diff --git a/library/algebra/complete_lattice.lean b/library/algebra/complete_lattice.lean index d42bd042c..f7c217fda 100644 --- a/library/algebra/complete_lattice.lean +++ b/library/algebra/complete_lattice.lean @@ -89,7 +89,7 @@ le_Inf (take x, suppose x ∈ '{a, b}, or.elim (eq_or_mem_of_mem_insert this) (suppose x = a, begin subst x, exact h₁ end) (suppose x ∈ '{b}, - assert x = b, from !eq_of_mem_singleton this, + have x = b, from !eq_of_mem_singleton this, begin subst x, exact h₂ end)) lemma le_sup_left (a b : A) : a ≤ a ⊔ b := @@ -104,7 +104,7 @@ Sup_le (take x, suppose x ∈ '{a, b}, or.elim (eq_or_mem_of_mem_insert this) (suppose x = a, by subst x; assumption) (suppose x ∈ '{b}, - assert x = b, from !eq_of_mem_singleton this, + have x = b, from !eq_of_mem_singleton this, by subst x; assumption)) end complete_lattice_Inf @@ -152,7 +152,7 @@ le_Inf (take x, suppose x ∈ '{a, b}, or.elim (eq_or_mem_of_mem_insert this) (suppose x = a, begin subst x, exact h₁ end) (suppose x ∈ '{b}, - assert x = b, from !eq_of_mem_singleton this, + have x = b, from !eq_of_mem_singleton this, begin subst x, exact h₂ end)) lemma le_sup_left (a b : A) : a ≤ a ⊔ b := @@ -167,7 +167,7 @@ Sup_le (take x, suppose x ∈ '{a, b}, or.elim (eq_or_mem_of_mem_insert this) (assume H : x = a, by subst x; exact h₁) (suppose x ∈ '{b}, - assert x = b, from !eq_of_mem_singleton this, + have x = b, from !eq_of_mem_singleton this, by subst x; exact h₂)) end complete_lattice_Sup @@ -265,12 +265,12 @@ lemma Inf_singleton {a : A} : ⨅'{a} = a := have ⨅'{a} ≤ a, from Inf_le !mem_insert, have a ≤ ⨅'{a}, from - le_Inf (take b, suppose b ∈ '{a}, assert b = a, from eq_of_mem_singleton this, by rewrite this; apply le.refl), + le_Inf (take b, suppose b ∈ '{a}, have b = a, from eq_of_mem_singleton this, by rewrite this; apply le.refl), le.antisymm `⨅'{a} ≤ a` `a ≤ ⨅'{a}` lemma Sup_singleton {a : A} : ⨆'{a} = a := have ⨆'{a} ≤ a, from - Sup_le (take b, suppose b ∈ '{a}, assert b = a, from eq_of_mem_singleton this, by rewrite this; apply le.refl), + Sup_le (take b, suppose b ∈ '{a}, have b = a, from eq_of_mem_singleton this, by rewrite this; apply le.refl), have a ≤ ⨆'{a}, from le_Sup !mem_insert, le.antisymm `⨆'{a} ≤ a` `a ≤ ⨆'{a}` diff --git a/library/algebra/field.lean b/library/algebra/field.lean index b296fbcf2..e224a5910 100644 --- a/library/algebra/field.lean +++ b/library/algebra/field.lean @@ -80,15 +80,15 @@ section division_ring division_ring.mul_ne_zero (and.right H2) (and.left H2) theorem eq_one_div_of_mul_eq_one (H : a * b = 1) : b = 1 / a := - assert a ≠ 0, from + have a ≠ 0, from suppose a = 0, have 0 = (1:A), by inst_simp, absurd this zero_ne_one, - assert b = (1 / a) * a * b, by inst_simp, + have b = (1 / a) * a * b, by inst_simp, show b = 1 / a, by inst_simp theorem eq_one_div_of_mul_eq_one_left (H : b * a = 1) : b = 1 / a := - assert a ≠ 0, from + have a ≠ 0, from suppose a = 0, have 0 = (1:A), by inst_simp, absurd this zero_ne_one, @@ -205,7 +205,7 @@ section field by rewrite [(division_ring.one_div_mul_one_div Ha Hb), mul.comm b] theorem field.div_mul_right (Hb : b ≠ 0) (H : a * b ≠ 0) : a / (a * b) = 1 / b := - assert a ≠ 0, from and.left (ne_zero_and_ne_zero_of_mul_ne_zero H), + have a ≠ 0, from and.left (ne_zero_and_ne_zero_of_mul_ne_zero H), symm (calc 1 / b = a * ((1 / a) * (1 / b)) : by inst_simp ... = a * (1 / (b * a)) : division_ring.one_div_mul_one_div this Hb @@ -222,7 +222,7 @@ section field by rewrite [mul.comm, (!div_mul_cancel Hb)] theorem one_div_add_one_div (Ha : a ≠ 0) (Hb : b ≠ 0) : 1 / a + 1 / b = (a + b) / (a * b) := - assert a * b ≠ 0, from (division_ring.mul_ne_zero Ha Hb), + have a * b ≠ 0, from (division_ring.mul_ne_zero Ha Hb), by rewrite [add.comm, -(field.div_mul_left Ha this), -(field.div_mul_right Hb this), *division.def, -right_distrib] diff --git a/library/algebra/group.lean b/library/algebra/group.lean index a6a69eccb..a6b9e02db 100644 --- a/library/algebra/group.lean +++ b/library/algebra/group.lean @@ -156,7 +156,7 @@ section group by simp theorem inv_eq_of_mul_eq_one {a b : A} (H : a * b = 1) : a⁻¹ = b := - assert a⁻¹ * 1 = b, by inst_simp, + have a⁻¹ * 1 = b, by inst_simp, by inst_simp theorem one_inv [simp] : 1⁻¹ = (1 : A) := @@ -166,14 +166,14 @@ section group inv_eq_of_mul_eq_one (mul.left_inv a) theorem inv.inj {a b : A} (H : a⁻¹ = b⁻¹) : a = b := - assert a = a⁻¹⁻¹, by simp_nohyps, + have a = a⁻¹⁻¹, by simp_nohyps, by inst_simp theorem inv_eq_inv_iff_eq (a b : A) : a⁻¹ = b⁻¹ ↔ a = b := iff.intro (assume H, inv.inj H) (by simp) theorem inv_eq_one_iff_eq_one (a : A) : a⁻¹ = 1 ↔ a = 1 := - assert a⁻¹ = 1⁻¹ ↔ a = 1, from inv_eq_inv_iff_eq a 1, + have a⁻¹ = 1⁻¹ ↔ a = 1, from inv_eq_inv_iff_eq a 1, by simp theorem eq_one_of_inv_eq_one (a : A) : a⁻¹ = 1 → a = 1 := @@ -186,11 +186,11 @@ section group iff.intro !eq_inv_of_eq_inv !eq_inv_of_eq_inv theorem eq_inv_of_mul_eq_one {a b : A} (H : a * b = 1) : a = b⁻¹ := - assert a⁻¹ = b, from inv_eq_of_mul_eq_one H, + have a⁻¹ = b, from inv_eq_of_mul_eq_one H, by inst_simp theorem mul.right_inv [simp] (a : A) : a * a⁻¹ = 1 := - assert a = a⁻¹⁻¹, by simp, + have a = a⁻¹⁻¹, by simp, by inst_simp theorem mul_inv_cancel_left [simp] (a b : A) : a * (a⁻¹ * b) = b := @@ -203,7 +203,7 @@ section group inv_eq_of_mul_eq_one (by inst_simp) theorem eq_of_mul_inv_eq_one {a b : A} (H : a * b⁻¹ = 1) : a = b := - assert a⁻¹ * 1 = a⁻¹, by inst_simp, + have a⁻¹ * 1 = a⁻¹, by inst_simp, by inst_simp theorem eq_mul_inv_of_mul_eq {a b c : A} (H : a * c = b) : a = b * c⁻¹ := @@ -237,11 +237,11 @@ section group iff.intro eq_mul_inv_of_mul_eq mul_eq_of_eq_mul_inv theorem mul_left_cancel {a b c : A} (H : a * b = a * c) : b = c := - assert a⁻¹ * (a * b) = b, by inst_simp, + have a⁻¹ * (a * b) = b, by inst_simp, by inst_simp theorem mul_right_cancel {a b c : A} (H : a * b = c * b) : a = c := - assert a * b * b⁻¹ = a, by inst_simp, + have a * b * b⁻¹ = a, by inst_simp, by inst_simp theorem mul_eq_one_of_mul_eq_one {a b : A} (H : b * a = 1) : a * b = 1 := @@ -277,7 +277,7 @@ section group lemma is_conj.symm (a b : A) : a ~ b → b ~ a := assume Pab, obtain x (Pconj : x ∘c b = a), from Pab, - assert Pxinv : x⁻¹ ∘c x ∘c b = x⁻¹ ∘c a, by simp, + have Pxinv : x⁻¹ ∘c x ∘c b = x⁻¹ ∘c a, by simp, exists.intro x⁻¹ (by simp) lemma is_conj.trans (a b c : A) : a ~ b → b ~ c → a ~ c := @@ -324,7 +324,7 @@ section add_group by simp theorem neg_eq_of_add_eq_zero {a b : A} (H : a + b = 0) : -a = b := - assert -a + 0 = b, by inst_simp, + have -a + 0 = b, by inst_simp, by inst_simp theorem neg_zero [simp] : -0 = (0 : A) := neg_eq_of_add_eq_zero (zero_add 0) @@ -332,11 +332,11 @@ section add_group theorem neg_neg [simp] (a : A) : -(-a) = a := neg_eq_of_add_eq_zero (add.left_inv a) theorem eq_neg_of_add_eq_zero {a b : A} (H : a + b = 0) : a = -b := - assert -a = b, from neg_eq_of_add_eq_zero H, + have -a = b, from neg_eq_of_add_eq_zero H, by inst_simp theorem neg.inj {a b : A} (H : -a = -b) : a = b := - assert a = -(-a), by simp_nohyps, + have a = -(-a), by simp_nohyps, by inst_simp theorem neg_eq_neg_iff_eq (a b : A) : -a = -b ↔ a = b := @@ -346,7 +346,7 @@ section add_group iff.mp !neg_eq_neg_iff_eq theorem neg_eq_zero_iff_eq_zero (a : A) : -a = 0 ↔ a = 0 := - assert -a = -0 ↔ a = 0, from neg_eq_neg_iff_eq a 0, + have -a = -0 ↔ a = 0, from neg_eq_neg_iff_eq a 0, by simp theorem eq_zero_of_neg_eq_zero {a : A} : -a = 0 → a = 0 := @@ -359,7 +359,7 @@ section add_group iff.intro !eq_neg_of_eq_neg !eq_neg_of_eq_neg theorem add.right_inv [simp] (a : A) : a + -a = 0 := - assert a = -(-a), by simp, + have a = -(-a), by simp, by inst_simp theorem add_neg_cancel_left [simp] (a b : A) : a + (-a + b) = b := @@ -403,11 +403,11 @@ section add_group iff.intro eq_add_neg_of_add_eq add_eq_of_eq_add_neg theorem add_left_cancel {a b c : A} (H : a + b = a + c) : b = c := - assert -a + (a + b) = b, by inst_simp, + have -a + (a + b) = b, by inst_simp, by inst_simp theorem add_right_cancel {a b c : A} (H : a + b = c + b) : a = c := - assert a + b + -b = a, by inst_simp, + have a + b + -b = a, by inst_simp, by inst_simp definition add_group.to_left_cancel_semigroup [trans_instance] : add_left_cancel_semigroup A := @@ -459,7 +459,7 @@ section add_group by rewrite [sub_eq_add_neg, add.assoc, -sub_eq_add_neg] theorem eq_of_sub_eq_zero {a b : A} (H : a - b = 0) : a = b := - assert -a + 0 = -a, by inst_simp, + have -a + 0 = -a, by inst_simp, by inst_simp theorem eq_iff_sub_eq_zero (a b : A) : a = b ↔ a - b = 0 := diff --git a/library/algebra/group_bigops.lean b/library/algebra/group_bigops.lean index 48e6d3bcb..18c7d21b8 100644 --- a/library/algebra/group_bigops.lean +++ b/library/algebra/group_bigops.lean @@ -136,7 +136,7 @@ section monoid ∀ {l : list A} b, (∀ a, a ∈ l → f a = b) → Prodl l f = b ^ length l | nil := take b, assume Pconst, by rewrite [length_nil, {b^0}pow_zero] | (a::l) := take b, assume Pconst, - assert Pconstl : ∀ a', a' ∈ l → f a' = b, + have Pconstl : ∀ a', a' ∈ l → f a' = b, from take a' Pa'in, Pconst a' (mem_cons_of_mem a Pa'in), by rewrite [Prodl_cons f, Pconst a !mem_cons, Prodl_eq_pow_of_const b Pconstl, length_cons, add_one, pow_succ b] @@ -304,9 +304,9 @@ namespace finset (take x s', assume H1 : x ∉ s', assume IH : (∀ {x : A}, x ∈ s' → f x = g x) → Prod s' f = Prod s' g, assume H2 : ∀{y}, y ∈ insert x s' → f y = g y, - assert H3 : ∀y, y ∈ s' → f y = g y, from + have H3 : ∀y, y ∈ s' → f y = g y, from take y, assume H', H2 (mem_insert_of_mem _ H'), - assert H4 : f x = g x, from H2 !mem_insert, + have H4 : f x = g x, from H2 !mem_insert, by rewrite [Prod_insert_of_not_mem f H1, Prod_insert_of_not_mem g H1, IH H3, H4]) theorem Prod_singleton (a : A) (f : A → B) : Prod '{a} f = f a := @@ -412,15 +412,15 @@ section Prod Prod (insert a s) f = Prod s f := by_cases (suppose finite s, - assert (#finset a ∈ set.to_finset s), by rewrite mem_to_finset_eq; apply H, + have (#finset a ∈ set.to_finset s), by rewrite mem_to_finset_eq; apply H, by rewrite [↑Prod, to_finset_insert, finset.Prod_insert_of_mem f this]) (assume nfs : ¬ finite s, - assert ¬ finite (insert a s), from assume H, nfs (finite_of_finite_insert H), + have ¬ finite (insert a s), from assume H, nfs (finite_of_finite_insert H), by rewrite [Prod_of_not_finite nfs, Prod_of_not_finite this]) theorem Prod_insert_of_not_mem (f : A → B) {a : A} {s : set A} [finite s] (H : a ∉ s) : Prod (insert a s) f = f a * Prod s f := - assert (#finset a ∉ set.to_finset s), by rewrite mem_to_finset_eq; apply H, + have (#finset a ∉ set.to_finset s), by rewrite mem_to_finset_eq; apply H, by rewrite [↑Prod, to_finset_insert, finset.Prod_insert_of_not_mem f this] theorem Prod_union (f : A → B) {s₁ s₂ : set A} [finite s₁] [finite s₂] diff --git a/library/algebra/group_power.lean b/library/algebra/group_power.lean index 1667bf748..35fe8ff49 100644 --- a/library/algebra/group_power.lean +++ b/library/algebra/group_power.lean @@ -108,7 +108,7 @@ theorem inv_pow (a : A) : ∀n, (a⁻¹)^n = (a^n)⁻¹ | (succ n) := by rewrite [pow_succ, pow_succ', inv_pow, mul_inv] theorem pow_sub (a : A) {m n : ℕ} (H : m ≥ n) : a^(m - n) = a^m * (a^n)⁻¹ := -assert H1 : m - n + n = m, from nat.sub_add_cancel H, +have H1 : m - n + n = m, from nat.sub_add_cancel H, have H2 : a^(m - n) * a^n = a^m, by rewrite [-pow_add, H1], eq_mul_inv_of_mul_eq H2 @@ -132,7 +132,7 @@ private lemma gpow_add_aux (a : A) (m n : nat) : gpow a ((of_nat m) + -[1+n]) = gpow a (of_nat m) * gpow a (-[1+n]) := or.elim (nat.lt_or_ge m (nat.succ n)) (assume H : (m < nat.succ n), - assert H1 : (#nat nat.succ n - m > nat.zero), from nat.sub_pos_of_lt H, + have H1 : (#nat nat.succ n - m > nat.zero), from nat.sub_pos_of_lt H, calc gpow a ((of_nat m) + -[1+n]) = gpow a (sub_nat_nat m (nat.succ n)) : rfl ... = gpow a (-[1+ nat.pred (nat.sub (nat.succ n) m)]) : {sub_nat_nat_of_lt H} diff --git a/library/algebra/ordered_field.lean b/library/algebra/ordered_field.lean index 93d6197df..396359c3b 100644 --- a/library/algebra/ordered_field.lean +++ b/library/algebra/ordered_field.lean @@ -170,16 +170,16 @@ section linear_ordered_field theorem div_lt_div_of_mul_sub_mul_div_neg (Hc : c ≠ 0) (Hd : d ≠ 0) (H : (a * d - b * c) / (c * d) < 0) : a / c < b / d := - assert H1 : (a * d - c * b) / (c * d) < 0, by rewrite [mul.comm c b]; exact H, - assert H2 : a / c - b / d < 0, by rewrite [!div_sub_div Hc Hd]; exact H1, - assert H3 : a / c - b / d + b / d < 0 + b / d, from add_lt_add_right H2 _, + have H1 : (a * d - c * b) / (c * d) < 0, by rewrite [mul.comm c b]; exact H, + have H2 : a / c - b / d < 0, by rewrite [!div_sub_div Hc Hd]; exact H1, + have H3 : a / c - b / d + b / d < 0 + b / d, from add_lt_add_right H2 _, begin rewrite [zero_add at H3, sub_eq_add_neg at H3, neg_add_cancel_right at H3], exact H3 end theorem div_le_div_of_mul_sub_mul_div_nonpos (Hc : c ≠ 0) (Hd : d ≠ 0) (H : (a * d - b * c) / (c * d) ≤ 0) : a / c ≤ b / d := - assert H1 : (a * d - c * b) / (c * d) ≤ 0, by rewrite [mul.comm c b]; exact H, - assert H2 : a / c - b / d ≤ 0, by rewrite [!div_sub_div Hc Hd]; exact H1, - assert H3 : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _, + have H1 : (a * d - c * b) / (c * d) ≤ 0, by rewrite [mul.comm c b]; exact H, + have H2 : a / c - b / d ≤ 0, by rewrite [!div_sub_div Hc Hd]; exact H1, + have H3 : a / c - b / d + b / d ≤ 0 + b / d, from add_le_add_right H2 _, begin rewrite [zero_add at H3, sub_eq_add_neg at H3, neg_add_cancel_right at H3], exact H3 end theorem div_pos_of_pos_of_pos (Ha : 0 < a) (Hb : 0 < b) : 0 < a / b := @@ -351,12 +351,12 @@ section linear_ordered_field theorem exists_add_lt_and_pos_of_lt (H : b < a) : ∃ c : A, b + c < a ∧ c > 0 := exists.intro ((a - b) / (1 + 1)) - (and.intro (assert H2 : a + a > (b + b) + (a - b), from calc + (and.intro (have H2 : a + a > (b + b) + (a - b), from calc a + a > b + a : add_lt_add_right H ... = b + a + b - b : add_sub_cancel ... = b + b + a - b : add.right_comm ... = (b + b) + (a - b) : add_sub, - assert H3 : (a + a) / 2 > ((b + b) + (a - b)) / 2, + have H3 : (a + a) / 2 > ((b + b) + (a - b)) / 2, from div_lt_div_of_lt_of_pos H2 two_pos, by rewrite [one_add_one_eq_two, sub_eq_add_neg, add_self_div_two at H3, -div_add_div_same at H3, add_self_div_two at H3]; exact H3) @@ -423,7 +423,7 @@ section discrete_linear_ordered_field ), le_of_one_le_div Hb H' theorem le_of_one_div_le_one_div_of_neg (H : b < 0) (Hl : 1 / a ≤ 1 / b) : b ≤ a := - assert Ha : a ≠ 0, from ne_of_lt (neg_of_one_div_neg (calc + have Ha : a ≠ 0, from ne_of_lt (neg_of_one_div_neg (calc 1 / a ≤ 1 / b : Hl ... < 0 : one_div_neg_of_neg H)), have H' : -b > 0, from neg_pos_of_neg H, diff --git a/library/algebra/ordered_group.lean b/library/algebra/ordered_group.lean index d80a4a1e4..7ac936b12 100644 --- a/library/algebra/ordered_group.lean +++ b/library/algebra/ordered_group.lean @@ -241,12 +241,12 @@ structure ordered_comm_group [class] (A : Type) extends add_comm_group A, order_ theorem ordered_comm_group.le_of_add_le_add_left [ordered_comm_group A] {a b c : A} (H : a + b ≤ a + c) : b ≤ c := -assert H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _, +have H' : -a + (a + b) ≤ -a + (a + c), from ordered_comm_group.add_le_add_left _ _ H _, by rewrite *neg_add_cancel_left at H'; exact H' theorem ordered_comm_group.lt_of_add_lt_add_left [ordered_comm_group A] {a b c : A} (H : a + b < a + c) : b < c := -assert H' : -a + (a + b) < -a + (a + c), from ordered_comm_group.add_lt_add_left _ _ H _, +have H' : -a + (a + b) < -a + (a + c), from ordered_comm_group.add_lt_add_left _ _ H _, by rewrite *neg_add_cancel_left at H'; exact H' definition ordered_comm_group.to_ordered_cancel_comm_monoid [trans_instance] [s : ordered_comm_group A] : ordered_cancel_comm_monoid A := @@ -386,7 +386,7 @@ section iff.mp !add_le_iff_le_sub_right theorem le_add_iff_neg_add_le : a ≤ b + c ↔ -b + a ≤ c := - assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), + have H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), by rewrite neg_add_cancel_left at H; exact H theorem le_add_of_neg_add_le {a b c : A} : -b + a ≤ c → a ≤ b + c := @@ -405,7 +405,7 @@ section iff.mp !le_add_iff_sub_left_le theorem le_add_iff_sub_right_le : a ≤ b + c ↔ a - c ≤ b := - assert H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff), + have H: a ≤ b + c ↔ a - c ≤ b + c - c, from iff.symm (!add_le_add_right_iff), by rewrite [sub_eq_add_neg (b+c) c at H, add_neg_cancel_right at H]; exact H theorem le_add_of_sub_right_le {a b c : A} : a - c ≤ b → a ≤ b + c := @@ -415,7 +415,7 @@ section iff.mp !le_add_iff_sub_right_le theorem le_add_iff_neg_add_le_left : a ≤ b + c ↔ -b + a ≤ c := - assert H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), + have H: a ≤ b + c ↔ -b + a ≤ -b + (b + c), from iff.symm (!add_le_add_left_iff), by rewrite neg_add_cancel_left at H; exact H theorem le_add_of_neg_add_le_left {a b c : A} : -b + a ≤ c → a ≤ b + c := @@ -434,8 +434,8 @@ section iff.mp !le_add_iff_neg_add_le_right theorem le_add_iff_neg_le_sub_left : c ≤ a + b ↔ -a ≤ b - c := - assert H : c ≤ a + b ↔ -a + c ≤ b, from !le_add_iff_neg_add_le, - assert H' : -a + c ≤ b ↔ -a ≤ b - c, from !add_le_iff_le_sub_right, + have H : c ≤ a + b ↔ -a + c ≤ b, from !le_add_iff_neg_add_le, + have H' : -a + c ≤ b ↔ -a ≤ b - c, from !add_le_iff_le_sub_right, iff.trans H H' theorem le_add_of_neg_le_sub_left {a b c : A} : -a ≤ b - c → c ≤ a + b := @@ -454,7 +454,7 @@ section iff.mp !le_add_iff_neg_le_sub_right theorem add_lt_iff_lt_neg_add_left : a + b < c ↔ b < -a + c := - assert H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff), + have H: a + b < c ↔ -a + (a + b) < -a + c, from iff.symm (!add_lt_add_left_iff), begin rewrite neg_add_cancel_left at H, exact H end theorem add_lt_of_lt_neg_add_left {a b c : A} : b < -a + c → a + b < c := @@ -485,7 +485,7 @@ section iff.mp !add_lt_iff_lt_sub_left theorem add_lt_iff_lt_sub_right : a + b < c ↔ a < c - b := - assert H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff), + have H: a + b < c ↔ a + b - b < c - b, from iff.symm (!add_lt_add_right_iff), by rewrite [sub_eq_add_neg at H, add_neg_cancel_right at H]; exact H theorem add_lt_of_lt_sub_right {a b c : A} : a < c - b → a + b < c := @@ -495,7 +495,7 @@ section iff.mp !add_lt_iff_lt_sub_right theorem lt_add_iff_neg_add_lt_left : a < b + c ↔ -b + a < c := - assert H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff), + have H: a < b + c ↔ -b + a < -b + (b + c), from iff.symm (!add_lt_add_left_iff), by rewrite neg_add_cancel_left at H; exact H theorem lt_add_of_neg_add_lt_left {a b c : A} : -b + a < c → a < b + c := @@ -760,7 +760,7 @@ section ... = abs a + b : by rewrite (abs_of_nonneg H2) ... = abs a + abs b : by rewrite (abs_of_nonneg H3)) (assume H3 : ¬ b ≥ 0, - assert H4 : b ≤ 0, from le_of_lt (lt_of_not_ge H3), + have H4 : b ≤ 0, from le_of_lt (lt_of_not_ge H3), calc abs (a + b) = a + b : by rewrite (abs_of_nonneg H1) ... = abs a + b : by rewrite (abs_of_nonneg H2) @@ -791,8 +791,8 @@ section or.elim (le.total 0 (a + b)) (assume H2 : 0 ≤ a + b, aux2 H2) (assume H2 : a + b ≤ 0, - assert H3 : -a + -b = -(a + b), by rewrite neg_add, - assert H4 : -(a + b) ≥ 0, from iff.mpr (neg_nonneg_iff_nonpos (a+b)) H2, + have H3 : -a + -b = -(a + b), by rewrite neg_add, + have H4 : -(a + b) ≥ 0, from iff.mpr (neg_nonneg_iff_nonpos (a+b)) H2, have H5 : -a + -b ≥ 0, begin rewrite -H3 at H4, exact H4 end, calc abs (a + b) = abs (-a + -b) : by rewrite [-abs_neg, neg_add] diff --git a/library/algebra/ordered_ring.lean b/library/algebra/ordered_ring.lean index e3ed6ca41..c349149dd 100644 --- a/library/algebra/ordered_ring.lean +++ b/library/algebra/ordered_ring.lean @@ -209,7 +209,7 @@ structure ordered_ring [class] (A : Type) theorem ordered_ring.mul_le_mul_of_nonneg_left [s : ordered_ring A] {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) : c * a ≤ c * b := have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab, -assert H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1, +have H2 : 0 ≤ c * (b - a), from ordered_ring.mul_nonneg _ _ Hc H1, begin rewrite mul_sub_left_distrib at H2, exact (iff.mp !sub_nonneg_iff_le H2) @@ -218,7 +218,7 @@ end theorem ordered_ring.mul_le_mul_of_nonneg_right [s : ordered_ring A] {a b c : A} (Hab : a ≤ b) (Hc : 0 ≤ c) : a * c ≤ b * c := have H1 : 0 ≤ b - a, from iff.elim_right !sub_nonneg_iff_le Hab, -assert H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc, +have H2 : 0 ≤ (b - a) * c, from ordered_ring.mul_nonneg _ _ H1 Hc, begin rewrite mul_sub_right_distrib at H2, exact (iff.mp !sub_nonneg_iff_le H2) @@ -227,7 +227,7 @@ end theorem ordered_ring.mul_lt_mul_of_pos_left [s : ordered_ring A] {a b c : A} (Hab : a < b) (Hc : 0 < c) : c * a < c * b := have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab, -assert H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1, +have H2 : 0 < c * (b - a), from ordered_ring.mul_pos _ _ Hc H1, begin rewrite mul_sub_left_distrib at H2, exact (iff.mp !sub_pos_iff_lt H2) @@ -236,7 +236,7 @@ end theorem ordered_ring.mul_lt_mul_of_pos_right [s : ordered_ring A] {a b c : A} (Hab : a < b) (Hc : 0 < c) : a * c < b * c := have H1 : 0 < b - a, from iff.elim_right !sub_pos_iff_lt Hab, -assert H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc, +have H2 : 0 < (b - a) * c, from ordered_ring.mul_pos _ _ H1 Hc, begin rewrite mul_sub_right_distrib at H2, exact (iff.mp !sub_pos_iff_lt H2) @@ -264,7 +264,7 @@ section theorem mul_le_mul_of_nonpos_left (H : b ≤ a) (Hc : c ≤ 0) : c * a ≤ c * b := have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc, - assert H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc', + have H1 : -c * b ≤ -c * a, from mul_le_mul_of_nonneg_left H Hc', have H2 : -(c * b) ≤ -(c * a), begin rewrite [-*neg_mul_eq_neg_mul at H1], @@ -274,7 +274,7 @@ section theorem mul_le_mul_of_nonpos_right (H : b ≤ a) (Hc : c ≤ 0) : a * c ≤ b * c := have Hc' : -c ≥ 0, from iff.mpr !neg_nonneg_iff_nonpos Hc, - assert H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc', + have H1 : b * -c ≤ a * -c, from mul_le_mul_of_nonneg_right H Hc', have H2 : -(b * c) ≤ -(a * c), begin rewrite [-*neg_mul_eq_mul_neg at H1], @@ -291,7 +291,7 @@ section theorem mul_lt_mul_of_neg_left (H : b < a) (Hc : c < 0) : c * a < c * b := have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc, - assert H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc', + have H1 : -c * b < -c * a, from mul_lt_mul_of_pos_left H Hc', have H2 : -(c * b) < -(c * a), begin rewrite [-*neg_mul_eq_neg_mul at H1], @@ -301,7 +301,7 @@ section theorem mul_lt_mul_of_neg_right (H : b < a) (Hc : c < 0) : a * c < b * c := have Hc' : -c > 0, from iff.mpr !neg_pos_iff_neg Hc, - assert H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc', + have H1 : b * -c < a * -c, from mul_lt_mul_of_pos_right H Hc', have H2 : -(b * c) < -(a * c), begin rewrite [-*neg_mul_eq_mul_neg at H1], diff --git a/library/algebra/ring.lean b/library/algebra/ring.lean index e37433a94..e372057c4 100644 --- a/library/algebra/ring.lean +++ b/library/algebra/ring.lean @@ -374,7 +374,7 @@ section (suppose a * a = b * b, have (a - b) * (a + b) = 0, by rewrite [mul.comm, -mul_self_sub_mul_self_eq, this, sub_self], - assert a - b = 0 ∨ a + b = 0, from !eq_zero_or_eq_zero_of_mul_eq_zero this, + have a - b = 0 ∨ a + b = 0, from !eq_zero_or_eq_zero_of_mul_eq_zero this, or.elim this (suppose a - b = 0, or.inl (eq_of_sub_eq_zero this)) (suppose a + b = 0, or.inr (eq_neg_of_add_eq_zero this))) @@ -383,7 +383,7 @@ section (suppose a = -b, by rewrite [this, neg_mul_neg])) theorem mul_self_eq_one_iff (a : A) : a * a = 1 ↔ a = 1 ∨ a = -1 := - assert a * a = 1 * 1 ↔ a = 1 ∨ a = -1, from mul_self_eq_mul_self_iff a 1, + have a * a = 1 * 1 ↔ a = 1 ∨ a = -1, from mul_self_eq_mul_self_iff a 1, by rewrite mul_one at this; exact this -- TODO: c - b * c → c = 0 ∨ b = 1 and variants diff --git a/library/algebra/ring_power.lean b/library/algebra/ring_power.lean index 43ebf84f8..b0febfaf1 100644 --- a/library/algebra/ring_power.lean +++ b/library/algebra/ring_power.lean @@ -110,11 +110,11 @@ begin end theorem pow_ge_one {x : A} (i : ℕ) (xge1 : x ≥ 1) : x^i ≥ 1 := -assert H : x^i ≥ 1^i, from pow_le_pow_of_le i (le_of_lt zero_lt_one) xge1, +have H : x^i ≥ 1^i, from pow_le_pow_of_le i (le_of_lt zero_lt_one) xge1, by rewrite one_pow at H; exact H theorem pow_gt_one {x : A} {i : ℕ} (xgt1 : x > 1) (ipos : i > 0) : x^i > 1 := -assert xpos : x > 0, from lt.trans zero_lt_one xgt1, +have xpos : x > 0, from lt.trans zero_lt_one xgt1, begin induction i with [i, ih], {exfalso, exact !lt.irrefl ipos}, diff --git a/library/data/bag.lean b/library/data/bag.lean index 644a4bdd1..ac5a101ad 100644 --- a/library/data/bag.lean +++ b/library/data/bag.lean @@ -107,17 +107,17 @@ protected lemma ext {b₁ b₂ : bag A} : (∀ a, count a b₁ = count a b₂) quot.induction_on₂ b₁ b₂ (λ l₁ l₂ (h : ∀ a, count a ⟦l₁⟧ = count a ⟦l₂⟧), have gen : ∀ (l₁ l₂ : list A), (∀ a, list.count a l₁ = list.count a l₂) → l₁ ~ l₂ | [] [] h₁ := !perm.refl - | [] (a₂::s₂) h₁ := assert list.count a₂ [] = list.count a₂ (a₂::s₂), from h₁ a₂, by rewrite [count_nil at this, count_cons_eq at this]; contradiction + | [] (a₂::s₂) h₁ := have list.count a₂ [] = list.count a₂ (a₂::s₂), from h₁ a₂, by rewrite [count_nil at this, count_cons_eq at this]; contradiction | (a::s₁) s₂ h₁ := - assert g₁ : list.count a (a::s₁) > 0, from count_gt_zero_of_mem !mem_cons, - assert list.count a (a::s₁) = list.count a s₂, from h₁ a, - assert list.count a s₂ > 0, by rewrite [-this]; exact g₁, + have g₁ : list.count a (a::s₁) > 0, from count_gt_zero_of_mem !mem_cons, + have list.count a (a::s₁) = list.count a s₂, from h₁ a, + have