refactor(library/logic/connectives): rename theorems
This commit is contained in:
Jeremy Avigad
parent
6903118900
commit
3e9a484851
23 changed files with 195 additions and 190 deletions
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@ -537,7 +537,7 @@ We now use =not_intro= and =absurd= to produce a proof term for
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import logic
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constants a b : Prop
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check fun (Hab : a → b) (Hnb : ¬ b),
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not_intro (fun Ha : a, absurd (Hab Ha) Hnb)
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not.intro (fun Ha : a, absurd (Hab Ha) Hnb)
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#+END_SRC
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@ -252,7 +252,7 @@ strong_order_pair.mk strict_order_with_le.le
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(assume H : a ≤ b ∧ a ≠ b,
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have H1 : a < b ∨ a = b,
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from iff.mp !strict_order_with_le.le_iff_lt_or_eq (and.elim_left H),
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show a < b, from or.resolve_left H1 (and.elim_right H)))
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show a < b, from or_resolve_left H1 (and.elim_right H)))
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strict_order_with_le.le_iff_lt_or_eq
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@ -321,13 +321,13 @@ section
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theorem mul.cancel_right {a b c : A} (Ha : a ≠ 0) (H : b * a = c * a) : b = c :=
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have H1 : b * a - c * a = 0, from iff.mp !eq_iff_sub_eq_zero H,
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have H2 : (b - c) * a = 0, from eq.trans !mul_sub_right_distrib H1,
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have H3 : b - c = 0, from or.resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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have H3 : b - c = 0, from or_resolve_left (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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iff.elim_right !eq_iff_sub_eq_zero H3
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theorem mul.cancel_left {a b c : A} (Ha : a ≠ 0) (H : a * b = a * c) : b = c :=
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have H1 : a * b - a * c = 0, from iff.mp !eq_iff_sub_eq_zero H,
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have H2 : a * (b - c) = 0, from eq.trans !mul_sub_left_distrib H1,
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have H3 : b - c = 0, from or.resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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have H3 : b - c = 0, from or_resolve_right (eq_zero_or_eq_zero_of_mul_eq_zero H2) Ha,
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iff.elim_right !eq_iff_sub_eq_zero H3
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-- TODO: do we want the iff versions?
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@ -42,7 +42,7 @@ open eq.ops
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namespace nat
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theorem succ_pred_of_pos {n : ℕ} (H : n > 0) : succ (pred n) = n :=
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(or.resolve_right (zero_or_succ_pred n) (ne.symm (lt_imp_ne H))⁻¹)
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(or_resolve_right (zero_or_succ_pred n) (ne.symm (lt_imp_ne H))⁻¹)
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theorem sub_pos_of_gt {m n : ℕ} (H : n > m) : n - m > 0 :=
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have H1 : n = n - m + m, from (add_sub_ge_left (lt_imp_le H))⁻¹,
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@ -866,23 +866,23 @@ have H2 : (nat_abs a) * (nat_abs b) = 0, from
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... = (nat_abs 0) : {H}
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... = 0 : nat_abs_of_nat 0,
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have H3 : (nat_abs a) = 0 ∨ (nat_abs b) = 0, from mul.eq_zero H2,
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or.imp_or H3
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or_of_or_of_imp_of_imp H3
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(assume H : (nat_abs a) = 0, nat_abs_eq_zero H)
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(assume H : (nat_abs b) = 0, nat_abs_eq_zero H)
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theorem mul_cancel_left_or {a b c : ℤ} (H : a * b = a * c) : a = 0 ∨ b = c :=
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have H2 : a * (b - c) = 0, by simp,
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have H3 : a = 0 ∨ b - c = 0, from mul_eq_zero H2,
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or.imp_or_right H3 (assume H4 : b - c = 0, sub_eq_zero H4)
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or_of_or_of_imp_right H3 (assume H4 : b - c = 0, sub_eq_zero H4)
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theorem mul_cancel_left {a b c : ℤ} (H1 : a ≠ 0) (H2 : a * b = a * c) : b = c :=
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or.resolve_right (mul_cancel_left_or H2) H1
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or_resolve_right (mul_cancel_left_or H2) H1
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theorem mul_cancel_right_or {a b c : ℤ} (H : b * a = c * a) : a = 0 ∨ b = c :=
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mul_cancel_left_or ((H ▸ (mul_comm b a)) ▸ mul_comm c a)
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theorem mul_cancel_right {a b c : ℤ} (H1 : c ≠ 0) (H2 : a * c = b * c) : a = b :=
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or.resolve_right (mul_cancel_right_or H2) H1
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or_resolve_right (mul_cancel_right_or H2) H1
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theorem mul_ne_zero {a b : ℤ} (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 :=
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(assume H : a * b = 0,
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@ -18,7 +18,7 @@ open int eq.ops
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namespace int
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theorem nonneg_elim {a : ℤ} : nonneg a → ∃n : ℕ, a = n :=
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cases_on a (take n H, exists_intro n rfl) (take n' H, false_elim H)
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cases_on a (take n H, exists_intro n rfl) (take n' H, false.elim H)
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theorem le_intro {a b : ℤ} {n : ℕ} (H : a + n = b) : a ≤ b :=
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have H1 : b - a = n, from add_imp_sub_right (!add_comm ▸ H),
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@ -262,7 +262,7 @@ theorem le_imp_lt_or_eq {a b : ℤ} (H : a ≤ b) : a < b ∨ a = b :=
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le_imp_succ_le_or_eq H
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theorem le_ne_imp_lt {a b : ℤ} (H1 : a ≤ b) (H2 : a ≠ b) : a < b :=
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or.resolve_left (le_imp_lt_or_eq H1) H2
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or_resolve_left (le_imp_lt_or_eq H1) H2
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theorem le_imp_lt_succ {a b : ℤ} (H : a ≤ b) : a < b + 1 :=
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add_le_right H 1
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@ -379,26 +379,26 @@ by_cases a
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or.inl (neg_le_pos n m))
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(take m : ℕ,
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show -n ≤ -succ m ∨ -n > -succ m, from
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or.imp_or le_or_gt
