21 lines
1.1 KiB
Text
21 lines
1.1 KiB
Text
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import logic
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variable matrix.{l} : Type.{l} → Type.{l}
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variable same_dim {A : Type} : matrix A → matrix A → Prop
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variable add {A : Type} (m1 m2 : matrix A) {H : same_dim m1 m2} : matrix A
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theorem same_dim_irrel {A : Type} {m1 m2 : matrix A} {H1 H2 : same_dim m1 m2} : @add A m1 m2 H1 = @add A m1 m2 H2 :=
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have eq : H1 = H2, from rfl,
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subst eq rfl
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theorem same_dim_eq_args {A : Type} {m1 m2 m1' m2' : matrix A} (H1 : m1 = m1') (H2 : m2 = m2') (H : same_dim m1 m2) : same_dim m1' m2' :=
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subst H1 (subst H2 H)
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theorem add_congr {A : Type} (m1 m2 m1' m2' : matrix A) (H1 : m1 = m1') (H2 : m2 = m2') (H : same_dim m1 m2) : @add A m1 m2 H = @add A m1' m2' (same_dim_eq_args H1 H2 H) :=
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have base : ∀ (H1 : m1 = m1) (H2 : m2 = m2), @add A m1 m2 H = @add A m1 m2 (eq_rec (eq_rec H H1) H2), from
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assume H1 H2, rfl,
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have general : ∀ (H1 : m1 = m1') (H2 : m2 = m2'), @add A m1 m2 H = @add A m1' m2' (eq_rec (eq_rec H H1) H2), from
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subst H1 (subst H2 base),
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calc @add A m1 m2 H = @add A m1' m2' (eq_rec (eq_rec H H1) H2) : general H1 H2
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... = @add A m1' m2' (same_dim_eq_args H1 H2 H) : same_dim_irrel
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