list.count a s₂ > 0, by rewrite [-this]; exact g₁, have a ∈ s₂, from mem_of_count_gt_zero this, have ∃ l r, s₂ = l++(a::r), from mem_split this, obtain l r (e₁ : s₂ = l++(a::r)), from this, have ∀ a, list.count a s₁ = list.count a (l++r), from take a₁, - assert e₂ : list.count a₁ (a::s₁) = list.count a₁ (l++(a::r)), by rewrite -e₁; exact h₁ a₁, + have e₂ : list.count a₁ (a::s₁) = list.count a₁ (l++(a::r)), by rewrite -e₁; exact h₁ a₁, by_cases (suppose a₁ = a, begin rewrite [-this at e₂, list.count_append at e₂, *count_cons_eq at e₂, add_succ at e₂], @@ -134,7 +134,7 @@ quot.induction_on₂ b₁ b₂ (λ l₁ l₂ (h : ∀ a, count a ⟦l₁⟧ = co definition insert.inj {a : A} {b₁ b₂ : bag A} : insert a b₁ = insert a b₂ → b₁ = b₂ := assume h, bag.ext (take x, - assert e : count x (insert a b₁) = count x (insert a b₂), by rewrite h, + have e : count x (insert a b₁) = count x (insert a b₂), by rewrite h, by_cases (suppose x = a, begin subst x, rewrite [*count_insert at e], injection e, assumption end) (suppose x ≠ a, begin rewrite [*count_insert_of_ne this at e], assumption end)) @@ -274,8 +274,8 @@ private definition min_count (l₁ l₂ : list A) : list A → list A private lemma not_mem_max_count_of_not_mem (l₁ l₂ : list A) : ∀ {a l}, a ∉ l → a ∉ max_count l₁ l₂ l | a [] h := !not_mem_nil | a (b::l) h := - assert ih : a ∉ max_count l₁ l₂ l, from not_mem_max_count_of_not_mem (not_mem_of_not_mem_cons h), - assert a ≠ b, from ne_of_not_mem_cons h, + have ih : a ∉ max_count l₁ l₂ l, from not_mem_max_count_of_not_mem (not_mem_of_not_mem_cons h), + have a ≠ b, from ne_of_not_mem_cons h, by_cases (suppose list.count b l₁ ≥ list.count b l₂, begin unfold max_count, rewrite [if_pos this], @@ -289,12 +289,12 @@ private lemma not_mem_max_count_of_not_mem (l₁ l₂ : list A) : ∀ {a l}, a private lemma max_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a ∈ l → nodup l → list.count a (max_count l₁ l₂ l) = max (list.count a l₁) (list.count a l₂) | a [] h₁ h₂ := absurd h₁ !not_mem_nil | a (b::l) h₁ h₂ := - assert nodup l, from nodup_of_nodup_cons h₂, - assert b ∉ l, from not_mem_of_nodup_cons h₂, + have nodup l, from nodup_of_nodup_cons h₂, + have b ∉ l, from not_mem_of_nodup_cons h₂, or.elim (eq_or_mem_of_mem_cons h₁) (suppose a = b, have a ∉ l, by rewrite this; assumption, - assert a ∉ max_count l₁ l₂ l, from not_mem_max_count_of_not_mem l₁ l₂ this, + have a ∉ max_count l₁ l₂ l, from not_mem_max_count_of_not_mem l₁ l₂ this, by_cases (suppose i : list.count a l₁ ≥ list.count a l₂, begin unfold max_count, subst b, @@ -305,8 +305,8 @@ private lemma max_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a rewrite [if_neg i, list.count_append, count_gen, max_eq_right_of_lt (lt_of_not_ge i), count_eq_zero_of_not_mem `a ∉ max_count l₁ l₂ l`] end)) (suppose a ∈ l, - assert a ≠ b, from suppose a = b, begin subst b, contradiction end, - assert ih : list.count a (max_count l₁ l₂ l) = max (list.count a l₁) (list.count a l₂), from + have a ≠ b, from suppose a = b, begin subst b, contradiction end, + have ih : list.count a (max_count l₁ l₂ l) = max (list.count a l₁) (list.count a l₂), from max_count_eq `a ∈ l` `nodup l`, by_cases (suppose i : list.count b l₁ ≥ list.count b l₂, begin @@ -321,8 +321,8 @@ private lemma max_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a private lemma not_mem_min_count_of_not_mem (l₁ l₂ : list A) : ∀ {a l}, a ∉ l → a ∉ min_count l₁ l₂ l | a [] h := !not_mem_nil | a (b::l) h := - assert ih : a ∉ min_count l₁ l₂ l, from not_mem_min_count_of_not_mem (not_mem_of_not_mem_cons h), - assert a ≠ b, from ne_of_not_mem_cons h, + have ih : a ∉ min_count l₁ l₂ l, from not_mem_min_count_of_not_mem (not_mem_of_not_mem_cons h), + have a ≠ b, from ne_of_not_mem_cons h, by_cases (suppose list.count b l₁ ≤ list.count b l₂, begin unfold min_count, rewrite [if_pos this], @@ -336,12 +336,12 @@ private lemma not_mem_min_count_of_not_mem (l₁ l₂ : list A) : ∀ {a l}, a private lemma min_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a ∈ l → nodup l → list.count a (min_count l₁ l₂ l) = min (list.count a l₁) (list.count a l₂) | a [] h₁ h₂ := absurd h₁ !not_mem_nil | a (b::l) h₁ h₂ := - assert nodup l, from nodup_of_nodup_cons h₂, - assert b ∉ l, from not_mem_of_nodup_cons h₂, + have nodup l, from nodup_of_nodup_cons h₂, + have b ∉ l, from not_mem_of_nodup_cons h₂, or.elim (eq_or_mem_of_mem_cons h₁) (suppose a = b, have a ∉ l, by rewrite this; assumption, - assert a ∉ min_count l₁ l₂ l, from not_mem_min_count_of_not_mem l₁ l₂ this, + have a ∉ min_count l₁ l₂ l, from not_mem_min_count_of_not_mem l₁ l₂ this, by_cases (suppose i : list.count a l₁ ≤ list.count a l₂, begin unfold min_count, subst b, @@ -352,8 +352,8 @@ private lemma min_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a rewrite [if_neg i, list.count_append, count_gen, min_eq_right (le_of_lt (lt_of_not_ge i)), count_eq_zero_of_not_mem `a ∉ min_count l₁ l₂ l`] end)) (suppose a ∈ l, - assert a ≠ b, from suppose a = b, by subst b; contradiction, - assert ih : list.count a (min_count l₁ l₂ l) = min (list.count a l₁) (list.count a l₂), from min_count_eq `a ∈ l` `nodup l`, + have a ≠ b, from suppose a = b, by subst b; contradiction, + have ih : list.count a (min_count l₁ l₂ l) = min (list.count a l₁) (list.count a l₂), from min_count_eq `a ∈ l` `nodup l`, by_cases (suppose i : list.count b l₁ ≤ list.count b l₂, begin unfold min_count, @@ -367,8 +367,8 @@ private lemma min_count_eq (l₁ l₂ : list A) : ∀ {a : A} {l : list A}, a private lemma perm_max_count_left {l₁ l₂ l₃ l₄ : list A} (h₁ : l₁ ~ l₃) (h₂ : l₂ ~ l₄) : ∀ l, max_count l₁ l₂ l ~ max_count l₃ l₄ l | [] := by esimp | (a::l) := - assert e₁ : list.count a l₁ = list.count a l₃, from count_eq_of_perm h₁ a, - assert e₂ : list.count a l₂ = list.count a l₄, from count_eq_of_perm h₂ a, + have e₁ : list.count a l₁ = list.count a l₃, from count_eq_of_perm h₁ a, + have e₂ : list.count a l₂ = list.count a l₄, from count_eq_of_perm h₂ a, by_cases (suppose list.count a l₁ ≥ list.count a l₂, begin unfold max_count, rewrite [-e₁, -e₂, *if_pos this], exact perm_app !perm.refl !perm_max_count_left end) @@ -408,8 +408,8 @@ calc max_count l₁ l₂ l₃ ~ max_count r₁ r₂ l₃ : perm_max_count_left p private lemma perm_min_count_left {l₁ l₂ l₃ l₄ : list A} (h₁ : l₁ ~ l₃) (h₂ : l₂ ~ l₄) : ∀ l, min_count l₁ l₂ l ~ min_count l₃ l₄ l | [] := by esimp | (a::l) := - assert e₁ : list.count a l₁ = list.count a l₃, from count_eq_of_perm h₁ a, - assert e₂ : list.count a l₂ = list.count a l₄, from count_eq_of_perm h₂ a, + have e₁ : list.count a l₁ = list.count a l₃, from count_eq_of_perm h₁ a, + have e₂ : list.count a l₂ = list.count a l₄, from count_eq_of_perm h₂ a, by_cases (suppose list.count a l₁ ≤ list.count a l₂, begin unfold min_count, rewrite [-e₁, -e₂, *if_pos this], exact perm_app !perm.refl !perm_min_count_left end) @@ -455,9 +455,9 @@ lemma count_union (a : A) (b₁ b₂ : bag A) : count a (b₁ ∪ b₂) = max (c quot.induction_on₂ b₁ b₂ (λ l₁ l₂, by_cases (suppose a ∈ union_list l₁ l₂, !max_count_eq this !nodup_union_list) (suppose ¬ a ∈ union_list l₁ l₂, - assert ¬ a ∈ l₁, from not_mem_of_not_mem_union_list_left `¬ a ∈ union_list l₁ l₂`, - assert ¬ a ∈ l₂, from not_mem_of_not_mem_union_list_right `¬ a ∈ union_list l₁ l₂`, - assert n : ¬ a ∈ max_count l₁ l₂ (union_list l₁ l₂), from not_mem_max_count_of_not_mem l₁ l₂ `¬ a ∈ union_list l₁ l₂`, + have ¬ a ∈ l₁, from not_mem_of_not_mem_union_list_left `¬ a ∈ union_list l₁ l₂`, + have ¬ a ∈ l₂, from not_mem_of_not_mem_union_list_right `¬ a ∈ union_list l₁ l₂`, + have n : ¬ a ∈ max_count l₁ l₂ (union_list l₁ l₂), from not_mem_max_count_of_not_mem l₁ l₂ `¬ a ∈ union_list l₁ l₂`, begin unfold [union, count], rewrite [count_eq_zero_of_not_mem `¬ a ∈ l₁`, count_eq_zero_of_not_mem `¬ a ∈ l₂`, max_self], @@ -468,9 +468,9 @@ lemma count_inter (a : A) (b₁ b₂ : bag A) : count a (b₁ ∩ b₂) = min (c quot.induction_on₂ b₁ b₂ (λ l₁ l₂, by_cases (suppose a ∈ union_list l₁ l₂, !min_count_eq this !nodup_union_list) (suppose ¬ a ∈ union_list l₁ l₂, - assert ¬ a ∈ l₁, from not_mem_of_not_mem_union_list_left `¬ a ∈ union_list l₁ l₂`, - assert ¬ a ∈ l₂, from not_mem_of_not_mem_union_list_right `¬ a ∈ union_list l₁ l₂`, - assert n : ¬ a ∈ min_count l₁ l₂ (union_list l₁ l₂), from not_mem_min_count_of_not_mem l₁ l₂ `¬ a ∈ union_list l₁ l₂`, + have ¬ a ∈ l₁, from not_mem_of_not_mem_union_list_left `¬ a ∈ union_list l₁ l₂`, + have ¬ a ∈ l₂, from not_mem_of_not_mem_union_list_right `¬ a ∈ union_list l₁ l₂`, + have n : ¬ a ∈ min_count l₁ l₂ (union_list l₁ l₂), from not_mem_min_count_of_not_mem l₁ l₂ `¬ a ∈ union_list l₁ l₂`, begin unfold [inter, count], rewrite [count_eq_zero_of_not_mem `¬ a ∈ l₁`, count_eq_zero_of_not_mem `¬ a ∈ l₂`, min_self], @@ -653,10 +653,10 @@ private lemma all_of_subcount_eq_tt : ∀ {l₁ l₂ : list A}, subcount l₁ l | [] l₂ h := take x, !zero_le | (a::l₁) l₂ h := take x, have subcount l₁ l₂ = tt, from by_contradiction (suppose subcount l₁ l₂ ≠ tt, - assert subcount l₁ l₂ = ff, from eq_ff_of_ne_tt this, + have subcount l₁ l₂ = ff, from eq_ff_of_ne_tt this, begin unfold subcount at h, rewrite [this at h, if_t_t at h], contradiction end), - assert ih : ∀ a, list.count a l₁ ≤ list.count a l₂, from all_of_subcount_eq_tt this, - assert i : list.count a (a::l₁) ≤ list.count a l₂, from by_contradiction (suppose ¬ list.count a (a::l₁) ≤ list.count a l₂, + have ih : ∀ a, list.count a l₁ ≤ list.count a l₂, from all_of_subcount_eq_tt this, + have i : list.count a (a::l₁) ≤ list.count a l₂, from by_contradiction (suppose ¬ list.count a (a::l₁) ≤ list.count a l₂, begin unfold subcount at h, rewrite [if_neg this at h], contradiction end), by_cases (suppose x = a, by rewrite this; apply i) @@ -667,7 +667,7 @@ private lemma ex_of_subcount_eq_ff : ∀ {l₁ l₂ : list A}, subcount l₁ l | (a::l₁) l₂ h := by_cases (suppose i : list.count a (a::l₁) ≤ list.count a l₂, have subcount l₁ l₂ = ff, from by_contradiction (suppose subcount l₁ l₂ ≠ ff, - assert subcount l₁ l₂ = tt, from eq_tt_of_ne_ff this, + have subcount l₁ l₂ = tt, from eq_tt_of_ne_ff this, begin unfold subcount at h, rewrite [if_pos i at h, this at h], diff --git a/library/data/encodable.lean b/library/data/encodable.lean index c3f310173..3b8d2fdba 100644 --- a/library/data/encodable.lean +++ b/library/data/encodable.lean @@ -19,7 +19,7 @@ definition countable_of_encodable {A : Type} : encodable A → countable A := assume e : encodable A, have injective encode, from λ (a₁ a₂ : A) (h : encode a₁ = encode a₂), - assert decode A (encode a₁) = decode A (encode a₂), by rewrite h, + have decode A (encode a₁) = decode A (encode a₂), by rewrite h, by rewrite [*encodek at this]; injection this; assumption, exists.intro encode this @@ -71,16 +71,16 @@ else open decidable private theorem decode_encode_sum : ∀ s : sum A B, decode_sum (encode_sum s) = some s | (sum.inl a) := - assert aux : 2 > (0:nat), from dec_trivial, + have aux : 2 > (0:nat), from dec_trivial, begin esimp [encode_sum, decode_sum], rewrite [mul_mod_right, if_pos (eq.refl (0 : nat)), nat.mul_div_cancel_left _ aux, encodable.encodek] end | (sum.inr b) := - assert aux₁ : 2 > (0:nat), from dec_trivial, - assert aux₂ : 1 % 2 = (1:nat), by rewrite [nat.mod_def], - assert aux₃ : 1 ≠ (0:nat), from dec_trivial, + have aux₁ : 2 > (0:nat), from dec_trivial, + have aux₂ : 1 % 2 = (1:nat), by rewrite [nat.mod_def], + have aux₃ : 1 ≠ (0:nat), from dec_trivial, begin esimp [encode_sum, decode_sum], rewrite [add.comm, add_mul_mod_self_left, aux₂, if_neg aux₃, nat.add_sub_cancel_left, @@ -225,9 +225,9 @@ private lemma enle.total (a b : A) : enle a b ∨ enle b a := private lemma enle.antisymm (a b : A) : enle a b → enle b a → a = b := assume h₁ h₂, -assert encode a = encode b, from le.antisymm h₁ h₂, -assert decode A (encode a) = decode A (encode b), by rewrite this, -assert some a = some b, by rewrite [*encodek at this]; exact this, +have encode a = encode b, from le.antisymm h₁ h₂, +have decode A (encode a) = decode A (encode b), by rewrite this, +have some a = some b, by rewrite [*encodek at this]; exact this, option.no_confusion this (λ e, e) private definition decidable_enle [instance] (a b : A) : decidable (enle a b) := @@ -248,7 +248,7 @@ quot.lift_on s (λ l, encode (ensort (elt_of l))) (λ l₁ l₂ p, have elt_of l₁ ~ elt_of l₂, from p, - assert ensort (elt_of l₁) = ensort (elt_of l₂), from sorted_eq_of_perm this, + have ensort (elt_of l₁) = ensort (elt_of l₂), from sorted_eq_of_perm this, by rewrite this) private definition decode_finset (n : nat) : option (finset A) := diff --git a/library/data/equiv.lean b/library/data/equiv.lean index 10bc82c8b..977cc0a2e 100644 --- a/library/data/equiv.lean +++ b/library/data/equiv.lean @@ -32,9 +32,9 @@ definition inv {A B : Type} [e : equiv A B] : B → A := lemma eq_of_to_fun_eq {A B : Type} : ∀ {e₁ e₂ : equiv A B}, fn e₁ = fn e₂ → e₁ = e₂ | (mk f₁ g₁ l₁ r₁) (mk f₂ g₂ l₂ r₂) h := - assert f₁ = f₂, from h, - assert g₁ = g₂, from funext (λ x, - assert f₁ (g₁ x) = f₂ (g₂ x), from eq.trans (r₁ x) (eq.symm (r₂ x)), + have f₁ = f₂, from h, + have g₁ = g₂, from funext (λ x, + have f₁ (g₁ x) = f₂ (g₂ x), from eq.trans (r₁ x) (eq.symm (r₂ x)), have f₁ (g₁ x) = f₁ (g₂ x), begin subst f₂, exact this end, show g₁ x = g₂ x, from injective_of_left_inverse l₁ this), by congruence; repeat assumption @@ -295,7 +295,7 @@ open decidable definition decidable_eq_of_equiv {A B : Type} [h : decidable_eq A] : A ≃ B → decidable_eq B | (mk f g l r) := take b₁ b₂, match h (g b₁) (g b₂) with - | inl he := inl (assert aux : f (g b₁) = f (g b₂), from congr_arg f he, + | inl he := inl (have aux : f (g b₁) = f (g b₂), from congr_arg f he, begin rewrite *r at aux, exact aux end) | inr hn := inr (λ b₁eqb₂, by subst b₁eqb₂; exact absurd rfl hn) end @@ -330,7 +330,7 @@ by_cases (suppose r = a, by_cases (suppose r = b, begin unfold swap_core, rewrite [if_pos `r = a`, if_pos (eq.refl b), -`r = a`, -`r = b`, if_pos (eq.refl r)] end) (suppose ¬ r = b, - assert b ≠ a, from assume h, begin rewrite h at this, contradiction end, + have b ≠ a, from assume h, begin rewrite h at this, contradiction end, begin unfold swap_core, rewrite [*if_pos `r = a`, if_pos (eq.refl b), if_neg `b ≠ a`, `r = a`] end)) (suppose ¬ r = a, by_cases (suppose r = b, begin unfold swap_core, rewrite [if_neg `¬ r = a`, *if_pos `r = b`, if_pos (eq.refl a), this] end) diff --git a/library/data/examples/vector.lean b/library/data/examples/vector.lean index 76d824887..d593dbe0b 100644 --- a/library/data/examples/vector.lean +++ b/library/data/examples/vector.lean @@ -257,11 +257,11 @@ namespace vector | 0 (m+1) [] (y::ys) h₁ h₂ := by contradiction | (n+1) 0 (x::xs) [] h₁ h₂ := by contradiction | (n+1) (m+1) (x::xs) (y::ys) h₁ h₂ := - assert e₁ : n = m, from succ.inj h₂, - assert e₂ : x = y, begin unfold to_list at h₁, injection h₁, assumption end, + have e₁ : n = m, from succ.inj h₂, + have e₂ : x = y, begin unfold to_list at h₁, injection h₁, assumption end, have e₃ : to_list xs = to_list ys, begin unfold to_list at h₁, injection h₁, assumption end, - assert xs == ys, from heq_of_list_eq e₃ e₁, - assert y :: xs == y :: ys, begin clear heq_of_list_eq h₁ h₂ e₃, revert xs ys this, induction e₁, intro xs ys h, rewrite [eq_of_heq h] end, + have xs == ys, from heq_of_list_eq e₃ e₁, + have y :: xs == y :: ys, begin clear heq_of_list_eq h₁ h₂ e₃, revert xs ys this, induction e₁, intro xs ys h, rewrite [eq_of_heq h] end, show x :: xs == y :: ys, by rewrite e₂; exact this theorem list_eq_of_heq {n m} {v₁ : vector A n} {v₂ : vector A m} : v₁ == v₂ → n = m → to_list v₁ = to_list v₂ := diff --git a/library/data/fin.lean b/library/data/fin.lean index 30338ca37..bc856c9f8 100644 --- a/library/data/fin.lean +++ b/library/data/fin.lean @@ -54,7 +54,7 @@ dmap_nodup_of_dinj (dinj_lt n) (list.nodup_upto n) lemma mem_upto (n : nat) : ∀ (i : fin n), i ∈ upto n := take i, fin.destruct i (take ival Piltn, - assert ival ∈ list.upto n, from mem_upto_of_lt Piltn, + have ival ∈ list.upto n, from mem_upto_of_lt Piltn, mem_dmap Piltn this) lemma upto_zero : upto 0 = [] := @@ -138,7 +138,7 @@ by intro hlt he; substvars; exact absurd hlt (lt.irrefl n) lemma lt_max_of_ne_max {i : fin (succ n)} : i ≠ maxi → i < n := assume hne : i ≠ maxi, -assert vne : val i ≠ n, from +have vne : val i ≠ n, from assume he, have val (@maxi n) = n, from rfl, have val i = val (@maxi n), from he ⬝ this⁻¹, @@ -160,7 +160,7 @@ take i j, destruct i (destruct j (take iv ilt jv jlt Pmkeq, lemma lt_of_inj_of_max (f : fin (succ n) → fin (succ n)) : injective f → (f maxi = maxi) → ∀ i : fin (succ n), i < n → f i < n := assume Pinj Peq, take i, assume Pilt, -assert P1 : f i = f maxi → i = maxi, from assume Peq, Pinj i maxi Peq, +have P1 : f i = f maxi → i = maxi, from assume Peq, Pinj i maxi Peq, have f i ≠ maxi, from begin rewrite -Peq, intro P2, apply absurd (P1 P2) (ne_max_of_lt_max Pilt) end, lt_max_of_ne_max this @@ -188,7 +188,7 @@ end lemma lift_fun_of_inj {f : fin n → fin n} : injective f → injective (lift_fun f) := assume Pinj, take i j, -assert Pdi : decidable (i = maxi), from _, assert Pdj : decidable (j = maxi), from _, +have Pdi : decidable (i = maxi), from _, have Pdj : decidable (j = maxi), from _, begin cases Pdi with Pimax Pinmax, cases Pdj with Pjmax Pjnmax, @@ -205,7 +205,7 @@ end lemma lift_fun_inj : injective (@lift_fun n) := take f₁ f₂ Peq, funext (λ i, -assert lift_fun f₁ (lift_succ i) = lift_fun f₂ (lift_succ i), from congr_fun Peq _, +have lift_fun f₁ (lift_succ i) = lift_fun f₂ (lift_succ i), from congr_fun Peq _, begin revert this, rewrite [*lift_fun_eq], apply lift_succ_inj end) lemma lower_inj_apply {f Pinj Pmax} (i : fin n) : @@ -299,7 +299,7 @@ begin let vj := nat.pred vk in have vk = vj+1, from eq.symm (succ_pred_of_pos HT), - assert vj < n, from + have vj < n, from lt_of_succ_lt_succ (eq.subst `vk = vj+1` pk), have succ (mk vj `vj < n`) = mk vk pk, from val_inj (eq.symm `vk = vj+1`), @@ -405,7 +405,7 @@ definition fin_one_equiv_unit : fin 1 ≃ unit := ⦄ definition fin_sum_equiv (n m : nat) : (fin n + fin m) ≃ fin (n+m) := -assert aux₁ : ∀ {v}, v < m → (v + n) < (n + m), from +have aux₁ : ∀ {v}, v < m → (v + n) < (n + m), from take v, suppose v < m, calc v + n < m + n : add_lt_add_of_lt_of_le this !le.refl ... = n + m : add.comm, @@ -425,7 +425,7 @@ assert aux₁ : ∀ {v}, v < m → (v + n) < (n + m), from { cases f₂ with v hlt, esimp, have ¬ v + n < n, from suppose v + n < n, - assert v < n - n, from nat.lt_sub_of_add_lt this !le.refl, + have v < n - n, from nat.lt_sub_of_add_lt this !le.refl, have v < 0, by rewrite [nat.sub_self at this]; exact this, absurd this !not_lt_zero, rewrite [dif_neg this], congruence, congruence, rewrite [nat.add_sub_cancel] } @@ -439,17 +439,17 @@ assert aux₁ : ∀ {v}, v < m → (v + n) < (n + m), from definition fin_prod_equiv_of_pos (n m : nat) : n > 0 → (fin n × fin m) ≃ fin (n*m) := suppose n > 0, -assert aux₁ : ∀ {v₁ v₂}, v₁ < n → v₂ < m → v₁ + v₂ * n < n*m, from +have aux₁ : ∀ {v₁ v₂}, v₁ < n → v₂ < m → v₁ + v₂ * n < n*m, from take v₁ v₂, assume h₁ h₂, have nat.succ v₂ ≤ m, from succ_le_of_lt h₂, - assert nat.succ v₂ * n ≤ m * n, from mul_le_mul_right _ this, + have nat.succ v₂ * n ≤ m * n, from mul_le_mul_right _ this, have v₂ * n + n ≤ n * m, by rewrite [-add_one at this, right_distrib at this, one_mul at this, mul.comm m n at this]; exact this, - assert v₁ + (v₂ * n + n) < n + n * m, from add_lt_add_of_lt_of_le h₁ this, + have v₁ + (v₂ * n + n) < n + n * m, from add_lt_add_of_lt_of_le h₁ this, have v₁ + v₂ * n + n < n * m + n, by rewrite [add.assoc, add.comm (n*m) n]; exact this, lt_of_add_lt_add_right this, -assert aux₂ : ∀ v, v % n < n, from +have aux₂ : ∀ v, v % n < n, from take v, mod_lt _ `n > 0`, -assert aux₃ : ∀ {v}, v < n * m → v / n < m, from +have aux₃ : ∀ {v}, v < n * m → v / n < m, from take v, assume h, by rewrite mul.comm at h; exact nat.div_lt_of_lt_mul h, ⦃ equiv, to_fun := λ p : (fin n × fin m), match p with (mk v₁ hlt₁, mk v₂ hlt₂) := mk (v₁ + v₂ * n) (aux₁ hlt₁ hlt₂) end, diff --git a/library/data/finset/basic.lean b/library/data/finset/basic.lean index 82164ccec..1b2f6a2d4 100644 --- a/library/data/finset/basic.lean +++ b/library/data/finset/basic.lean @@ -52,7 +52,7 @@ definition to_finset [decidable_eq A] (l : list A) : finset A := lemma to_finset_eq_of_nodup [decidable_eq A] {l : list A} (n : nodup l) : to_finset_of_nodup l n = to_finset l := -assert P : to_nodup_list_of_nodup n = to_nodup_list l, from +have P : to_nodup_list_of_nodup n = to_nodup_list l, from begin rewrite [↑to_nodup_list, ↑to_nodup_list_of_nodup], congruence, @@ -237,11 +237,11 @@ quot.induction_on s (assume nodup_l, H1) (take a l', assume IH nodup_al', - assert aux₁: a ∉ l', from not_mem_of_nodup_cons nodup_al', - assert e : list.insert a l' = a :: l', from insert_eq_of_not_mem aux₁, - assert nodup l', from nodup_of_nodup_cons nodup_al', - assert P (quot.mk (subtype.tag l' this)), from IH this, - assert P (insert a (quot.mk (subtype.tag l' _))), from H2 aux₁ this, + have aux₁: a ∉ l', from not_mem_of_nodup_cons nodup_al', + have e : list.insert a l' = a :: l', from insert_eq_of_not_mem aux₁, + have nodup l', from nodup_of_nodup_cons nodup_al', + have P (quot.mk (subtype.tag l' this)), from IH this, + have P (insert a (quot.mk (subtype.tag l' _))), from H2 aux₁ this, begin revert nodup_al', rewrite [-e], @@ -319,7 +319,7 @@ theorem erase_insert {a : A} {s : finset A} : a ∉ s → erase a (insert a s) = (λ bin, by subst b; contradiction)) (λ bnea : b ≠ a, iff.intro (λ bin, - assert b ∈ insert a s, from mem_of_mem_erase bin, + have b ∈ insert a s, from mem_of_mem_erase bin, mem_of_mem_insert_of_ne this bnea) (λ bin, have b ∈ insert a s, from mem_insert_of_mem _ bin, diff --git a/library/data/finset/card.lean b/library/data/finset/card.lean index 24bcd3a1a..fff8184b4 100644 --- a/library/data/finset/card.lean +++ b/library/data/finset/card.lean @@ -22,7 +22,7 @@ begin show card s₁ + card (insert a s₂) = card (s₁ ∪ (insert a s₂)) + card (s₁ ∩ (insert a s₂)), from decidable.by_cases (assume as1 : a ∈ s₁, - assert H : a ∉ s₁ ∩ s₂, from assume H', ans2 (mem_of_mem_inter_right H'), + have H : a ∉ s₁ ∩ s₂, from assume H', ans2 (mem_of_mem_inter_right H'), begin rewrite [card_insert_of_not_mem ans2, union_comm, -insert_union, union_comm], rewrite [insert_union, insert_eq_of_mem as1, insert_eq, inter_distrib_left, inter_comm], @@ -30,7 +30,7 @@ begin rewrite IH end) (assume ans1 : a ∉ s₁, - assert H : a ∉ s₁ ∪ s₂, from assume H', + have H : a ∉ s₁ ∪ s₂, from assume H', or.elim (mem_or_mem_of_mem_union H') (assume as1, ans1 as1) (assume as2, ans2 as2), begin rewrite [card_insert_of_not_mem ans2, union_comm, -insert_union, union_comm], @@ -92,8 +92,8 @@ lemma card_le_of_inj_on (a : finset A) (b : finset B) (Pex : ∃ f : A → B, set.inj_on f (ts a) ∧ (image f a ⊆ b)): card a ≤ card b := obtain f Pinj, from Pex, -assert Psub : _, from and.right Pinj, -assert Ple : card (image f a) ≤ card b, from card_le_card_of_subset Psub, +have Psub : _, from and.right Pinj, +have Ple : card (image f a) ≤ card b, from card_le_card_of_subset Psub, by rewrite [(card_image_eq_of_inj_on (and.left Pinj))⁻¹]; exact Ple theorem card_image_le (f : A → B) (s : finset A) : card (image f s) ≤ card s := @@ -167,7 +167,7 @@ theorem eq_of_card_eq_of_subset {s₁ s₂ : finset A} (Hcard : card s₁ = card s₁ = s₂ := have H : card s₁ + 0 = card s₁ + card (s₂ \ s₁), by rewrite [Hcard at {1}, card_eq_card_add_card_diff Hsub], -assert H1 : s₂ \ s₁ = ∅, from eq_empty_of_card_eq_zero (add.left_cancel H)⁻¹, +have H1 : s₂ \ s₁ = ∅, from eq_empty_of_card_eq_zero (add.left_cancel H)⁻¹, by rewrite [-union_diff_cancel Hsub, H1, union_empty] lemma exists_two_of_card_gt_one {s : finset A} : 1 < card s → ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b := @@ -210,12 +210,12 @@ finset.induction_on s have H2 : ∀ a₁ a₂ : A, a₁ ∈ s' → a₂ ∈ s' → a₁ ≠ a₂ → f a₁ ∩ f a₂ = ∅, from take a₁ a₂, assume H3 H4 H5, H1 (!mem_insert_of_mem H3) (!mem_insert_of_mem H4) H5, - assert H6 : card (⋃ (x : A) ∈ s', f x) = ∑ (x : A) ∈ s', card (f x), from IH H2, - assert H7 : ∀ x, x ∈ s' → f a ∩ f x = ∅, from + have H6 : card (⋃ (x : A) ∈ s', f x) = ∑ (x : A) ∈ s', card (f x), from IH H2, + have H7 : ∀ x, x ∈ s' → f a ∩ f x = ∅, from take x, assume xs', have anex : a ≠ x, from assume aex, (eq.subst aex H) xs', H1 !mem_insert (!mem_insert_of_mem xs') anex, - assert H8 : f a ∩ (⋃ (x : A) ∈ s', f x) = ∅, from + have H8 : f a ∩ (⋃ (x : A) ∈ s', f x) = ∅, from calc f a ∩ (⋃ (x : A) ∈ s', f x) = (⋃ (x : A) ∈ s', f a ∩ f x) : by rewrite inter_Union ... = (⋃ (x : A) ∈ s', ∅) : by rewrite [Union_ext H7] diff --git a/library/data/finset/comb.lean b/library/data/finset/comb.lean index bc7b16df4..672967979 100644 --- a/library/data/finset/comb.lean +++ b/library/data/finset/comb.lean @@ -460,7 +460,7 @@ begin apply yps end)) (assume H : x ⊆ insert a s, - assert H' : erase a x ⊆ s, from erase_subset_of_subset_insert H, + have H' : erase a x ⊆ s, from erase_subset_of_subset_insert H, decidable.by_cases (suppose a ∈ x, or.inr (exists.intro (erase a x) diff --git a/library/data/finset/equiv.lean b/library/data/finset/equiv.lean index 27a7d6caa..92cc2d4b5 100644 --- a/library/data/finset/equiv.lean +++ b/library/data/finset/equiv.lean @@ -35,8 +35,8 @@ private lemma mem_of_nat_of_odd {n : nat} {s : nat} : odd (s / 2^n) → n ∈ of assume h, have 2^n < succ s, from by_contradiction (suppose ¬(2^n < succ s), - assert 2^n > s, from lt_of_succ_le (le_of_not_gt this), - assert s / 2^n = 0, from div_eq_zero_of_lt this, + have 2^n > s, from lt_of_succ_le (le_of_not_gt this), + have s / 2^n = 0, from div_eq_zero_of_lt this, by rewrite