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or_of_or_of_imp_of_imp le_or_gt
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(assume H : succ m ≤ n,
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le_neg (iff.elim_left (iff.symm (le_of_nat (succ m) n)) H))
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(assume H : succ m > n,
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lt_neg (iff.elim_left (iff.symm (lt_of_nat n (succ m))) H))))
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theorem trichotomy_alt (a b : ℤ) : (a < b ∨ a = b) ∨ a > b :=
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or.imp_or_left (le_or_gt a b) (assume H : a ≤ b, le_imp_lt_or_eq H)
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or_of_or_of_imp_left (le_or_gt a b) (assume H : a ≤ b, le_imp_lt_or_eq H)
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theorem trichotomy (a b : ℤ) : a < b ∨ a = b ∨ a > b :=
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iff.elim_left or.assoc (trichotomy_alt a b)
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theorem le_total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
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or.imp_or_right (le_or_gt a b) (assume H : b < a, lt_imp_le H)
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or_of_or_of_imp_right (le_or_gt a b) (assume H : b < a, lt_imp_le H)
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theorem not_lt_imp_le {a b : ℤ} (H : ¬ a < b) : b ≤ a :=
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or.resolve_left (le_or_gt b a) H
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or_resolve_left (le_or_gt b a) H
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theorem not_le_imp_lt {a b : ℤ} (H : ¬ a ≤ b) : b < a :=
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or.resolve_right (le_or_gt a b) H
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or_resolve_right (le_or_gt a b) H
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-- (non)positivity and (non)negativity
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-- -------------------------------------
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@ -448,7 +448,7 @@ calc
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... = -a : (neg_move (Hn⁻¹))⁻¹
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theorem nat_abs_cases (a : ℤ) : a = (nat_abs a) ∨ a = - (nat_abs a) :=
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or.imp_or (le_total 0 a)
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or_of_or_of_imp_of_imp (le_total 0 a)
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(assume H : a ≥ 0, (nat_abs_nonneg_eq H)⁻¹)
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(assume H : a ≤ 0, (neg_move ((nat_abs_negative H)⁻¹))⁻¹)
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@ -564,7 +564,7 @@ have H2 : (nat_abs a) * (nat_abs b) = 1, from
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... = (nat_abs 1) : {H}
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... = 1 : nat_abs_of_nat 1,
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have H3 : (nat_abs a) = 1, from mul_eq_one_left H2,
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or.imp_or (nat_abs_cases a)
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or_of_or_of_imp_of_imp (nat_abs_cases a)
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(assume H4 : a = (nat_abs a), H3 ▸ H4)
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(assume H4 : a = - (nat_abs a), H3 ▸ H4)
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@ -146,12 +146,12 @@ induction_on s or.inr
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assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
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assume H1 : x ∈ y::s ++ t,
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have H2 : x = y ∨ x ∈ s ++ t, from H1,
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have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or.imp_or_right H2 IH,
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have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_of_or_of_imp_right H2 IH,
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iff.elim_right or.assoc H3)
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theorem mem.or_imp_concat {x : T} {s t : list T} : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
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induction_on s
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(take H, or.elim H false_elim (assume H, H))
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(take H, or.elim H false.elim (assume H, H))
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(take y s,
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assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
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assume H : x ∈ y::s ∨ x ∈ t,
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@ -167,7 +167,7 @@ iff.intro mem.concat_imp_or mem.or_imp_concat
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theorem mem.split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
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induction_on l
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(take H : x ∈ nil, false_elim (iff.elim_left !mem.nil H))
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(take H : x ∈ nil, false.elim (iff.elim_left !mem.nil H))
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(take y l,
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assume IH : x ∈ l → ∃s t : list T, l = s ++ (x::t),
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assume H : x ∈ y::l,
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@ -226,7 +226,7 @@ rec_on l
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assume iH : ¬x ∈ l → find x l = length l,
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assume P₁ : ¬x ∈ y::l,
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have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (iff.flip_sign !mem.cons) P₁,
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have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or P₂),
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have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or_iff_not_and_not P₂),
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calc
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find x (y::l) = if x = y then 0 else succ (find x l) : !find.cons
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... = succ (find x l) : if_neg (and.elim_left P₃)
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@ -63,7 +63,7 @@ induction_on n
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(show succ m = succ (pred (succ m)), from congr_arg succ !pred.succ⁻¹))
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theorem zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
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or.imp_or (zero_or_succ_pred n) (assume H, H)
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or_of_or_of_imp_of_imp (zero_or_succ_pred n) (assume H, H)
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(assume H : n = succ (pred n), exists_intro (pred n) H)
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theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n :=
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@ -34,10 +34,10 @@ congr_fun (fix_eq div.F x) y
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notation a div b := divide a b
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theorem div_zero (a : ℕ) : a div 0 = 0 :=
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divide_def a 0 ⬝ if_neg (!and.not_left (lt.irrefl 0))
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divide_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
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theorem div_less {a b : ℕ} (h : a < b) : a div b = 0 :=
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divide_def a b ⬝ if_neg (!and.not_right (lt_imp_not_ge h))
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divide_def a b ⬝ if_neg (!not_and_of_not_right (lt_imp_not_ge h))
