35 lines
1 KiB
Text
35 lines
1 KiB
Text
Set: pp::colors
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Set: pp::unicode
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Assumed: cast
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Assumed: CastEq
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Assumed: CastApp
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Assumed: DomInj
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Assumed: RanInj
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Set: pp::colors
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Defined: TypeM
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Defined: TypeU
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λ (A A' : TypeM)
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(B : A → TypeM)
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(B' : A' → TypeM)
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(f : Π x : A, B x)
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(g : Π x : A', B' x)
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(a : A)
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(b : A')
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(H1 : (Π x : A, B x) == (Π x : A', B' x))
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(H2 : f == g)
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(H3 : a == b),
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let S1 : (Π x : A', B' x) == (Π x : A, B x) := Symm H1,
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L2 : A' == A := DomInj S1,
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b' : A := cast L2 b,
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L3 : b == b' := CastEq L2 b,
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L4 : a == b' := HTrans H3 L3,
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L5 : f a == f b' := Congr2 f L4,
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g' : Π x : A, B x := cast S1 g,
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L6 : g == g' := CastEq S1 g,
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L7 : f == g' := HTrans H2 L6,
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L8 : f b' == g' b' := Congr1 b' L7,
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L9 : f a == g' b' := HTrans L5 L8,
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L10 : g' b' == g b := CastApp S1 L2 g b
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in HTrans L9 L10 :
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Π (A A' : TypeM) (B : A → TypeM) (B' : A' → TypeM) (f : Π x : A, B x) (g : Π x : A', B' x) (a : A) (b : A'),
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(Π x : A, B x) == (Π x : A', B' x) → f == g → a == b → f a == g b
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