this at h; exact absurd h dec_trivial), have n < succ s, from calc n ≤ 2^n : le_pow_self dec_trivial n @@ -47,12 +47,12 @@ mem_sep_of_mem this h private lemma succ_mem_of_nat (n : nat) (s : nat) : succ n ∈ of_nat s ↔ n ∈ of_nat (s / 2) := iff.intro (suppose succ n ∈ of_nat s, - assert odd (s / 2^(succ n)), from odd_of_mem_of_nat this, + have odd (s / 2^(succ n)), from odd_of_mem_of_nat this, have odd ((s / 2) / (2 ^ n)), by rewrite [pow_succ' at this, nat.div_div_eq_div_mul, mul.comm]; assumption, show n ∈ of_nat (s / 2), from mem_of_nat_of_odd this) (suppose n ∈ of_nat (s / 2), - assert odd ((s / 2) / (2 ^ n)), from odd_of_mem_of_nat this, - assert odd (s / 2^(succ n)), by rewrite [pow_succ', mul.comm, -nat.div_div_eq_div_mul]; assumption, + have odd ((s / 2) / (2 ^ n)), from odd_of_mem_of_nat this, + have odd (s / 2^(succ n)), by rewrite [pow_succ', mul.comm, -nat.div_div_eq_div_mul]; assumption, show succ n ∈ of_nat s, from mem_of_nat_of_odd this) private lemma odd_of_zero_mem (s : nat) : 0 ∈ of_nat s ↔ odd s := @@ -78,7 +78,7 @@ finset.induction_on s dec_trivial suppose even (2^a + to_nat s), by_cases (suppose e : even (2^a), by_cases (suppose even (to_nat s), - assert 0 ∉ s, from iff.mp ih this, + have 0 ∉ s, from iff.mp ih this, suppose 0 ∈ insert a s, or.elim (eq_or_mem_of_mem_insert this) (suppose 0 = a, begin rewrite [-this at e], exact absurd e not_even_one end) (by contradiction)) @@ -90,7 +90,7 @@ finset.induction_on s dec_trivial have even (to_nat s), from iff.mpr ih (by rewrite -this at nains; exact nains), absurd this `odd (to_nat s)`) (suppose 0 ∈ s, - assert a ≠ 0, from suppose a = 0, by subst a; contradiction, + have a ≠ 0, from suppose a = 0, by subst a; contradiction, begin cases a with a, exact absurd rfl `0 ≠ 0`, have odd (2*2^a), by rewrite [pow_succ' at o, mul.comm]; exact o, @@ -111,9 +111,9 @@ finset.induction_on s dec_trivial private lemma of_nat_eq_insert_zero {s : nat} : 0 ∉ of_nat s → of_nat (2^0 + s) = insert 0 (of_nat s) := assume h : 0 ∉ of_nat s, -assert even s, from iff.mp (even_of_not_zero_mem s) h, +have even s, from iff.mp (even_of_not_zero_mem s) h, have odd (s+1), from odd_succ_of_even this, -assert zmem : 0 ∈ of_nat (s+1), from iff.mpr (odd_of_zero_mem (s+1)) this, +have zmem : 0 ∈ of_nat (s+1), from iff.mpr (odd_of_zero_mem (s+1)) this, obtain w (hw : s = 2*w), from exists_of_even `even s`, begin rewrite [pow_zero, add.comm, hw], @@ -122,8 +122,8 @@ begin match n with | 0 := iff.intro (λ h, !mem_insert) (λ h, by rewrite [hw at zmem]; exact zmem) | succ m := - assert d₁ : 1 / 2 = (0:nat), from dec_trivial, - assert aux : _, from calc + have d₁ : 1 / 2 = (0:nat), from dec_trivial, + have aux : _, from calc succ m ∈ of_nat (2 * w + 1) ↔ m ∈ of_nat ((2*w+1) / 2) : succ_mem_of_nat ... ↔ m ∈ of_nat w : by rewrite [add.comm, add_mul_div_self_left _ _ (dec_trivial : 2 > 0), d₁, zero_add] ... ↔ m ∈ of_nat (2*w / 2) : by rewrite [mul.comm, nat.mul_div_cancel _ (dec_trivial : 2 > 0)] @@ -141,7 +141,7 @@ private lemma of_nat_eq_insert : ∀ {n s : nat}, n ∉ of_nat s → of_nat (2^n | (succ n) s h := have n ∉ of_nat (s / 2), from iff.mp (not_iff_not_of_iff !succ_mem_of_nat) h, - assert ih : of_nat (2^n + s / 2) = insert n (of_nat (s / 2)), from of_nat_eq_insert this, + have ih : of_nat (2^n + s / 2) = insert n (of_nat (s / 2)), from of_nat_eq_insert this, finset.ext (λ x, have gen : ∀ m, m ∈ of_nat (2^(succ n) + s) ↔ m ∈ insert (succ n) (of_nat s) | zero := @@ -163,13 +163,13 @@ private lemma of_nat_eq_insert : ∀ {n s : nat}, n ∉ of_nat s → of_nat (2^n ... ↔ odd s : aux₁ ... ↔ 0 ∈ insert (succ n) (of_nat s) : aux₂ | (succ m) := - assert aux : m ∈ insert n (of_nat (s / 2)) ↔ succ m ∈ insert (succ n) (of_nat s), from iff.intro + have aux : m ∈ insert n (of_nat (s / 2)) ↔ succ m ∈ insert (succ n) (of_nat s), from iff.intro (assume hl, or.elim (eq_or_mem_of_mem_insert hl) (suppose m = n, by subst m; apply mem_insert) (suppose m ∈ of_nat (s / 2), finset.mem_insert_of_mem _ (iff.mpr !succ_mem_of_nat this))) (assume hr, or.elim (eq_or_mem_of_mem_insert hr) (suppose succ m = succ n, - assert m = n, by injection this; assumption, + have m = n, by injection this; assumption, by subst m; apply mem_insert) (suppose succ m ∈ of_nat s, finset.mem_insert_of_mem _ (iff.mp !succ_mem_of_nat this))), calc @@ -189,7 +189,7 @@ private definition predimage (s : finset nat) : finset nat := private lemma mem_image_pred_of_succ_mem {n : nat} {s : finset nat} : succ n ∈ s → n ∈ image pred s := assume h, - assert pred (succ n) ∈ image pred s, from mem_image_of_mem _ h, + have pred (succ n) ∈ image pred s, from mem_image_of_mem _ h, begin rewrite [pred_succ at this], assumption end private lemma mem_predimage_of_succ_mem {n : nat} {s : finset nat} : succ n ∈ s → n ∈ predimage s := diff --git a/library/data/finset/partition.lean b/library/data/finset/partition.lean index 5bbf78036..89a194f68 100644 --- a/library/data/finset/partition.lean +++ b/library/data/finset/partition.lean @@ -32,8 +32,8 @@ lemma equiv_class_disjoint (f : partition) (a1 a2 : finset A) (Pa1 : a1 ∈ equi (Pa2 : a2 ∈ equiv_classes f) : a1 ≠ a2 → a1 ∩ a2 = ∅ := assume Pne, -assert Pe1 : _, from exists_of_mem_image Pa1, obtain g1 Pg1, from Pe1, -assert Pe2 : _, from exists_of_mem_image Pa2, obtain g2 Pg2, from Pe2, +have Pe1 : _, from exists_of_mem_image Pa1, obtain g1 Pg1, from Pe1, +have Pe2 : _, from exists_of_mem_image Pa2, obtain g2 Pg2, from Pe2, begin apply inter_eq_empty_of_disjoint, apply disjoint.intro, @@ -99,7 +99,7 @@ lemma binary_inter_empty_Union_disjoint_sets {P : finset A → Prop} [decP : dec assume Pds, inter_eq_empty (take a, assume Pa nPa, obtain s Psin Pains, from iff.elim_left !mem_Union_iff Pa, obtain t Ptin Paint, from iff.elim_left !mem_Union_iff nPa, - assert s ≠ t, + have s ≠ t, from assume Peq, absurd (Peq ▸ of_mem_sep Psin) (of_mem_sep Ptin), have e₁ : s ∩ t = empty, from Pds s t (mem_of_mem_sep Psin) (mem_of_mem_sep Ptin) `s ≠ t`, have a ∈ s ∩ t, from mem_inter Pains Paint, diff --git a/library/data/fintype/basic.lean b/library/data/fintype/basic.lean index 8c795d36e..ba0305769 100644 --- a/library/data/fintype/basic.lean +++ b/library/data/fintype/basic.lean @@ -23,7 +23,7 @@ definition fintype_of_equiv {A B : Type} [h : fintype A] : A ≃ B → fintype B (nodup_map (injective_of_left_inverse l) !fintype.unique) (λ b, have g b ∈ elements_of A, from fintype.complete (g b), - assert f (g b) ∈ map f (elements_of A), from mem_map f this, + have f (g b) ∈ map f (elements_of A), from mem_map f this, by rewrite r at this; exact this) end @@ -70,7 +70,7 @@ theorem ne_of_find_discr_eq_some {f g : A → B} {a : A} : ∀ {l}, find_discr f have find_discr f g l = some a, by rewrite [find_discr_cons_of_eq l this at e]; exact e, ne_of_find_discr_eq_some this) (assume h : f x ≠ g x, - assert some x = some a, by rewrite [find_discr_cons_of_ne l h at e]; exact e, + have some x = some a, by rewrite [find_discr_cons_of_ne l h at e]; exact e, by clear ne_of_find_discr_eq_some; injection this; subst a; exact h) theorem all_eq_of_find_discr_eq_none {f g : A → B} : ∀ {l}, find_discr f g l = none → ∀ a, a ∈ l → f a = g a @@ -151,7 +151,7 @@ match h₁ with obtain x px, from ex, absurd px (all_of_check_pred_eq_tt h (c x))) | ff := λ h : check_pred (λ a, ¬ p a) e = ff, inl ( - assert ∃ x, ¬¬p x, from ex_of_check_pred_eq_ff h, + have ∃ x, ¬¬p x, from ex_of_check_pred_eq_ff h, obtain x nnpx, from this, exists.intro x (not_not_elim nnpx)) end rfl end diff --git a/library/data/fintype/card.lean b/library/data/fintype/card.lean index 683895bab..7873bffbf 100644 --- a/library/data/fintype/card.lean +++ b/library/data/fintype/card.lean @@ -25,25 +25,25 @@ lemma card_le_of_inj (A : Type) [finA : fintype A] [deceqA : decidable_eq A] (B : Type) [finB : fintype B] [deceqB : decidable_eq B] : (∃ f : A → B, injective f) → card A ≤ card B := assume Pex, obtain f Pinj, from Pex, -assert Pinj_on_univ : _, from iff.mp !set.injective_iff_inj_on_univ Pinj, -assert Pinj_ts : set.inj_on f (ts univ), from to_set_univ⁻¹ ▸ Pinj_on_univ, -assert Psub : (image f univ) ⊆ univ, from !subset_univ, +have Pinj_on_univ : _, from iff.mp !set.injective_iff_inj_on_univ Pinj, +have Pinj_ts : set.inj_on f (ts univ), from to_set_univ⁻¹ ▸ Pinj_on_univ, +have Psub : (image f univ) ⊆ univ, from !subset_univ, finset.card_le_of_inj_on univ univ (exists.intro f (and.intro Pinj_ts Psub)) -- used to prove that inj ∧ eq card => surj lemma univ_of_card_eq_univ {A : Type} [finA : fintype A] [deceqA : decidable_eq A] {s : finset A} : finset.card s = card A → s = univ := assume Pcardeq, ext (take a, -assert D : decidable (a ∈ s), from decidable_mem a s, begin +have D : decidable (a ∈ s), from decidable_mem a s, begin apply iff.intro, intro ain, apply mem_univ, intro ain, cases D with Pin Pnin, exact Pin, - assert Pplus1 : finset.card (insert a s) = finset.card s + 1, - exact card_insert_of_not_mem Pnin, + have Pplus1 : finset.card (insert a s) = finset.card s + 1, + from card_insert_of_not_mem Pnin, rewrite Pcardeq at Pplus1, - assert Ple : finset.card (insert a s) ≤ card A, - apply card_le_card_of_subset, apply subset_univ, + have Ple : finset.card (insert a s) ≤ card A, + begin apply card_le_card_of_subset, apply subset_univ end, rewrite Pplus1 at Ple, exact false.elim (not_succ_le_self Ple) end) diff --git a/library/data/fintype/function.lean b/library/data/fintype/function.lean index a0d5cf923..b936ec60f 100644 --- a/library/data/fintype/function.lean +++ b/library/data/fintype/function.lean @@ -75,12 +75,12 @@ lemma nodup_mem_all_nodups [deceqA : decidable_eq A] {n : nat} ⦃l : list A⦄ lemma length_mem_all_lists : ∀ {n : nat} ⦃l : list A⦄, l ∈ all_lists_of_len n → length l = n | 0 [] := assume P, rfl -| 0 (a::l) := assume Pin, assert Peq : (a::l) = [], from mem_singleton Pin, +| 0 (a::l) := assume Pin, have Peq : (a::l) = [], from mem_singleton Pin, by contradiction | (succ n) [] := assume Pin, obtain pr Pprin Ppr, from exists_of_mem_map Pin, by contradiction | (succ n) (a::l) := assume Pin, obtain pr Pprin Ppr, from exists_of_mem_map Pin, - assert Pl : l ∈ all_lists_of_len n, + have Pl : l ∈ all_lists_of_len n, from mem_of_mem_product_right ((pair_of_cons Ppr) ▸ Pprin), by rewrite [length_cons, length_mem_all_lists Pl] @@ -138,8 +138,8 @@ lemma eq_of_kth_eq [deceqA : decidable_eq A] rewrite *kth_succ_of_cons at keq, exact keq end, - assert ih : l₁ = l₂, from eq_of_kth_eq ih₁ ih₂, - assert k₁ : a₁ = a₂, + have ih : l₁ = l₂, from eq_of_kth_eq ih₁ ih₂, + have k₁ : a₁ = a₂, begin have lt₁ : 0 < length (a₁::l₁), from !zero_lt_succ, have lt₂ : 0 < length (a₂::l₂), from !zero_lt_succ, @@ -163,7 +163,7 @@ lemma kth_find [deceqA : decidable_eq A] : ∀ {l : list A} {a} P, kth (find a l) l P = a | [] := take a, assume P, absurd P !not_lt_zero | (x::l) := take a, begin - assert Pd : decidable (a = x), {apply deceqA}, + have Pd : decidable (a = x), begin apply deceqA end, cases Pd with Pe Pne, rewrite [find_cons_of_eq l Pe], intro P, rewrite [kth_zero_of_cons, Pe], rewrite [find_cons_of_ne l Pne], intro P, rewrite [kth_succ_of_cons], @@ -179,8 +179,8 @@ lemma find_kth [deceqA : decidable_eq A] : end | (succ k) (a::l) := assume P, begin rewrite [kth_succ_of_cons], - assert Pd : decidable ((kth k l (lt_of_succ_lt_succ P)) = a), - {apply deceqA}, + have Pd : decidable ((kth k l (lt_of_succ_lt_succ P)) = a), + begin apply deceqA end, cases Pd with Pe Pne, rewrite [find_cons_of_eq l Pe], apply zero_lt_succ, rewrite [find_cons_of_ne l Pne], apply succ_lt_succ, apply find_kth @@ -193,10 +193,10 @@ lemma find_kth_of_nodup [deceqA : decidable_eq A] : by rewrite [kth_zero_of_cons, find_cons_of_eq l rfl] | (succ k) (a::l) := assume Plt Pnodup, begin rewrite [kth_succ_of_cons], - assert Pd : decidable ((kth k l (lt_of_succ_lt_succ Plt)) = a), - {apply deceqA}, + have Pd : decidable ((kth k l (lt_of_succ_lt_succ Plt)) = a), + begin apply deceqA end, cases Pd with Pe Pne, - assert Pin : a ∈ l, {rewrite -Pe, apply kth_mem}, + have Pin : a ∈ l, begin rewrite -Pe, apply kth_mem end, exact absurd Pin (not_mem_of_nodup_cons Pnodup), rewrite [find_cons_of_ne l Pne], apply congr (eq.refl succ), apply find_kth_of_nodup (lt_of_succ_lt_succ Plt) (nodup_of_nodup_cons Pnodup) @@ -265,10 +265,10 @@ include deceqB lemma fun_eq_list_to_fun_map (f : A → B) : ∀ P, f = list_to_fun (map f (elems A)) P := assume Pleq, funext (take a, - assert Plt : _, from Pleq⁻¹ ▸ find_lt_length (complete a), begin + have Plt : _, from Pleq⁻¹ ▸ find_lt_length (complete a), begin rewrite [list_to_fun_apply _ Pleq a (Pleq⁻¹ ▸ find_lt_length (complete a))], - assert Pmlt : find a (elems A) < length (map f (elems A)), - {rewrite length_map, exact Plt}, + have Pmlt : find a (elems A) < length (map f (elems A)), + begin rewrite length_map, exact Plt end, rewrite [@kth_of_map A B f (find a (elems A)) (elems A) Plt _, kth_find] end) @@ -277,9 +277,9 @@ lemma list_eq_map_list_to_fun (l : list B) (leq : length l = card A) begin apply eq_of_kth_eq, rewrite length_map, apply leq, intro k Plt Plt2, - assert Plt1 : k < length (elems A), {apply leq ▸ Plt}, - assert Plt3 : find (kth k (elems A) Plt1) (elems A) < length l, - {rewrite leq, apply find_kth}, + have Plt1 : k < length (elems A), begin apply leq ▸ Plt end, + have Plt3 : find (kth k (elems A) Plt1) (elems A) < length l, + begin rewrite leq, apply find_kth end, rewrite [kth_of_map Plt1 Plt2, list_to_fun_apply l leq _ Plt3], congruence, rewrite [find_kth_of_nodup Plt1 (unique A)] @@ -303,9 +303,9 @@ lemma nodup_all_funs : nodup (@all_funs A B _ _ _) := dmap_nodup_of_dinj dinj_list_to_fun nodup_all_lists lemma all_funs_complete (f : A → B) : f ∈ all_funs := - assert Plin : map f (elems A) ∈ all_lists_of_len (card A), + have Plin : map f (elems A) ∈ all_lists_of_len (card A), from mem_all_lists (by rewrite length_map), - assert Plfin : list_to_fun (map f (elems A)) (length_map_of_fintype f) ∈ all_funs, + have Plfin : list_to_fun (map f (elems A)) (length_map_of_fintype f) ∈ all_funs, from mem_dmap _ Plin, begin rewrite [fun_eq_list_to_fun_map f (length_map_of_fintype f)], apply Plfin end @@ -342,7 +342,7 @@ lemma found_of_surj {f : A → B} (surj : surjective f) : ∀ b, let elts := elems A, k := find b (map f elts) in k < length elts := λ b, let elts := elems A, img := map f elts, k := find b img in have Pin : b ∈ img, from mem_map_of_surj surj b, - assert Pfound : k < length img, from find_lt_length (mem_map_of_surj surj b), + have Pfound : k < length img, from find_lt_length (mem_map_of_surj surj b), length_map f elts ▸ Pfound definition right_inv {f : A → B} (surj : surjective f) : B → A := @@ -395,12 +395,12 @@ decidable.rec_on decidable_forall_finite (assume P : surjective f, P) (assume Pnsurj : ¬surjective f, obtain b Pne, from exists_not_of_not_forall Pnsurj, - assert Pall : ∀ a, f a ≠ b, from forall_not_of_not_exists Pne, - assert Pbnin : b ∉ image f univ, from λ Pin, + have Pall : ∀ a, f a ≠ b, from forall_not_of_not_exists Pne, + have Pbnin : b ∉ image f univ, from λ Pin, obtain a Pa, from exists_of_mem_image Pin, absurd (and.right Pa) (Pall a), - assert Puniv : finset.card (image f univ) = card A, from card_eq_card_image_of_inj Pinj, - assert Punivb : finset.card (image f univ) = card B, from eq.trans Puniv Peqcard, - assert P : image f univ = univ, from univ_of_card_eq_univ Punivb, + have Puniv : finset.card (image f univ) = card A, from card_eq_card_image_of_inj Pinj, + have Punivb : finset.card (image f univ) = card B, from eq.trans Puniv Peqcard, + have P : image f univ = univ, from univ_of_card_eq_univ Punivb, absurd (P⁻¹▸ mem_univ b) Pbnin) end inj @@ -422,9 +422,9 @@ lemma nodup_all_injs : nodup (all_injs A) := lemma all_injs_complete {f : A → A} : injective f → f ∈ (all_injs A) := assume Pinj, - assert Plin : map f (elems A) ∈ all_nodups_of_len (card A), + have Plin : map f (elems A) ∈ all_nodups_of_len (card A), from begin apply mem_all_nodups, apply length_map, apply nodup_of_inj Pinj end, - assert Plfin : list_to_fun (map f (elems A)) (length_map_of_fintype f) ∈ !all_injs, + have Plfin : list_to_fun (map f (elems A)) (length_map_of_fintype f) ∈ !all_injs, from mem_dmap _ Plin, begin rewrite [fun_eq_list_to_fun_map f (length_map_of_fintype f)], apply Plfin end @@ -439,13 +439,13 @@ lemma univ_of_leq_univ_of_nodup {l : list A} (n : nodup l) (leq : length l = car lemma inj_of_mem_all_injs {f : A → A} : f ∈ (all_injs A) → injective f := assume Pfin, obtain l Pex, from exists_of_mem_dmap Pfin, obtain leq Pin Peq, from Pex, - assert Pmap : map f (elems A) = l, from Peq⁻¹ ▸ list_to_fun_to_list l leq, + have Pmap : map f (elems A) = l, from Peq⁻¹ ▸ list_to_fun_to_list l leq, begin apply inj_of_nodup, rewrite Pmap, apply nodup_mem_all_nodups Pin end lemma perm_of_inj {f : A → A} : injective f → perm (map f (elems A)) (elems A) := assume Pinj, - assert P1 : univ = to_finset_of_nodup (elems A) (unique A), from rfl, - assert P2 : to_finset_of_nodup (map f (elems A)) (nodup_of_inj Pinj) = univ, + have P1 : univ = to_finset_of_nodup (elems A) (unique A), from rfl, + have P2 : to_finset_of_nodup (map f (elems A)) (nodup_of_inj Pinj) = univ, from univ_of_leq_univ_of_nodup _ !length_map, quot.exact (P1 ▸ P2) diff --git a/library/data/hf.lean b/library/data/hf.lean index a75fc7819..89937e0a1 100644 --- a/library/data/hf.lean +++ b/library/data/hf.lean @@ -113,8 +113,8 @@ begin unfold [mem, insert], rewrite to_finset_of_finset, intros, apply mem_of_me protected theorem ext {s₁ s₂ : hf} : (∀ a, a ∈ s₁ ↔ a ∈ s₂) → s₁ = s₂ := assume h, -assert to_finset s₁ = to_finset s₂, from finset.ext h, -assert of_finset (to_finset s₁) = of_finset (to_finset s₂), by rewrite this, +have to_finset s₁ = to_finset s₂, from finset.ext h, +have of_finset (to_finset s₁) = of_finset (to_finset s₂), by rewrite this, by rewrite [*of_finset_to_finset at this]; exact this theorem insert_eq_of_mem {a : hf} {s : hf} : a ∈ s → insert a s = s := @@ -122,7 +122,7 @@ begin unfold mem, intro h, unfold [mem, insert], rewrite (finset.insert_eq_of_me protected theorem induction [recursor 4] {P : hf → Prop} (h₁ : P empty) (h₂ : ∀ (a s : hf), a ∉ s → P s → P (insert a s)) (s : hf) : P s := -assert P (of_finset (to_finset s)), from +have P (of_finset (to_finset s)), from @finset.induction _ _ _ h₁ (λ a s nain ih, begin @@ -388,16 +388,16 @@ begin revert s₂, induction s₁ with a s₁ nain ih, take s₂, suppose ∅ ⊆ s₂, !zero_le, take s₂, suppose insert a s₁ ⊆ s₂, - assert a ∈ s₂, from mem_of_subset_of_mem this !mem_insert, - have a ∉ erase a s₂, from !not_mem_erase, - have s₁ ⊆ erase a s₂, from subset_of_forall + have a ∈ s₂, from mem_of_subset_of_mem this !mem_insert, + have a ∉ erase a s₂, from !not_mem_erase, + have s₁ ⊆ erase a s₂, from subset_of_forall (take x xin, by_cases (suppose x = a, by subst x; contradiction) (suppose x ≠ a, have x ∈ s₂, from mem_of_subset_of_mem `insert a s₁ ⊆ s₂` (mem_insert_of_mem _ `x ∈ s₁`), mem_erase_of_ne_of_mem `x ≠ a` `x ∈ s₂`)), - have s₁ ≤ erase a s₂, from ih _ this, - assert insert a s₁ ≤ insert a (erase a s₂), from + have s₁ ≤ erase a s₂, from ih _ this, + have insert a s₁ ≤ insert a (erase a s₂), from insert_le_insert_of_le (or.inr `a ∉ erase a s₂`) this, by rewrite [insert_erase `a ∈ s₂` at this]; exact this end @@ -465,7 +465,7 @@ begin obtain w h₁ h₂, from this, begin subst x, rewrite to_finset_of_finset, exact iff.mp !finset.mem_powerset_iff_subset h₁ end, suppose finset.subset (to_finset x) (to_finset s), - assert finset.mem (to_finset x) (finset.powerset (to_finset s)), from iff.mpr !finset.mem_powerset_iff_subset this, + have finset.mem (to_finset x) (finset.powerset (to_finset s)), from iff.mpr !finset.mem_powerset_iff_subset this, exists.intro (to_finset x) (and.intro this (of_finset_to_finset x)) end diff --git a/library/data/int/basic.lean b/library/data/int/basic.lean index 7fd6c9567..16e23d7f0 100644 --- a/library/data/int/basic.lean +++ b/library/data/int/basic.lean @@ -544,7 +544,7 @@ has_dvd.mk has_dvd.dvd /- additional properties -/ theorem of_nat_sub {m n : ℕ} (H : m ≥ n) : of_nat (m - n) = of_nat m - of_nat n := -assert m - n + n = m, from nat.sub_add_cancel H, +have m - n + n = m, from nat.sub_add_cancel H, begin symmetry, apply sub_eq_of_eq_add, diff --git a/library/data/int/div.lean b/library/data/int/div.lean index 4b21ddb2f..02016221b 100644 --- a/library/data/int/div.lean +++ b/library/data/int/div.lean @@ -105,7 +105,7 @@ protected theorem div_zero (a : ℤ) : a / 0 = 0 := by rewrite [div_def, sign_zero, zero_mul] protected theorem div_one (a : ℤ) : a / 1 = a := -assert (1 : int) > 0, from dec_trivial, +have (1 : int) > 0, from dec_trivial, int.cases_on a (take m : nat, by rewrite [-of_nat_one, -of_nat_div, nat.div_one]) (take m : nat, by rewrite [!neg_succ_of_nat_div this, -of_nat_one, -of_nat_div, nat.div_one]) @@ -133,7 +133,7 @@ int.cases_on a theorem div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) : a / b = 0 := lt.by_cases (suppose b < 0, - assert a < -b, from abs_of_neg this ▸ H2, + have a < -b, from abs_of_neg this ▸ H2, calc a / b = - (a / -b) : by rewrite [int.div_neg, neg_neg] ... = 0 : by rewrite [div_eq_zero_of_lt H1 this, neg_zero]) @@ -155,8 +155,8 @@ private theorem add_mul_div_self_aux2 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a < 0) obtain m (Hm : a = -[1+m]), from exists_eq_neg_succ_of_nat H1, or.elim (nat.lt_or_ge m (n * k)) (assume m_lt_nk : m < n * k, - assert H3 : m + 1 ≤ n * k, from nat.succ_le_of_lt m_lt_nk, - assert H4 : m / k + 1 ≤ n, + have H3 : m + 1 ≤ n * k, from nat.succ_le_of_lt m_lt_nk, + have H4 : m / k + 1 ≤ n, from nat.succ_le_of_lt (nat.div_lt_of_lt_mul m_lt_nk), have (-[1+m] + n * k) / k = -[1+m] / k + n, from calc (-[1+m] + n * k) / k @@ -371,7 +371,7 @@ by rewrite [add.comm, add_mod_eq_add_mod_right _ H, add.comm] theorem mod_eq_mod_of_add_mod_eq_add_mod_right {m n k i : ℤ} (H : (m + i) % n = (k + i) % n) : m % n = k % n := -assert H1 : (m + i + (-i)) % n = (k + i + (-i)) % n, from add_mod_eq_add_mod_right _ H, +have H1 : (m + i + (-i)) % n = (k + i + (-i)) % n, from add_mod_eq_add_mod_right _ H, by rewrite [*add_neg_cancel_right at H1]; apply H1 theorem mod_eq_mod_of_add_mod_eq_add_mod_left {m n k i : ℤ} : @@ -666,7 +666,7 @@ lt_of_mul_lt_mul_right (le_of_lt H) protected theorem lt_mul_of_div_lt {a b c : ℤ} (H1 : c > 0) (H2 : a / c < b) : a < b * c := -assert H3 : (a / c + 1) * c ≤ b * c, +have H3 : (a / c + 1) * c ≤ b * c, from !mul_le_mul_of_nonneg_right (add_one_le_of_lt H2) (le_of_lt H1), have H4 : a / c * c + c ≤ b * c, by rewrite [right_distrib at H3, one_mul at H3]; apply H3, calc diff --git a/library/data/int/gcd.lean b/library/data/int/gcd.lean index 44d63ee3b..32cd10376 100644 --- a/library/data/int/gcd.lean +++ b/library/data/int/gcd.lean @@ -46,7 +46,7 @@ open nat theorem gcd_of_ne_zero (a : ℤ) {b : ℤ} (H : b ≠ 0) : gcd a b = gcd b (abs a % abs b) := have nat_abs b ≠ 0, from assume H', H (eq_zero_of_nat_abs_eq_zero H'), have nat_abs b > 0, from pos_of_ne_zero this, -assert nat.gcd (nat_abs a) (nat_abs b) = (nat.gcd (nat_abs b) (nat_abs a % nat_abs b)), +have nat.gcd (nat_abs a) (nat_abs b) = (nat.gcd (nat_abs b) (nat_abs a % nat_abs b)), from @nat.gcd_of_pos (nat_abs a) (nat_abs b) this, calc gcd a b = nat.gcd (nat_abs b) (nat_abs a % nat_abs b) : by rewrite [↑gcd, this] @@ -156,8 +156,8 @@ or.elim (le_or_gt 0 a₁) div_gcd_eq_div_gcd_of_nonneg H (ne_of_gt H1) (ne_of_gt H2) H3 H5) (assume H3 : a₁ < 0, have H4 : a₂ * b₁ < 0, by rewrite -H; apply mul_neg_of_neg_of_pos H3 H2, - assert H5 : a₂ < 0, from neg_of_mul_neg_right H4 (le_of_lt H1), - assert H6 : abs a₁ / (gcd (abs a₁) (abs b₁)) = abs a₂ / (gcd (abs a₂) (abs b₂)), + have H5 : a₂ < 0, from neg_of_mul_neg_right H4 (le_of_lt H1), + have H6 : abs a₁ / (gcd (abs a₁) (abs b₁)) = abs a₂ / (gcd (abs a₂) (abs b₂)), begin apply div_gcd_eq_div_gcd_of_nonneg, rewrite [abs_of_pos H1, abs_of_pos H2, abs_of_neg H3, abs_of_neg H5], @@ -249,12 +249,12 @@ theorem coprime_swap {a b : ℤ} (H : coprime b a) : coprime a b := !gcd.comm ▸ H theorem dvd_of_coprime_of_dvd_mul_right {a b c : ℤ} (H1 : coprime c b) (H2 : c ∣ a * b) : c ∣ a := -assert H3 : gcd (a * c) (a * b) = abs a, from +have H3 : gcd (a * c) (a * b) = abs a, from calc gcd (a * c) (a * b) = abs a * gcd c b : gcd_mul_left ... = abs a * 1 : H1 ... = abs a : mul_one, -assert H4 : (c ∣ gcd (a * c) (a * b)), from dvd_gcd !dvd_mul_left H2, +have H4 : (c ∣ gcd (a * c) (a * b)), from dvd_gcd !dvd_mul_left H2, by rewrite [-dvd_abs_iff, -H3]; apply H4 theorem dvd_of_coprime_of_dvd_mul_left {a b c : ℤ} (H1 : coprime c a) (H2 : c ∣ a * b) : c ∣ b := @@ -291,7 +291,7 @@ calc theorem not_coprime_of_dvd_of_dvd {m n d : ℤ} (dgt1 : d > 1) (Hm : d ∣ m) (Hn : d ∣ n) : ¬ coprime m n := assume co : coprime m n, -assert d ∣ gcd m n, from dvd_gcd Hm Hn, +have d ∣ gcd m n, from dvd_gcd Hm Hn, have d ∣ 1, by rewrite [↑coprime at co, co at this]; apply this, have d ≤ 1, from le_of_dvd dec_trivial this, show false, from not_lt_of_ge `d ≤ 1` `d > 1` diff --git a/library/data/int/order.lean b/library/data/int/order.lean index a128432e5..81052110d 100644 --- a/library/data/int/order.lean +++ b/library/data/int/order.lean @@ -122,7 +122,7 @@ have a + of_nat (n + m) = a + 0, from ... = a + 0 : by rewrite add_zero, have of_nat (n + m) = of_nat 0, from add.left_cancel this, have n + m = 0, from of_nat.inj this, -assert n = 0, from nat.eq_zero_of_add_eq_zero_right this, +have n = 0, from nat.eq_zero_of_add_eq_zero_right this, show a = b, from calc a = a + 0 : add_zero @@ -341,7 +341,7 @@ or.elim (le_or_gt a 1) (suppose a ≤ 1, show a = 1, from le.antisymm this (add_one_le_of_lt `a > 0`)) (suppose