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theorem zero_div (b : ℕ) : 0 div b = 0 :=
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divide_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt.of_lt_of_le l r) (lt.irrefl 0)))
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@ -74,10 +74,10 @@ theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (
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congr_fun (fix_eq mod.F x) y
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theorem mod_zero (a : ℕ) : a mod 0 = a :=
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modulo_def a 0 ⬝ if_neg (!and.not_left (lt.irrefl 0))
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modulo_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
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theorem mod_less {a b : ℕ} (h : a < b) : a mod b = a :=
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modulo_def a b ⬝ if_neg (!and.not_right (lt_imp_not_ge h))
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modulo_def a b ⬝ if_neg (!not_and_of_not_right (lt_imp_not_ge h))
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theorem zero_mod (b : ℕ) : 0 mod b = 0 :=
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modulo_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt.of_lt_of_le l r) (lt.irrefl 0)))
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@ -152,10 +152,10 @@ discriminate
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or.inl Hlt)
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theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m :=
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or.resolve_left (le_imp_succ_le_or_eq H1) H2
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or_resolve_left (le_imp_succ_le_or_eq H1) H2
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theorem le_succ_imp_le_or_eq {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m :=
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or.imp_or_left (le_imp_succ_le_or_eq H)
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or_of_or_of_imp_left (le_imp_succ_le_or_eq H)
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(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
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theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m :=
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@ -210,7 +210,7 @@ discriminate
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have H3 : k ≤ m, from H2 ▸ H,
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have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
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show n ≤ m ∨ n = succ m, from
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or.imp_or H4
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or_of_or_of_imp_of_imp H4
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(take H5 : succ k ≤ m, show n ≤ m, from Hn⁻¹ ▸ H5)
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(take H5 : k = m, show n = succ m, from H5 ▸ Hn))
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@ -275,7 +275,7 @@ theorem le_imp_lt_or_eq {n m : ℕ} (H : n ≤ m) : n < m ∨ n = m :=
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or.swap (eq_or_lt_of_le H)
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theorem le_ne_imp_lt {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : n < m :=
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or.resolve_left (le_imp_lt_or_eq H1) H2
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or_resolve_left (le_imp_lt_or_eq H1) H2
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theorem lt_succ_imp_le {n m : ℕ} (H : n < succ m) : n ≤ m :=
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succ_le_cancel (succ_le_of_lt H)
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@ -347,13 +347,13 @@ theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ n > m :=
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lt.trichotomy n m
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theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n :=
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or.imp_or_right le_or_gt (assume H : m < n, lt_imp_le H)
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or_of_or_of_imp_right le_or_gt (assume H : m < n, lt_imp_le H)
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theorem not_lt_imp_ge {n m : ℕ} (H : ¬ n < m) : n ≥ m :=
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or.resolve_left le_or_gt H
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or_resolve_left le_or_gt H
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theorem not_le_imp_gt {n m : ℕ} (H : ¬ n ≤ m) : n > m :=
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or.resolve_right le_or_gt H
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or_resolve_right le_or_gt H
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-- ### misc
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@ -396,7 +396,7 @@ theorem case_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀ {y : nat
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case y H0 (take y, H1 !succ_pos)
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theorem zero_or_pos {n : ℕ} : n = 0 ∨ n > 0 :=
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or.imp_or_left
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or_of_or_of_imp_left
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(or.swap (le_imp_lt_or_eq !zero_le))
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(take H : 0 = n, H⁻¹)
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@ -493,7 +493,7 @@ have H5 : k ≤ m, from mul_le_cancel_left Hn H3,
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le_antisym H4 H5
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theorem mul_cancel_left_or {n m k : ℕ} (H : n * m = n * k) : n = 0 ∨ m = k :=
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or.imp_or_right zero_or_pos
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or_of_or_of_imp_right zero_or_pos
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(assume Hn : n > 0, mul_cancel_left Hn H)
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theorem mul_cancel_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k :=
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@ -36,9 +36,9 @@ theorem propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
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or.elim (prop_complete a)
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(assume Hat, or.elim (prop_complete b)
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(assume Hbt, Hat ⬝ Hbt⁻¹)
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(assume Hbf, false_elim (Hbf ▸ (Hab (of_eq_true Hat)))))
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(assume Hbf, false.elim (Hbf ▸ (Hab (of_eq_true Hat)))))
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(assume Haf, or.elim (prop_complete b)
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(assume Hbt, false_elim (Haf ▸ (Hba (of_eq_true Hbt))))
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(assume Hbt, false.elim (Haf ▸ (Hba (of_eq_true Hbt))))
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(assume Hbf, Haf ⬝ Hbf⁻¹))
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theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b :=
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@ -1,32 +1,40 @@
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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Jeremy Avigad, Leonardo de Moura
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/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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definition imp (a b : Prop) : Prop := a → b
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Module: logic.connectives
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Authors: Jeremy Avigad, Leonardo de Moura
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The propositional connectives. See also init.datatypes.
