fix(library/nat/div): remove unnecessary '_''s
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
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1 changed files with 8 additions and 8 deletions
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@ -73,8 +73,8 @@ let f := rec_measure default measure rec_val in
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case_strong_induction_on m
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(take x,
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have H1 : f' 0 x = default, from rfl,
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have H2 [fact]: ¬ measure x < 0, from not_lt_zero _,
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have H3 : restrict default measure f 0 x = default, from if_neg H2 _ _,
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have H2 [fact]: ¬ measure x < 0, from not_lt_zero,
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have H3 : restrict default measure f 0 x = default, from if_neg H2,
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show f' 0 x = restrict default measure f 0 x, from trans H1 (symm H3))
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(take m,
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assume IH: ∀n, n ≤ m → ∀x, f' n x = restrict default measure f n x,
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@ -86,12 +86,12 @@ case_strong_induction_on m
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take z,
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assume Hzx : measure z < measure x,
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calc
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f' m z = restrict default measure f m z : IH m (le_refl m) z
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f' m z = restrict default measure f m z : IH m le_refl z
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... = f z : restrict_lt_eq _ _ _ _ _ (lt_le_trans Hzx (lt_succ_imp_le H1)),
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have H2 [fact] : f' (succ m) x = rec_val x f, from
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calc
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f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
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... = rec_val x (f' m) : if_pos H1 _ _
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... = rec_val x (f' m) : if_pos H1
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... = rec_val x f : rec_decreasing (f' m) f x H2a,
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let m' := measure x in
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have H3a : ∀z, measure z < m' → f' m' z = f z, from
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@ -102,10 +102,10 @@ case_strong_induction_on m
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... = f z : restrict_lt_eq _ _ _ _ _ Hzx,
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have H3 : restrict default measure f (succ m) x = rec_val x f, from
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calc
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restrict default measure f (succ m) x = f x : if_pos H1 _ _
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restrict default measure f (succ m) x = f x : if_pos H1
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... = f' (succ m') x : refl _
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... = if measure x < succ m' then rec_val x (f' m') else default : rfl
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... = rec_val x (f' m') : if_pos (self_lt_succ _) _ _
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... = rec_val x (f' m') : if_pos self_lt_succ
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... = rec_val x f : rec_decreasing _ _ _ H3a,
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show f' (succ m) x = restrict default measure f (succ m) x,
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from trans H2 (symm H3))
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@ -113,9 +113,9 @@ case_strong_induction_on m
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have H2 : f' (succ m) x = default, from
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calc
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f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
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... = default : if_neg H1 _ _,
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... = default : if_neg H1,
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have H3 : restrict default measure f (succ m) x = default,
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from if_neg H1 _ _,
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from if_neg H1,
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show f' (succ m) x = restrict default measure f (succ m) x,
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from trans H2 (symm H3)))
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