a > 1, - assert a * b ≥ 2 * 1, + have a * b ≥ 2 * 1, from mul_le_mul (add_one_le_of_lt `a > 1`) (add_one_le_of_lt `b > 0`) trivial H, have false, by rewrite [H' at this]; exact this, false.elim this) diff --git a/library/data/list/basic.lean b/library/data/list/basic.lean index 2d7bd1c37..bd96fa248 100644 --- a/library/data/list/basic.lean +++ b/library/data/list/basic.lean @@ -559,7 +559,7 @@ list.induction_on l (λ h : a ∈ nil, absurd h (not_mem_nil a)) (λ x xs r ainxxs, or.elim (eq_or_mem_of_mem_cons ainxxs) (λ aeqx : a = x, - assert aux : ∃ l, x::xs≈x|l, from + have aux : ∃ l, x::xs≈x|l, from exists.intro xs (qhead x xs), by rewrite aeqx; exact aux) (λ ainxs : a ∈ xs, diff --git a/library/data/list/comb.lean b/library/data/list/comb.lean index c891f9f7e..402bb15c3 100644 --- a/library/data/list/comb.lean +++ b/library/data/list/comb.lean @@ -424,14 +424,14 @@ theorem product_nil : ∀ (l : list A), product l (@nil B) = [] theorem eq_of_mem_map_pair₁ {a₁ a : A} {b₁ : B} {l : list B} : (a₁, b₁) ∈ map (λ b, (a, b)) l → a₁ = a := assume ain, -assert pr1 (a₁, b₁) ∈ map pr1 (map (λ b, (a, b)) l), from mem_map pr1 ain, -assert a₁ ∈ map (λb, a) l, by revert this; rewrite [map_map, ↑pr1]; intro this; assumption, +have pr1 (a₁, b₁) ∈ map pr1 (map (λ b, (a, b)) l), from mem_map pr1 ain, +have a₁ ∈ map (λb, a) l, by revert this; rewrite [map_map, ↑pr1]; intro this; assumption, eq_of_map_const this theorem mem_of_mem_map_pair₁ {a₁ a : A} {b₁ : B} {l : list B} : (a₁, b₁) ∈ map (λ b, (a, b)) l → b₁ ∈ l := assume ain, -assert pr2 (a₁, b₁) ∈ map pr2 (map (λ b, (a, b)) l), from mem_map pr2 ain, -assert b₁ ∈ map (λx, x) l, by rewrite [map_map at this, ↑pr2 at this]; exact this, +have pr2 (a₁, b₁) ∈ map pr2 (map (λ b, (a, b)) l), from mem_map pr2 ain, +have b₁ ∈ map (λx, x) l, by rewrite [map_map at this, ↑pr2 at this]; exact this, by rewrite [map_id at this]; exact this theorem mem_product {a : A} {b : B} : ∀ {l₁ l₂}, a ∈ l₁ → b ∈ l₂ → (a, b) ∈ product l₁ l₂ @@ -439,10 +439,10 @@ theorem mem_product {a : A} {b : B} : ∀ {l₁ l₂}, a ∈ l₁ → b ∈ l₂ | (x::l₁) l₂ h₁ h₂ := or.elim (eq_or_mem_of_mem_cons h₁) (assume aeqx : a = x, - assert (a, b) ∈ map (λ b, (a, b)) l₂, from mem_map _ h₂, + have (a, b) ∈ map (λ b, (a, b)) l₂, from mem_map _ h₂, begin rewrite [-aeqx, product_cons], exact mem_append_left _ this end) (assume ainl₁ : a ∈ l₁, - assert (a, b) ∈ product l₁ l₂, from mem_product ainl₁ h₂, + have (a, b) ∈ product l₁ l₂, from mem_product ainl₁ h₂, begin rewrite [product_cons], exact mem_append_right _ this end) theorem mem_of_mem_product_left {a : A} {b : B} : ∀ {l₁ l₂}, (a, b) ∈ product l₁ l₂ → a ∈ l₁ @@ -450,7 +450,7 @@ theorem mem_of_mem_product_left {a : A} {b : B} : ∀ {l₁ l₂}, (a, b) ∈ pr | (x::l₁) l₂ h := or.elim (mem_or_mem_of_mem_append h) (suppose (a, b) ∈ map (λ b, (x, b)) l₂, - assert a = x, from eq_of_mem_map_pair₁ this, + have a = x, from eq_of_mem_map_pair₁ this, by rewrite this; exact !mem_cons) (suppose (a, b) ∈ product l₁ l₂, have a ∈ l₁, from mem_of_mem_product_left this, @@ -468,7 +468,7 @@ theorem mem_of_mem_product_right {a : A} {b : B} : ∀ {l₁ l₂}, (a, b) ∈ p theorem length_product : ∀ (l₁ : list A) (l₂ : list B), length (product l₁ l₂) = length l₁ * length l₂ | [] l₂ := by rewrite [length_nil, zero_mul] | (x::l₁) l₂ := - assert length (product l₁ l₂) = length l₁ * length l₂, from length_product l₁ l₂, + have length (product l₁ l₂) = length l₁ * length l₂, from length_product l₁ l₂, by rewrite [product_cons, length_append, length_cons, length_map, this, right_distrib, one_mul, add.comm] end product @@ -521,14 +521,14 @@ lemma exists_of_mem_dmap : ∀ {l : list A} {b : B}, b ∈ dmap p f l → ∃ a rewrite [dmap_cons_of_pos Pa, mem_cons_iff], intro Pb, cases Pb with Peq Pin, exact exists.intro a (exists.intro Pa (and.intro !mem_cons Peq)), - assert Pex : ∃ (a : A) (P : p a), a ∈ l ∧ b = f a P, exact exists_of_mem_dmap Pin, + have Pex : ∃ (a : A) (P : p a), a ∈ l ∧ b = f a P, from exists_of_mem_dmap Pin, cases Pex with a' Pex', cases Pex' with Pa' P', exact exists.intro a' (exists.intro Pa' (and.intro (mem_cons_of_mem a (and.left P')) (and.right P'))) end) (assume nPa, begin rewrite [dmap_cons_of_neg nPa], intro Pin, - assert Pex : ∃ (a : A) (P : p a), a ∈ l ∧ b = f a P, exact exists_of_mem_dmap Pin, + have Pex : ∃ (a : A) (P : p a), a ∈ l ∧ b = f a P, from exists_of_mem_dmap Pin, cases Pex with a' Pex', cases Pex' with Pa' P', exact exists.intro a' (exists.intro Pa' (and.intro (mem_cons_of_mem a (and.left P')) (and.right P'))) end) @@ -537,8 +537,8 @@ lemma map_dmap_of_inv_of_pos {g : B → A} (Pinv : ∀ a (Pa : p a), g (f a Pa) ∀ {l : list A}, (∀ ⦃a⦄, a ∈ l → p a) → map g (dmap p f l) = l | [] := assume Pl, by rewrite [dmap_nil, map_nil] | (a::l) := assume Pal, - assert Pa : p a, from Pal a !mem_cons, - assert Pl : ∀ a, a ∈ l → p a, + have Pa : p a, from Pal a !mem_cons, + have Pl : ∀ a, a ∈ l → p a, from take x Pxin, Pal x (mem_cons_of_mem a Pxin), by rewrite [dmap_cons_of_pos Pa, map_cons, Pinv, map_dmap_of_inv_of_pos Pl] diff --git a/library/data/list/perm.lean b/library/data/list/perm.lean index a963a4824..24be55f54 100644 --- a/library/data/list/perm.lean +++ b/library/data/list/perm.lean @@ -239,22 +239,22 @@ include Ha definition decidable_perm_aux : ∀ (n : nat) (l₁ l₂ : list A), length l₁ = n → length l₂ = n → decidable (l₁ ~ l₂) | 0 l₁ l₂ H₁ H₂ := - assert l₁n : l₁ = [], from eq_nil_of_length_eq_zero H₁, - assert l₂n : l₂ = [], from eq_nil_of_length_eq_zero H₂, + have l₁n : l₁ = [], from eq_nil_of_length_eq_zero H₁, + have l₂n : l₂ = [], from eq_nil_of_length_eq_zero H₂, by rewrite [l₁n, l₂n]; exact (inl perm.nil) | (n+1) (x::t₁) l₂ H₁ H₂ := by_cases (assume xinl₂ : x ∈ l₂, let t₂ : list A := erase x l₂ in have len_t₁ : length t₁ = n, begin injection H₁ with e, exact e end, - assert length t₂ = pred (length l₂), from length_erase_of_mem xinl₂, - assert length t₂ = n, by rewrite [this, H₂], + have length t₂ = pred (length l₂), from length_erase_of_mem xinl₂, + have length t₂ = n, by rewrite [this, H₂], match decidable_perm_aux n t₁ t₂ len_t₁ this with | inl p := inl (calc x::t₁ ~ x::(erase x l₂) : skip x p ... ~ l₂ : perm_erase xinl₂) | inr np := inr (λ p : x::t₁ ~ l₂, - assert erase x (x::t₁) ~ erase x l₂, from erase_perm_erase_of_perm x p, + have erase x (x::t₁) ~ erase x l₂, from erase_perm_erase_of_perm x p, have t₁ ~ erase x l₂, by rewrite [erase_cons_head at this]; exact this, absurd this np) end) @@ -325,7 +325,7 @@ perm_induction_on p' obtain (s₂₁ s₂₂ : list A) (C₂₁ : s₂ = s₂₁ ++ s₂₂) (C₂₂ : x::t₂ = s₂₁++(a::s₂₂)), from qeq_split e₂, discr C₁₂ (λ (s₁₁_eq : s₁₁ = []) (x_eq_a : x = a) (t₁_eq : t₁ = s₁₂), - assert s₁_p : s₁ ~ t₂, from calc + have s₁_p : s₁ ~ t₂, from calc s₁ = s₁₁ ++ s₁₂ : C₁₁ ... = t₁ : by rewrite [-t₁_eq, s₁₁_eq, append_nil_left] ... ~ t₂ : p, @@ -345,8 +345,8 @@ perm_induction_on p' ... = s₂ : by rewrite C₂₁ qed)) (λ (ts₁₁ : list A) (s₁₁_eq : s₁₁ = x::ts₁₁) (t₁_eq : t₁ = ts₁₁++(a::s₁₂)), - assert t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app, - assert s₁_eq : s₁ = x::(ts₁₁++s₁₂), from calc + have t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app, + have s₁_eq : s₁ = x::(ts₁₁++s₁₂), from calc s₁ = s₁₁ ++ s₁₂ : C₁₁ ... = x::(ts₁₁++ s₁₂) : by rewrite s₁₁_eq, discr C₂₂ @@ -359,7 +359,7 @@ perm_induction_on p' ... = s₂ : by rewrite [t₂_eq, C₂₁, s₂₁_eq, append_nil_left] qed) (λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)), - assert t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app, + have t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app, proof calc s₁ = x::(ts₁₁++s₁₂) : s₁_eq ... ~ x::(ts₂₁++s₂₂) : skip x (r t₁_qeq t₂_qeq) @@ -370,7 +370,7 @@ perm_induction_on p' obtain (s₂₁ s₂₂ : list A) (C₂₁ : s₂ = s₂₁ ++ s₂₂) (C₂₂ : x::y::t₂ = s₂₁++(a::s₂₂)), from qeq_split e₂, discr₂ C₁₂ (λ (s₁₁_eq : s₁₁ = []) (s₁₂_eq : s₁₂ = x::t₁) (y_eq_a : y = a), - assert s₁_p : s₁ ~ x::t₂, from calc + have s₁_p : s₁ ~ x::t₂, from calc s₁ = s₁₁ ++ s₁₂ : C₁₁ ... = x::t₁ : by rewrite [s₁₂_eq, s₁₁_eq, append_nil_left] ... ~ x::t₂ : skip x p, @@ -396,7 +396,7 @@ perm_induction_on p' ... = s₂ : by rewrite C₂₁ qed)) (λ (s₁₁_eq : s₁₁ = [y]) (x_eq_a : x = a) (t₁_eq : t₁ = s₁₂), - assert s₁_p : s₁ ~ y::t₂, from calc + have s₁_p : s₁ ~ y::t₂, from calc s₁ = y::t₁ : by rewrite [C₁₁, s₁₁_eq, t₁_eq] ... ~ y::t₂ : skip y p, discr₂ C₂₂ @@ -422,7 +422,7 @@ perm_induction_on p' ... = s₂ : by rewrite C₂₁ qed)) (λ (ts₁₁ : list A) (s₁₁_eq : s₁₁ = y::x::ts₁₁) (t₁_eq : t₁ = ts₁₁++(a::s₁₂)), - assert s₁_eq : s₁ = y::x::(ts₁₁++s₁₂), by rewrite [C₁₁, s₁₁_eq], + have s₁_eq : s₁ = y::x::(ts₁₁++s₁₂), by rewrite [C₁₁, s₁₁_eq], discr₂ C₂₂ (λ (s₂₁_eq : s₂₁ = []) (s₂₂_eq : s₂₂ = y::t₂) (x_eq_a : x = a), proof calc @@ -445,9 +445,9 @@ perm_induction_on p' ... = s₂ : by rewrite C₂₁ qed) (λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::y::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)), - assert t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app, - assert t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app, - assert p_aux : ts₁₁++s₁₂ ~ ts₂₁++s₂₂, from r t₁_qeq t₂_qeq, + have t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app, + have t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app, + have p_aux : ts₁₁++s₁₂ ~ ts₂₁++s₂₂, from r t₁_qeq t₂_qeq, proof calc s₁ = y::x::(ts₁₁++s₁₂) : by rewrite s₁_eq ... ~ y::x::(ts₂₁++s₂₂) : skip y (skip x p_aux) @@ -521,39 +521,39 @@ assume p, perm_induction_on p nil (λ x t₁ t₂ p r, by_cases (λ xint₁ : x ∈ t₁, - assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)), + have xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)), by rewrite [erase_dup_cons_of_mem xint₁, erase_dup_cons_of_mem xint₂]; exact r) (λ nxint₁ : x ∉ t₁, - assert nxint₂ : x ∉ t₂, from + have nxint₂ : x ∉ t₂, from assume xint₂ : x ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (perm.symm r) (mem_erase_dup xint₂))) nxint₁, by rewrite [erase_dup_cons_of_not_mem nxint₂, erase_dup_cons_of_not_mem nxint₁]; exact (skip x r))) (λ y x t₁ t₂ p r, by_cases (λ xinyt₁ : x ∈ y::t₁, by_cases (λ yint₁ : y ∈ t₁, - assert yint₂ : y ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁)), - assert yinxt₂ : y ∈ x::t₂, from or.inr (yint₂), + have yint₂ : y ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁)), + have yinxt₂ : y ∈ x::t₂, from or.inr (yint₂), or.elim (eq_or_mem_of_mem_cons xinyt₁) (λ xeqy : x = y, - assert xint₂ : x ∈ t₂, by rewrite [-xeqy at yint₂]; exact yint₂, + have xint₂ : x ∈ t₂, by rewrite [-xeqy at yint₂]; exact yint₂, begin rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂, erase_dup_cons_of_mem yint₁, erase_dup_cons_of_mem xint₂], exact r end) (λ xint₁ : x ∈ t₁, - assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)), + have xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)), begin rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂, erase_dup_cons_of_mem yint₁, erase_dup_cons_of_mem xint₂], exact r end)) (λ nyint₁ : y ∉ t₁, - assert nyint₂ : y ∉ t₂, from + have nyint₂ : y ∉ t₂, from assume yint₂ : y ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (perm.symm r) (mem_erase_dup yint₂))) nyint₁, by_cases (λ xeqy : x = y, - assert nxint₂ : x ∉ t₂, by rewrite [-xeqy at nyint₂]; exact nyint₂, - assert yinxt₂ : y ∈ x::t₂, by rewrite [xeqy]; exact !mem_cons, + have nxint₂ : x ∉ t₂, by rewrite [-xeqy at nyint₂]; exact nyint₂, + have yinxt₂ : y ∈ x::t₂, by rewrite [xeqy]; exact !mem_cons, begin rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂, erase_dup_cons_of_not_mem nyint₁, erase_dup_cons_of_not_mem nxint₂, xeqy], @@ -561,8 +561,8 @@ assume p, perm_induction_on p end) (λ xney : x ≠ y, have x ∈ t₁, from or_resolve_right xinyt₁ xney, - assert x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup this)), - assert y ∉ x::t₂, from + have x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup this)), + have y ∉ x::t₂, from suppose y ∈ x::t₂, or.elim (eq_or_mem_of_mem_cons this) (λ h, absurd h (ne.symm xney)) (λ h, absurd h nyint₂), @@ -572,20 +572,20 @@ assume p, perm_induction_on p exact skip y r end))) (λ nxinyt₁ : x ∉ y::t₁, - have xney : x ≠ y, from ne_of_not_mem_cons nxinyt₁, - have nxint₁ : x ∉ t₁, from not_mem_of_not_mem_cons nxinyt₁, - assert nxint₂ : x ∉ t₂, from + have xney : x ≠ y, from ne_of_not_mem_cons nxinyt₁, + have nxint₁ : x ∉ t₁, from not_mem_of_not_mem_cons nxinyt₁, + have nxint₂ : x ∉ t₂, from assume xint₂ : x ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (perm.symm r) (mem_erase_dup xint₂))) nxint₁, by_cases (λ yint₁ : y ∈ t₁, - assert yinxt₂ : y ∈ x::t₂, from or.inr (mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁))), + have yinxt₂ : y ∈ x::t₂, from or.inr (mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁))), begin rewrite [erase_dup_cons_of_not_mem nxinyt₁, erase_dup_cons_of_mem yinxt₂, erase_dup_cons_of_mem yint₁, erase_dup_cons_of_not_mem nxint₂], exact skip x r end) (λ nyint₁ : y ∉ t₁, - assert nyinxt₂ : y ∉ x::t₂, from + have nyinxt₂ : y ∉ x::t₂, from assume yinxt₂ : y ∈ x::t₂, or.elim (eq_or_mem_of_mem_cons yinxt₂) (λ h, absurd h (ne.symm xney)) (λ h, absurd (mem_of_mem_erase_dup (mem_perm (perm.symm r) (mem_erase_dup h))) nyint₁), @@ -624,10 +624,10 @@ list.induction_on l (λ p, by rewrite [*union_nil]; exact p) (λ x xs r p, by_cases (λ xint₁ : x ∈ t₁, - assert xint₂ : x ∈ t₂, from mem_perm p xint₁, + have xint₂ : x ∈ t₂, from mem_perm p xint₁, by rewrite [union_cons_of_mem _ xint₁, union_cons_of_mem _ xint₂]; exact (r p)) (λ nxint₁ : x ∉ t₁, - assert nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁, + have nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁, by rewrite [union_cons_of_not_mem _ nxint₁, union_cons_of_not_mem _ nxint₂]; exact (skip _ (r p)))) theorem perm_union [congr] {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (union l₁ t₁) ~ (union l₂ t₂) := @@ -641,10 +641,10 @@ include H theorem perm_insert [congr] (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → (insert a l₁) ~ (insert a l₂) := assume p, by_cases (λ ainl₁ : a ∈ l₁, - assert ainl₂ : a ∈ l₂, from mem_perm p ainl₁, + have ainl₂ : a ∈ l₂, from mem_perm p ainl₁, by rewrite [insert_eq_of_mem ainl₁, insert_eq_of_mem ainl₂]; exact p) (λ nainl₁ : a ∉ l₁, - assert nainl₂ : a ∉ l₂, from not_mem_perm p nainl₁, + have nainl₂ : a ∉ l₂, from not_mem_perm p nainl₁, by rewrite [insert_eq_of_not_mem nainl₁, insert_eq_of_not_mem nainl₂]; exact (skip _ p)) end perm_insert @@ -678,10 +678,10 @@ list.induction_on l (λ p, by rewrite [*inter_nil]) (λ x xs r p, by_cases (λ xint₁ : x ∈ t₁, - assert xint₂ : x ∈ t₂, from mem_perm p xint₁, + have xint₂ : x ∈ t₂, from mem_perm p xint₁, by rewrite [inter_cons_of_mem _ xint₁, inter_cons_of_mem _ xint₂]; exact (skip _ (r p))) (λ nxint₁ : x ∉ t₁, - assert nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁, + have nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁, by rewrite [inter_cons_of_not_mem _ nxint₁, inter_cons_of_not_mem _ nxint₂]; exact (r p))) theorem perm_inter [congr] {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (inter l₁ t₁) ~ (inter l₂ t₂) := @@ -704,13 +704,13 @@ theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a have eqv : ∀a, a ∈ t₁ ↔ a ∈ s₁++s₂, from take a, iff.intro (suppose a ∈ t₁, - assert a ∈ a₂::t₂, from iff.mp (e a) (mem_cons_of_mem _ this), - have a ∈ s₁++(a₁::s₂), by rewrite [t₂_eq at this]; exact this, + have a ∈ a₂::t₂, from iff.mp (e a) (mem_cons_of_mem _ this), + have a ∈ s₁++(a₁::s₂), by rewrite [t₂_eq at this]; exact this, or.elim (mem_or_mem_of_mem_append this) (suppose a ∈ s₁, mem_append_left s₂ this) (suppose a ∈ a₁::s₂, or.elim (eq_or_mem_of_mem_cons this) (suppose a = a₁, - assert a₁ ∉ t₁, from not_mem_of_nodup_cons d₁, + have a₁ ∉ t₁, from not_mem_of_nodup_cons d₁, by subst a; contradiction) (suppose a ∈ s₂, mem_append_right s₁ this))) (suppose a ∈ s₁ ++ s₂, or.elim (mem_or_mem_of_mem_append this) @@ -719,8 +719,8 @@ theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a have a ∈ a₁::t₁, from iff.mpr (e a) this, or.elim (eq_or_mem_of_mem_cons this) (suppose a = a₁, - have a₁ ∉ s₁++s₂, from not_mem_of_nodup_cons dt₂', - assert a₁ ∉ s₁, from not_mem_of_not_mem_append_left this, + have a₁ ∉ s₁++s₂, from not_mem_of_nodup_cons dt₂', + have a₁ ∉ s₁, from not_mem_of_not_mem_append_left this, by subst a; contradiction) (suppose a ∈ t₁, this)) (suppose a ∈ s₂, @@ -728,8 +728,8 @@ theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a have a ∈ a₁::t₁, from iff.mpr (e a) this, or.elim (eq_or_mem_of_mem_cons this) (suppose a = a₁, - have a₁ ∉ s₁++s₂, from not_mem_of_nodup_cons dt₂', - assert a₁ ∉ s₂, from not_mem_of_not_mem_append_right this, + have a₁ ∉ s₁++s₂, from not_mem_of_nodup_cons dt₂', + have a₁ ∉ s₂, from not_mem_of_not_mem_append_right this, by subst a; contradiction) (suppose a ∈ t₁, this))), have ds₁s₂ : nodup (s₁++s₂), from nodup_of_nodup_cons dt₂', diff --git a/library/data/list/set.lean b/library/data/list/set.lean index 770464b31..6d3641bb2 100644 --- a/library/data/list/set.lean +++ b/library/data/list/set.lean @@ -41,14 +41,14 @@ lemma length_erase_of_mem {a : A} : ∀ {l}, a ∈ l → length (erase a l) = pr by_cases (suppose a = x, by rewrite [this, erase_cons_head]) (suppose a ≠ x, - assert ainyxs : a ∈ y::xs, from or_resolve_right h this, + have ainyxs : a ∈ y::xs, from or_resolve_right h this, by rewrite [erase_cons_tail _ this, *length_cons, length_erase_of_mem ainyxs]) lemma length_erase_of_not_mem {a : A} : ∀ {l}, a ∉ l → length (erase a l) = length l | [] h := rfl | (x::xs) h := - assert anex : a ≠ x, from λ aeqx : a = x, absurd (or.inl aeqx) h, - assert aninxs : a ∉ xs, from λ ainxs : a ∈ xs, absurd (or.inr ainxs) h, + have anex : a ≠ x, from λ aeqx : a = x, absurd (or.inl aeqx) h, + have aninxs : a ∉ xs, from λ ainxs : a ∈ xs, absurd (or.inr ainxs) h, by rewrite [erase_cons_tail _ anex, length_cons, length_erase_of_not_mem aninxs] lemma erase_append_left {a : A} : ∀ {l₁} (l₂), a ∈ l₁ → erase a (l₁++l₂) = erase a l₁ ++ l₂ @@ -57,7 +57,7 @@ lemma erase_append_left {a : A} : ∀ {l₁} (l₂), a ∈ l₁ → erase a (l by_cases (λ aeqx : a = x, by rewrite [aeqx, append_cons, *erase_cons_head]) (λ anex : a ≠ x, - assert ainxs : a ∈ xs, from mem_of_ne_of_mem anex h, + have ainxs : a ∈ xs, from mem_of_ne_of_mem anex h, by rewrite [append_cons, *erase_cons_tail _ anex, erase_append_left l₂ ainxs]) lemma erase_append_right {a : A} : ∀ {l₁} (l₂), a ∉ l₁ → erase a (l₁++l₂) = l₁ ++ erase a l₂ @@ -66,7 +66,7 @@ lemma erase_append_right {a : A} : ∀ {l₁} (l₂), a ∉ l₁ → erase a (l by_cases (λ aeqx : a = x, by rewrite aeqx at h; exact (absurd !mem_cons h)) (λ anex : a ≠ x, - assert nainxs : a ∉ xs, from not_mem_of_not_mem_cons h, + have nainxs : a ∉ xs, from not_mem_of_not_mem_cons h, by rewrite [append_cons, *erase_cons_tail _ anex, erase_append_right l₂ nainxs]) lemma erase_sub (a : A) : ∀ l, erase a l ⊆ l @@ -75,31 +75,31 @@ lemma erase_sub (a : A) : ∀ l, erase a l ⊆ l by_cases (λ aeqx : a = x, by rewrite [aeqx at xine, erase_cons_head at xine]; exact (or.inr xine)) (λ anex : a ≠ x, - assert yinxe : y ∈ x :: erase a xs, by rewrite [erase_cons_tail _ anex at xine]; exact xine, - assert subxs : erase a xs ⊆ xs, from erase_sub xs, + have yinxe : y ∈ x :: erase a xs, by rewrite [erase_cons_tail _ anex at xine]; exact xine, + have subxs : erase a xs ⊆ xs, from erase_sub xs, by_cases (λ yeqx : y = x, by rewrite yeqx; apply mem_cons) (λ ynex : y ≠ x, - assert yine : y ∈ erase a xs, from mem_of_ne_of_mem ynex yinxe, - assert yinxs : y ∈ xs, from subxs yine, + have yine : y ∈ erase a xs, from mem_of_ne_of_mem ynex yinxe, + have yinxs : y ∈ xs, from subxs yine, or.inr yinxs)) theorem mem_erase_of_ne_of_mem {a b : A} : ∀ {l : list A}, a ≠ b → a ∈ l → a ∈ erase b l | [] n i := absurd i !not_mem_nil | (c::l) n i := by_cases (λ beqc : b = c, - assert ainl : a ∈ l, from or.elim (eq_or_mem_of_mem_cons i) + have ainl : a ∈ l, from or.elim (eq_or_mem_of_mem_cons i) (λ aeqc : a = c, absurd aeqc (beqc ▸ n)) (λ ainl : a ∈ l, ainl), by rewrite [beqc, erase_cons_head]; exact ainl) (λ bnec : b ≠ c, by_cases (λ aeqc : a = c, - assert aux : a ∈ c :: erase b l, by rewrite [aeqc]; exact !mem_cons, + have aux : a ∈ c :: erase b l, by rewrite [aeqc]; exact !mem_cons, by rewrite [erase_cons_tail _ bnec]; exact aux) (λ anec : a ≠ c, have ainl : a ∈ l, from mem_of_ne_of_mem anec i, have ainel : a ∈ erase b l, from mem_erase_of_ne_of_mem n ainl, - assert aux : a ∈ c :: erase b l, from mem_cons_of_mem _ ainel, + have aux : a ∈ c :: erase b l, from mem_cons_of_mem _ ainel, by rewrite [erase_cons_tail _ bnec]; exact aux)) -- theorem mem_of_mem_erase {a b : A} : ∀ {l}, a ∈ erase b l → a ∈ l @@ -117,10 +117,10 @@ theorem mem_of_mem_erase {a b : A} : ∀ {l}, a ∈ erase b l → a ∈ l theorem all_erase_of_all {p : A → Prop} (a : A) : ∀ {l}, all l p → all (erase a l) p | [] h := by rewrite [erase_nil]; exact h | (b::l) h := - assert h₁ : all l p, from all_of_all_cons h, + have h₁ : all l p, from all_of_all_cons h, have h₂ : all (erase a l) p, from all_erase_of_all h₁, have pb : p b, from of_all_cons h, - assert h₃ : all (b :: erase a l) p, from all_cons_of_all pb h₂, + have h₃ : all (b :: erase a l) p, from all_cons_of_all pb h₂, by_cases (λ aeqb : a = b, by rewrite [aeqb, erase_cons_head]; exact h₁) (λ aneb : a ≠ b, by rewrite [erase_cons_tail _ aneb]; exact h₃) @@ -280,13 +280,13 @@ nodup_append_of_nodup_of_nodup_of_disjoint d₂ d₄ disj₂ theorem nodup_map {f : A → B} (inj : injective f) : ∀ {l : list A}, nodup l → nodup (map f l) | [] n := begin rewrite [map_nil], apply nodup_nil end | (x::xs) n := - assert nxinxs : x ∉ xs, from not_mem_of_nodup_cons n, - assert ndxs : nodup xs, from nodup_of_nodup_cons n, - assert ndmfxs : nodup (map f xs), from nodup_map ndxs, - assert nfxinm : f x ∉ map f xs, from + have nxinxs : x ∉ xs, from not_mem_of_nodup_cons n, + have ndxs : nodup xs, from nodup_of_nodup_cons n, + have ndmfxs : nodup (map f xs), from nodup_map ndxs, + have nfxinm : f x ∉ map f xs, from λ ab : f x ∈ map f xs, obtain (y : A) (yinxs : y ∈ xs) (fyfx : f y = f x), from exists_of_mem_map ab, - assert yeqx : y = x, from inj fyfx, + have yeqx : y = x, from inj fyfx, by subst y; contradiction, nodup_cons nfxinm ndmfxs @@ -299,7 +299,7 @@ theorem nodup_erase_of_nodup [decidable_eq A] (a : A) : ∀ {l}, nodup l → nod have ndl : nodup l, from nodup_of_nodup_cons n, have ndeal : nodup (erase a l), from nodup_erase_of_nodup ndl, have nbineal : b ∉ erase a l, from λ i, absurd (erase_sub _ _ i) nbinl, - assert aux : nodup (b :: erase a l), from nodup_cons nbineal ndeal, + have aux : nodup (b :: erase a l), from nodup_cons nbineal ndeal, by rewrite [erase_cons_tail _ aneb]; exact aux) theorem mem_erase_of_nodup [decidable_eq A] (a : A) : ∀ {l}, nodup l → a ∉ erase a l @@ -307,11 +307,11 @@ theorem mem_erase_of_nodup [decidable_eq A] (a : A) : ∀ {l}, nodup l → a ∉ | (b::l) n := have ndl : nodup l, from nodup_of_nodup_cons n, have naineal : a ∉ erase a l, from mem_erase_of_nodup ndl, - assert nbinl : b ∉ l, from not_mem_of_nodup_cons n, + have nbinl : b ∉ l, from not_mem_of_nodup_cons n, by_cases (λ aeqb : a = b, by rewrite [aeqb, erase_cons_head]; exact nbinl) (λ aneb : a ≠ b, - assert aux : a ∉ b :: erase a l, from + have aux : a ∉ b :: erase a l, from assume ainbeal : a ∈ b :: erase a l, or.elim (eq_or_mem_of_mem_cons ainbeal) (λ aeqb : a = b, absurd aeqb aneb) (λ aineal : a ∈ erase a l, absurd aineal naineal), @@ -366,16 +366,16 @@ theorem nodup_erase_dup [decidable_eq A] : ∀ l : list A, nodup (erase_dup l) | (a::l) := by_cases (λ ainl : a ∈ l, by rewrite [erase_dup_cons_of_mem ainl]; exact (nodup_erase_dup l)) (λ nainl : a ∉ l, - assert r : nodup (erase_dup l), from nodup_erase_dup l, - assert nin : a ∉ erase_dup l, from + have r : nodup (erase_dup l), from nodup_erase_dup l, + have nin : a ∉ erase_dup l, from assume ab : a ∈ erase_dup l, absurd (mem_of_mem_erase_dup ab) nainl, by rewrite [erase_dup_cons_of_not_mem nainl]; exact (nodup_cons nin r)) theorem erase_dup_eq_of_nodup [decidable_eq A] : ∀ {l : list A}, nodup l → erase_dup l = l | [] d := rfl | (a::l) d := - assert nainl : a ∉ l, from not_mem_of_nodup_cons d, - assert dl : nodup l, from nodup_of_nodup_cons d, + have nainl : a ∉ l, from not_mem_of_nodup_cons d, + have dl : nodup l, from nodup_of_nodup_cons d, by rewrite [erase_dup_cons_of_not_mem nainl, erase_dup_eq_of_nodup dl] definition decidable_nodup [instance] [decidable_eq A] : ∀ (l : list A), decidable (nodup l) @@ -421,8 +421,8 @@ theorem nodup_filter (p : A → Prop) [decidable_pred p] : ∀ {l : list A}, nod | (a::l) nd := have nainl : a ∉ l, from not_mem_of_nodup_cons nd, have ndl : nodup l, from nodup_of_nodup_cons nd, - assert ndf : nodup (filter p l), from nodup_filter ndl, - assert nainf : a ∉ filter p l, from + have ndf : nodup (filter p l), from nodup_filter ndl, + have nainf : a ∉ filter p l, from assume ainf, absurd (mem_of_mem_filter ainf) nainl, by_cases (λ pa : p a, by rewrite [filter_cons_of_pos _ pa]; exact (nodup_cons nainf ndf)) @@ -575,14 +575,14 @@ by_cases theorem nodup_union_of_nodup_of_nodup : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → nodup (union l₁ l₂) | [] l₂ n₁ nl₂ := by rewrite nil_union; exact nl₂ | (a::l₁) l₂ nal₁ nl₂ := - assert nl₁ : nodup l₁, from nodup_of_nodup_cons nal₁, - assert nl₁l₂ : nodup (union l₁ l₂), from nodup_union_of_nodup_of_nodup nl₁ nl₂, + have nl₁ : nodup l₁, from nodup_of_nodup_cons nal₁, + have nl₁l₂ : nodup (union l₁ l₂), from nodup_union_of_nodup_of_nodup nl₁ nl₂, by_cases (λ ainl₂ : a ∈ l₂, by rewrite [union_cons_of_mem l₁ ainl₂]; exact nl₁l₂) (λ nainl₂ : a ∉ l₂, have nainl₁ : a ∉ l₁, from