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-/
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variables {a b c d : Prop}
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/- implies -/
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definition imp (a b : Prop) : Prop := a → b
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theorem mt (H1 : a → b) (H2 : ¬b) : ¬a :=
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assume Ha : a, absurd (H1 Ha) H2
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-- make c explicit and rename to false.elim
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theorem false_elim {c : Prop} (H : false) : c :=
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/- false -/
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theorem false.elim {c : Prop} (H : false) : c :=
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false.rec c H
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-- not
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-- ---
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/- not -/
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theorem not_elim (H1 : ¬a) (H2 : a) : false := H1 H2
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theorem not.elim (H1 : ¬a) (H2 : a) : false := H1 H2
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theorem not_intro (H : a → false) : ¬a := H
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theorem not.intro (H : a → false) : ¬a := H
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theorem not_not_intro (Ha : a) : ¬¬a :=
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assume Hna : ¬a, absurd Ha Hna
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theorem not_implies_left (H : ¬(a → b)) : ¬¬a :=
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theorem not_not_of_not_implies (H : ¬(a → b)) : ¬¬a :=
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assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
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theorem not_implies_right (H : ¬(a → b)) : ¬b :=
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theorem not_of_not_implies (H : ¬(a → b)) : ¬b :=
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assume Hb : b, absurd (assume Ha : a, Hb) H
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theorem not_not_em : ¬¬(a ∨ ¬a) :=
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|
@ -35,121 +43,111 @@ assume not_em : ¬(a ∨ ¬a),
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assume Hp : a, absurd (or.inl Hp) not_em,
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absurd (or.inr Hnp) not_em
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-- and
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-- ---
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/- and -/
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namespace and
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definition not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (elim_left H) Hna
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definition not_and_of_not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (and.elim_left H) Hna
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definition not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (elim_right H) Hnb
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definition not_and_of_not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (and.elim_right H) Hnb
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theorem swap (H : a ∧ b) : b ∧ a :=
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intro (elim_right H) (elim_left H)
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theorem and.swap (H : a ∧ b) : b ∧ a :=
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and.intro (and.elim_right H) (and.elim_left H)
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theorem imp_and (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
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elim H₁ (assume Ha : a, assume Hb : b, intro (H₂ Ha) (H₃ Hb))
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theorem and_of_and_of_imp_of_imp (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
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and.elim H₁ (assume Ha : a, assume Hb : b, and.intro (H₂ Ha) (H₃ Hb))
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theorem imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
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elim H₁ (assume Ha : a, assume Hc : c, intro (H Ha) Hc)
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theorem and_of_and_of_imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
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and.elim H₁ (assume Ha : a, assume Hc : c, and.intro (H Ha) Hc)
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theorem imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
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elim H₁ (assume Hc : c, assume Ha : a, intro Hc (H Ha))
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theorem and_of_and_of_imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
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and.elim H₁ (assume Hc : c, assume Ha : a, and.intro Hc (H Ha))
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theorem comm : a ∧ b ↔ b ∧ a :=
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iff.intro (λH, swap H) (λH, swap H)
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theorem and.comm : a ∧ b ↔ b ∧ a :=
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iff.intro (λH, and.swap H) (λH, and.swap H)
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theorem assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
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theorem and.assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
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iff.intro
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(assume H, intro
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(elim_left (elim_left H))
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(intro (elim_right (elim_left H)) (elim_right H)))
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(assume H, intro
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(intro (elim_left H) (elim_left (elim_right H)))
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(elim_right (elim_right H)))
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end and
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(assume H, and.intro
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(and.elim_left (and.elim_left H))
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(and.intro (and.elim_right (and.elim_left H)) (and.elim_right H)))
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(assume H, and.intro
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(and.intro (and.elim_left H) (and.elim_left (and.elim_right H)))
