feat(logic/basic): When a dependent If-Then-Else is not one of its ar…

…guments (#10924)

This is the dependent version of #10800.

src/logic/basic.lean

@@ -340,6 +340,8 @@ is classically true but not constructively. -/
 
 theorem of_not_not : ¬¬a → a := by_contra
 
+lemma not_ne_iff {α : Sort*} {a b : α} : ¬ a ≠ b ↔ a = b := not_not
+
 -- See Note [decidable namespace]
 protected theorem decidable.of_not_imp [decidable a] (h : ¬ (a → b)) : a :=
 decidable.by_contradiction (not_not_of_not_imp h)
@@ -1087,6 +1089,9 @@ theorem forall_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∀ h' : p, q
 theorem exists_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃ h' : p, q h') ↔ q h :=
 @exists_const (q h) p ⟨h⟩
 
+lemma exists_iff_of_forall {p : Prop} {q : p → Prop} (h : ∀ h, q h) : (∃ h, q h) ↔ p :=
+⟨Exists.fst, λ H, ⟨H, h H⟩⟩
+
 theorem exists_unique_prop_of_true {p : Prop} {q : p → Prop} (h : p) : (∃! h' : p, q h') ↔ q h :=
 @exists_unique_const (q h) p ⟨h⟩ _
 
@@ -1316,38 +1321,57 @@ end classical
 
 section ite
 variables {α β γ : Sort*} {σ : α → Sort*} (f : α → β) {P Q : Prop} [decidable P] [decidable Q]
-  {a b c : α}
+  {a b c : α} {A : P → α} {B : ¬ P → α}
+
+lemma dite_eq_iff : dite P A B = c ↔ (∃ h, A h = c) ∨ ∃ h, B h = c := by by_cases P; simp *
+lemma ite_eq_iff : ite P a b = c ↔ P ∧ a = c ∨ ¬ P ∧ b = c :=
+dite_eq_iff.trans $ by rw [exists_prop, exists_prop]
+
+@[simp] lemma dite_eq_left_iff : dite P (λ _, a) B = a ↔ ∀ h, B h = a := by by_cases P; simp *
+@[simp] lemma dite_eq_right_iff : dite P A (λ _, b) = b ↔ ∀ h, A h = b := by by_cases P; simp *
+@[simp] lemma ite_eq_left_iff : ite P a b = a ↔ (¬ P → b = a) := dite_eq_left_iff
+@[simp] lemma ite_eq_right_iff : ite P a b = b ↔ (P → a = b) := dite_eq_right_iff
 
-lemma ite_eq_iff : ite P a b = c ↔ P ∧ a = c ∨ ¬ P ∧ b = c := by by_cases P; simp *
+lemma dite_ne_left_iff : dite P (λ _, a) B ≠ a ↔ ∃ h, a ≠ B h :=
+by { rw [ne.def, dite_eq_left_iff, not_forall], exact exists_congr (λ h, by rw ne_comm) }
 
-@[simp] lemma ite_eq_left_iff : ite P a b = a ↔ (¬ P → b = a) := by by_cases P; simp *
-@[simp] lemma ite_eq_right_iff : ite P a b = b ↔ (P → a = b) := by by_cases P; simp *
+lemma dite_ne_right_iff : dite P A (λ _, b) ≠ b ↔ ∃ h, A h ≠ b :=
+by simp only [ne.def, dite_eq_right_iff, not_forall]
 
-lemma ite_ne_left_iff : ite P a b ≠ a ↔ ¬ P ∧ a ≠ b :=
-by rw [ne.def, ite_eq_left_iff, ne_comm, not_imp]
+lemma ite_ne_left_iff : ite P a b ≠ a ↔ ¬ P ∧ a ≠ b := dite_ne_left_iff.trans $ by rw exists_prop
+lemma ite_ne_right_iff : ite P a b ≠ b ↔ P ∧ a ≠ b := dite_ne_right_iff.trans $ by rw exists_prop
 
-lemma ite_ne_right_iff : ite P a b ≠ b ↔ P ∧ a ≠ b := by rw [ne.def, ite_eq_right_iff, not_imp]
+protected lemma ne.dite_eq_left_iff (h : ∀ h, a ≠ B h) : dite P (λ _, a) B = a ↔ P :=
+dite_eq_left_iff.trans $ ⟨λ H, of_not_not $ λ h', h h' (H h').symm, λ h H, (H h).elim⟩
 
-protected lemma ne.ite_eq_left_iff (h : a ≠ b) : ite P a b = a ↔ P :=
-ite_eq_left_iff.trans $ not_imp_comm.trans $ imp_iff_right h.symm
+protected lemma ne.dite_eq_right_iff (h : ∀ h, A h ≠ b) : dite P A (λ _, b) = b ↔ ¬ P :=
+dite_eq_right_iff.trans $ ⟨λ H h', h h' (H h'), λ h' H, (h' H).elim⟩
 
+protected lemma ne.ite_eq_left_iff (h : a ≠ b) : ite P a b = a ↔ P := ne.dite_eq_left_iff $ λ _, h
 protected lemma ne.ite_eq_right_iff (h : a ≠ b) : ite P a b = b ↔ ¬ P :=
-ite_eq_right_iff.trans $ imp_iff_not h
+ne.dite_eq_right_iff $ λ _, h
 
-protected lemma ne.ite_ne_left_iff (h : a ≠ b) : ite P a b ≠ a ↔ ¬ P :=
-ite_ne_left_iff.trans $ and_iff_left h
+protected lemma ne.dite_ne_left_iff (h : ∀ h, a ≠ B h) : dite P (λ _, a) B ≠ a ↔ ¬ P :=
+dite_ne_left_iff.trans $ exists_iff_of_forall h
 
-protected lemma ne.ite_ne_right_iff (h : a ≠ b) : ite P a b ≠ b ↔ P :=
-ite_ne_right_iff.trans $ and_iff_left h
+protected lemma ne.dite_ne_right_iff (h : ∀ h, A h ≠ b) : dite P A (λ _, b) ≠ b ↔ P :=
+dite_ne_right_iff.trans $ exists_iff_of_forall h
 
-variables (P Q) (a b)
+protected lemma ne.ite_ne_left_iff (h : a ≠ b) : ite P a b ≠ a ↔ ¬ P := ne.dite_ne_left_iff $ λ _, h
 
-lemma ite_eq_or_eq : ite P a b = a ∨ ite P a b = b :=
-decidable.by_cases (λ h, or.inl (if_pos h)) (λ h, or.inr (if_neg h))
+protected lemma ne.ite_ne_right_iff (h : a ≠ b) : ite P a b ≠ b ↔ P := ne.dite_ne_right_iff $ λ _, h
+
+variables (P Q) (a b)
 
 /-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/
 @[simp] lemma dite_eq_ite : dite P (λ h, a) (λ h, b) = ite P a b := rfl
 
+lemma dite_eq_or_eq : (∃ h, dite P A B = A h) ∨ ∃ h, dite P A B = B h :=
+decidable.by_cases (λ h, or.inl ⟨h, dif_pos h⟩) (λ h, or.inr ⟨h, dif_neg h⟩)
+
+lemma ite_eq_or_eq : ite P a b = a ∨ ite P a b = b :=
+decidable.by_cases (λ h, or.inl (if_pos h)) (λ h, or.inr (if_neg h))
+
 /-- A function applied to a `dite` is a `dite` of that function applied to each of the branches. -/
 lemma apply_dite (x : P → α) (y : ¬P → α) : f (dite P x y) = dite P (λ h, f (x h)) (λ h, f (y h)) :=
 by by_cases h : P; simp [h]