not_mem_of_nodup_cons nal₁, - assert nainl₁l₂ : a ∉ union l₁ l₂, from + have nainl₁l₂ : a ∉ union l₁ l₂, from assume ainl₁l₂ : a ∈ union l₁ l₂, or.elim (mem_or_mem_of_mem_union ainl₁l₂) (λ ainl₁, absurd ainl₁ nainl₁) (λ ainl₂, absurd ainl₂ nainl₂), @@ -591,8 +591,8 @@ theorem nodup_union_of_nodup_of_nodup : ∀ {l₁ l₂ : list A}, nodup l₁ → theorem union_eq_append : ∀ {l₁ l₂ : list A}, disjoint l₁ l₂ → union l₁ l₂ = append l₁ l₂ | [] l₂ d := rfl | (a::l₁) l₂ d := - assert nainl₂ : a ∉ l₂, from disjoint_left d !mem_cons, - assert d₁ : disjoint l₁ l₂, from disjoint_of_disjoint_cons_left d, + have nainl₂ : a ∉ l₂, from disjoint_left d !mem_cons, + have d₁ : disjoint l₁ l₂, from disjoint_of_disjoint_cons_left d, by rewrite [union_cons_of_not_mem _ nainl₂, append_cons, union_eq_append d₁] theorem all_union {p : A → Prop} : ∀ {l₁ l₂ : list A}, all l₁ p → all l₂ p → all (union l₁ l₂) p @@ -600,8 +600,8 @@ theorem all_union {p : A → Prop} : ∀ {l₁ l₂ : list A}, all l₁ p → al | (a::l₁) l₂ h₁ h₂ := have h₁' : all l₁ p, from all_of_all_cons h₁, have pa : p a, from of_all_cons h₁, - assert au : all (union l₁ l₂) p, from all_union h₁' h₂, - assert au' : all (a :: union l₁ l₂) p, from all_cons_of_all pa au, + have au : all (union l₁ l₂) p, from all_union h₁' h₂, + have au' : all (a :: union l₁ l₂) p, from all_cons_of_all pa au, by_cases (λ ainl₂ : a ∈ l₂, by rewrite [union_cons_of_mem _ ainl₂]; exact au) (λ nainl₂ : a ∉ l₂, by rewrite [union_cons_of_not_mem _ nainl₂]; exact au') @@ -757,10 +757,10 @@ theorem mem_inter_of_mem_of_mem : ∀ {l₁ l₂} {a : A}, a ∈ l₁ → a ∈ theorem nodup_inter_of_nodup : ∀ {l₁ : list A} (l₂), nodup l₁ → nodup (inter l₁ l₂) | [] l₂ d := nodup_nil | (a::l₁) l₂ d := - have d₁ : nodup l₁, from nodup_of_nodup_cons d, - assert d₂ : nodup (inter l₁ l₂), from nodup_inter_of_nodup _ d₁, - have nainl₁ : a ∉ l₁, from not_mem_of_nodup_cons d, - assert naini : a ∉ inter l₁ l₂, from λ i, absurd (mem_of_mem_inter_left i) nainl₁, + have d₁ : nodup l₁, from nodup_of_nodup_cons d, + have d₂ : nodup (inter l₁ l₂), from nodup_inter_of_nodup _ d₁, + have nainl₁ : a ∉ l₁, from not_mem_of_nodup_cons d, + have naini : a ∉ inter l₁ l₂, from λ i, absurd (mem_of_mem_inter_left i) nainl₁, by_cases (λ ainl₂ : a ∈ l₂, by rewrite [inter_cons_of_mem _ ainl₂]; exact (nodup_cons naini d₂)) (λ nainl₂ : a ∉ l₂, by rewrite [inter_cons_of_not_mem _ nainl₂]; exact d₂) @@ -768,17 +768,17 @@ theorem nodup_inter_of_nodup : ∀ {l₁ : list A} (l₂), nodup l₁ → nodup theorem inter_eq_nil_of_disjoint : ∀ {l₁ l₂ : list A}, disjoint l₁ l₂ → inter l₁ l₂ = [] | [] l₂ d := rfl | (a::l₁) l₂ d := - assert aux_eq : inter l₁ l₂ = [], from inter_eq_nil_of_disjoint (disjoint_of_disjoint_cons_left d), - assert nainl₂ : a ∉ l₂, from disjoint_left d !mem_cons, + have aux_eq : inter l₁ l₂ = [], from inter_eq_nil_of_disjoint (disjoint_of_disjoint_cons_left d), + have nainl₂ : a ∉ l₂, from disjoint_left d !mem_cons, by rewrite [inter_cons_of_not_mem _ nainl₂, aux_eq] theorem all_inter_of_all_left {p : A → Prop} : ∀ {l₁} (l₂), all l₁ p → all (inter l₁ l₂) p | [] l₂ h := trivial | (a::l₁) l₂ h := - have h₁ : all l₁ p, from all_of_all_cons h, - assert h₂ : all (inter l₁ l₂) p, from all_inter_of_all_left _ h₁, - have pa : p a, from of_all_cons h, - assert h₃ : all (a :: inter l₁ l₂) p, from all_cons_of_all pa h₂, + have h₁ : all l₁ p, from all_of_all_cons h, + have h₂ : all (inter l₁ l₂) p, from all_inter_of_all_left _ h₁, + have pa : p a, from of_all_cons h, + have h₃ : all (a :: inter l₁ l₂) p, from all_cons_of_all pa h₂, by_cases (λ ainl₂ : a ∈ l₂, by rewrite [inter_cons_of_mem _ ainl₂]; exact h₃) (λ nainl₂ : a ∉ l₂, by rewrite [inter_cons_of_not_mem _ nainl₂]; exact h₂) @@ -786,11 +786,11 @@ theorem all_inter_of_all_left {p : A → Prop} : ∀ {l₁} (l₂), all l₁ p theorem all_inter_of_all_right {p : A → Prop} : ∀ (l₁) {l₂}, all l₂ p → all (inter l₁ l₂) p | [] l₂ h := trivial | (a::l₁) l₂ h := - assert h₁ : all (inter l₁ l₂) p, from all_inter_of_all_right _ h, + have h₁ : all (inter l₁ l₂) p, from all_inter_of_all_right _ h, by_cases (λ ainl₂ : a ∈ l₂, - have pa : p a, from of_mem_of_all ainl₂ h, - assert h₂ : all (a :: inter l₁ l₂) p, from all_cons_of_all pa h₁, + have pa : p a, from of_mem_of_all ainl₂ h, + have h₂ : all (a :: inter l₁ l₂) p, from all_cons_of_all pa h₁, by rewrite [inter_cons_of_mem _ ainl₂]; exact h₂) (λ nainl₂ : a ∉ l₂, by rewrite [inter_cons_of_not_mem _ nainl₂]; exact h₁) diff --git a/library/data/list/sort.lean b/library/data/list/sort.lean index f5d9803b5..09f2c534a 100644 --- a/library/data/list/sort.lean +++ b/library/data/list/sort.lean @@ -33,34 +33,34 @@ lemma min_core_lemma : ∀ {b l} a, b ∈ l ∨ b = a → R (min_core R l a) b | b [] a h := or.elim h (suppose b ∈ [], absurd this !not_mem_nil) (suppose b = a, - assert R a a, from rf a, + have R a a, from rf a, begin subst b, unfold min_core, assumption end) | b (c::l) a h := or.elim h (suppose b ∈ c :: l, or.elim (eq_or_mem_of_mem_cons this) (suppose b = c, or.elim (em (R c a)) (suppose R c a, - assert R (min_core R l b) b, from min_core_lemma _ (or.inr rfl), + have R (min_core R l b) b, from min_core_lemma _ (or.inr rfl), begin unfold min_core, rewrite [if_pos `R c a`], subst c, assumption end) (suppose ¬ R c a, - assert R a c, from or_resolve_right (to c a) this, - assert R (min_core R l a) a, from min_core_lemma _ (or.inr rfl), - assert R (min_core R l a) c, from tr this `R a c`, + have R a c, from or_resolve_right (to c a) this, + have R (min_core R l a) a, from min_core_lemma _ (or.inr rfl), + have R (min_core R l a) c, from tr this `R a c`, begin unfold min_core, rewrite [if_neg `¬ R c a`], subst b, exact `R (min_core R l a) c` end)) (suppose b ∈ l, or.elim (em (R c a)) (suppose R c a, - assert R (min_core R l c) b, from min_core_lemma _ (or.inl `b ∈ l`), + have R (min_core R l c) b, from min_core_lemma _ (or.inl `b ∈ l`), begin unfold min_core, rewrite [if_pos `R c a`], assumption end) (suppose ¬ R c a, - assert R (min_core R l a) b, from min_core_lemma _ (or.inl `b ∈ l`), + have R (min_core R l a) b, from min_core_lemma _ (or.inl `b ∈ l`), begin unfold min_core, rewrite [if_neg `¬ R c a`], assumption end))) (suppose b = a, - assert R (min_core R l a) b, from min_core_lemma _ (or.inr this), + have R (min_core R l a) b, from min_core_lemma _ (or.inr this), or.elim (em (R c a)) (suppose R c a, - assert R (min_core R l c) c, from min_core_lemma _ (or.inr rfl), - assert R (min_core R l c) a, from tr this `R c a`, + have R (min_core R l c) c, from min_core_lemma _ (or.inr rfl), + have R (min_core R l c) a, from tr this `R c a`, begin unfold min_core, rewrite [if_pos `R c a`], subst b, exact `R (min_core R l c) a` end) (suppose ¬ R c a, begin unfold min_core, rewrite [if_neg `¬ R c a`], assumption end)) @@ -76,10 +76,10 @@ lemma min_lemma : ∀ {l} (h : l ≠ nil), all l (R (min R l h)) all_of_forall (take x, suppose x ∈ b::l, or.elim (eq_or_mem_of_mem_cons this) (suppose x = b, - assert R (min_core R l b) b, from min_core_le to tr rf b, + have R (min_core R l b) b, from min_core_le to tr rf b, begin subst x, unfold min, assumption end) (suppose x ∈ l, - assert R (min_core R l b) x, from min_core_le_of_mem to tr rf _ this, + have R (min_core R l b) x, from min_core_le_of_mem to tr rf _ this, begin unfold min, assumption end)) variable (R) @@ -135,7 +135,7 @@ assume h₁ h₂, by rewrite h₂ at h₁; contradiction include decR lemma sort_aux_lemma {l n} (h : length l = succ n) : length (erase (min R l (ne_nil h)) l) = n := have min R l _ ∈ l, from min_mem R l (ne_nil h), -assert length (erase (min R l _) l) = pred (length l), from length_erase_of_mem this, +have length (erase (min R l _) l) = pred (length l), from length_erase_of_mem this, by rewrite h at this; exact this definition sort_aux : Π (n : nat) (l : list A), length l = n → list A @@ -154,7 +154,7 @@ lemma sort_aux_perm : ∀ {n : nat} {l : list A} (h : length l = n), sort_aux R | 0 l h := by rewrite [↑sort_aux, eq_nil_of_length_eq_zero h] | (succ n) l h := let m := min R l (ne_nil h) in - assert leq : length (erase m l) = n, from sort_aux_lemma R h, + have leq : length (erase m l) = n, from sort_aux_lemma R h, calc m :: sort_aux R n (erase m l) leq ~ m :: erase m l : perm.skip m (sort_aux_perm leq) ... ~ l : perm_erase (min_mem _ _ _) @@ -166,10 +166,10 @@ lemma strongly_sorted_sort_aux : ∀ {n : nat} {l : list A} (h : length l = n), | 0 l h := !strongly_sorted.base | (succ n) l h := let m := min R l (ne_nil h) in - assert leq : length (erase m l) = n, from sort_aux_lemma R h, - assert ss : strongly_sorted R (sort_aux R n (erase m l) leq), from strongly_sorted_sort_aux leq, - assert all l (R m), from min_lemma to tr rf (ne_nil h), - assert hall : all (sort_aux R n (erase m l) leq) (R m), from + have leq : length (erase m l) = n, from sort_aux_lemma R h, + have ss : strongly_sorted R (sort_aux R n (erase m l) leq), from strongly_sorted_sort_aux leq, + have all l (R m), from min_lemma to tr rf (ne_nil h), + have hall : all (sort_aux R n (erase m l) leq) (R m), from all_of_forall (take x, suppose x ∈ sort_aux R n (erase m l) leq, have x ∈ erase m l, from mem_perm (sort_aux_perm R leq) this, diff --git a/library/data/list/sorted.lean b/library/data/list/sorted.lean index fafaae626..38cbfb791 100644 --- a/library/data/list/sorted.lean +++ b/library/data/list/sorted.lean @@ -104,10 +104,10 @@ lemma eq_of_sorted_of_perm (tr : transitive R) (anti : anti_symmetric R) : ∀ { | (a::l₁) l₂ h₁ h₂ h₃ := have aux : ∀ {t}, l₂ = a::t → a::l₁ = l₂, from take t, suppose l₂ = a::t, - have l₁ ~ t, by rewrite [this at h₁]; apply perm_cons_inv h₁, - have sorted R l₁, from and.right (sorted_inv h₂), - have sorted R t, by rewrite [`l₂ = a::t` at h₃]; exact and.right (sorted_inv h₃), - assert l₁ = t, from eq_of_sorted_of_perm `l₁ ~ t` `sorted R l₁` `sorted R t`, + have l₁ ~ t, by rewrite [this at h₁]; apply perm_cons_inv h₁, + have sorted R l₁, from and.right (sorted_inv h₂), + have sorted R t, by rewrite [`l₂ = a::t` at h₃]; exact and.right (sorted_inv h₃), + have l₁ = t, from eq_of_sorted_of_perm `l₁ ~ t` `sorted R l₁` `sorted R t`, show a :: l₁ = l₂, by rewrite [`l₂ = a::t`, this], have a ∈ l₂, from mem_perm h₁ !mem_cons, obtain s t (e₁ : l₂ = s ++ (a::t)), from mem_split this, @@ -125,8 +125,8 @@ lemma eq_of_sorted_of_perm (tr : transitive R) (anti : anti_symmetric R) : ∀ { apply or.elim (eq_or_mem_of_mem_cons `b ∈ a::l₁`), suppose b = a, by rewrite this at e₁; exact aux e₁, suppose b ∈ l₁, - have R a b, from of_mem_of_all this hall₁, - assert b = a, from anti `R b a` `R a b`, + have R a b, from of_mem_of_all this hall₁, + have b = a, from anti `R b a` `R a b`, by rewrite this at e₁; exact aux e₁ } end end list diff --git a/library/data/nat/basic.lean b/library/data/nat/basic.lean index dd73228de..f2af9cc46 100644 --- a/library/data/nat/basic.lean +++ b/library/data/nat/basic.lean @@ -148,7 +148,7 @@ nat.induction_on n (by simp) (take a iH, -- TODO(Leo): replace with forward reasoning after we add strategies for it. - assert succ (a + m) = succ (a + k) → a + m = a + k, from !succ.inj, + have succ (a + m) = succ (a + k) → a + m = a + k, from !succ.inj, by inst_simp) protected theorem add_right_cancel {n m k : ℕ} (H : n + m = k + m) : n = k := diff --git a/library/data/nat/div.lean b/library/data/nat/div.lean index ecf99e4f8..a2048b8c8 100644 --- a/library/data/nat/div.lean +++ b/library/data/nat/div.lean @@ -165,15 +165,15 @@ begin intro x IH, show succ x = succ x / y * y + succ x % y, from if H1 : succ x < y then - assert H2 : succ x / y = 0, from div_eq_zero_of_lt H1, - assert H3 : succ x % y = succ x, from mod_eq_of_lt H1, + have H2 : succ x / y = 0, from div_eq_zero_of_lt H1, + have H3 : succ x % y = succ x, from mod_eq_of_lt H1, begin rewrite [H2, H3, zero_mul, zero_add] end else have H2 : y ≤ succ x, from le_of_not_gt H1, - assert H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2, - assert H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2, + have H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2, + have H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2, have H5 : succ x - y < succ x, from sub_lt !succ_pos H, - assert H6 : succ x - y ≤ x, from le_of_lt_succ H5, + have H6 : succ x - y ≤ x, from le_of_lt_succ H5, (calc succ x / y * y + succ x % y = succ ((succ x - y) / y) * y + succ x % y : by rewrite H3 @@ -210,7 +210,7 @@ by_cases_zero_pos n assume npos : n > 0, assume H1 : (m + i) % n = (k + i) % n, have H2 : (m + i % n) % n = (k + i % n) % n, by rewrite [*add_mod_mod, H1], - assert H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n, + have H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n, from add_mod_eq_add_mod_right _ H2, begin revert H3, @@ -298,11 +298,11 @@ theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) % k = (m * (n % k)) % k : !mul.comm ▸ !mul.comm ▸ !mul_mod_eq_mod_mul_mod protected theorem div_one (n : ℕ) : n / 1 = n := -assert n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹, +have n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹, begin rewrite [-this at {2}, mul_one, mod_one] end protected theorem div_self {n : ℕ} (H : n > 0) : n / n = 1 := -assert (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H, +have (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H, by rewrite [nat.div_one at this, -this, *mul_one] theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : m / n * n = m := @@ -448,11 +448,11 @@ end lemma le_of_dvd {m n : nat} : n > 0 → m ∣ n → m ≤ n := assume (h₁ : n > 0) (h₂ : m ∣ n), -assert h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂, +have h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂, by_contradiction (λ nle : ¬ m ≤ n, have h₄ : m > n, from lt_of_not_ge nle, - assert h₅ : n % m = n, from mod_eq_of_lt h₄, + have h₅ : n % m = n, from mod_eq_of_lt h₄, begin rewrite h₃ at h₅, subst n, exact absurd h₁ (lt.irrefl 0) @@ -513,7 +513,7 @@ lt_of_mul_lt_mul_right (calc ... < n * k : H) protected theorem lt_mul_of_div_lt {m n k : ℕ} (H1 : k > 0) (H2 : m / k < n) : m < n * k := -assert H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2), +have H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2), have H4 : m / k * k + k ≤ n * k, by rewrite [succ_mul at H3]; apply H3, calc m = m / k * k + m % k : eq_div_mul_add_mod @@ -564,29 +564,29 @@ nat.strong_induction_on a (λ a ih, let k₁ := a / (b*c) in let k₂ := a %(b*c) in - assert bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`, - assert k₂ < b * c, from mod_lt _ bc_pos, - assert k₂ ≤ a, from !mod_le, + have bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`, + have k₂ < b * c, from mod_lt _ bc_pos, + have k₂ ≤ a, from !mod_le, or.elim (eq_or_lt_of_le this) (suppose k₂ = a, - assert i₁ : a < b * c, by rewrite -this; assumption, - assert k₁ = 0, from div_eq_zero_of_lt i₁, - assert a / b < c, by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁, + have i₁ : a < b * c, by rewrite -this; assumption, + have k₁ = 0, from div_eq_zero_of_lt i₁, + have a / b < c, by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁, begin rewrite [`k₁ = 0`], show (a / b) / c = 0, from div_eq_zero_of_lt `a / b < c` end) (suppose k₂ < a, - assert a = k₁*(b*c) + k₂, from eq_div_mul_add_mod a (b*c), - assert a / b = k₁*c + k₂ / b, by + have a = k₁*(b*c) + k₂, from eq_div_mul_add_mod a (b*c), + have a / b = k₁*c + k₂ / b, by rewrite [this at {1}, mul.comm b c at {2}, -mul.assoc, add.comm, add_mul_div_self `b > 0`, add.comm], - assert e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by + have e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by rewrite [this, add.comm, add_mul_div_self `c > 0`, add.comm], - assert e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`, - assert e₃ : k₂ / (b * c) = 0, from div_eq_zero_of_lt `k₂ < b * c`, - assert (k₂ / b) / c = 0, by rewrite [e₂, e₃], - show (a / b) / c = k₁, by rewrite [e₁, this])) + have e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`, + have e₃ : k₂ / (b * c) = 0, from div_eq_zero_of_lt `k₂ < b * c`, + have (k₂ / b) / c = 0, by rewrite [e₂, e₃], + show (a / b) / c = k₁, by rewrite [e₁, this])) protected lemma div_div_eq_div_mul (a b c : nat) : (a / b) / c = a / (b * c) := begin diff --git a/library/data/nat/examples/partial_sum.lean b/library/data/nat/examples/partial_sum.lean index 5e697b95d..4c0a3cadb 100644 --- a/library/data/nat/examples/partial_sum.lean +++ b/library/data/nat/examples/partial_sum.lean @@ -28,6 +28,6 @@ lemma two_mul_partial_sum_eq : ∀ n, 2 * partial_sum n = (succ n) * n theorem partial_sum_eq : ∀ n, partial_sum n = ((n + 1) * n) / 2 := take n, -assert h₁ : (2 * partial_sum n) / 2 = ((succ n) * n) / 2, by rewrite two_mul_partial_sum_eq, -assert h₂ : (2:nat) > 0, from dec_trivial, +have h₁ : (2 * partial_sum n) / 2 = ((succ n) * n) / 2, by rewrite two_mul_partial_sum_eq, +have h₂ : (2:nat) > 0, from dec_trivial, by rewrite [nat.mul_div_cancel_left _ h₂ at h₁]; exact h₁ diff --git a/library/data/nat/find.lean b/library/data/nat/find.lean index bba8d29a5..f4b54412d 100644 --- a/library/data/nat/find.lean +++ b/library/data/nat/find.lean @@ -58,7 +58,7 @@ acc.intro y (λ (z : nat) (l : z ≺ y), private lemma acc_of_acc_of_lt : ∀ {x y : nat}, acc gtb x → y < x → acc gtb y | 0 y a0 ylt0 := absurd ylt0 !not_lt_zero | (succ x) y asx yltsx := - assert acc gtb x, from acc_of_acc_succ asx, + have acc gtb x, from acc_of_acc_succ asx, by_cases (suppose y = x, by substvars; assumption) (suppose y ≠ x, acc_of_acc_of_lt `acc gtb x` (lt_of_le_of_ne (le_of_lt_succ yltsx) this)) diff --git a/library/data/nat/gcd.lean b/library/data/nat/gcd.lean index 2ef6aaaca..e67884123 100644 --- a/library/data/nat/gcd.lean +++ b/library/data/nat/gcd.lean @@ -306,7 +306,7 @@ calc theorem not_coprime_of_dvd_of_dvd {m n d : ℕ} (dgt1 : d > 1) (Hm : d ∣ m) (Hn : d ∣ n) : ¬ coprime m n := assume co : coprime m n, -assert d ∣ gcd m n, from dvd_gcd Hm Hn, +have d ∣ gcd m n, from dvd_gcd Hm Hn, have d ∣ 1, by rewrite [↑coprime at co, co at this]; apply this, have d ≤ 1, from le_of_dvd dec_trivial this, show false, from not_lt_of_ge `d ≤ 1` `d > 1` diff --git a/library/data/nat/order.lean b/library/data/nat/order.lean index 766579146..d6129dfd8 100644 --- a/library/data/nat/order.lean +++ b/library/data/nat/order.lean @@ -275,7 +275,7 @@ exists_eq_succ_of_lt H theorem pos_of_dvd_of_pos {m n : ℕ} (H1 : m ∣ n) (H2 : n > 0) : m > 0 := pos_of_ne_zero (suppose m = 0, - assert n = 0, from eq_zero_of_zero_dvd (this ▸ H1), + have n = 0, from eq_zero_of_zero_dvd (this ▸ H1), ne_of_lt H2 (by subst n)) /- multiplication -/ @@ -367,11 +367,11 @@ or.elim !lt_or_ge protected theorem min_add_add_left (a b c : ℕ) : min (a + b) (a + c) = a + min b c := decidable.by_cases (suppose b ≤ c, - assert a + b ≤ a + c, from add_le_add_left this _, + have a + b ≤ a + c, from add_le_add_left this _, by rewrite [min_eq_left `b ≤ c`, min_eq_left this]) (suppose ¬ b ≤ c, - assert c ≤ b, from le_of_lt (lt_of_not_ge this), - assert a + c ≤ a + b, from add_le_add_left this _, + have c ≤ b, from le_of_lt (lt_of_not_ge this), + have a + c ≤ a + b, from add_le_add_left this _, by rewrite [min_eq_right `c ≤ b`, min_eq_right this]) protected theorem min_add_add_right (a b c : ℕ) : min (a + c) (b + c) = min a b + c := @@ -380,11 +380,11 @@ by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply nat.min_add_add_lef protected theorem max_add_add_left (a b c : ℕ) : max (a + b) (a + c) = a + max b c := decidable.by_cases (suppose b ≤ c, - assert a + b ≤ a + c, from add_le_add_left this _, + have a + b ≤ a + c, from add_le_add_left this _, by rewrite [max_eq_right `b ≤ c`, max_eq_right this]) (suppose ¬ b ≤ c, - assert c ≤ b, from le_of_lt (lt_of_not_ge this), - assert a + c ≤ a + b, from add_le_add_left this _, + have c ≤ b, from le_of_lt (lt_of_not_ge this), + have a + c ≤ a + b, from add_le_add_left this _, by rewrite [max_eq_left `c ≤ b`, max_eq_left this]) protected theorem max_add_add_right (a b c : ℕ) : max (a + c) (b + c) = max a b + c := diff --git a/library/data/nat/pairing.lean b/library/data/nat/pairing.lean index 3af9a1d59..0ef8474b6 100644 --- a/library/data/nat/pairing.lean +++ b/library/data/nat/pairing.lean @@ -27,19 +27,19 @@ by_cases rewrite [if_pos this, add_sub_of_le (sqrt_lower n)] end) (suppose h₁ : ¬ n - s*s < s, - have s ≤ n - s*s, from le_of_not_gt h₁, - assert s + s*s ≤ n - s*s + s*s, from add_le_add_right this (s*s), - assert s*s + s ≤ n, by rewrite [nat.sub_add_cancel (sqrt_lower n) at this, + have s ≤ n - s*s, from le_of_not_gt h₁, + have s + s*s ≤ n - s*s + s*s, from add_le_add_right this (s*s), + have s*s + s ≤ n, by rewrite [nat.sub_add_cancel (sqrt_lower n) at this, add.comm at this]; assumption, - have n ≤ s*s + s + s, from sqrt_upper n, - have n - s*s ≤ s + s, from calc + have n ≤ s*s + s + s, from sqrt_upper n, + have n - s*s ≤ s + s, from calc n - s*s ≤ (s*s + s + s) - s*s : nat.sub_le_sub_right this (s*s) ... = (s*s + (s+s)) - s*s : by rewrite add.assoc ... = s + s : by rewrite nat.add_sub_cancel_left, have n - s*s - s ≤ s, from calc n - s*s - s ≤ (s + s) - s : nat.sub_le_sub_right this s ... = s : by rewrite nat.add_sub_cancel_left, - assert h₂ : ¬ s < n - s*s - s, from not_lt_of_ge this, + have h₂ : ¬ s < n - s*s - s, from not_lt_of_ge this, begin esimp [unpair], rewrite [if_neg h₁], esimp, @@ -50,7 +50,7 @@ by_cases theorem unpair_mkpair (a b : nat) : unpair (mkpair a b) = (a, b) := by_cases (suppose a < b, - assert a ≤ b + b, from calc + have a ≤ b + b, from calc a ≤ b : le_of_lt this ... ≤ b+b : !le_add_right, begin @@ -61,9 +61,9 @@ by_cases end) (suppose ¬ a < b, have b ≤ a, from le_of_not_gt this, - assert a + b ≤ a + a, from add_le_add_left this a, + have a + b ≤ a + a, from add_le_add_left this a, have a + b ≥ a, from !le_add_right, - assert ¬ a + b < a, from not_lt_of_ge this, + have ¬ a + b < a, from not_lt_of_ge this, begin esimp [mkpair], rewrite [if_neg `¬ a < b`], @@ -82,9 +82,9 @@ or.elim (eq_or_lt_of_le this) let s := sqrt n in by_cases (suppose h : n - s*s < s, - assert n > 0, from lt_of_succ_lt `n > 1`, - assert sqrt n > 0, from sqrt_pos_of_pos this, - assert sqrt n * sqrt n > 0, from mul_pos this this, + have n > 0, from lt_of_succ_lt `n > 1`, + have sqrt n > 0, from sqrt_pos_of_pos this, + have sqrt n * sqrt n > 0, from mul_pos this this, begin unfold unpair, rewrite [if_pos h], esimp, exact sub_lt `n > 0` `sqrt n * sqrt n > 0` end) (suppose ¬ n - s*s < s, begin unfold unpair, rewrite [if_neg this], esimp, apply sqrt_lt `n > 1` end)) diff --git a/library/data/nat/parity.lean b/library/data/nat/parity.lean index 1c2a1d17e..2b7843d6c 100644 --- a/library/data/nat/parity.lean +++ b/library/data/nat/parity.lean @@ -68,7 +68,7 @@ have h : n+1 ≡ 1 [mod 2], from this, by_contradiction (suppose ¬ odd (succ n), have n+1 ≡ 0 [mod 2], from even_of_not_odd this, have 1 ≡ 0 [mod 2], from eq.trans (eq.symm h) this, - assert 1 = 0, from this, + have 1 = 0, from this, by contradiction) lemma eq_1_of_ne_0_lt_2 : ∀ {n : nat}, n ≠ 0 → n < 2 → n = 1 @@ -85,13 +85,13 @@ suppose odd n, lemma odd_of_mod_eq {n} : n % 2 = 1 → odd n := suppose n % 2 = 1, by_contradiction (suppose ¬ odd n, - assert n % 2 = 0, from even_of_not_odd this, + have n % 2 = 0, from even_of_not_odd this, by rewrite this at *; contradiction) lemma even_succ_of_odd {n} : odd n → even (succ n) := suppose odd n, - assert n % 2 = 1 % 2, from mod_eq_of_odd this, - assert (n+1) % 2 = 2 % 2, from add_mod_eq_add_mod_right 1 this, + have n % 2 = 1 % 2, from mod_eq_of_odd this, + have (n+1) % 2 = 2 % 2, from add_mod_eq_add_mod_right 1 this, by rewrite mod_self at this; exact this lemma odd_succ_succ_of_odd {n} : odd n → odd (succ (succ n)) := @@ -174,7 +174,7 @@ assume h, by_contradiction (λ hn, have ∃ k, n = 2 * k, from exists_of_even this, obtain k₁ (hk₁ : n = 2 * k₁ + 1), from h, obtain k₂ (hk₂ : n = 2 * k₂), from this, - assert (2 * k₁ + 1) % 2 = (2 * k₂) % 2, by rewrite [-hk₁, -hk₂], + have (2 * k₁ + 1) % 2 = (2 * k₂) % 2, by rewrite [-hk₁, -hk₂], begin rewrite [mul_mod_right at this, add.comm at this, add_mul_mod_self_left at this], contradiction @@ -188,19 +188,19 @@ even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, left_distrib lemma even_add_of_odd_of_odd {n m} : odd n → odd m → even (n+m) := suppose odd n, suppose odd m, -assert even (succ n + succ m), +have even (succ n + succ m), from even_add_of_even_of_even (even_succ_of_odd `odd n`) (even_succ_of_odd `odd m`), have even(succ (succ (n + m))), by rewrite [add_succ at this, succ_add at this]; exact this, even_of_even_succ_succ this lemma odd_add_of_even_of_odd {n m} : even n → odd m → odd (n+m) := suppose even n, suppose odd m, -assert even (n + succ m), from even_add_of_even_of_even `even n` (even_succ_of_odd `odd m`), +have even (n + succ m), from even_add_of_even_of_even `even n` (even_succ_of_odd `odd m`), odd_of_even_succ this lemma odd_add_of_odd_of_even {n m} : odd n → even m → odd (n+m) := suppose odd n, suppose even m, -assert odd (m+n), from odd_add_of_even_of_odd `even m` `odd n`, +have odd (m+n), from odd_add_of_even_of_odd `even m` `odd n`, by rewrite add.comm at this; exact this lemma even_mul_of_even_left {n} (m) : even n → even (n*m) := @@ -210,15 +210,15 @@ even_of_exists (exists.intro (k*m) (by rewrite [hk, mul.assoc])) lemma even_mul_of_even_right {n} (m) : even n → even (m*n) := suppose even n, -assert even (n*m), from even_mul_of_even_left _ this, +have even (n*m), from even_mul_of_even_left _ this, by rewrite mul.comm at this; exact this lemma odd_mul_of_odd_of_odd {n m} : odd n → odd m → odd (n*m) := suppose odd n, suppose odd m, -assert even (n * succ m), from