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(and.elim_right (and.elim_right H)))
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-- or
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-- --
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/- or -/
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namespace or
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definition not_intro (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
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definition not_or (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
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assume H : a ∨ b, or.rec_on H
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(assume Ha, absurd Ha Hna)
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(assume Hb, absurd Hb Hnb)
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theorem imp_or (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
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elim H₁
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(assume Ha : a, inl (H₂ Ha))
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(assume Hb : b, inr (H₃ Hb))
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theorem or_of_or_of_imp_of_imp (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
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or.elim H₁
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(assume Ha : a, or.inl (H₂ Ha))
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(assume Hb : b, or.inr (H₃ Hb))
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theorem imp_or_left (H₁ : a ∨ c) (H : a → b) : b ∨ c :=
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elim H₁
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(assume H₂ : a, inl (H H₂))
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(assume H₂ : c, inr H₂)
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theorem or_of_or_of_imp_left (H₁ : a ∨ c) (H : a → b) : b ∨ c :=
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or.elim H₁
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(assume H₂ : a, or.inl (H H₂))
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(assume H₂ : c, or.inr H₂)
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theorem imp_or_right (H₁ : c ∨ a) (H : a → b) : c ∨ b :=
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elim H₁
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(assume H₂ : c, inl H₂)
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(assume H₂ : a, inr (H H₂))
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theorem or_of_or_of_imp_right (H₁ : c ∨ a) (H : a → b) : c ∨ b :=
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or.elim H₁
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(assume H₂ : c, or.inl H₂)
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(assume H₂ : a, or.inr (H H₂))
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theorem elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
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elim H Ha (assume H₂, elim H₂ Hb Hc)
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theorem or.elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
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or.elim H Ha (assume H₂, or.elim H₂ Hb Hc)
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theorem resolve_right (H₁ : a ∨ b) (H₂ : ¬a) : b :=
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elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
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theorem or_resolve_right (H₁ : a ∨ b) (H₂ : ¬a) : b :=
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or.elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
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theorem resolve_left (H₁ : a ∨ b) (H₂ : ¬b) : a :=
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elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
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theorem or_resolve_left (H₁ : a ∨ b) (H₂ : ¬b) : a :=
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or.elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
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theorem swap (H : a ∨ b) : b ∨ a :=
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elim H (assume Ha, inr Ha) (assume Hb, inl Hb)
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theorem or.swap (H : a ∨ b) : b ∨ a :=
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or.elim H (assume Ha, or.inr Ha) (assume Hb, or.inl Hb)
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theorem comm : a ∨ b ↔ b ∨ a :=
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iff.intro (λH, swap H) (λH, swap H)
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theorem or.comm : a ∨ b ↔ b ∨ a :=
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iff.intro (λH, or.swap H) (λH, or.swap H)
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theorem assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
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theorem or.assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
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iff.intro
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(assume H, elim H
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(assume H₁, elim H₁
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(assume Ha, inl Ha)
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(assume Hb, inr (inl Hb)))
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(assume Hc, inr (inr Hc)))
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(assume H, elim H
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(assume Ha, (inl (inl Ha)))
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(assume H₁, elim H₁
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(assume Hb, inl (inr Hb))
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(assume Hc, inr Hc)))
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end or
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(assume H, or.elim H
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(assume H₁, or.elim H₁
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(assume Ha, or.inl Ha)
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(assume Hb, or.inr (or.inl Hb)))
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(assume Hc, or.inr (or.inr Hc)))
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(assume H, or.elim H
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(assume Ha, (or.inl (or.inl Ha)))
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(assume H₁, or.elim H₁
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(assume Hb, or.inl (or.inr Hb))
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(assume Hc, or.inr Hc)))
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-- iff
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-- ---
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/- iff -/
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namespace iff
