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[Merged by Bors] - feat(data/part): Lemmas for get on binary function instances #12194

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[Merged by Bors] - feat(data/part): Lemmas for get on binary function instances #12194

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add part
BoltonBailey Feb 21, 2022
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Merge branch 'master' of github.com:leanprover-community/mathlib into…
BoltonBailey Feb 22, 2022
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81 changes: 81 additions & 0 deletions src/data/part.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -505,6 +505,19 @@ lemma one_mem_one [has_one α] : (1 : α) ∈ (1 : part α) := ⟨trivial, rfl
lemma mul_mem_mul [has_mul α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma * mb ∈ a * b := by tidy

@[to_additive]
lemma left_dom_of_mul_dom [has_mul α] {a b : part α} (hab : dom (a * b)) :
a.dom := by tidy

@[to_additive]
lemma right_dom_of_mul_dom [has_mul α] {a b : part α} (hab : dom (a * b)) :
b.dom := by tidy

@[to_additive, simp]
lemma mul_get_eq [has_mul α] (a b : part α) (hab : dom (a * b)) :
(a * b).get hab = a.get (left_dom_of_mul_dom hab) * b.get (right_dom_of_mul_dom hab) :=
by tidy

@[to_additive]
lemma some_mul_some [has_mul α] (a b : α) : some a * some b = some (a * b) := by tidy

Expand All @@ -518,32 +531,100 @@ lemma inv_some [has_inv α] (a : α) : (some a)⁻¹ = some (a⁻¹) := rfl
lemma div_mem_div [has_div α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma / mb ∈ a / b := by tidy

@[to_additive]
lemma left_dom_of_div_dom [has_div α] {a b : part α} (hab : dom (a / b)) :
a.dom := by tidy

@[to_additive]
lemma right_dom_of_div_dom [has_div α] {a b : part α} (hab : dom (a / b)) :
b.dom := by tidy

@[to_additive, simp]
lemma div_get_eq [has_div α] (a b : part α) (hab : dom (a / b)) :
(a / b).get hab = a.get (left_dom_of_div_dom hab) / b.get (right_dom_of_div_dom hab) :=
by tidy

@[to_additive]
lemma some_div_some [has_div α] (a b : α) : some a / some b = some (a / b) := by tidy

lemma mod_mem_mod [has_mod α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma % mb ∈ a % b := by tidy

lemma left_dom_of_mod_dom [has_mod α] {a b : part α} (hab : dom (a % b)) :
a.dom := by tidy

lemma right_dom_of_mod_dom [has_mod α] {a b : part α} (hab : dom (a % b)) :
b.dom := by tidy

@[simp]
lemma mod_get_eq [has_mod α] (a b : part α) (hab : dom (a % b)) :
(a % b).get hab = a.get (left_dom_of_mod_dom hab) % b.get (right_dom_of_mod_dom hab) :=
by tidy

lemma some_mod_some [has_mod α] (a b : α) : some a % some b = some (a % b) := by tidy

lemma append_mem_append [has_append α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma ++ mb ∈ a ++ b := by tidy

lemma left_dom_of_append_dom [has_append α] {a b : part α} (hab : dom (a ++ b)) :
a.dom := by tidy

lemma right_dom_of_append_dom [has_append α] {a b : part α} (hab : dom (a ++ b)) :
b.dom := by tidy

@[simp]
lemma append_get_eq [has_append α] (a b : part α) (hab : dom (a ++ b)) :
(a ++ b).get hab = a.get (left_dom_of_append_dom hab) ++ b.get (right_dom_of_append_dom hab) :=
by tidy

lemma some_append_some [has_append α] (a b : α) : some a ++ some b = some (a ++ b) := by tidy

lemma inter_mem_inter [has_inter α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma ∩ mb ∈ a ∩ b := by tidy

lemma left_dom_of_inter_dom [has_inter α] {a b : part α} (hab : dom (a ∩ b)) :
a.dom := by tidy

lemma right_dom_of_inter_dom [has_inter α] {a b : part α} (hab : dom (a ∩ b)) :
b.dom := by tidy

@[simp]
lemma inter_get_eq [has_inter α] (a b : part α) (hab : dom (a ∩ b)) :
(a ∩ b).get hab = a.get (left_dom_of_inter_dom hab) ∩ b.get (right_dom_of_inter_dom hab) :=
by tidy

lemma some_inter_some [has_inter α] (a b : α) : some a ∩ some b = some (a ∩ b) := by tidy

lemma union_mem_union [has_union α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma ∪ mb ∈ a ∪ b := by tidy

lemma left_dom_of_union_dom [has_union α] {a b : part α} (hab : dom (a ∪ b)) :
a.dom := by tidy

lemma right_dom_of_union_dom [has_union α] {a b : part α} (hab : dom (a ∪ b)) :
b.dom := by tidy

@[simp]
lemma union_get_eq [has_union α] (a b : part α) (hab : dom (a ∪ b)) :
(a ∪ b).get hab = a.get (left_dom_of_union_dom hab) ∪ b.get (right_dom_of_union_dom hab) :=
by tidy

lemma some_union_some [has_union α] (a b : α) : some a ∪ some b = some (a ∪ b) := by tidy

lemma sdiff_mem_sdiff [has_sdiff α] (a b : part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma \ mb ∈ a \ b := by tidy

lemma left_dom_of_sdiff_dom [has_sdiff α] {a b : part α} (hab : dom (a \ b)) :
a.dom := by tidy

lemma right_dom_of_sdiff_dom [has_sdiff α] {a b : part α} (hab : dom (a \ b)) :
b.dom := by tidy

@[simp]
lemma sdiff_get_eq [has_sdiff α] (a b : part α) (hab : dom (a \ b)) :
(a \ b).get hab = a.get (left_dom_of_sdiff_dom hab) \ b.get (right_dom_of_sdiff_dom hab) :=
by tidy

lemma some_sdiff_some [has_sdiff α] (a b : α) : some a \ some b = some (a \ b) := by tidy

end instances
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