equal.bend source

equal.bend on the hub · documented module

# bend-mathlib/equal.bend: equality lemmas (congruence, substitution, chains).import Base# Applying a two-argument function to equal arguments gives equal results.law cong2:  for -A: Type  for -B: Type  for -C: Type  for -f: A -> B -> C  for -a1: A  for -a2: A  for -b1: B  for -b2: B  for ea: {a1 == a2 : A}  for eb: {b1 == b2 : B}  {f(a1, b1) == f(a2, b2) : C}def cong2(A, B, C, f, a1, a2, b1, b2, ea, eb):  %ea : {f(a1, b1) == f(_, b2) : C}  %eb : {f(a1, b1) == f(a1, _) : C}  {==}# If a equals b, any property of a is also a property of b.law subst:  for -A: Type  for -P: A -> Type  for -a: A  for -b: A  for e: {a == b : A}  for p: P(a)  P(b)def subst(A, P, a, b, e, p):  %e : P(_)  p# Equality chains through three steps: a = b, b = c and c = d give a = d.law trans3:  for -A: Type  for -a: A  for -b: A  for -c: A  for -d: A  for ab: {a == b : A}  for bc: {b == c : A}  for cd: {c == d : A}  {a == d : A}def trans3(A, a, b, c, d, ab, bc, cd):  %cd : {a == _ : A}  %bc : {a == _ : A}  ab# Equal naturals have equal successors.law cong_succ:  for -a: Nat  for -b: Nat  for e: {a == b : Nat}  {1n+a == 1n+b : Nat}def cong_succ(a, b, e):  %e : {1n+a == 1n+_ : Nat}  {==}