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definition def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
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definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
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!eq.refl
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end iff
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-- exists_unique
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-- -------------
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/- exists_unique -/
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|
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definition exists_unique {A : Type} (p : A → Prop) :=
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∃x, p x ∧ ∀y, p y → y = x
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|
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notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
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|
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theorem exists_unique_intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) : ∃!x, p x :=
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theorem exists_unique.intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) :
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∃!x, p x :=
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exists_intro w (and.intro H1 H2)
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|
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theorem exists_unique_elim {A : Type} {p : A → Prop} {b : Prop}
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theorem exists_unique.elim {A : Type} {p : A → Prop} {b : Prop}
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(H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=
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obtain w Hw, from H2,
|
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H1 w (and.elim_left Hw) (and.elim_right Hw)
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|
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-- if-then-else
|
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-- ------------
|
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/- if-then-else -/
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|
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section
|
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open eq.ops
|
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|
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|
|
@ -161,8 +159,8 @@ section
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|||
definition if_false (t e : A) : (if false then t else e) = e :=
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if_neg not_false
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|
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theorem if_cond_congr [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) (t e : A)
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: (if c₁ then t else e) = (if c₂ then t else e) :=
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theorem if_congr_cond [H₁ : decidable c₁] [H₂ : decidable c₂] (Heq : c₁ ↔ c₂) (t e : A) :
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(if c₁ then t else e) = (if c₂ then t else e) :=
|
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decidable.rec_on H₁
|
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(λ Hc₁ : c₁, decidable.rec_on H₂
|
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(λ Hc₂ : c₂, if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
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|
|
@ -174,9 +172,10 @@ section
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|||
theorem if_congr_aux [H₁ : decidable c₁] [H₂ : decidable c₂] {t₁ t₂ e₁ e₂ : A}
|
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(Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
|
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(if c₁ then t₁ else e₁) = (if c₂ then t₂ else e₂) :=
|
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Ht ▸ He ▸ (if_cond_congr Hc t₁ e₁)
|
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Ht ▸ He ▸ (if_congr_cond Hc t₁ e₁)
|
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|
||||
theorem if_congr [H₁ : decidable c₁] {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
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||||
theorem if_congr [H₁ : decidable c₁] {t₁ t₂ e₁ e₂ : A} (Hc : c₁ ↔ c₂) (Ht : t₁ = t₂)
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(He : e₁ = e₂) :
|
||||
(if c₁ then t₁ else e₁) = (@ite c₂ (decidable.decidable_iff_equiv H₁ Hc) A t₂ e₂) :=
|
||||
have H2 [visible] : decidable c₂, from (decidable.decidable_iff_equiv H₁ Hc),
|
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if_congr_aux Hc Ht He
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|
|
|
|||
|
|
@ -185,7 +185,7 @@ assume Hem H1,
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|||
have Hx : ∀x, P x, from
|
||||
take x,
|
||||
have H1 : P x ∨ C, from H1 x,
|
||||
or.resolve_left H1 Hnc,
|
||||
or_resolve_left H1 Hnc,
|
||||
or.inl Hx)
|
||||
|
||||
theorem thm23a : (∃x, P x) ∧ C → (∃x, P x ∧ C) :=
|
||||
|
|
|
|||
|
|
@ -47,12 +47,13 @@ theorem not_not_elim {a : Prop} [D : decidable a] (H : ¬¬a) : a :=
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|||
iff.mp not_not_iff H
|
||||
|
||||
theorem not_true_iff_false : (¬true) ↔ false :=
|
||||
iff.intro (assume H, H trivial) false_elim
|
||||
iff.intro (assume H, H trivial) false.elim
|
||||
|
||||
theorem not_false_iff_true : (¬false) ↔ true :=
|
||||
iff.intro (assume H, trivial) (assume H H', H')
|
||||
|
||||
theorem not_or {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a ∨ b)) ↔ (¬a ∧ ¬b) :=
|
||||
theorem not_or_iff_not_and_not {a b : Prop} [Da : decidable a] [Db : decidable b] :
|
||||
(¬(a ∨ b)) ↔ (¬a ∧ ¬b) :=
|
||||
iff.intro
|
||||
(assume H, or.elim (em a)
|
||||
(assume Ha, absurd (or.inl Ha) H)
|
||||
|
|
@ -82,16 +83,16 @@ iff.intro
|
|||
(assume Ha : a, or.inr (H Ha))
|
||||
(assume Hna : ¬a, or.inl Hna)))
|
||||
(assume (H : ¬a ∨ b) (Ha : a),
|
||||
or.resolve_right H (not_not_iff⁻¹ ▸ Ha))
|
||||
or_resolve_right H (not_not_iff⁻¹ ▸ Ha))
|
||||
|
||||
theorem not_implies {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a → b)) ↔ (a ∧ ¬b) :=
|
||||
calc (¬(a → b)) ↔ (¬(¬a ∨ b)) : {imp_or}
|
||||
... ↔ (¬¬a ∧ ¬b) : not_or
|
||||
... ↔ (¬¬a ∧ ¬b) : not_or_iff_not_and_not
|
||||
... ↔ (a ∧ ¬b) : {not_not_iff}
|
||||
|
||||
theorem peirce {a b : Prop} [D : decidable a] : ((a → b) → a) → a :=
|
||||
assume H, by_contradiction (assume Hna : ¬a,
|
||||
have Hnna : ¬¬a, from not_implies_left (mt H Hna),
|
||||
have Hnna : ¬¬a, from not_not_of_not_implies (mt H Hna),
|
||||
absurd (not_not_elim Hnna) Hna)
|
||||
|
||||
theorem not_exists_forall {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
|
||||
|
|
@ -116,12 +117,12 @@ iff.intro
|
|||
