_body, s_outer) ]].
Decs[[ [block | blocks@[_|_]] ^ (s_outer, s_body) ]] :=
new s_dec,
s_dec -P-> s_outer,
Dec[[ block ^ (s_dec, s_outer) ]],
Decs[[ blocks ^ (s_dec, s_body) ]],
distinct/name D(s_dec).
s0
s0
s2
s3
s1
let
var x1 := 2
var y2 := z3 + 11
var z4 := x5 * y6
in z7 end

172. Mutually Recursive Functions
34
s1 odd1 even6
s2
s3
even11
x2
x7odd9
even4
x3
x8
Dec[[ FunDecs(fdecs) ^ (s, s_outer) ]] :=
Map2[[ fdecs ^ (s, s_outer) ]].
[[ FunDec(f, args, t, e) ^ (s, s_outer) ]] :=
new s_fun,
s_fun -P-> s,
distinct/name D(s_fun),
MapTs2[[ args ^ (s_fun, s_outer) ]],
s0
s0
s1
s2
s3
x5
x10
let
function odd1(x2 : int) : int =
if x3 > 0 then even4(x5-1) else 0
function even6(x7 : int) : int =
if x8 > 0 then odd9(x10-1) else 1
in
even11(34)
end

173. Namespaces
35
s1
s2
Type
foo1
Var
foo2
Type
foo4
Var
foo5
s3
Var
x3
Var
x6
[[ TypeDec(x, t) ^ (s) ]] :=
[[ t ^ (s) ]],
Type{x} <- s.
[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) ]],
[[ e ^ (s_outer) ]],
Var{x} <- s.
s0
s0
s1
s2
s3
let
type foo1 = int
var foo2 : int := 24
var x3 : foo4 := foo5
in
x6
end

174. Explicit versus Inferred Variable Type
36
let
var x : int := 20 + 1
var y := 21
in
x + y
end
[[ VarDec(x, t, e) ^ (s, s_outer) ]] :=
[[ t ^ (s_outer) : ty1 ]],
[[ e ^ (s_outer) : ty2 ]],
ty2 <? ty1,
Var{x} <- s,
Var{x} : ty1 !.
[[ VarDecNoType(x, e) ^ (s, s_outer) ]] :=
[[ e ^ (s_outer) : ty ]],
ty != NIL(),
Var{x} <- s,
Var{x} : ty !.

175. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s1
s3
s0
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

176. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s1
s3
s0
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

177. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s1
s3
s0
Type
point1
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

178. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s1
s3
s0
Type
point1
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

179. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s1
s3
s0
Type
point1
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

180. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

181. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

182. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

183. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

184. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

185. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

186. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

187. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

188. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
s3
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

189. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
s3
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

190. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

191. Record Definitions
37
[[ RecordTy(fields) ^ (s) : ty ]] :=
new s_rec,
ty == RECORD(s_rec),
NIL() <! ty,
distinct/name D(s_rec)/Field,
Map2[[ fields ^ (s_rec, s) ]].
[[ Field(x, t) ^ (s_rec, s_outer) ]] :=
Field{x} <- s_rec,
Field{x} : ty !,
[[ t ^ (s_outer) : ty ]].
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

192. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

193. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

194. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

195. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

196. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

197. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
Type
point5 s_rec
s1
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

198. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
Type
point5 s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

199. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
Type
point5 s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

200. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

201. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

202. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

203. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

204. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

205. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

206. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

207. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
Field
x6
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

208. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
Field
x6
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

209. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
Field
x6
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

210. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
Field
x6
Field
y7
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

211. Record Creation
38
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
Var
p4
INT
INT
s0
s2
s3
[[ r@Record(t, inits) ^ (s) : ty ]] :=
[[ t ^ (s) : ty ]],
ty == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
D(s_rec)/Field subseteq/name
R(s_use)/Field,
distinct/name R(s_use)/Field,
Map2[[ inits ^ (s_use, s) ]].
[[ InitField(x, e) ^ (s_use, s) ]] :=
Field{x} -> s_use,
Field{x} |-> d,
d : ty1,
[[ e ^ (s) : ty2 ]],
ty2 <? ty1.
s4
Type
point5
s4
Field
x6
Field
y7
s_rec
s1
RECORD(s_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

212. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty. RECORD(s2)
Var
p4
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

213. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty. RECORD(s2)
Var
p4
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

214. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty. RECORD(s2)
Var
p4
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

215. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty. RECORD(s2)
Var
p4
Var
p8
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

216. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty. RECORD(s2)
Var
p4
Var
p8
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

217. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5 s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

218. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5 s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

219. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5 s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

220. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5 s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

221. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

222. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

223. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

224. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

225. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

226. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

227. Record Field Access
39
s0
s2
s1
s3
s0
Type
point1
s1 s2
RECORD(s2)
Field
x2
Field
y3
s3
INT
INT
[[ FieldVar(e, f) ^ (s) : ty ]] :=
[[ e ^ (s) : ty_e ]],
ty_e == RECORD(s_rec),
new s_use, s_use -I-> s_rec,
Field{f} -> s_use,
Field{f} |-> d,
d : ty.
s5
RECORD(s2)
Var
p4
Var
p8
s5
Field
x9
s_rec
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

228. Alternative (Nominal) Record Encoding
40
s0
s1
s3
s0
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

229. Alternative (Nominal) Record Encoding
40
s0
s1
s3
s0
s1
s3
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

230. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

231. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

232. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

233. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

234. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

235. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

236. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

237. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

238. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

239. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

240. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

241. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

242. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

243. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

244. Alternative (Nominal) Record Encoding
40
s0
s2
s1
s3
s5
s4
s0
Type
point1
s1 s2
Field
x2
Field
y3
s3
INT
INT
Type
point5
s4
Field
x6
Field
y7
Var
p4
RECORD(point1)
Var
p8
s5
Field
x9
d_rec s_recRECORD(d_rec)
let
type point1 = { x2 : int, y3 : int }
var p4 := point5{ x6 = 4, y7 = 5 }
in
p8.x9
end

245. Full Tiger Specification Online
41
https://github.com/MetaBorgCube/metaborg-tiger
https://github.com/MetaBorgCube/metaborg-tiger/blob/master/org.metaborg.lang.tiger/trans/static-semantics/static-semantics.nabl2

246. Conclusion
42

247. Constraint-based type checking
- Language-specific specification
- Language-independent solver
- Support inference
NaBL2 specifications
- Meta-language for constraint generators
- Rich binding structure using scope graphs
- Many examples of constraint rules for Tiger language
Summary
43

248. Open research topics
- Extensions with new type system features
‣ Structural typing
‣ Type normalization
- Dealing with ambiguity
‣ Ambiguous references
‣ Overload resolution
- Parametric polymorphism
- Incremental analysis
Future Work
44

249. What are we discussing next week?
- How do we describe the meaning of constraints? (semantics)
- What are techniques to solve constraints?
‣ Unification
‣ Simplification and propagation
- Describe the relation between constraint semantics and solver
‣ Soundness
‣ Completeness
Next Week
45