even_mul_of_even_right _ (even_succ_of_odd `odd m`), -assert even (n * m + n), by rewrite mul_succ at this; exact this, +have even (n * succ m), from even_mul_of_even_right _ (even_succ_of_odd `odd m`), +have even (n * m + n), by rewrite mul_succ at this; exact this, by_contradiction (suppose ¬ odd (n*m), - assert even (n*m), from even_of_not_odd this, + have even (n*m), from even_of_not_odd this, absurd `even (n * m + n)` (not_even_of_odd (odd_add_of_even_of_odd this `odd n`))) lemma even_of_even_mul_self {n} : even (n * n) → even n := @@ -274,7 +274,7 @@ assume h₁ h₂, (suppose odd n, or.elim (em (even m)) (suppose even m, absurd `odd n` (not_odd_of_even (iff.mpr h₂ `even m`))) (suppose odd m, - assert d : 1 / 2 = (0:nat), from dec_trivial, + have d : 1 / 2 = (0:nat), from dec_trivial, obtain w₁ (hw₁ : n = 2*w₁ + 1), from exists_of_odd `odd n`, obtain w₂ (hw₂ : m = 2*w₂ + 1), from exists_of_odd `odd m`, begin diff --git a/library/data/nat/power.lean b/library/data/nat/power.lean index 1d422bcd0..a3cc62072 100644 --- a/library/data/nat/power.lean +++ b/library/data/nat/power.lean @@ -58,16 +58,16 @@ by krewrite [*pow_succ, *pow_zero, mul_one] theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a ^ b = a ^ c → b = c | a 0 0 h₁ h₂ := rfl | a (succ b) 0 h₁ h₂ := - assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂), - assert (1:nat) < 1, by rewrite [this at h₁]; exact h₁, + have a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂), + have (1:nat) < 1, by rewrite [this at h₁]; exact h₁, absurd `1 <[nat] 1` !lt.irrefl | a 0 (succ c) h₁ h₂ := - assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)), - assert (1:nat) < 1, by rewrite [this at h₁]; exact h₁, + have a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right (eq.symm h₂)), + have (1:nat) < 1, by rewrite [this at h₁]; exact h₁, absurd `1 <[nat] 1` !lt.irrefl | a (succ b) (succ c) h₁ h₂ := - assert a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial), - assert a^b = a^c, by rewrite [*pow_succ at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero this) h₂), + have a ≠ 0, from assume aeq0, by rewrite [aeq0 at h₁]; exact (absurd h₁ dec_trivial), + have a^b = a^c, by rewrite [*pow_succ at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero this) h₂), by rewrite [pow_cancel_left h₁ this] theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → (a ^ succ b) / a = a ^ b @@ -87,7 +87,7 @@ iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h) lemma pow_dvd_of_pow_succ_dvd {p i n : nat} : p^(succ i) ∣ n → p^i ∣ n := suppose p^(succ i) ∣ n, -assert p^i ∣ p^(succ i), +have p^i ∣ p^(succ i), by rewrite [pow_succ']; apply nat.dvd_of_eq_mul; apply rfl, dvd.trans `p^i ∣ p^(succ i)` `p^(succ i) ∣ n` diff --git a/library/data/nat/sqrt.lean b/library/data/nat/sqrt.lean index 18c5a3d05..63e62ef26 100644 --- a/library/data/nat/sqrt.lean +++ b/library/data/nat/sqrt.lean @@ -31,7 +31,7 @@ theorem sqrt_aux_le : ∀ (s n), sqrt_aux s n ≤ s | (succ s) n := or.elim (em ((succ s)*(succ s) ≤ n)) (λ h, begin unfold sqrt_aux, rewrite [if_pos h] end) (λ h, - assert sqrt_aux s n ≤ succ s, from le.step (sqrt_aux_le s n), + have sqrt_aux s n ≤ succ s, from le.step (sqrt_aux_le s n), begin unfold sqrt_aux, rewrite [if_neg h], assumption end) definition sqrt (n : nat) : nat := @@ -42,7 +42,7 @@ theorem sqrt_aux_lower : ∀ {s n : nat}, s ≤ n → sqrt_aux s n * sqrt_aux s | (succ s) n h := by_cases (λ h₁ : (succ s)*(succ s) ≤ n, by rewrite [sqrt_aux_succ_of_pos h₁]; exact h₁) (λ h₂ : ¬ (succ s)*(succ s) ≤ n, - assert aux : s ≤ n, from le_of_succ_le h, + have aux : s ≤ n, from le_of_succ_le h, by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_lower aux)) theorem sqrt_lower (n : nat) : sqrt n * sqrt n ≤ n := @@ -54,8 +54,8 @@ theorem sqrt_aux_upper : ∀ {s n : nat}, n ≤ s*s + s + s → n ≤ sqrt_aux s (λ h₁ : (succ s)*(succ s) ≤ n, by rewrite [sqrt_aux_succ_of_pos h₁]; exact h) (λ h₂ : ¬ (succ s)*(succ s) ≤ n, - assert h₃ : n < (succ s) * (succ s), from lt_of_not_ge h₂, - assert h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact le_of_lt_succ h₃, + have h₃ : n < (succ s) * (succ s), from lt_of_not_ge h₂, + have h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact le_of_lt_succ h₃, by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_upper h₄)) theorem sqrt_upper (n : nat) : n ≤ sqrt n * sqrt n + sqrt n + sqrt n := @@ -65,8 +65,8 @@ sqrt_aux_upper aux private theorem le_squared : ∀ (n : nat), n ≤ n*n | 0 := !le.refl | (succ n) := - have aux₁ : 1 ≤ succ n, from succ_le_succ !zero_le, - assert aux₂ : 1 * succ n ≤ succ n * succ n, from nat.mul_le_mul aux₁ !le.refl, + have aux₁ : 1 ≤ succ n, from succ_le_succ !zero_le, + have aux₂ : 1 * succ n ≤ succ n * succ n, from nat.mul_le_mul aux₁ !le.refl, by rewrite [one_mul at aux₂]; exact aux₂ private theorem lt_squared : ∀ {n : nat}, n > 1 → n < n * n @@ -74,7 +74,7 @@ private theorem lt_squared : ∀ {n : nat}, n > 1 → n < n * n | 1 h := absurd h dec_trivial | (succ (succ n)) h := have 1 < succ (succ n), from dec_trivial, - assert succ (succ n) * 1 < succ (succ n) * succ (succ n), from mul_lt_mul_of_pos_left this dec_trivial, + have succ (succ n) * 1 < succ (succ n) * succ (succ n), from mul_lt_mul_of_pos_left this dec_trivial, by rewrite [mul_one at this]; exact this theorem sqrt_le (n : nat) : sqrt n ≤ n := @@ -83,13 +83,13 @@ calc sqrt n ≤ sqrt n * sqrt n : le_squared theorem eq_zero_of_sqrt_eq_zero {n : nat} : sqrt n = 0 → n = 0 := suppose sqrt n = 0, -assert n ≤ sqrt n * sqrt n + sqrt n + sqrt n, from !sqrt_upper, -have n ≤ 0, by rewrite [*`sqrt n = 0` at this]; exact this, +have n ≤ sqrt n * sqrt n + sqrt n + sqrt n, from !sqrt_upper, +have n ≤ 0, by rewrite [*`sqrt n = 0` at this]; exact this, eq_zero_of_le_zero this theorem le_three_of_sqrt_eq_one {n : nat} : sqrt n = 1 → n ≤ 3 := suppose sqrt n = 1, -assert n ≤ sqrt n * sqrt n + sqrt n + sqrt n, from !sqrt_upper, +have n ≤ sqrt n * sqrt n + sqrt n + sqrt n, from !sqrt_upper, show n ≤ 3, by rewrite [*`sqrt n = 1` at this]; exact this theorem sqrt_lt : ∀ {n : nat}, n > 1 → sqrt n < n @@ -116,21 +116,21 @@ theorem sqrt_pos_of_pos {n : nat} : n > 0 → sqrt n > 0 := suppose n > 0, have sqrt n ≠ 0, from suppose sqrt n = 0, - assert n = 0, from eq_zero_of_sqrt_eq_zero this, + have n = 0, from eq_zero_of_sqrt_eq_zero this, by subst n; exact absurd `0 > 0` !lt.irrefl, pos_of_ne_zero this theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n → sqrt_aux s (n*n + k) = n | 0 h₂ := - assert neqz : n = 0, from eq_zero_of_le_zero h₂, + have neqz : n = 0, from eq_zero_of_le_zero h₂, by rewrite neqz | (succ s) h₂ := by_cases (λ hl : (succ s)*(succ s) ≤ n*n + k, - have l₁ : n*n + k ≤ n*n + n + n, from by rewrite [add.assoc]; exact (add_le_add_left h₁ (n*n)), - assert l₂ : n*n + k < n*n + n + n + 1, from lt_succ_of_le l₁, - have l₃ : n*n + k < (succ n)*(succ n), by rewrite [-succ_mul_succ_eq at l₂]; exact l₂, - assert l₄ : (succ s)*(succ s) < (succ n)*(succ n), from lt_of_le_of_lt hl l₃, - have ng : ¬ succ s > (succ n), from + have l₁ : n*n + k ≤ n*n + n + n, from by rewrite [add.assoc]; exact (add_le_add_left h₁ (n*n)), + have l₂ : n*n + k < n*n + n + n + 1, from lt_succ_of_le l₁, + have l₃ : n*n + k < (succ n)*(succ n), by rewrite [-succ_mul_succ_eq at l₂]; exact l₂, + have l₄ : (succ s)*(succ s) < (succ n)*(succ n), from lt_of_le_of_lt hl l₃, + have ng : ¬ succ s > (succ n), from assume g : succ s > succ n, have g₁ : (succ s)*(succ s) > (succ n)*(succ n), from mul_lt_mul_of_le_of_le g g, absurd (lt.trans g₁ l₄) !lt.irrefl, @@ -139,7 +139,7 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n assume sseqsn : succ s = succ n, by rewrite [sseqsn at l₄]; exact (absurd l₄ !lt.irrefl), have sslen : s < n, from lt_of_succ_lt_succ (lt_of_le_of_ne sslesn ssnesn), - assert sseqn : succ s = n, from le.antisymm sslen h₂, + have sseqn : succ s = n, from le.antisymm sslen h₂, by rewrite [sqrt_aux_succ_of_pos hl]; exact sseqn) (λ hg : ¬ (succ s)*(succ s) ≤ n*n + k, or.elim (eq_or_lt_of_le h₂) @@ -160,6 +160,6 @@ sqrt_offset_eq !zero_le theorem mul_square_cancel {a b : nat} : a*a = b*b → a = b := assume h, -assert aux : sqrt (a*a) = sqrt (b*b), by rewrite h, +have aux : sqrt (a*a) = sqrt (b*b), by rewrite h, by rewrite [*sqrt_eq at aux]; exact aux end nat diff --git a/library/data/nat/sub.lean b/library/data/nat/sub.lean index 652c074f0..b850b6208 100644 --- a/library/data/nat/sub.lean +++ b/library/data/nat/sub.lean @@ -212,8 +212,8 @@ sub.cases le.intro (add.right_cancel H4)) protected theorem sub_pos_of_lt {m n : ℕ} (H : m < n) : n - m > 0 := -assert H1 : n = n - m + m, from (nat.sub_add_cancel (le_of_lt H))⁻¹, -have H2 : 0 + m < n - m + m, begin rewrite [zero_add, -H1], exact H end, +have H1 : n = n - m + m, from (nat.sub_add_cancel (le_of_lt H))⁻¹, +have H2 : 0 + m < n - m + m, begin rewrite [zero_add, -H1], exact H end, !lt_of_add_lt_add_right H2 protected theorem lt_of_sub_pos {m n : ℕ} (H : n - m > 0) : m < n := @@ -242,7 +242,7 @@ sub.cases sub.cases (assume H : m ≤ k, have H2 : n - k ≤ n - m, from nat.sub_le_sub_left H n, - assert H3 : n - k ≤ mn, from nat.sub_eq_of_add_eq Hmn ▸ H2, + have H3 : n - k ≤ mn, from nat.sub_eq_of_add_eq Hmn ▸ H2, show n - k ≤ mn + 0, begin rewrite add_zero, assumption end) (take km : ℕ, assume Hkm : k + km = m, @@ -364,12 +364,12 @@ have (n - m) + (m - k) + ((k - m) + (m - n)) = (n - m) + (m - n) + ((m - k) + (k this ▸ add_le_add !nat.sub_lt_sub_add_sub !nat.sub_lt_sub_add_sub theorem dist_add_add_le_add_dist_dist (n m k l : ℕ) : dist (n + m) (k + l) ≤ dist n k + dist m l := -assert H : dist (n + m) (k + m) + dist (k + m) (k + l) = dist n k + dist m l, +have H : dist (n + m) (k + m) + dist (k + m) (k + l) = dist n k + dist m l, by rewrite [dist_add_add_left, dist_add_add_right], by rewrite -H; apply dist.triangle_inequality theorem dist_mul_right (n k m : ℕ) : dist (n * k) (m * k) = dist n m * k := -assert ∀ n m, dist n m = n - m + (m - n), from take n m, rfl, +have ∀ n m, dist n m = n - m + (m - n), from take n m, rfl, by rewrite [this, this n m, right_distrib, *nat.mul_sub_right_distrib] theorem dist_mul_left (k n m : ℕ) : dist (k * n) (k * m) = k * dist n m := diff --git a/library/data/pnat.lean b/library/data/pnat.lean index 897d106ff..4b4bf6f8d 100644 --- a/library/data/pnat.lean +++ b/library/data/pnat.lean @@ -308,7 +308,7 @@ begin end theorem pnat_cancel' (n m : ℕ+) : (n * n * m)⁻¹ * (rat_of_pnat n * rat_of_pnat n) = m⁻¹ := -assert hsimp : ∀ a b c : ℚ, (a * a * (b * b * c)) = (a * b) * (a * b) * c, +have hsimp : ∀ a b c : ℚ, (a * a * (b * b * c)) = (a * b) * (a * b) * c, begin intro a b c, rewrite[-*rat.mul_assoc], @@ -322,7 +322,7 @@ theorem pceil_helper {a : ℚ} {n : ℕ+} (H : pceil a ≤ n) (Ha : a > 0) : n le.trans (inv_ge_of_le H) (one_div_le_one_div_of_le Ha (ubound_ge a)) theorem inv_pceil_div (a b : ℚ) (Ha : a > 0) (Hb : b > 0) : (pceil (a / b))⁻¹ ≤ b / a := -assert (pceil (a / b))⁻¹ ≤ 1 / (1 / (b / a)), +have (pceil (a / b))⁻¹ ≤ 1 / (1 / (b / a)), begin apply one_div_le_one_div_of_le, show 0 < 1 / (b / a), from @@ -387,7 +387,7 @@ begin end theorem p_add_fractions (n : ℕ+) : (2 * n)⁻¹ + (2 * 3 * n)⁻¹ + (3 * n)⁻¹ = n⁻¹ := -assert T : 2⁻¹ + 2⁻¹ * 3⁻¹ + 3⁻¹ = 1, from dec_trivial, +have T : 2⁻¹ + 2⁻¹ * 3⁻¹ + 3⁻¹ = 1, from dec_trivial, by rewrite[*pnat.inv_mul_eq_mul_inv,-*right_distrib,T,rat.one_mul] theorem rat_power_two_le (k : ℕ+) : rat_of_pnat k ≤ 2^k~ := diff --git a/library/data/prod.lean b/library/data/prod.lean index 87141cbb3..ce4ac0943 100644 --- a/library/data/prod.lean +++ b/library/data/prod.lean @@ -37,6 +37,6 @@ namespace prod theorem eq_of_swap_eq {A : Type} : ∀ p₁ p₂ : A × A, swap p₁ = swap p₂ → p₁ = p₂ := take p₁ p₂, assume seqs, - assert swap (swap p₁) = swap (swap p₂), from congr_arg swap seqs, + have swap (swap p₁) = swap (swap p₂), from congr_arg swap seqs, by rewrite *swap_swap at this; exact this end prod diff --git a/library/data/rat/basic.lean b/library/data/rat/basic.lean index 96d3af46f..f813b7b1a 100644 --- a/library/data/rat/basic.lean +++ b/library/data/rat/basic.lean @@ -39,8 +39,8 @@ have num b * denom a > 0, from H ▸ this, show num b > 0, from pos_of_mul_pos_right this (le_of_lt (denom_pos a)) theorem num_neg_of_equiv {a b : prerat} (H : a ≡ b) (na_neg : num a < 0) : num b < 0 := -assert H₁ : num a * denom b = num b * denom a, from H, -assert num a * denom b < 0, from mul_neg_of_neg_of_pos na_neg (denom_pos b), +have H₁ : num a * denom b = num b * denom a, from H, +have num a * denom b < 0, from mul_neg_of_neg_of_pos na_neg (denom_pos b), have -(-num b * denom a) < 0, begin rewrite [neg_mul_eq_neg_mul, neg_neg, -H₁], exact this end, have -num b > 0, from pos_of_mul_pos_right (pos_of_neg_neg this) (le_of_lt (denom_pos a)), neg_of_neg_pos this @@ -314,11 +314,11 @@ theorem reduce_eq_reduce : ∀ {a b : prerat}, a ≡ b → reduce a = reduce b decidable.by_cases (assume anz : an = 0, have H' : bn * ad = 0, by rewrite [-H, anz, zero_mul], - assert bnz : bn = 0, + have bnz : bn = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H') (ne_of_gt adpos), by rewrite [↑reduce, if_pos anz, if_pos bnz]) (assume annz : an ≠ 0, - assert bnnz : bn ≠ 0, from + have bnnz : bn ≠ 0, from assume bnz, have H' : an * bd = 0, by rewrite [H, bnz, zero_mul], have anz : an = 0, @@ -541,7 +541,7 @@ decidable.by_cases (suppose a = 0, by substvars) (quot.induction_on a (take u H, - assert H' : prerat.num u ≠ 0, from take H'', H (quot.sound (prerat.equiv_zero_of_num_eq_zero H'')), + have H' : prerat.num u ≠ 0, from take H'', H (quot.sound (prerat.equiv_zero_of_num_eq_zero H'')), begin cases u with un ud udpos, rewrite [▸*, ↑num, ↑denom, ↑reduce, ↑prerat.reduce, if_neg H', ▸*], diff --git a/library/data/rat/order.lean b/library/data/rat/order.lean index fa7959d79..d20e3c00d 100644 --- a/library/data/rat/order.lean +++ b/library/data/rat/order.lean @@ -205,7 +205,7 @@ protected theorem le_refl (a : ℚ) : a ≤ a := by rewrite [rat.le_def, sub_self]; apply nonneg_zero protected theorem le_trans (H1 : a ≤ b) (H2 : b ≤ c) : a ≤ c := -assert H3 : nonneg (c - b + (b - a)), from nonneg_add H2 H1, +have H3 : nonneg (c - b + (b - a)), from nonneg_add H2 H1, begin revert H3, rewrite [rat.sub.def, add.assoc, sub_eq_add_neg, neg_add_cancel_left], @@ -256,14 +256,14 @@ have c + b - (c + a) = b - a, show nonneg (c + b - (c + a)), from this⁻¹ ▸ H protected theorem mul_nonneg (H1 : a ≥ (0 : ℚ)) (H2 : b ≥ (0 : ℚ)) : a * b ≥ (0 : ℚ) := -assert nonneg (a * b), from nonneg_mul (to_nonneg H1) (to_nonneg H2), +have nonneg (a * b), from nonneg_mul (to_nonneg H1) (to_nonneg H2), begin rewrite -sub_zero at this, exact this end private theorem to_pos : a > 0 → pos a := by intros; rewrite -sub_zero; eassumption protected theorem mul_pos (H1 : a > (0 : ℚ)) (H2 : b > (0 : ℚ)) : a * b > (0 : ℚ) := -assert pos (a * b), from pos_mul (to_pos H1) (to_pos H2), +have pos (a * b), from pos_mul (to_pos H1) (to_pos H2), begin rewrite -sub_zero at this, exact this end definition decidable_lt [instance] : decidable_rel rat.lt := @@ -322,7 +322,7 @@ protected definition discrete_linear_ordered_field [trans_instance] : add_lt_add_left := @rat.add_lt_add_left⦄ theorem of_nat_abs (a : ℤ) : abs (of_int a) = of_nat (int.nat_abs a) := -assert ∀ n : ℕ, of_int (int.neg_succ_of_nat n) = - of_nat (nat.succ n), from λ n, rfl, +have ∀ n : ℕ, of_int (int.neg_succ_of_nat n) = - of_nat (nat.succ n), from λ n, rfl, int.induction_on a (take b, abs_of_nonneg !of_nat_nonneg) (take b, by rewrite [this, abs_neg, abs_of_nonneg !of_nat_nonneg]) @@ -407,7 +407,7 @@ definition ubound : ℚ → ℕ := λ a : ℚ, nat.succ (int.nat_abs (num a)) theorem ubound_ge (a : ℚ) : of_nat (ubound a) ≥ a := have h : abs a * abs (of_int (denom a)) = abs (of_int (num a)), from !abs_mul ▸ !mul_denom ▸ rfl, -assert of_int (denom a) > 0, from of_int_lt_of_int_of_lt !denom_pos, +have of_int (denom a) > 0, from of_int_lt_of_int_of_lt !denom_pos, have 1 ≤ abs (of_int (denom a)), begin rewrite (abs_of_pos this), apply of_int_le_of_int_of_le, diff --git a/library/data/real/basic.lean b/library/data/real/basic.lean index 49bff3918..473948955 100644 --- a/library/data/real/basic.lean +++ b/library/data/real/basic.lean @@ -102,7 +102,7 @@ end private theorem ineq_helper (a b : ℚ) (k m n : ℕ+) (H : a ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹) (H2 : b ≤ (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹) : (rat_of_pnat k) * a + b * (rat_of_pnat k) ≤ m⁻¹ + n⁻¹ := -assert H3 : (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹ = (2 * k)⁻¹ * (m⁻¹ + n⁻¹), +have H3 : (k * 2 * m)⁻¹ + (k * 2 * n)⁻¹ = (2 * k)⁻¹ * (m⁻¹ + n⁻¹), begin rewrite [left_distrib, *pnat.inv_mul_eq_mul_inv], rewrite (mul.comm k⁻¹) @@ -304,15 +304,15 @@ definition K₂ (s t : seq) := max (K s) (K t) private theorem K₂_symm (s t : seq) : K₂ s t = K₂ t s := if H : K s < K t then - (assert H1 : K₂ s t = K t, from pnat.max_eq_right H, - assert H2 : K₂ t s = K t, from pnat.max_eq_left (pnat.not_lt_of_ge (pnat.le_of_lt H)), + (have H1 : K₂ s t = K t, from pnat.max_eq_right H, + have H2 : K₂ t s = K t, from pnat.max_eq_left (pnat.not_lt_of_ge (pnat.le_of_lt H)), by rewrite [H1, -H2]) else - (assert H1 : K₂ s t = K s, from pnat.max_eq_left H, + (have H1 : K₂ s t = K s, from pnat.max_eq_left H, if J : K t < K s then - (assert H2 : K₂ t s = K s, from pnat.max_eq_right J, by rewrite [H1, -H2]) + (have H2 : K₂ t s = K s, from pnat.max_eq_right J, by rewrite [H1, -H2]) else - (assert Heq : K t = K s, from + (have Heq : K t = K s, from pnat.eq_of_le_of_ge (pnat.le_of_not_gt H) (pnat.le_of_not_gt J), by rewrite [↑K₂, Heq])) diff --git a/library/data/real/complete.lean b/library/data/real/complete.lean index 4f9263825..c97490fb7 100644 --- a/library/data/real/complete.lean +++ b/library/data/real/complete.lean @@ -558,7 +558,7 @@ open nat theorem archimedean_small {ε : ℝ} (H : ε > 0) : ∃ (n : ℕ), 1 / succ n < ε := let n := int.nat_abs (ceil (2 / ε)) in -assert int.of_nat n ≥ ceil (2 / ε), +have int.of_nat n ≥ ceil (2 / ε), by rewrite of_nat_nat_abs; apply le_abs_self, have int.of_nat (succ n) ≥ ceil (2 / ε), begin apply le.trans, exact this, apply int.of_nat_le_of_nat_of_le, apply le_succ end, diff --git a/library/data/real/division.lean b/library/data/real/division.lean index 1c8b780c1..5d1acfb7c 100644 --- a/library/data/real/division.lean +++ b/library/data/real/division.lean @@ -422,10 +422,10 @@ theorem inv_well_defined {s t : seq} (Hs : regular s) (Ht : regular t) (Heq : s apply one_is_reg end) else - (assert H : s_inv Hs = zero, from funext (λ n, dif_neg Hsep), + (have H : s_inv Hs = zero, from funext (λ n, dif_neg Hsep), have Hsept : ¬ sep t zero, from assume H', Hsep (sep_of_equiv_sep Ht Hs (equiv.symm _ _ Heq) H'), - assert H' : s_inv Ht = zero, from funext (λ n, dif_neg Hsept), + have H' : s_inv Ht = zero, from funext (λ n, dif_neg Hsept), by rewrite [H', H]; apply equiv.refl) theorem s_neg_neg {s : seq} : sneg (sneg s) ≡ s := @@ -545,7 +545,7 @@ theorem s_le_of_equiv_le_right {s t u : seq} (Hs : regular s) (Ht : regular t) ( noncomputable definition r_inv (s : reg_seq) : reg_seq := reg_seq.mk (s_inv (reg_seq.is_reg s)) (if H : sep (reg_seq.sq s) zero then reg_inv_reg (reg_seq.is_reg s) H else - assert Hz : s_inv (reg_seq.is_reg s) = zero, from funext (λ n, dif_neg H), + have Hz : s_inv (reg_seq.is_reg s) = zero, from funext (λ n, dif_neg H), by rewrite Hz; apply zero_is_reg) theorem r_inv_zero : requiv (r_inv r_zero) r_zero := @@ -680,8 +680,8 @@ theorem eq_zero_of_nonneg_of_forall_le {x : ℝ} (xnonneg : x ≥ 0) (H : ∀ ε x = 0 := have ∀ ε : ℝ, ε > 0 → x < ε, from take ε, suppose ε > 0, - assert e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos, - assert ε / 2 < ε, from div_two_lt_of_pos `ε > 0`, + have e2pos : ε / 2 > 0, from div_pos_of_pos_of_pos `ε > 0` two_pos, + have ε / 2 < ε, from div_two_lt_of_pos `ε > 0`, begin apply lt_of_le_of_lt, apply H _ e2pos, apply this end, eq_zero_of_nonneg_of_forall_lt xnonneg this diff --git a/library/data/set/card.lean b/library/data/set/card.lean index aa735489c..05f3dcbbe 100644 --- a/library/data/set/card.lean +++ b/library/data/set/card.lean @@ -27,7 +27,7 @@ theorem card_insert_of_mem {a : A} {s : set A} (H : a ∈ s) : card (insert a s) if fins : finite s then (by rewrite [↑card, to_finset_insert, -mem_to_finset_eq at H, finset.card_insert_of_mem H]) else - (assert ¬ finite (insert a s), from suppose _, absurd (!finite_of_finite_insert this) fins, + (have ¬ finite (insert a s), from suppose _, absurd (!finite_of_finite_insert this) fins, by rewrite [card_of_not_finite fins, card_of_not_finite this]) theorem card_insert_of_not_mem {a : A} {s : set A} [finite s] (H : a ∉ s) : @@ -142,7 +142,7 @@ begin end theorem exists_two_of_card_gt_one {s : set A} (H : 1 < card s) : ∃ a b, a ∈ s ∧ b ∈ s ∧ a ≠ b := -assert fins : finite s, from +have fins : finite s, from by_contradiction (assume nfins, by rewrite [card_of_not_finite nfins at H]; apply !not_succ_le_zero H), by rewrite [-to_set_to_finset s]; apply finset.exists_two_of_card_gt_one H diff --git a/library/data/set/filter.lean b/library/data/set/filter.lean index 87ad7139e..238fa00aa 100644 --- a/library/data/set/filter.lean +++ b/library/data/set/filter.lean @@ -201,7 +201,7 @@ namespace complete_lattice λ a b Ha Hb, obtain a₁ [a₁F₁ [a₂ [a₂F₂ (Ha' : a ⊇ a₁ ∩ a₂)]]], from Ha, obtain b₁ [b₁F₁ [b₂ [b₂F₂ (Hb' : b ⊇ b₁ ∩ b₂)]]], from Hb, - assert a₁ ∩ b₁ ∩ (a₂ ∩ b₂) = a₁ ∩ a₂ ∩ (b₁ ∩ b₂), + have a₁ ∩ b₁ ∩ (a₂ ∩ b₂) = a₁ ∩ a₂ ∩ (b₁ ∩ b₂), by rewrite [*inter_assoc, inter_left_comm b₁], have a ∩ b ⊇ a₁ ∩ b₁ ∩ (a₂ ∩ b₂), begin diff --git a/library/data/set/finite.lean b/library/data/set/finite.lean index df1546892..e64525877 100644 --- a/library/data/set/finite.lean +++ b/library/data/set/finite.lean @@ -168,14 +168,14 @@ theorem finite_iff_finite_of_bij_on {B : Type} {f : A → B} {s : set A} {t : se iff.intro (assume fs, finite_of_bij_on bijf) (assume ft, finite_of_bij_on' bijf) theorem finite_powerset (s : set A) [finite s] : finite 𝒫 s := -assert H : 𝒫 s = finset.to_set ' (finset.to_set (#finset 𝒫 (to_finset s))), +have H : 𝒫 s = finset.to_set ' (finset.to_set (#finset 𝒫 (to_finset s))), from ext (take t, iff.intro (suppose t ∈ 𝒫 s, - assert t ⊆ s, from this, - assert finite t, from finite_subset this, - assert (#finset to_finset t ∈ 𝒫 (to_finset s)), + have t ⊆ s, from this, + have finite t, from finite_subset this, + have (#finset to_finset t ∈ 𝒫 (to_finset s)), by rewrite [finset.mem_powerset_iff_subset, to_finset_subset_to_finset_eq]; apply `t ⊆ s`, - assert to_finset t ∈ (finset.to_set (finset.powerset (to_finset s))), from this, + have to_finset t ∈ (finset.to_set (finset.powerset (to_finset s))), from this, mem_image this (by rewrite to_set_to_finset)) (assume H', obtain t' [(tmem : (#finset t' ∈ 𝒫 (to_finset s))) (teq : finset.to_set t' = t)], diff --git a/library/data/stream.lean b/library/data/stream.lean index 4638c89b0..ae936ca9b 100644 --- a/library/data/stream.lean +++ b/library/data/stream.lean @@ -634,7 +634,7 @@ obtain (i₁ : nat) hlt₁ he₁, from h₁, obtain (i₂ : nat) hlt₂ he₂, from h₂, lt.by_cases (λ i₁lti₂ : i₁ < i₂, - assert aux : nth i₁ s₂ = nth i₁ s₃, from he₂ _ i₁lti₂, + have aux : nth i₁ s₂ = nth i₁ s₃, from he₂ _ i₁lti₂, begin existsi i₁, split, {rewrite -aux, exact hlt₁}, @@ -650,7 +650,7 @@ lt.by_cases exact !he₂ jlti₁ end) (λ i₂lti₁ : i₂ < i₁, - assert nth i₂ s₁ = nth i₂ s₂, from he₁ _ i₂lti₁, + have nth i₂ s₁ = nth i₂ s₂, from he₁ _ i₂lti₁, begin existsi i₂, split, {rewrite this, exact hlt₂}, diff --git a/library/init/simplifier.lean b/library/init/simplifier.lean index de6804522..aa2cdcebb 100644 --- a/library/init/simplifier.lean +++ b/library/init/simplifier.lean @@ -75,12 +75,12 @@ iff.intro (assume H, and.intro (assume a, implies_of_if_pos H a) (assume H, and.rec_on H (assume Hab Hnac, decidable.rec_on A_dec (assume a, - assert rw : @decidable.inl A a = A_dec, from + have rw : @decidable.inl A a = A_dec, from subsingleton.rec_on (subsingleton_decidable A) (assume H, H (@decidable.inl A a) A_dec), by rewrite [rw, if_pos a] ; exact Hab a) (assume na, - assert rw : @decidable.inr A na = A_dec, from + have rw : @decidable.inr A na = A_dec, from subsingleton.rec_on (subsingleton_decidable A) (assume H, H (@decidable.inr A na) A_dec), by rewrite [rw, if_neg na] ; exact Hnac na))) diff --git a/library/logic/examples/colog88.lean b/library/logic/examples/colog88.lean index 55e450283..469be3e75 100644 --- a/library/logic/examples/colog88.lean +++ b/library/logic/examples/colog88.lean @@ -42,7 +42,7 @@ lemma betaA (p : Phi A) : matchA (introA p) = p := -- As in all inductive datatypes, we would be able to prove that constructors are injective. lemma introA_injective : ∀ {p p' : Phi A}, introA p = introA p' → p = p' := λ p p' h, - assert aux : matchA (introA p) = matchA (introA p'), by rewrite h, + have aux : matchA (introA p) = matchA (introA p'), by rewrite h, by rewrite [*betaA at aux]; exact aux -- For any type T, there is an injection from T to (T → Prop) @@ -80,9 +80,9 @@ rfl lemma not_P0_x0 : ¬ P0 x0 := λ h : P0 x0, obtain (P : A → Prop) (hp : f P = x0 ∧ ¬ P x0), from h, - have fp_eq : f P = f P0, from and.elim_left hp, - assert p_eq : P = P0, from f_injective fp_eq, - have nh : ¬ P0 x0, by rewrite [p_eq at hp]; exact (and.elim_right hp), + have fp_eq : f P = f P0, from and.elim_left hp, + have p_eq : P = P0, from f_injective fp_eq, + have nh : ¬ P0 x0, by rewrite [p_eq at hp]; exact (and.elim_right hp), absurd h nh lemma P0_x0 : P0 x0 := diff --git a/library/logic/examples/cont.lean b/library/logic/examples/cont.lean index a63ab59bd..9e4db4b60 100644 --- a/library/logic/examples/cont.lean +++ b/library/logic/examples/cont.lean @@ -58,12 +58,12 @@ lemma β0_eq (β : nat → nat) : ∀ α, zω =[f β] α → β 0 = β (α m) := lemma not_all_continuous : false := let β := znkω (M f + 1) 1 in let α := znkω m (M f + 1) in -assert βeq₁ : zω =[M f + 1] β, from +have βeq₁ : zω =[M f + 1] β, from λ (a : nat) (h : a < M f + 1), begin unfold zω, unfold znkω, rewrite [if_pos h] end, -assert βeq₂ : zω =[M f] β, from pred_beq βeq₁, -assert m_eq_fβ : m = f β, from M_spec f β βeq₂, -assert aux : ∀ α, zω =[m] α → β 0 = β (α m), by rewrite m_eq_fβ at {1}; exact (β0_eq β), -assert zero_eq_one : 0 = 1, from calc +have βeq₂ : zω =[M f] β, from pred_beq βeq₁, +have m_eq_fβ : m = f β, from M_spec f β βeq₂, +have aux : ∀ α, zω =[m] α → β 0 = β (α m), by rewrite m_eq_fβ at {1}; exact (β0_eq β), +have zero_eq_one : 0 = 1, from calc 0 = β 0 : by rewrite znkω_succ ... = β (α m) : aux α (zω_eq_znkω m (M f + 1)) ... = β (M f + 1) : by rewrite znkω_bound diff --git a/library/logic/examples/leftinv_of_inj.lean b/library/logic/examples/leftinv_of_inj.lean index ac4735670..a079a9874 100644 --- a/library/logic/examples/leftinv_of_inj.lean +++ b/library/logic/examples/leftinv_of_inj.lean @@ -20,8 +20,8 @@ assume inj : ∀ a₁ a₂, f a₁ = f a₂ → a₁ = a₂, let finv : B → A := mk_left_inv f in have linv : left_inverse finv f, from λ a, - assert ex : ∃ a₁ : A, f a₁ = f a, from exists.intro a rfl, - assert h₁ : f (some ex) = f a, from !some_spec, + have ex : ∃ a₁ : A, f a₁ = f a, from exists.intro a rfl, + have h₁ : f (some ex) = f a, from !some_spec, begin esimp [mk_left_inv, compose, id], rewrite [dif_pos ex], diff --git a/library/logic/examples/propositional/soundness.lean b/library/logic/examples/propositional/soundness.lean index 75fb9fad5..867dbbb05 100644 --- a/library/logic/examples/propositional/soundness.lean +++ b/library/logic/examples/propositional/soundness.lean @@ -123,13 +123,13 @@ namespace PropF have aux : bnot (TrueQ v A) = tt, by rewrite (eq_ff_of_ne_tt f), bor_inl aux)) (λ Γ A B H₁ H₂ r₁ r₂ v s, - assert aux₁ : bnot (TrueQ v A) || TrueQ v B = tt, from r₁ v s, - assert aux₂ : TrueQ v A = tt, from r₂ v s, + have aux₁ : bnot (TrueQ v A) || TrueQ v B = tt, from r₁ v s, + have aux₂ : TrueQ v A = tt, from r₂ v s, by rewrite [aux₂ at aux₁, bnot_true at aux₁, ff_bor at aux₁]; exact aux₁) (λ Γ A H r v s, by_contradiction (λ n : TrueQ v A ≠ tt, - assert aux₁ : TrueQ v A = ff, from eq_ff_of_ne_tt n, - assert aux₂ : TrueQ v (~A) = tt, begin change (bnot (TrueQ v A) || ff = tt), rewrite aux₁ end, + have aux₁ : TrueQ v A = ff, from eq_ff_of_ne_tt n, + have aux₂ : TrueQ v (~A) = tt, begin change (bnot (TrueQ v A) || ff = tt), rewrite aux₁ end, have aux₃ : Satisfies v ((~A)::Γ), from Satisfies_cons s aux₂, have aux₄ : TrueQ v ⊥ = tt, from r v aux₃, absurd aux₄ ff_ne_tt)) diff --git a/library/logic/examples/propositional/soundness_type.lean b/library/logic/examples/propositional/soundness_type.lean index dc74502cd..91a4c6fa4 100644 --- a/library/logic/examples/propositional/soundness_type.lean +++ b/library/logic/examples/propositional/soundness_type.lean @@ -129,13 +129,13 @@ namespace PropF have aux : bnot (TrueQ v A) = tt, by rewrite (eq_ff_of_ne_tt f), bor_inl aux) | ⌞B⌟ Γ (ImpE Γ A B H₁ H₂) s := - assert aux₁ : bnot (TrueQ v A) || TrueQ v B = tt, from Soundness_general H₁ s, - assert aux₂ : TrueQ v A = tt, from Soundness_general H₂ s, + have aux₁ : bnot (TrueQ v A) || TrueQ v B = tt, from Soundness_general H₁ s, + have aux₂ : TrueQ v A = tt, from Soundness_general H₂ s, by rewrite [aux₂ at aux₁, bnot_true at aux₁, ff_bor at aux₁]; exact aux₁ | ⌞A⌟ Γ (BotC Γ A H) s := by_contradiction (λ n : TrueQ v A ≠ tt, - assert aux₁ : TrueQ v A = ff, from eq_ff_of_ne_tt n, - assert aux₂ : TrueQ v (~A) = tt, begin change (bnot (TrueQ v A) || ff = tt), rewrite aux₁ end, + have aux₁ : TrueQ v A = ff, from eq_ff_of_ne_tt n, + have aux₂ : TrueQ v (~A) = tt, begin change (bnot (TrueQ v A) || ff = tt), rewrite aux₁ end, have aux₃ : Satisfies v ((~A)::Γ), from Satisfies_cons s aux₂, have aux₄ : TrueQ v ⊥ = tt, from Soundness_general H aux₃, absurd aux₄ ff_ne_tt) diff --git a/library/logic/weak_fan.lean b/library/logic/weak_fan.lean index cce9f592f..07d38f218 100644 --- a/library/logic/weak_fan.lean +++ b/library/logic/weak_fan.lean @@ -53,7 +53,7 @@ private lemma Y_unique : ∀ {P l₁ l₂}, length l₁ = length l₂ → Y P l have n₁ : length l₁ = length l₂, by rewrite [*length_cons at h₁]; apply nat.add_right_cancel h₁, have n₂ : Y P l₁, from and.elim_left h₂, have n₃ : Y P l₂, from and.elim_left h₃, - assert ih : l₁ = l₂, from Y_unique n₁ n₂ n₃, + have ih : l₁ = l₂, from Y_unique n₁ n₂ n₃, begin clear Y_unique, subst l₂, congruence, show a₁ = a₂, @@ -94,7 +94,7 @@ private lemma Y_approx : ∀ {P l}, approx (X P) l → Y P l theorem weak_fan : ∀ {P}, barred P → inductively_barred P [] := λ P Hbar, obtain l Hd HP, from Hbar (X P), -assert ib : inductively_barred P l, from inductively_barred.base l HP, +have ib : inductively_barred P l, from inductively_barred.base l HP, begin clear Hbar HP, induction l with a l ih, diff --git a/library/theories/analysis/inner_product.lean b/library/theories/analysis/inner_product.lean index 9501b1e4d..6334b0d2d 100644 --- a/library/theories/analysis/inner_product.lean +++ b/library/theories/analysis/inner_product.lean @@ -212,8 +212,8 @@ theorem cauchy_schwartz' (u v : V) : ⟨u, v⟩ ≤ ∥ u ∥ * ∥ v ∥ := ip_ theorem eq_proj_on_cauchy_schwartz {u v : V} (H : v ≠ 0) (H₁ : abs ⟨u, v⟩ = ∥ u ∥ * ∥ v ∥) : u = proj_on u H := -assert ∥ v ∥ ≠ 0, from assume H', H (eq_zero_of_norm_eq_zero H'), -assert ∥ u ∥ = ∥ proj_on u H ∥, by rewrite [norm_proj_on_eq, H₁, mul_div_cancel _ this], +have ∥ v ∥ ≠ 0, from assume H', H (eq_zero_of_norm_eq_zero H'), +have ∥ u ∥ = ∥ proj_on u H ∥, by rewrite [norm_proj_on_eq, H₁, mul_div_cancel _ this], have ∥ u - proj_on u H ∥^2 + ∥ u ∥^2 = 0 + ∥ u ∥^2, by rewrite [zero_add, norm_squared_pythagorean u H at {2}, this], have ∥ u - proj_on u H ∥^2 = 0, from eq_of_add_eq_add_right this, diff --git a/library/theories/analysis/real_limit.lean b/library/theories/analysis/real_limit.lean index b839100f0..c8bf2b606 100644 --- a/library/theories/analysis/real_limit.lean +++ b/library/theories/analysis/real_limit.lean @@ -529,16 +529,16 @@ let aX := (λ n, (abs x)^n), have noninc_aX : nonincreasing aX, from nonincreasing_of_forall_succ_le (take i, - assert (abs x) * (abs x)^i ≤ 1 * (abs x)^i, + have (abs x) * (abs x)^i ≤ 1 * (abs x)^i, from mul_le_mul_of_nonneg_right (le_of_lt H) (!pow_nonneg_of_nonneg !abs_nonneg), - assert (abs x) * (abs x)^i ≤ (abs x)^i, by krewrite one_mul at this; exact this, + have (abs x) * (abs x)^i ≤ (abs x)^i, by krewrite one_mul at this; exact this, show (abs x) ^ (succ i) ≤ (abs x)^i, by rewrite pow_succ; apply this), have bdd_aX : ∀ i, 0 ≤ aX i, from take i, !pow_nonneg_of_nonneg !abs_nonneg, -assert aXconv : aX ⟶ iaX in ℕ, proof converges_to_seq_inf_of_nonincreasing noninc_aX bdd_aX qed, +have aXconv : aX ⟶ iaX in ℕ, proof converges_to_seq_inf_of_nonincreasing noninc_aX bdd_aX qed, have asXconv : asX ⟶ iaX in ℕ, from converges_to_seq_offset_succ aXconv, have asXconv' : asX ⟶ (abs x) * iaX in ℕ, from mul_left_converges_to_seq (abs x) aXconv, have iaX = (abs x) * iaX, from converges_to_seq_unique asXconv asXconv', -assert iaX = 0, from eq_zero_of_mul_eq_self_left (ne_of_lt H) (eq.symm this), +have iaX = 0, from eq_zero_of_mul_eq_self_left (ne_of_lt H) (eq.symm this), show aX ⟶ 0 in ℕ, begin rewrite -this, exact aXconv end --from this ▸ aXconv end xn diff --git a/library/theories/combinatorics/choose.lean b/library/theories/combinatorics/choose.lean index 3ed918c22..c3ac5076b 100644 --- a/library/theories/combinatorics/choose.lean +++ b/library/theories/combinatorics/choose.lean @@ -36,7 +36,7 @@ nat.induction_on n take k, suppose succ n' < k, obtain k' (keq : k = succ k'), from exists_eq_succ_of_lt this, - assert n' < k', by rewrite keq at this; apply lt_of_succ_lt_succ this, + have n' < k', by rewrite keq at this; apply lt_of_succ_lt_succ this, by rewrite [keq, choose_succ_succ, IH _ this, IH _ (lt.trans this !lt_succ_self)]) theorem choose_self (n : ℕ) : choose n n = 1 := @@ -71,7 +71,7 @@ begin cases k with k, {intros, rewrite [choose_zero_right], apply zero_lt_one}, suppose succ k ≤ succ n, - assert k ≤ n, from le_of_succ_le_succ this, + have k ≤ n, from le_of_succ_le_succ this, by rewrite [choose_succ_succ]; apply add_pos_right (ih this) end @@ -129,16 +129,16 @@ include deceqA private theorem aux₀ (s : finset A) : {t ∈ powerset s | card t = 0} = '{∅} := ext (take t, iff.intro (assume H, - assert t = ∅, from eq_empty_of_card_eq_zero (of_mem_sep H), + have t = ∅, from eq_empty_of_card_eq_zero (of_mem_sep H), show t ∈ '{ ∅ }, by rewrite [this, mem_singleton_iff]) (assume H, - assert t = ∅, by rewrite mem_singleton_iff at H; assumption, + have t = ∅, by rewrite mem_singleton_iff at H; assumption, by substvars; exact mem_sep_of_mem !empty_mem_powerset rfl)) private theorem aux₁ (k : ℕ) : {t ∈ powerset (∅ : finset A) | card t = succ k} = ∅ := eq_empty_of_forall_not_mem (take t, assume H, - assert t ∈ powerset ∅, from mem_of_mem_sep H, - assert t = ∅, by rewrite [powerset_empty at this, mem_singleton_iff at this]; assumption, + have t ∈ powerset ∅, from mem_of_mem_sep H, + have t = ∅, by rewrite [powerset_empty at this, mem_singleton_iff at this]; assumption, have card (∅ : finset A) = succ k, by rewrite -this; apply of_mem_sep H, nat.no_confusion this) @@ -154,18 +154,18 @@ iff.intro (assume H, obtain H' cardt, from H, obtain t' [(t'pows : t' ∈ powerset s) (teq : insert a t' = t)], from exists_of_mem_image H', - assert aint : a ∈ t, by rewrite -teq; apply mem_insert, - assert anint' : a ∉ t', from + have aint : a ∈ t, by rewrite -teq; apply mem_insert, + have anint' : a ∉ t', from (assume aint', have a ∈ s, from mem_of_subset_of_mem (subset_of_mem_powerset t'pows) aint', anins this), - assert t' = erase a t, by rewrite [-teq, erase_insert (aux₂ anins t'pows)], + have t' = erase a t, by rewrite [-teq, erase_insert (aux₂ anins t'pows)], have card t' = k, by rewrite [this, card_erase_of_mem aint, cardt], mem_image (mem_sep_of_mem t'pows this) teq) (assume H, obtain t' [Ht' (teq : insert a t' = t)], from exists_of_mem_image H, - assert t'pows : t' ∈ powerset s, from mem_of_mem_sep Ht', - assert cardt' : card t' = k, from of_mem_sep Ht', + have t'pows : t' ∈ powerset s, from mem_of_mem_sep Ht', + have cardt' : card t' = k, from of_mem_sep Ht', and.intro (show t ∈ (insert a) ' (powerset s), from mem_image t'pows teq) (show card t = succ k, @@ -196,9 +196,9 @@ have set.inj_on (insert a) (ts {t ∈ powerset s| card t = k}), from take t₁ t₂, assume Ht₁ Ht₂, assume Heq : insert a t₁ = insert a t₂, have t₁ ∈ powerset s, from mem_of_mem_sep Ht₁, - assert anint₁ : a ∉ t₁, from aux₂ anins this, + have anint₁ : a ∉ t₁, from aux₂ anins this, have t₂ ∈ powerset s, from mem_of_mem_sep Ht₂, - assert anint₂ : a ∉ t₂, from aux₂ anins this, + have anint₂ : a ∉ t₂, from aux₂ anins this, calc t₁ = erase a (insert a t₁) : by rewrite (erase_insert anint₁) ... = erase a (insert a t₂) : Heq diff --git a/library/theories/group_theory/action.lean b/library/theories/group_theory/action.lean index f1e6ee460..2a3631f0a 100644 --- a/library/theories/group_theory/action.lean +++ b/library/theories/group_theory/action.lean @@ -51,7 +51,7 @@ lemma exists_of_orbit {b : S} : b ∈ orbit hom H a → ∃ h, h ∈ H ∧ hom h assume Pb, obtain p (Pp₁ : p ∈ image hom H) (Pp₂ : move_by a p = b), from exists_of_mem_image Pb, obtain h (Ph₁ : h ∈ H) (Ph₂ : hom h = p), from exists_of_mem_image Pp₁, - assert Phab : hom h a = b, from calc + have Phab : hom h a = b, from calc hom h a = p a : Ph₂ ... = b : Pp₂, exists.intro h (and.intro Ph₁ Phab) @@ -111,7 +111,7 @@ rfl lemma stab_lmul {f g : G} : g ∈ stab hom H a → hom (f*g) a = hom f a := assume Pgstab, -assert hom g a = a, from of_mem_sep Pgstab, calc +have hom g a = a, from of_mem_sep Pgstab, calc hom (f*g) a = perm.f ((hom f) * (hom g)) a : is_hom hom ... = ((hom f) ∘ (hom g)) a : by rewrite perm_f_mul ... = (hom f) a : by unfold compose; rewrite this @@ -123,7 +123,7 @@ lemma stab_subset : stab hom H a ⊆ H := lemma reverse_move {h g : G} : g ∈ moverset hom H a (hom h a) → hom (h⁻¹*g) a = a := assume Pg, -assert hom g a = hom h a, from of_mem_sep Pg, calc +have hom g a = hom h a, from of_mem_sep Pg, calc hom (h⁻¹*g) a = perm.f ((hom h⁻¹) * (hom g)) a : by rewrite (is_hom hom) ... = ((hom h⁻¹) ∘ hom g) a : by rewrite perm_f_mul ... = perm.f ((hom h)⁻¹ * hom h) a : by unfold compose; rewrite [this, perm_f_mul, hom_map_inv hom h] @@ -133,7 +133,7 @@ assert hom g a = hom h a, from of_mem_sep Pg, calc lemma moverset_inj_on_orbit : set.inj_on (moverset hom H a) (ts (orbit hom H a)) := take b1 b2, assume Pb1, obtain h1 Ph1₁ Ph1₂, from exists_of_orbit Pb1, - assert Ph1b1 : h1 ∈ moverset hom H a b1, + have Ph1b1 : h1 ∈ moverset hom H a b1, from mem_sep_of_mem Ph1₁ Ph1₂, assume Psetb2 Pmeq, begin subst b1, @@ -147,7 +147,7 @@ include finsubgH lemma subg_stab_of_move {h g : G} : h ∈ H → g ∈ moverset hom H a (hom h a) → h⁻¹*g ∈ stab hom H a := assume Ph Pg, - assert Phinvg : h⁻¹*g ∈ H, from begin + have Phinvg : h⁻¹*g ∈ H, from begin apply finsubg_mul_closed H, apply finsubg_has_inv H, assumption, apply mem_of_mem_sep Pg @@ -155,31 +155,31 @@ lemma subg_stab_of_move {h g : G} : mem_sep_of_mem Phinvg (reverse_move Pg) lemma subg_stab_closed : finset_mul_closed_on (stab hom H a) := - take f g, assume Pfstab, assert Pf : hom f a = a, from of_mem_sep Pfstab, + take f g, assume Pfstab, have Pf : hom f a = a, from of_mem_sep Pfstab, assume Pgstab, - assert Pfg : hom (f*g) a = a, from calc + have Pfg : hom (f*g) a = a, from calc hom (f*g) a = (hom f) a : stab_lmul Pgstab ... = a : Pf, - assert PfginH : (f*g) ∈ H, + have PfginH : (f*g) ∈ H, from finsubg_mul_closed H (mem_of_mem_sep Pfstab) (mem_of_mem_sep Pgstab), mem_sep_of_mem PfginH Pfg lemma subg_stab_has_one : 1 ∈ stab hom H a := - assert P : hom 1 a = a, from calc + have P : hom 1 a = a, from calc hom 1 a = perm.f (1 : perm S) a : {hom_map_one hom} ... = a : rfl, - assert PoneinH : 1 ∈ H, from finsubg_has_one H, + have PoneinH : 1 ∈ H, from finsubg_has_one H, mem_sep_of_mem PoneinH P lemma subg_stab_has_inv : finset_has_inv (stab hom H a) := - take f, assume Pfstab, assert Pf : hom f a = a, from of_mem_sep Pfstab, - assert Pfinv : hom f⁻¹ a = a, from calc + take f, assume Pfstab, have Pf : hom f a = a, from of_mem_sep Pfstab, + have Pfinv : hom f⁻¹ a = a, from calc hom f⁻¹ a = hom f⁻¹ ((hom f) a) : by rewrite Pf ... = perm.f ((hom f⁻¹) * (hom f)) a : by rewrite perm_f_mul ... = hom (f⁻¹ * f) a : by rewrite (is_hom hom) ... = hom 1 a : by rewrite mul.left_inv ... = perm.f (1 : perm S) a : by rewrite (hom_map_one hom), - assert PfinvinH : f⁻¹ ∈ H, from finsubg_has_inv H (mem_of_mem_sep Pfstab), + have PfinvinH : f⁻¹ ∈ H, from finsubg_has_inv H (mem_of_mem_sep Pfstab), mem_sep_of_mem PfinvinH Pfinv definition subg_stab_is_finsubg [instance] : @@ -190,14 +190,14 @@ lemma subg_lcoset_eq_moverset {h : G} : h ∈ H → fin_lcoset (stab hom H a) h = moverset hom H a (hom h a) := assume Ph, ext (take g, iff.intro (assume Pl, obtain f (Pf₁ : f ∈ stab hom H a) (Pf₂ : h*f = g), from exists_of_mem_image Pl, - assert Pfstab : hom f a = a, from of_mem_sep Pf₁, - assert PginH : g ∈ H, begin + have Pfstab : hom f a = a, from of_mem_sep Pf₁, + have PginH : g ∈ H, begin subst Pf₂, apply finsubg_mul_closed H, assumption, apply mem_of_mem_sep Pf₁ end, - assert Pga : hom g a = hom h a, from calc + have Pga : hom g a = hom h a, from calc hom g a = hom (h*f) a : by subst g ... = hom h a : stab_lmul Pf₁, mem_sep_of_mem PginH Pga) @@ -214,7 +214,7 @@ lemma subg_moverset_of_orbit_is_lcoset_of_stab (b : S) : assume Porb, obtain p (Pp₁ : p ∈ image hom H) (Pp₂ : move_by a p = b), from exists_of_mem_image Porb, obtain h (Ph₁ : h ∈ H) (Ph₂ : hom h = p), from exists_of_mem_image Pp₁, - assert Phab : hom h a = b, from by subst p; assumption, + have Phab : hom h a = b, from by subst p; assumption, exists.intro h (and.intro Ph₁ (Phab ▸ subg_lcoset_eq_moverset Ph₁)) lemma subg_lcoset_of_stab_is_moverset_of_orbit (h : G) : @@ -270,8 +270,8 @@ orbit_of_exists (exists.intro (h*g) (and.intro lemma in_orbit_symm {a b : S} : a ∈ orbit hom H b → b ∈ orbit hom H a := assume Painb, obtain h PhinH Phba, from exists_of_orbit Painb, -assert perm.f (hom h)⁻¹ a = b, by rewrite [-Phba, -perm_f_mul, mul.left_inv], -assert (hom h⁻¹) a = b, by rewrite [hom_map_inv, this], +have perm.f (hom h)⁻¹ a = b, by rewrite [-Phba, -perm_f_mul, mul.left_inv], +have (hom h⁻¹) a = b, by rewrite [hom_map_inv, this], orbit_of_exists (exists.intro h⁻¹ (and.intro (finsubg_has_inv H PhinH) this)) lemma orbit_is_partition : is_partition (orbit hom H) := @@ -422,7 +422,7 @@ lemma aol_fixed_point_subset_normalizer (J : lcoset_type univ H) : is_fixed_point (action_on_lcoset H) H J → elt_of J ⊆ normalizer H := obtain j Pjin Pj, from exists_of_lcoset_type J, assume Pfp, -assert PH : ∀ {h}, h ∈ H → fin_lcoset (fin_lcoset H j) h = fin_lcoset H j, +have PH : ∀ {h}, h ∈ H → fin_lcoset (fin_lcoset H j) h = fin_lcoset H j, from take h, assume Ph, by rewrite [Pj, -action_on_lcoset_eq, Pfp h Ph], subset_of_forall take g, begin rewrite [-Pj, fin_lcoset_same, -inv_inv at {2}], @@ -431,7 +431,7 @@ subset_of_forall take g, begin apply finsubg_has_inv, apply mem_sep_of_mem !mem_univ, intro h Ph, - assert Phg : fin_lcoset (fin_lcoset H g) h = fin_lcoset H g, exact PH Ph, + have Phg : fin_lcoset (fin_lcoset H g) h = fin_lcoset H g, exact PH Ph, revert Phg, rewrite [↑conj_by, inv_inv, mul.assoc, fin_lcoset_compose, -fin_lcoset_same, ↑fin_lcoset, mem_image_iff, ↑lmul_by], intro Pex, cases Pex with k Pand, cases Pand with Pkin Pk, @@ -494,8 +494,8 @@ lemma lift_lower_eq : ∀ {p : perm (fin (succ n))} (P : p maxi = maxi), | (perm.mk pf Pinj) := assume Pmax, begin rewrite [↑lift_perm], congruence, apply funext, intro i, - assert Pfmax : pf maxi = maxi, apply Pmax, - assert Pd : decidable (i = maxi), + have Pfmax : pf maxi = maxi, apply Pmax, + have Pd : decidable (i = maxi), exact _, cases Pd with Pe Pne, rewrite [Pe, Pfmax], apply lift_fun_max, @@ -513,13 +513,13 @@ eq.symm to_set_univ ▸ iff.elim_left set.injective_iff_inj_on_univ lift_perm_in lemma lift_to_stab : image (@lift_perm n) univ = stab id univ maxi := ext (take (pp : perm (fin (succ n))), iff.intro (assume Pimg, obtain p P_ Pp, from exists_of_mem_image Pimg, - assert Ppp : pp maxi = maxi, from calc + have Ppp : pp maxi = maxi, from calc pp maxi = lift_perm p maxi : {eq.symm Pp} ... = lift_fun p maxi : rfl ... = maxi : lift_fun_max, mem_sep_of_mem !mem_univ Ppp) (assume Pstab, - assert Ppp : pp maxi = maxi, from of_mem_sep Pstab, + have Ppp : pp maxi = maxi, from of_mem_sep Pstab, mem_image !mem_univ (lift_lower_eq Ppp))) definition move_from_max_to (i : fin (succ n)) : perm (fin (succ n)) := diff --git a/library/theories/group_theory/cyclic.lean b/library/theories/group_theory/cyclic.lean index 3f861bafb..d0bb37a80 100644 --- a/library/theories/group_theory/cyclic.lean +++ b/library/theories/group_theory/cyclic.lean @@ -22,7 +22,7 @@ include ambG lemma pow_mod {a : A} {n m : nat} : a ^ m = 1 → a ^ n = a ^ (n % m) := assume Pid, -assert a ^ (n / m * m) = 1, from calc +have a ^ (n / m * m) = 1, from calc a ^ (n / m * m) = a ^ (m * (n / m)) : by rewrite (mul.comm (n / m) m) ... = (a ^ m) ^ (n / m) : by rewrite pow_mul ... = 1 ^ (n / m) : by rewrite Pid @@ -64,13 +64,13 @@ include deceqA lemma exists_pow_eq_one (a : A) : ∃ n, n < card A ∧ a ^ (succ n) = 1 := let f := (λ i : fin (succ (card A)), a ^ i) in -assert Pninj : ¬(injective f), from assume Pinj, +have Pninj : ¬(injective f), from assume Pinj, absurd (card_le_of_inj _ _ (exists.intro f Pinj)) (begin rewrite [card_fin], apply not_succ_le_self end), obtain i₁ P₁, from exists_not_of_not_forall Pninj, obtain i₂ P₂, from exists_not_of_not_forall P₁, obtain Pfe Pne, from and_not_of_not_implies P₂, -assert Pvne : val i₁ ≠ val i₂, from assume Pveq, absurd (eq_of_veq Pveq) Pne, +have Pvne : val i₁ ≠ val i₂, from assume Pveq, absurd (eq_of_veq Pveq) Pne, exists.intro (pred (dist i₁ i₂)) (begin rewrite [succ_pred_of_pos (dist_pos_of_ne Pvne)], apply and.intro, apply lt_of_succ_lt_succ, @@ -123,7 +123,7 @@ take g, assume Pgin, obtain n Plt Pe, from exists_pow_eq_one a, obtain i Pilt Pig, from of_mem_sep Pgin, let ni := -(mk_mod n i) in -assert Pinv : g*a^ni = 1, by +have Pinv : g*a^ni = 1, by rewrite [-Pig, mk_pow_mod Pe, -(pow_madd Pe), add.right_inv], begin rewrite [inv_eq_of_mul_eq_one Pinv], @@ -144,7 +144,7 @@ lemma mem_cyc (a : A) : ∀ {n : nat}, a^n ∈ cyc a lemma order_le {a : A} {n : nat} : a^(succ n) = 1 → order a ≤ succ n := assume Pe, let s := image (pow_nat a) (upto (succ n)) in -assert Psub: cyc a ⊆ s, from subset_of_forall +have Psub: cyc a ⊆ s, from subset_of_forall (take g, assume Pgin, obtain i Pilt Pig, from of_mem_sep Pgin, begin rewrite [-Pig, pow_mod Pe], apply mem_image, @@ -170,7 +170,7 @@ have dist i j = 0, from eq_of_veq (eq_of_dist_eq_zero this) lemma cyc_eq_cyc (a : A) (n : nat) : cyc_pow_fin a n = cyc a := -assert Psub : cyc_pow_fin a n ⊆ cyc a, from subset_of_forall +have Psub : cyc_pow_fin a n ⊆ cyc a, from subset_of_forall (take g, assume Pgin, obtain i Pin Pig, from exists_of_mem_image Pgin, by rewrite [-Pig]; apply mem_cyc), eq_of_card_eq_of_subset (begin apply eq.trans, @@ -193,14 +193,14 @@ lemma order_of_min_pow {a : A} {n : nat} (Pone : a^(succ n) = 1) (Pmin : ∀ i, i < n → a^(succ i) ≠ 1) : order a = succ n := or.elim (eq_or_lt_of_le (order_le Pone)) (λ P, P) (λ P : order a < succ n, begin - assert Pn : a^(order a) ≠ 1, + have Pn : a^(order a) ≠ 1, rewrite [-(succ_pred_of_pos (order_pos a))], apply Pmin, apply nat.lt_of_succ_lt_succ, rewrite [succ_pred_of_pos !order_pos], assumption, exact absurd (pow_order a) Pn end) lemma order_dvd_of_pow_eq_one {a : A} {n : nat} (Pone : a^n = 1) : order a ∣ n := -assert Pe : a^(n % order a) = 1, from +have Pe : a^(n % order a) = 1, from begin revert Pone, rewrite [eq_div_mul_add_mod n (order a) at {1}, pow_add, mul.comm _ (order a), pow_mul, pow_order, one_pow, one_mul], @@ -243,7 +243,7 @@ local attribute group_of_add_group [instance] lemma pow_eq_mul {n : nat} {i : fin (succ n)} : ∀ {k : nat}, i^k = mk_mod n (i*k) | 0 := by rewrite [pow_zero] | (succ k) := begin - assert Psucc : i^(succ k) = madd (i^k) i, apply pow_succ', + have Psucc : i^(succ k) = madd (i^k) i, apply pow_succ', rewrite [Psucc, pow_eq_mul], apply eq_of_veq, rewrite [mul_succ, val_madd, ↑mk_mod, mod_add_mod] @@ -268,7 +268,7 @@ lemma rotl_zero : ∀ {n : nat}, @rotl n 0 = id lemma rotl_id : ∀ {n : nat}, @rotl n n = id | 0 := funext take i, elim0 i | (nat.succ n) := - assert P : mk_mod n (n * succ n) = mk_mod n 0, + have P : mk_mod n (n * succ n) = mk_mod n 0, from eq_of_veq (by rewrite [↑mk_mod, mul_mod_left]), begin rewrite [rotl_succ', P], apply rotl_zero end diff --git a/library/theories/group_theory/finsubg.lean b/library/theories/group_theory/finsubg.lean index 51d1d74c3..e0ad7c70c 100644 --- a/library/theories/group_theory/finsubg.lean +++ b/library/theories/group_theory/finsubg.lean @@ -120,12 +120,12 @@ lemma fin_lcoset_same (x a : A) : x ∈ (fin_lcoset H a) = (fin_lcoset H x = fin end lemma fin_mem_lcoset (g : A) : g ∈ fin_lcoset H g := have P : g ∈ g ∘> ts H, from and.left (subg_in_coset_refl g), - assert P1 : g ∈ ts (fin_lcoset H g), from eq.symm (fin_lcoset_eq g) ▸ P, + have P1 : g ∈ ts (fin_lcoset H g), from eq.symm (fin_lcoset_eq g) ▸ P, eq.symm (mem_eq_mem_to_set _ g) ▸ P1 lemma fin_lcoset_subset {S : finset A} (Psub : S ⊆ H) : ∀ x, x ∈ H → fin_lcoset S x ⊆ H := - assert Psubs : set.subset (ts S) (ts H), from subset_eq_to_set_subset S H ▸ Psub, + have Psubs : set.subset (ts S) (ts H), from subset_eq_to_set_subset S H ▸ Psub, take x, assume Pxs : x ∈ ts H, - assert Pcoset : set.subset (x ∘> ts S) (ts H), from subg_lcoset_subset_subg Psubs x Pxs, + have Pcoset : set.subset (x ∘> ts S) (ts H), from subg_lcoset_subset_subg Psubs x Pxs, by rewrite [subset_eq_to_set_subset, fin_lcoset_eq x]; exact Pcoset lemma finsubg_lcoset_id {a : A} : a ∈ H → fin_lcoset H a = H := @@ -261,7 +261,7 @@ ext (take S, iff.intro lemma length_all_lcosets : length (all_lcosets G H) = card (fin_lcosets H G) := eq.trans (show length (all_lcosets G H) = length (list_lcosets G H), from - assert Pmap : map elt_of (all_lcosets G H) = list_lcosets G H, from + have Pmap : map elt_of (all_lcosets G H) = list_lcosets G H, from map_dmap_of_inv_of_pos (λ S P, rfl) (λ S, is_lcoset_of_mem_list_lcosets), by rewrite[-Pmap, length_map]) (by rewrite fin_lcosets_eq) @@ -409,9 +409,11 @@ obtain j Pjin Pj, from has_property J, obtain k Pkin Pk, from has_property K, Union_const (lcoset_not_empty J) begin rewrite [-Pk], intro h Phin, - assert Phinn : h ∈ normalizer H, - apply mem_of_subset_of_mem (lcoset_subset_normalizer_of_mem Pjin), - rewrite Pj, assumption, + have Phinn : h ∈ normalizer H, + begin + apply mem_of_subset_of_mem (lcoset_subset_normalizer_of_mem Pjin), + rewrite Pj, assumption + end, revert Phin Pgin, rewrite [-Pj, *fin_lcoset_same], intro Pheq Pgeq, @@ -454,7 +456,7 @@ lemma fin_coset_mul_assoc (J K L : lcoset_type (normalizer H) H) : J ^ K ^ L = J ^ (K ^ L) := obtain j Pjin Pj, from exists_of_lcoset_type J, obtain k Pkin Pk, from exists_of_lcoset_type K, -assert Pjk : j*k ∈ elt_of (J ^ K), from mul_mem_lcoset_mul J K Pjin Pkin, +have Pjk : j*k ∈ elt_of (J ^ K), from mul_mem_lcoset_mul J K Pjin Pkin, obtain l Plin Pl, from has_property L, subtype.eq (begin rewrite [fin_coset_mul_eq_lcoset (J ^ K) _ Pjk, @@ -479,7 +481,7 @@ end lemma fin_coset_left_inv (J : lcoset_type (normalizer H) H) : (fin_coset_inv J) ^ J = fin_coset_one := obtain j Pjin Pj, from exists_of_lcoset_type J, -assert Pjinv : j⁻¹ ∈ elt_of (fin_coset_inv J), from inv_mem_fin_inv Pjin, +have Pjinv : j⁻¹ ∈ elt_of (fin_coset_inv J), from inv_mem_fin_inv Pjin, subtype.eq begin rewrite [↑fin_coset_one, fin_coset_mul_eq_lcoset _ _ Pjinv, -Pj, fin_lcoset_inv] end @@ -514,7 +516,7 @@ lemma fcU_mul_closed : finset_mul_closed_on (fin_coset_Union Hc) := take j k, assume Pjin Pkin, obtain J PJ PjJ, from iff.elim_left !mem_Union_iff Pjin, obtain K PK PkK, from iff.elim_left !mem_Union_iff Pkin, -assert Pjk : j*k ∈ elt_of (J*K), from mul_mem_lcoset_mul J K PjJ PkK, +have Pjk : j*k ∈ elt_of (J*K), from mul_mem_lcoset_mul J K PjJ PkK, iff.elim_right !mem_Union_iff (exists.intro (J*K) (and.intro (finsubg_mul_closed Hc PJ PK) Pjk)) diff --git a/library/theories/group_theory/hom.lean b/library/theories/group_theory/hom.lean index a6ff0c9de..eb32fc10d 100644 --- a/library/theories/group_theory/hom.lean +++ b/library/theories/group_theory/hom.lean @@ -69,10 +69,10 @@ theorem hom_map_one : f 1 = 1 := eq.symm (mul.right_inv (f 1) ▸ (mul_inv_eq_of_eq_mul P)) theorem hom_map_inv (a : A) : f a⁻¹ = (f a)⁻¹ := - assert P : f 1 = 1, from hom_map_one f, - assert P1 : f (a⁻¹ * a) = 1, from (eq.symm (mul.left_inv a)) ▸ P, - assert P2 : (f a⁻¹) * (f a) = 1, from (is_hom f a⁻¹ a) ▸ P1, - assert P3 : (f a⁻¹) * (f a) = (f a)⁻¹ * (f a), from eq.symm (mul.left_inv (f a)) ▸ P2, + have P : f 1 = 1, from hom_map_one f, + have P1 : f (a⁻¹ * a) = 1, from (eq.symm (mul.left_inv a)) ▸ P, + have P2 : (f a⁻¹) * (f a) = 1, from (is_hom f a⁻¹ a) ▸ P1, + have P3 : (f a⁻¹) * (f a) = (f a)⁻¹ * (f a), from eq.symm (mul.left_inv (f a)) ▸ P2, mul_right_cancel P3 theorem hom_map_mul_closed (H : set A) : mul_closed_on H → mul_closed_on (f ' H) := @@ -80,8 +80,8 @@ theorem hom_map_mul_closed (H : set A) : mul_closed_on H → mul_closed_on (f ' assume Pb1 : b1 ∈ f ' H, assume Pb2 : b2 ∈ f ' H, obtain a1 (Pa1 : a1 ∈ H ∧ f a1 = b1), from Pb1, obtain a2 (Pa2 : a2 ∈ H ∧ f a2 = b2), from Pb2, - assert Pa1a2 : a1 * a2 ∈ H, from Pclosed a1 a2 (and.left Pa1) (and.left Pa2), - assert Pb1b2 : f (a1 * a2) = b1 * b2, from calc + have Pa1a2 : a1 * a2 ∈ H, from Pclosed a1 a2 (and.left Pa1) (and.left Pa2), + have Pb1b2 : f (a1 * a2) = b1 * b2, from calc f (a1 * a2) = f a1 * f a2 : is_hom f a1 a2 ... = b1 * f a2 : {and.right Pa1} ... = b1 * b2 : {and.right Pa2}, @@ -118,11 +118,11 @@ include is_subgH theorem hom_map_subgroup : is_subgroup (f ' H) := have Pone : 1 ∈ f ' H, from mem_image (@subg_has_one _ _ H _) (hom_map_one f), have Pclosed : mul_closed_on (f ' H), from hom_map_mul_closed f H subg_mul_closed, - assert Pinv : ∀ b, b ∈ f ' H → b⁻¹ ∈ f ' H, from + have Pinv : ∀ b, b ∈ f ' H → b⁻¹ ∈ f ' H, from assume b, assume Pimg, obtain a (Pa : a ∈ H ∧ f a = b), from Pimg, - assert Painv : a⁻¹ ∈ H, from subg_has_inv a (and.left Pa), - assert Pfainv : (f a)⁻¹ ∈ f ' H, from mem_image Painv (hom_map_inv f a), + have Painv : a⁻¹ ∈ H, from subg_has_inv a (and.left Pa), + have Pfainv : (f a)⁻¹ ∈ f ' H, from mem_image Painv (hom_map_inv f a), and.right Pa ▸ Pfainv, is_subgroup.mk Pone Pclosed Pinv end @@ -149,8 +149,8 @@ definition quot_over_ker [instance] : group (coset_of (ker f)) := mk_quotient_gr example (a x : A) : (x ∈ a ∘> ker f) = (f (a⁻¹*x) = 1) := rfl lemma ker_coset_same_val (a b : A): same_lcoset (ker f) a b → f a = f b := assume Psame, - assert Pin : f (b⁻¹*a) = 1, from subg_same_lcoset_in_lcoset a b Psame, - assert P : (f b)⁻¹ * (f a) = 1, from calc + have Pin : f (b⁻¹*a) = 1, from subg_same_lcoset_in_lcoset a b Psame, + have P : (f b)⁻¹ * (f a) = 1, from calc (f b)⁻¹ * (f a) = (f b⁻¹) * (f a) : (hom_map_inv f) ... = f (b⁻¹*a) : by rewrite [is_hom f] ... = 1 : by rewrite Pin, @@ -173,7 +173,7 @@ lemma ker_map_is_hom : homomorphic (ker_natural_map : coset_of (ker f) → B) := lemma ker_coset_inj (a b : A) : (ker_natural_map ⟦a⟧ = ker_natural_map ⟦b⟧) → ⟦a⟧ = ⟦b⟧ := assume Pfeq : f a = f b, - assert Painb : a ∈ b ∘> ker f, from calc + have Painb : a ∈ b ∘> ker f, from calc f (b⁻¹*a) = (f b⁻¹) * (f a) : by rewrite [is_hom f] ... = (f b)⁻¹ * (f a) : by rewrite (hom_map_inv f) ... = (f a)⁻¹ * (f a) : by rewrite Pfeq diff --git a/library/theories/group_theory/perm.lean b/library/theories/group_theory/perm.lean index 4ff686b6e..48449bdc5 100644 --- a/library/theories/group_theory/perm.lean +++ b/library/theories/group_theory/perm.lean @@ -76,7 +76,7 @@ lemma nodup_all_perms : nodup (@all_perms A _ _) := lemma all_perms_complete : ∀ p : perm A, p ∈ all_perms := take p, perm.destruct p (take f Pinj, - assert Pin : f ∈ all_injs A, from all_injs_complete Pinj, + have Pin : f ∈ all_injs A, from all_injs_complete Pinj, mem_dmap Pinj Pin) definition perm_is_fintype [instance] : fintype (perm A) := diff --git a/library/theories/group_theory/pgroup.lean b/library/theories/group_theory/pgroup.lean index 58d2a26c6..6570bac9a 100644 --- a/library/theories/group_theory/pgroup.lean +++ b/library/theories/group_theory/pgroup.lean @@ -43,10 +43,12 @@ lemma card_mod_eq_of_action_by_psubg {p : nat} : cases Pa with Pain Porb, substvars, cases Ppsubg with Pprime PcardH, - assert Pdvd : card (orbit hom H a) ∣ p ^ (succ m), - rewrite -PcardH, - apply dvd_of_eq_mul (finset.card (stab hom H a)), - apply orbit_stabilizer_theorem, + have Pdvd : card (orbit hom H a) ∣ p ^ (succ m), + begin + rewrite -PcardH, + apply dvd_of_eq_mul (finset.card (stab hom H a)), + apply orbit_stabilizer_theorem + end, apply or.elim (eq_one_or_dvd_of_dvd_prime_pow Pprime Pdvd), intro Pcardeq, contradiction, intro Ppdvd, exact Ppdvd @@ -131,7 +133,7 @@ Prodl_map lemma prodseq_eq_pow_of_constseq {n : nat} (s : seq A (succ n)) : constseq s → prodseq s = (s !zero) ^ succ n := -assume Pc, assert Pcl : ∀ i, i ∈ upto (succ n) → s i = s !zero, +assume Pc, have Pcl : ∀ i, i ∈ upto (succ n) → s i = s !zero, from take i, assume Pin, Pc i, by rewrite [↑prodseq, Prodl_eq_pow_of_const _ Pcl, fin.length_upto] @@ -142,7 +144,7 @@ assume Pc₁ Pc₂ Peq, funext take i, by rewrite [Pc₁ i, Pc₂ i, Peq] lemma peo_const_one : ∀ {n : nat}, peo (λ i : fin n, (1 : A)) | 0 := rfl | (succ n) := let s := λ i : fin (succ n), (1 : A) in - assert Pconst : constseq s, from take i, rfl, + have Pconst : constseq s, from take i, rfl, calc prodseq s = (s !zero) ^ succ n : prodseq_eq_pow_of_constseq s Pconst ... = (1 : A) ^ succ n : rfl ... = 1 : one_pow @@ -174,13 +176,13 @@ by rewrite [prodseq_eq, Peq, list_to_fun_to_list, prodl_eq_one_of_mem_all_prodl_ lemma all_prodseq_eq_one_complete {n : nat} {s : seq A (succ n)} : prodseq s = 1 → s ∈ all_prodseq_eq_one A n := assume Peq, -assert Plin : map s (elems (fin (succ n))) ∈ all_prodl_eq_one A n, +have Plin : map s (elems (fin (succ n))) ∈ all_prodl_eq_one A n, from begin apply all_prodl_eq_one_complete, rewrite [length_map], exact length_upto (succ n), rewrite prodseq_eq at Peq, exact Peq end, -assert Psin : list_to_fun (map s (elems (fin (succ n)))) (length_map_of_fintype s) ∈ all_prodseq_eq_one A n, +have Psin : list_to_fun (map s (elems (fin (succ n)))) (length_map_of_fintype s) ∈ all_prodseq_eq_one A n, from mem_dmap _ Plin, by rewrite [fun_eq_list_to_fun_map s (length_map_of_fintype s)]; apply Psin @@ -196,7 +198,7 @@ local attribute perm.f [coercion] lemma rotl_perm_peo_of_peo {n : nat} : ∀ {m} {s : seq A n}, peo s → peo (rotl_perm A n m s) | 0 := begin rewrite [↑rotl_perm, rotl_seq_zero], intros, assumption end | (succ m) := take s, - assert Pmul : rotl_perm A n (m + 1) s = rotl_fun 1 (rotl_perm A n m s), from + have Pmul : rotl_perm A n (m + 1) s = rotl_fun 1 (rotl_perm A n m s), from calc s ∘ (rotl (m + 1)) = s ∘ ((rotl m) ∘ (rotl 1)) : rotl_compose ... = s ∘ (rotl m) ∘ (rotl 1) : compose.assoc, begin @@ -211,18 +213,18 @@ dmap_nodup_of_dinj (dinj_tag peo) nodup_all_prodseq_eq_one lemma all_peo_seqs_complete {n : nat} : ∀ s : peo_seq A n, s ∈ all_peo_seqs A n := take ps, subtype.destruct ps (take s, assume Ps, - assert Pin : s ∈ all_prodseq_eq_one A n, from all_prodseq_eq_one_complete Ps, + have Pin : s ∈ all_prodseq_eq_one A n, from all_prodseq_eq_one_complete Ps, mem_dmap Ps Pin) lemma length_all_peo_seqs {n : nat} : length (all_peo_seqs A n) = (card A)^n := eq.trans (eq.trans (show length (all_peo_seqs A n) = length (all_prodseq_eq_one A n), from - assert Pmap : map elt_of (all_peo_seqs A n) = all_prodseq_eq_one A n, + have Pmap : map elt_of (all_peo_seqs A n) = all_prodseq_eq_one A n, from map_dmap_of_inv_of_pos (λ s P, rfl) (λ s, prodseq_eq_one_of_mem_all_prodseq_eq_one), by rewrite [-Pmap, length_map]) (show length (all_prodseq_eq_one A n) = length (all_prodl_eq_one A n), from - assert Pmap : map fun_to_list (all_prodseq_eq_one A n) = all_prodl_eq_one A n, + have Pmap : map fun_to_list (all_prodseq_eq_one A n) = all_prodl_eq_one A n, from map_dmap_of_inv_of_pos list_to_fun_to_list (λ l Pin, by rewrite [length_of_mem_all_prodl_eq_one Pin, card_fin]), by rewrite [-Pmap, length_map])) @@ -319,8 +321,8 @@ lemma generator_of_prime_of_dvd_order {p : nat} : prime p → p ∣ card A → ∃ g : A, g ≠ 1 ∧ g^p = 1 := assume Pprime Pdvd, let pp := nat.pred p, spp := nat.succ pp in -assert Peq : spp = p, from succ_pred_prime Pprime, -assert Ppsubg : psubg (@univ (fin spp) _) spp 1, +have Peq : spp = p, from succ_pred_prime Pprime, +have Ppsubg : psubg (@univ (fin spp) _) spp 1, from and.intro (eq.symm Peq ▸ Pprime) (by rewrite [Peq, card_fin, pow_one]), have (pow_nat (card A) pp) % spp = (card (fixed_points (rotl_perm_ps A pp) univ)) % spp, by rewrite -card_peo_seq; apply card_mod_eq_of_action_by_psubg Ppsubg, @@ -335,7 +337,7 @@ have Pfpcardgt1 : card (fixed_points (rotl_perm_ps A pp) univ) > 1, obtain s₁ s₂ Pin₁ Pin₂ Psnes, from exists_two_of_card_gt_one Pfpcardgt1, decidable.by_cases (λ Pe₁ : elt_of s₁ !zero = 1, - assert Pne₂ : elt_of s₂ !zero ≠ 1, + have Pne₂ : elt_of s₂ !zero ≠ 1, from assume Pe₂, absurd (subtype.eq (seq_eq_of_constseq_of_eq @@ -352,7 +354,7 @@ end theorem cauchy_theorem {p : nat} : prime p → p ∣ card A → ∃ g : A, order g = p := assume Pprime Pdvd, obtain g Pne Pgpow, from generator_of_prime_of_dvd_order Pprime Pdvd, -assert Porder : order g ∣ p, from order_dvd_of_pow_eq_one Pgpow, +have Porder : order g ∣ p, from order_dvd_of_pow_eq_one Pgpow, or.elim (eq_one_or_eq_self_of_prime_of_dvd Pprime Porder) (λ Pe, absurd (eq_one_of_order_eq_one Pe) Pne) (λ Porderp, exists.intro g Porderp) @@ -375,10 +377,10 @@ theorem first_sylow_theorem {p : nat} (Pp : prime p) : (by rewrite [pow_zero])) | (succ n) := assume Pdvd, obtain H PfinsubgH PcardH, from first_sylow_theorem n (pow_dvd_of_pow_succ_dvd Pdvd), - assert Ppsubg : psubg H p n, from and.intro Pp PcardH, - assert Ppowsucc : p^(succ n) ∣ (card (lcoset_type univ H) * p^n), + have Ppsubg : psubg H p n, from and.intro Pp PcardH, + have Ppowsucc : p^(succ n) ∣ (card (lcoset_type univ H) * p^n), by rewrite [-PcardH, -(lagrange_theorem' !subset_univ)]; exact Pdvd, - assert Ppdvd : p ∣ card (lcoset_type (normalizer H) H), from + have Ppdvd : p ∣ card (lcoset_type (normalizer H) H), from dvd_of_mod_eq_zero (by rewrite [-(card_psubg_cosets_mod_eq Ppsubg), -dvd_iff_mod_eq_zero]; exact dvd_of_pow_succ_dvd_mul_pow (pos_of_prime Pp) Ppowsucc), diff --git a/library/theories/group_theory/subgroup.lean b/library/theories/group_theory/subgroup.lean index 86875f5fe..7b7d986c5 100644 --- a/library/theories/group_theory/subgroup.lean +++ b/library/theories/group_theory/subgroup.lean @@ -112,7 +112,7 @@ lemma grcoset_eq_rcoset a (H : set A) : H <∘ a = coset.r a H := end lemma glcoset_sub a (S H : set A) : S ⊆ H → (a ∘> S) ⊆ (a ∘> H) := assume Psub, - assert P : _, from coset.l_sub a S H Psub, + have P : _, from coset.l_sub a S H Psub, eq.symm (glcoset_eq_lcoset a S) ▸ eq.symm (glcoset_eq_lcoset a H) ▸ P lemma glcoset_compose (a b : A) (H : set A) : a ∘> b ∘> H = a*b ∘> H := begin @@ -174,8 +174,8 @@ lemma closed_rcontract a (H : set A) : mul_closed_on H → a ∈ H → H <∘ a end lemma closed_lcontract_set a (H G : set A) : mul_closed_on G → H ⊆ G → a∈G → a∘>H ⊆ G := assume Pclosed, assume PHsubG, assume PainG, - assert PaGsubG : a ∘> G ⊆ G, from closed_lcontract a G Pclosed PainG, - assert PaHsubaG : a ∘> H ⊆ a ∘> G, from + have PaGsubG : a ∘> G ⊆ G, from closed_lcontract a G Pclosed PainG, + have PaHsubaG : a ∘> H ⊆ a ∘> G, from eq.symm (glcoset_eq_lcoset a H) ▸ eq.symm (glcoset_eq_lcoset a G) ▸ (coset.l_sub a H G PHsubG), subset.trans PaHsubaG PaGsubG definition subgroup.has_inv H := ∀ (a : A), a ∈ H → a⁻¹ ∈ H @@ -207,9 +207,9 @@ lemma subg_mul_closed : mul_closed_on H := @is_subgroup.mul_closed A s H is_subg lemma subg_has_inv : subgroup.has_inv H := @is_subgroup.has_inv A s H is_subg lemma subgroup_coset_id : ∀ a, a ∈ H → (a ∘> H = H ∧ H <∘ a = H) := take a, assume PHa : H a, - assert Pl : a ∘> H ⊆ H, from closed_lcontract a H subg_mul_closed PHa, - assert Pr : H <∘ a ⊆ H, from closed_rcontract a H subg_mul_closed PHa, - assert PHainv : H a⁻¹, from subg_has_inv a PHa, + have Pl : a ∘> H ⊆ H, from closed_lcontract a H subg_mul_closed PHa, + have Pr : H <∘ a ⊆ H, from closed_rcontract a H subg_mul_closed PHa, + have PHainv : H a⁻¹, from subg_has_inv a PHa, and.intro (ext (assume x, begin @@ -232,9 +232,9 @@ lemma subgroup_rcoset_id : ∀ a, a ∈ H → H <∘ a = H := take a, assume PHa : H a, and.right (subgroup_coset_id a PHa) lemma subg_in_coset_refl (a : A) : a ∈ a ∘> H ∧ a ∈ H <∘ a := - assert PH1 : H 1, from subg_has_one, - assert PHinvaa : H (a⁻¹*a), from (eq.symm (mul.left_inv a)) ▸ PH1, - assert PHainva : H (a*a⁻¹), from (eq.symm (mul.right_inv a)) ▸ PH1, + have PH1 : H 1, from subg_has_one, + have PHinvaa : H (a⁻¹*a), from (eq.symm (mul.left_inv a)) ▸ PH1, + have PHainva : H (a*a⁻¹), from (eq.symm (mul.right_inv a)) ▸ PH1, and.intro PHinvaa PHainva end set_reducible lemma subg_in_lcoset_same_lcoset (a b : A) : in_lcoset H a b → same_lcoset H a b := @@ -245,7 +245,7 @@ lemma subg_in_lcoset_same_lcoset (a b : A) : in_lcoset H a b → same_lcoset H a mul_inv_cancel_left b a ▸ Pbbinva lemma subg_same_lcoset_in_lcoset (a b : A) : same_lcoset H a b → in_lcoset H a b := assume Psame : a∘>H = b∘>H, - assert Pa : a ∈ a∘>H, from and.left (subg_in_coset_refl a), + have Pa : a ∈ a∘>H, from and.left (subg_in_coset_refl a), by exact (Psame ▸ Pa) lemma subg_lcoset_same (a b : A) : in_lcoset H a b = (a∘>H = b∘>H) := propext(iff.intro (subg_in_lcoset_same_lcoset a b) (subg_same_lcoset_in_lcoset a b)) @@ -257,7 +257,7 @@ lemma subg_rcoset_same (a b : A) : in_rcoset H a b = (H<∘a = H<∘b) := have Pabinvb : H <∘ a*b⁻¹*b = H <∘ b, from grcoset_compose (a*b⁻¹) b H ▸ Pabinv_b, inv_mul_cancel_right a b ▸ Pabinvb) (assume Psame, - assert Pa : a ∈ H<∘a, from and.right (subg_in_coset_refl a), + have Pa : a ∈ H<∘a, from and.right (subg_in_coset_refl a), by exact (Psame ▸ Pa))) lemma subg_same_lcoset.refl (a : A) : same_lcoset H a a := rfl lemma subg_same_rcoset.refl (a : A) : same_rcoset H a a := rfl diff --git a/library/theories/number_theory/bezout.lean b/library/theories/number_theory/bezout.lean index d0492322e..a31369632 100644 --- a/library/theories/number_theory/bezout.lean +++ b/library/theories/number_theory/bezout.lean @@ -97,7 +97,7 @@ decidable.by_cases (suppose ¬ p ∣ x, have cpx : coprime p x, from coprime_of_prime_of_not_dvd pp this, obtain (a b : ℤ) (Hab : a * p + b * x = gcd p x), from Bezout_aux p x, - assert a * p * y + b * x * y = y, + have a * p * y + b * x * y = y, by krewrite [-right_distrib, Hab, ↑coprime at cpx, cpx, int.one_mul], have p ∣ y, begin diff --git a/library/theories/number_theory/irrational_roots.lean b/library/theories/number_theory/irrational_roots.lean index fb06f6e94..46ea68a95 100644 --- a/library/theories/number_theory/irrational_roots.lean +++ b/library/theories/number_theory/irrational_roots.lean @@ -29,7 +29,7 @@ section from even_of_exists (exists.intro _ (eq.symm this)), have even b, from even_of_even_pow this, - assert 2 ∣ gcd a b, + have 2 ∣ gcd a b, from dvd_gcd (dvd_of_even `even a`) (dvd_of_even `even b`), have (2:nat) ∣ 1, begin rewrite [gcd_eq_one_of_coprime co at this], exact this end, @@ -51,7 +51,7 @@ section (suppose b = 0, have a^n = 0, by rewrite [H, this, zero_pow npos], - assert a = 0, + have a = 0, from eq_zero_of_pow_eq_zero this, show false, from ne_of_lt `0 < a` this⁻¹), @@ -59,7 +59,7 @@ section take p, suppose prime p, suppose p ∣ b, - assert p ∣ b^n, + have p ∣ b^n, from dvd_pow_of_dvd_of_pos `p ∣ b` `n > 0`, have p ∣ a^n, by rewrite H; apply dvd_mul_of_dvd_right this, @@ -156,7 +156,7 @@ section (suppose p = 0, show false, by note H := (pos_of_prime primep); rewrite this at H; exfalso; exact !lt.irrefl H), - assert agtz : a > 0, from pos_of_ne_zero + have agtz : a > 0, from pos_of_ne_zero (suppose a = 0, show false, by revert peq; rewrite [this, zero_pow npos]; exact pnez), have n * mult p a = 1, from calc @@ -197,7 +197,7 @@ section example {a b c : ℤ} (co : coprime a b) (apos : a > 0) (bpos : b > 0) (H : a * a = c * (b * b)) : b = 1 := - assert H₁ : gcd (c * b) a = gcd c a, + have H₁ : gcd (c * b) a = gcd c a, from gcd_mul_right_cancel_of_coprime _ (coprime_swap co), have a * a = c * b * b, by rewrite -mul.assoc at H; apply H, diff --git a/library/theories/number_theory/prime_factorization.lean b/library/theories/number_theory/prime_factorization.lean index a7a9a4106..87649d592 100644 --- a/library/theories/number_theory/prime_factorization.lean +++ b/library/theories/number_theory/prime_factorization.lean @@ -93,8 +93,8 @@ begin (take n', suppose n = p * n', have p > 0, from lt.trans zero_lt_one pgt1, - assert n / p = n', from !nat.div_eq_of_eq_mul_right this `n = p * n'`, - assert n' < n, + have n / p = n', from !nat.div_eq_of_eq_mul_right this `n = p * n'`, + have n' < n, by rewrite -this; apply mult_rec_decreasing pgt1 npos, begin rewrite [mult_rec pgt1 npos pdvdn, `n / p = n'`, pow_succ], subst n, @@ -104,14 +104,14 @@ begin end theorem mult_one_right (p : ℕ) : mult p 1 = 0:= -assert H : p^(mult p 1) = 1, from eq_one_of_dvd_one !pow_mult_dvd, +have H : p^(mult p 1) = 1, from eq_one_of_dvd_one !pow_mult_dvd, or.elim (le_or_gt p 1) (suppose p ≤ 1, by rewrite [!mult_eq_zero_of_le_one this]) (suppose p > 1, by_contradiction (suppose mult p 1 ≠ 0, have mult p 1 > 0, from pos_of_ne_zero this, - assert p^(mult p 1) > 1, from pow_gt_one `p > 1` this, + have p^(mult p 1) > 1, from pow_gt_one `p > 1` this, show false, by rewrite H at this; apply !lt.irrefl this)) private theorem mult_pow_mul {p n : ℕ} (i : ℕ) (pgt1 : p > 1) (npos : n > 0) : @@ -136,13 +136,13 @@ theorem le_mult {p i n : ℕ} (pgt1 : p > 1) (npos : n > 0) (pidvd : p^i ∣ n) dvd.elim pidvd (take m, suppose n = p^i * m, - assert m > 0, from pos_of_mul_pos_left (this ▸ npos), + have m > 0, from pos_of_mul_pos_left (this ▸ npos), by subst n; rewrite [mult_pow_mul i pgt1 this]; apply le_add_right) theorem not_dvd_div_pow_mult {p n : ℕ} (pgt1 : p > 1) (npos : n > 0) : ¬ p ∣ n / p^(mult p n) := assume pdvd : p ∣ n / p^(mult p n), obtain m (H : n / p^(mult p n) = p * m), from exists_eq_mul_right_of_dvd pdvd, -assert n = p^(succ (mult p n)) * m, from +have n = p^(succ (mult p n)) * m, from calc n = p^mult p n * (n / p^mult p n) : by rewrite (nat.mul_div_cancel' !pow_mult_dvd) ... = p^(succ (mult p n)) * m : by rewrite [H, pow_succ', mul.assoc], @@ -153,16 +153,16 @@ show false, from !not_succ_le_self this theorem mult_mul {p m n : ℕ} (primep : prime p) (mpos : m > 0) (npos : n > 0) : mult p (m * n) = mult p m + mult p n := let m' := m / p^mult p m, n' := n / p^mult p n in -assert p > 1, from gt_one_of_prime primep, -assert meq : m = p^mult p m * m', by rewrite (nat.mul_div_cancel' !pow_mult_dvd), -assert neq : n = p^mult p n * n', by rewrite (nat.mul_div_cancel' !pow_mult_dvd), +have p > 1, from gt_one_of_prime primep, +have meq : m = p^mult p m * m', by rewrite (nat.mul_div_cancel' !pow_mult_dvd), +have neq : n = p^mult p n * n', by rewrite (nat.mul_div_cancel' !pow_mult_dvd), have m'pos : m' > 0, from pos_of_mul_pos_left (meq ▸ mpos), have n'pos : n' > 0, from pos_of_mul_pos_left (neq ▸ npos), have npdvdm' : ¬ p ∣ m', from !not_dvd_div_pow_mult `p > 1` mpos, have npdvdn' : ¬ p ∣ n', from !not_dvd_div_pow_mult `p > 1` npos, -assert npdvdm'n' : ¬ p ∣ m' * n', from not_dvd_mul_of_prime primep npdvdm' npdvdn', -assert m'n'pos : m' * n' > 0, from mul_pos m'pos n'pos, -assert multm'n' : mult p (m' * n') = 0, from mult_eq_zero_of_not_dvd npdvdm'n', +have npdvdm'n' : ¬ p ∣ m' * n', from not_dvd_mul_of_prime primep npdvdm' npdvdn', +have m'n'pos : m' * n' > 0, from mul_pos m'pos n'pos, +have multm'n' : mult p (m' * n') = 0, from mult_eq_zero_of_not_dvd npdvdm'n', calc mult p (m * n) = mult p (p^(mult p m + mult p n) * (m' * n')) : by rewrite [pow_add, mul.right_comm, -mul.assoc, -meq, mul.assoc, @@ -196,8 +196,8 @@ begin {intros; rewrite nz; apply dvd_zero}, assume H : ∀ {p : ℕ}, prime p → mult p m ≤ mult p n, obtain m' (meq : m = p * m'), from exists_eq_mul_right_of_dvd pdvdm, - assert pgt1 : p > 1, from gt_one_of_prime primep, - assert m'pos : m' > 0, from pos_of_ne_zero + have pgt1 : p > 1, from gt_one_of_prime primep, + have m'pos : m' > 0, from pos_of_ne_zero (assume m'z, by revert mpos; rewrite [meq, m'z, mul_zero]; apply not_lt_zero), have m'ltm : m' < m, by rewrite [meq, -one_mul m' at {1}]; apply mul_lt_mul_of_lt_of_le m'pos pgt1 !le.refl, @@ -205,7 +205,7 @@ begin have multpn : mult p n ≥ 1, from le.trans multpm (H primep), obtain n' (neq : n = p * n'), from exists_eq_mul_right_of_dvd (dvd_of_mult_pos (lt_of_succ_le multpn)), - assert n'pos : n' > 0, from pos_of_ne_zero + have n'pos : n' > 0, from pos_of_ne_zero (assume n'z, by revert npos; rewrite [neq, n'z, mul_zero]; apply not_lt_zero), have ∀q, prime q → mult q m' ≤ mult q n', from (take q, @@ -215,7 +215,7 @@ begin have multqn : mult q n = mult q p + mult q n', by rewrite [neq, mult_mul primeq (pos_of_prime primep) n'pos], show mult q m' ≤ mult q n', from le_of_add_le_add_left (multqm ▸ multqn ▸ H primeq)), - assert m'dvdn' : m' ∣ n', from ih m' m'ltm m'pos n' this, + have m'dvdn' : m' ∣ n', from ih m' m'ltm m'pos n' this, show m ∣ n, by rewrite [meq, neq]; apply mul_dvd_mul !dvd.refl m'dvdn' end @@ -293,7 +293,7 @@ end theorem eq_prime_factorization {n : ℕ} (npos : n > 0) : n = (∏ p ∈ prime_factors n, p^(mult p n)) := let nprod := ∏ p ∈ prime_factors n, p^(mult p n) in -assert primefactors : ∀ p, p ∈ prime_factors n → prime p, +have primefactors : ∀ p, p ∈ prime_factors n → prime p, from take p, @prime_of_mem_prime_factors p n, have prodpos : (∏ q ∈ prime_factors n, q^(mult q n)) > 0, from Prod_pos (take q, assume qpf, @@ -306,6 +306,6 @@ eq_of_forall_prime_mult_eq npos prodpos eq.symm (mult_prod_pow_of_mem primep primefactors (λ p, mult p n) pprimefactors)) (assume pnprimefactors : p ∉ prime_factors n, have ¬ p ∣ n, from assume H, pnprimefactors (mem_prime_factors npos primep H), - assert mult p n = 0, from mult_eq_zero_of_not_dvd this, + have mult p n = 0, from mult_eq_zero_of_not_dvd this, by rewrite [this, mult_prod_pow_of_not_mem primep primefactors _ pnprimefactors])) end nat diff --git a/library/theories/number_theory/primes.lean b/library/theories/number_theory/primes.lean index ca5aa22bb..c329d0049 100644 --- a/library/theories/number_theory/primes.lean +++ b/library/theories/number_theory/primes.lean @@ -90,7 +90,7 @@ definition sub_dvd_of_not_prime2 {n : nat} : n ≥ 2 → ¬ prime n → {m | m assume h₁ h₂, have n ≠ 0, from assume h, begin subst n, exact absurd h₁ dec_trivial end, obtain m m_dvd_n m_ne_1 m_ne_n, from sub_dvd_of_not_prime h₁ h₂, -assert m_ne_0 : m ≠ 0, from assume h, begin subst m, exact absurd (eq_zero_of_zero_dvd m_dvd_n) `n ≠ 0` end, +have m_ne_0 : m ≠ 0, from assume h, begin subst m, exact absurd (eq_zero_of_zero_dvd m_dvd_n) `n ≠ 0` end, begin existsi m, split, assumption, split, @@ -145,7 +145,7 @@ lemma odd_of_prime {p : nat} : prime p → p > 2 → odd p := λ pp p_gt_2, by_contradiction (λ hn, have even p, from even_of_not_odd hn, obtain k `p = 2*k`, from exists_of_even this, - assert 2 ∣ p, by rewrite [`p = 2*k`]; apply dvd_mul_right, + have 2 ∣ p, by rewrite [`p = 2*k`]; apply dvd_mul_right, or.elim (eq_one_or_eq_self_of_prime_of_dvd pp this) (suppose 2 = 1, absurd this dec_trivial) (suppose 2 = p, by subst this; exact absurd p_gt_2 !lt.irrefl)) @@ -190,8 +190,8 @@ lemma dvd_or_dvd_of_prime_of_dvd_mul {p m n : nat} : prime p → p ∣ m * n → lemma dvd_of_prime_of_dvd_pow {p m : nat} : ∀ {n}, prime p → p ∣ m^n → p ∣ m | 0 hp hd := - assert p = 1, from eq_one_of_dvd_one hd, - have (1:nat) ≥ 2, begin rewrite -this at {1}, apply ge_two_of_prime hp end, + have p = 1, from eq_one_of_dvd_one hd, + have (1:nat) ≥ 2, begin rewrite -this at {1}, apply ge_two_of_prime hp end, absurd this dec_trivial | (succ n) hp hd := have p ∣ (m^n)*m, by rewrite [pow_succ' at hd]; exact hd, @@ -204,12 +204,12 @@ lemma coprime_pow_of_prime_of_not_dvd {p m a : nat} : prime p → ¬ p ∣ a → lemma coprime_primes {p q : nat} : prime p → prime q → p ≠ q → coprime p q := λ hp hq hn, - assert gcd p q ∣ p, from !gcd_dvd_left, + have gcd p q ∣ p, from !gcd_dvd_left, or.elim (eq_one_or_eq_self_of_prime_of_dvd hp this) (suppose gcd p q = 1, this) (assume h : gcd p q = p, - assert gcd p q ∣ q, from !gcd_dvd_right, - have p ∣ q, by rewrite -h; exact this, + have gcd p q ∣ q, from !gcd_dvd_right, + have p ∣ q, by rewrite -h; exact this, or.elim (eq_one_or_eq_self_of_prime_of_dvd hq this) (suppose p = 1, by subst p; exact absurd hp not_prime_one) (suppose p = q, by contradiction))