theorem iff_false_intro {a : Prop} (H : ¬a) : a ↔ false :=
|
||||
iff.intro
|
||||
(assume H1 : a, absurd H1 H)
|
||||
(assume H2 : false, false_elim H2)
|
||||
(assume H2 : false, false.elim H2)
|
||||
|
||||
theorem a_neq_a {A : Type} (a : A) : (a ≠ a) ↔ false :=
|
||||
iff.intro
|
||||
(assume H, false.of_ne H)
|
||||
(assume H, false_elim H)
|
||||
(assume H, false.elim H)
|
||||
|
||||
theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
|
||||
iff_true_intro rfl
|
||||
|
|
@ -134,7 +135,7 @@ iff.intro
|
|||
(assume H,
|
||||
have H' : ¬a, from assume Ha, (H ▸ Ha) Ha,
|
||||
H' (H⁻¹ ▸ H'))
|
||||
(assume H, false_elim H)
|
||||
(assume H, false.elim H)
|
||||
|
||||
theorem true_eq_false : (true ↔ false) ↔ false :=
|
||||
not_true_iff_false ▸ (a_iff_not_a true)
|
||||
|
|
|
|||
|
|
@ -35,16 +35,16 @@ is_congruence2.mk
|
|||
(take a1 b1 a2 b2,
|
||||
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
|
||||
iff.intro
|
||||
(assume H3 : a1 ∧ a2, and.imp_and H3 (iff.elim_left H1) (iff.elim_left H2))
|
||||
(assume H3 : b1 ∧ b2, and.imp_and H3 (iff.elim_right H1) (iff.elim_right H2)))
|
||||
(assume H3 : a1 ∧ a2, and_of_and_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
|
||||
(assume H3 : b1 ∧ b2, and_of_and_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
|
||||
|
||||
theorem is_congruence_or : is_congruence2 iff iff iff or :=
|
||||
is_congruence2.mk
|
||||
(take a1 b1 a2 b2,
|
||||
assume H1 : a1 ↔ b1, assume H2 : a2 ↔ b2,
|
||||
iff.intro
|
||||
(assume H3 : a1 ∨ a2, or.imp_or H3 (iff.elim_left H1) (iff.elim_left H2))
|
||||
(assume H3 : b1 ∨ b2, or.imp_or H3 (iff.elim_right H1) (iff.elim_right H2)))
|
||||
(assume H3 : a1 ∨ a2, or_of_or_of_imp_of_imp H3 (iff.elim_left H1) (iff.elim_left H2))
|
||||
(assume H3 : b1 ∨ b2, or_of_or_of_imp_of_imp H3 (iff.elim_right H1) (iff.elim_right H2)))
|
||||
|
||||
theorem is_congruence_imp : is_congruence2 iff iff iff imp :=
|
||||
is_congruence2.mk
|
||||
|
|
@ -87,8 +87,8 @@ relation.mp_like.mk (λa b (H : a ↔ b), iff.elim_left H)
|
|||
/- support for calculations with iff -/
|
||||
|
||||
namespace iff
|
||||
theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop} (H : a ↔ b) (H1 : P a) :
|
||||
P b :=
|
||||
theorem subst {P : Prop → Prop} [C : is_congruence iff iff P] {a b : Prop}
|
||||
(H : a ↔ b) (H1 : P a) : P b :=
|
||||
@general_subst.subst Prop iff P C a b H H1
|
||||
end iff
|
||||
|
||||
|
|
|
|||
|
|
@ -1,6 +1,11 @@
|
|||
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Authors: Leonardo de Moura, Jeremy Avigad
|
||||
/-
|
||||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
|
||||
Module: logic.quantifiers
|
||||
Authors: Leonardo de Moura, Jeremy Avigad
|
||||
-/
|
||||
|
||||
open inhabited nonempty
|
||||
|
||||
theorem exists_not_forall {A : Type} {p : A → Prop} (H : ∃x, p x) : ¬∀x, ¬p x :=
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ definition Prop := Type.{0}
|
|||
definition false := ∀x : Prop, x
|
||||
check false
|
||||
|
||||
theorem false_elim (C : Prop) (H : false) : C
|
||||
theorem false.elim (C : Prop) (H : false) : C
|
||||
:= H C
|
||||
|
||||
definition eq {A : Type} (a b : A)
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ definition Prop := Type.{0}
|
|||
definition false : Prop := ∀x : Prop, x
|
||||
check false
|
||||
|
||||
theorem false_elim (C : Prop) (H : false) : C
|
||||
theorem false.elim (C : Prop) (H : false) : C
|
||||
:= H C
|
||||
|
||||
definition eq {A : Type} (a b : A)
|
||||
|
|
|
|||
|
|
@ -4,7 +4,7 @@ definition Prop := Type.{0}
|
|||
definition false : Prop := ∀x : Prop, x
|
||||
check false
|
||||
|
||||
theorem false_elim (C : Prop) (H : false) : C
|
||||
theorem false.elim (C : Prop) (H : false) : C
|
||||
:= H C
|
||||
|
||||
definition eq {A : Type} (a b : A)
|
||||
|
|
|
|||
|
|
@ -14,7 +14,7 @@ theorem is_nil_nil (A : Type) : is_nil (@nil A)
|
|||
:= of_eq_true (refl true)
|
||||
|
||||
theorem cons_ne_nil {A : Type} (a : A) (l : list A) : ¬ cons a l = nil
|
||||
:= not_intro (assume H : cons a l = nil,
|
||||
:= not.intro (assume H : cons a l = nil,
|
||||
absurd
|
||||
(calc true = is_nil (@nil A) : refl _
|
||||
... = is_nil (cons a l) : { symm H }
|
||||
|
|
|
|||
|
|
@ -14,7 +14,7 @@ section
|
|||
notation `∅` := empty
|
||||
|
||||
theorem mem_empty (x : T) : ¬ (x ∈ ∅)
|
||||
:= not_intro (λH : x ∈ ∅, absurd H ff_ne_tt)
|
||||
:= not.intro (λH : x ∈ ∅, absurd H ff_ne_tt)
|
||||
end
|
||||
|
||||
end set
|
||||
|
|
|
|||
|
|
@ -48,13 +48,13 @@ rec_on s1
|
|||
(take a2, show decidable (inl B a1 = inl B a2), from H1 a1 a2)
|
||||
(take b2,
|
||||
have H3 : (inl B a1 = inr A b2) ↔ false,
|
||||
from iff.intro inl_neq_inr (assume H4, false_elim H4),
|
||||
from iff.intro inl_neq_inr (assume H4, false.elim H4),
|
||||
show decidable (inl B a1 = inr A b2), from decidable_iff_equiv _ (iff.symm H3)))
|
||||
(take b1, show decidable (inr A b1 = s2), from
|
||||
rec_on s2
|
||||
(take a2,
|
||||
have H3 : (inr A b1 = inl B a2) ↔ false,
|
||||
from iff.intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false_elim H4),
|
||||
from iff.intro (assume H4, inl_neq_inr (symm H4)) (assume H4, false.elim H4),
|
||||
show decidable (inr A b1 = inl B a2), from decidable_iff_equiv _ (iff.symm H3))
|
||||
(take b2, show decidable (inr A b1 = inr A b2), from H2 b1 b2))
|
||||
|
||||
|
|
|
|||
|
|
@ -60,7 +60,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
|
|||
(show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
|
||||
|
||||
theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
|
||||
:= or.imp_or (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
|
||||
:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
|
||||
|
||||
theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
|
||||
:= induction_on n H1 (take m IH, H2 m)
|
||||
|
|
@ -502,10 +502,10 @@ theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
|
|||
or_intro_left _ Hlt)
|
||||
|
||||
theorem succ_le_left {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
|
||||
:= or.resolve_left (succ_le_left_or H1) H2
|
||||
:= or_resolve_left (succ_le_left_or H1) H2
|
||||
|
||||
theorem succ_le_right_inv {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m
|
||||
:= or.imp_or (succ_le_left_or H)
|
||||
:= or_of_or_of_imp_of_imp (succ_le_left_or H)
|
||||
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
|
||||
(take H2 : n = succ m, H2)
|
||||
|
||||
|
|
@ -561,7 +561,7 @@ theorem pred_le_left_inv {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m
|
|||
have H3 : k ≤ m, from subst H2 H,
|
||||
have H4 : succ k ≤ m ∨ k = m, from succ_le_left_or H3,
|
||||
show n ≤ m ∨ n = succ m, from
|
||||
or.imp_or H4
|
||||
or_of_or_of_imp_of_imp H4
|
||||
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
|
||||
(take H5 : k = m, show n = succ m, from subst H5 Hn))
|
||||
|
||||
|
|
@ -589,7 +589,7 @@ theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
|
|||
or_intro_left _ Hlt)
|
||||
|
||||
theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
|
||||
:= or.resolve_left (le_imp_succ_le_or_eq H1) H2
|
||||
