lean2/tests/lean/cast4.lean

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Import cast
SetOption pp::colors false
Definition TypeM := (Type M)
Definition TypeU := (Type U)
Check fun (A A': TypeM)
(B : A -> TypeM)
(B' : A' -> TypeM)
(f : Pi x : A, B x)
(g : Pi x : A', B' x)
(a : A)
(b : A')
(H1 : (Pi x : A, B x) == (Pi x : A', B' x))
(H2 : f == g)
(H3 : a == b),
let
S1 : (Pi x : A', B' x) == (Pi x : A, B x) := Symm H1,
L2 : A' == A := DomInj S1,
b' : A := cast L2 b,
L3 : b == b' := CastEq L2 b,
L4 : a == b' := HTrans H3 L3,
L5 : f a == f b' := Congr2 f L4,
g' : (Pi x : A, B x) := cast S1 g,
L6 : g == g' := CastEq S1 g,
L7 : f == g' := HTrans H2 L6,
L8 : f b' == g' b' := Congr1 b' L7,
L9 : f a == g' b' := HTrans L5 L8,
L10 : g' b' == g b := CastApp S1 L2 g b
in HTrans L9 L10