:= or_resolve_left (le_imp_succ_le_or_eq H1) H2
|
||||
|
||||
theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
|
||||
:=
|
||||
|
|
@ -620,7 +620,7 @@ theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = su
|
|||
have H3 : k ≤ m, from subst H2 H,
|
||||
have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
|
||||
show n ≤ m ∨ n = succ m, from
|
||||
or.imp_or H4
|
||||
or_of_or_of_imp_of_imp H4
|
||||
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
|
||||
(take H5 : k = m, show n = succ m, from subst H5 Hn))
|
||||
|
||||
|
|
@ -806,13 +806,13 @@ theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
|
|||
(assume H : m < k, or_intro_right _ (succ_lt_right H)))
|
||||
|
||||
theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ m < n
|
||||
:= or.imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
|
||||
:= or_of_or_of_imp_of_imp (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
|
||||
|
||||
theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ m < n
|
||||
:= iff.elim_left or.assoc (trichotomy_alt n m)
|
||||
|
||||
theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
|
||||
:= or.imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
|
||||
:= or_of_or_of_imp_of_imp (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
|
||||
|
||||
-- interaction with mul under "positivity"
|
||||
|
||||
|
|
@ -871,7 +871,7 @@ theorem add_eq_self {n m : ℕ} (H : n + m = n) : m = 0
|
|||
---------- basic
|
||||
|
||||
theorem zero_or_positive (n : ℕ) : n = 0 ∨ n > 0
|
||||
:= or.imp_or (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
|
||||
:= or_of_or_of_imp_of_imp (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
|
||||
|
||||
theorem succ_positive {n m : ℕ} (H : n = succ m) : n > 0
|
||||
:= subst (symm H) (lt_zero m)
|
||||
|
|
@ -954,7 +954,7 @@ theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k
|
|||
n * m = n * 0 : H
|
||||
... = 0 : mul_zero_right n,
|
||||
have H3 : n = 0 ∨ m = 0, from mul_eq_zero H2,
|
||||
or.resolve_right H3 (ne.symm (lt_ne Hn)))
|
||||
or_resolve_right H3 (ne.symm (lt_ne Hn)))
|
||||
(take (l : ℕ),
|
||||
assume (IH : ∀ m, n * m = n * l → m = l),
|
||||
take (m : ℕ),
|
||||
|
|
|
|||
|
|
@ -54,7 +54,7 @@ theorem zero_or_succ (n : ℕ) : n = 0 ∨ n = succ (pred n)
|
|||
(show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
|
||||
|
||||
theorem zero_or_succ2 (n : ℕ) : n = 0 ∨ ∃k, n = succ k
|
||||
:= or.imp_or (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
|
||||
:= or_of_or_of_imp_of_imp (zero_or_succ n) (assume H, H) (assume H : n = succ (pred n), exists_intro (pred n) H)
|
||||
|
||||
theorem case {P : ℕ → Prop} (n : ℕ) (H1: P 0) (H2 : ∀m, P (succ m)) : P n
|
||||
:= induction_on n H1 (take m IH, H2 m)
|
||||
|
|
@ -496,10 +496,10 @@ theorem succ_le_left_or {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
|
|||
or.intro_left _ Hlt)
|
||||
|
||||
theorem succ_le_left {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
|
||||
:= or.resolve_left (succ_le_left_or H1) H2
|
||||
:= or_resolve_left (succ_le_left_or H1) H2
|
||||
|
||||
theorem succ_le_right_inv {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m
|
||||
:= or.imp_or (succ_le_left_or H)
|
||||
:= or_of_or_of_imp_of_imp (succ_le_left_or H)
|
||||
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
|
||||
(take H2 : n = succ m, H2)
|
||||
|
||||
|
|
@ -555,7 +555,7 @@ theorem pred_le_left_inv {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m
|
|||
have H3 : k ≤ m, from subst H2 H,
|
||||
have H4 : succ k ≤ m ∨ k = m, from succ_le_left_or H3,
|
||||
show n ≤ m ∨ n = succ m, from
|
||||
or.imp_or H4
|
||||
or_of_or_of_imp_of_imp H4
|
||||
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
|
||||
(take H5 : k = m, show n = succ m, from subst H5 Hn))
|
||||
|
||||
|
|
@ -583,10 +583,10 @@ theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m
|
|||
or.intro_left _ Hlt)
|
||||
|
||||
theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m
|
||||
:= or.resolve_left (le_imp_succ_le_or_eq H1) H2
|
||||
:= or_resolve_left (le_imp_succ_le_or_eq H1) H2
|
||||
|
||||
theorem le_succ_imp_le_or_eq {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m
|
||||
:= or.imp_or_left (le_imp_succ_le_or_eq H)
|
||||
:= or_of_or_of_imp_left (le_imp_succ_le_or_eq H)
|
||||
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
|
||||
|
||||
theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m
|
||||
|
|
@ -618,7 +618,7 @@ theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = su
|
|||
have H3 : k ≤ m, from subst H2 H,
|
||||
have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
|
||||
show n ≤ m ∨ n = succ m, from
|
||||
or.imp_or H4
|
||||
or_of_or_of_imp_of_imp H4
|
||||
(take H5 : succ k ≤ m, show n ≤ m, from subst (symm Hn) H5)
|
||||
(take H5 : k = m, show n = succ m, from subst H5 Hn))
|
||||
|
||||
|
|
@ -804,13 +804,13 @@ theorem le_or_lt (n m : ℕ) : n ≤ m ∨ m < n
|
|||
(assume H : m < k, or.intro_right _ (succ_lt_right H)))
|
||||
|
||||
theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ m < n
|
||||
:= or.imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
|
||||
:= or_of_or_of_imp_of_imp (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
|
||||
|
||||
theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ m < n
|
||||
:= iff.elim_left or.assoc (trichotomy_alt n m)
|
||||
|
||||
theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
|
||||
:= or.imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
|
||||
:= or_of_or_of_imp_of_imp (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
|
||||
|
||||
-- interaction with mul under "positivity"
|
||||
|
||||
|
|
@ -869,7 +869,7 @@ theorem add_eq_self {n m : ℕ} (H : n + m = n) : m = 0
|
|||
---------- basic
|
||||
|
||||
theorem zero_or_positive (n : ℕ) : n = 0 ∨ n > 0
|
||||
:= or.imp_or (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
|
||||
:= or_of_or_of_imp_of_imp (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H) (take H : n > 0, H)
|
||||
|
||||
theorem succ_positive {n m : ℕ} (H : n = succ m) : n > 0
|
||||
:= subst (symm H) (lt_zero m)
|
||||
|
|
@ -906,7 +906,7 @@ theorem case_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀y, y > 0
|
|||
:= case y H0 (take y', H1 _ (succ_pos _))
|
||||
|
||||
theorem zero_or_pos (n : ℕ) : n = 0 ∨ n > 0
|
||||
:= or.imp_or_left (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H)
|
||||
:= or_of_or_of_imp_left (or.swap (le_imp_lt_or_eq (zero_le n))) (take H : 0 = n, symm H)
|
||||
|
||||
theorem succ_imp_pos {n m : ℕ} (H : n = succ m) : n > 0
|
||||
:= subst (symm H) (succ_pos m)
|
||||
|
|
@ -958,7 +958,7 @@ theorem mul_left_inj {n m k : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k
|
|||
n * m = n * 0 : H
|
||||
... = 0 : mul_zero_right n,
|
||||
have H3 : n = 0 ∨ m = 0, from mul_eq_zero H2,
|
||||
or.resolve_right H3 (ne.symm (lt_ne Hn)))
|
||||
or_resolve_right H3 (ne.symm (lt_ne Hn)))
|
||||
(take (l : ℕ),
|
||||
assume (IH : ∀ m, n * m = n * l → m = l),
|
||||
take (m : ℕ),
|
||||
|
|
|
|||
Loading…
Reference in a new issue