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stringclasses 147
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stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
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stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
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|---|---|---|---|---|---|---|---|---|
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | have real_set := pw.1 | case val
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
x : ℝ
pw : ↑Set.univ
⊢ binary | case val
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
x : ℝ
pw : ↑Set.univ
real_set : Set ℝ
⊢ binary |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | haveI := Classical.dec (x ∈ real_set) | case val
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
x : ℝ
pw : ↑Set.univ
real_set : Set ℝ
⊢ binary | case val
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
x : ℝ
pw : ↑Set.univ
real_set : Set ℝ
this : Decidable (x ∈ real_set)
⊢ binary |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | exact (ite (x ∈ real_set) binary.one binary.zero) | case val
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
x : ℝ
pw : ↑Set.univ
real_set : Set ℝ
this : Decidable (x ∈ real_set)
⊢ binary | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | intro hite | case h.a.mp
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
⊢ (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one → x ∈ ↑pw | case h.a.mp
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hite : (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one
⊢ x ∈ ↑pw |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | split_ifs at hite with hi | case h.a.mp
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hite : (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one
⊢ x ∈ ↑pw | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hi : x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
hite : binary.one = binary.one
⊢ x ∈ ↑pw |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | simp only [Set.powerset_univ, set_coe_cast] at hi | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hi : x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
hite : binary.one = binary.one
⊢ x ∈ ↑pw | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hite : binary.one = binary.one
hi : x ∈ ↑pw
⊢ x ∈ ↑pw |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | exact hi | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
hite : binary.one = binary.one
hi : x ∈ ↑pw
⊢ x ∈ ↑pw | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | intro el | case h.a.mpr
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
⊢ x ∈ ↑pw → (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one | case h.a.mpr
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
⊢ (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | split_ifs with hi | case h.a.mpr
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
⊢ (if x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw) then binary.one else binary.zero) = binary.one | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
⊢ binary.one = binary.one
case neg
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∉ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | rfl | case pos
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∈ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
⊢ binary.one = binary.one | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | simp only [Set.powerset_univ, set_coe_cast] at hi | case neg
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∉ ↑(cast (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw)
⊢ False | case neg
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∉ ↑pw
⊢ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | indicator_card_eq_powerset_card_bij | [123, 1] | [178, 20] | exact hi el | case neg
f : ↑indicator' → ↑(𝒫 Set.univ) :=
Eq.mpr (_ : (↑indicator' → ↑(𝒫 Set.univ)) = (↑Set.univ → ↑(𝒫 Set.univ))) fun a =>
Subtype.casesOn a fun fn property =>
{ val := fn ⁻¹' {binary.one}, property := (_ : fn ⁻¹' {binary.one} ∈ 𝒫 Set.univ) }
pw : ↑(𝒫 Set.univ)
a : ↑indicator' :=
Eq.mpr (_ : ↑indicator' = ↑Set.univ)
{
val := fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero,
property :=
(_ :
(fun x =>
let_fun real_set := ↑(Eq.mp (_ : ↑(𝒫 Set.univ) = ↑Set.univ) pw);
if x ∈ real_set then binary.one else binary.zero) ∈
Set.univ) }
x : ℝ
el : x ∈ ↑pw
hi : x ∉ ↑pw
⊢ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat | [181, 1] | [187, 40] | simp only [Cardinal.mk_powerset,
Cardinal.mk_univ,
Cardinal.mk_eq_aleph0,
Cardinal.two_power_aleph0] | ⊢ Cardinal.mk ↑(𝒫 Set.univ) = Cardinal.mk ↑(𝒫 Set.univ) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | apply_fun Nat.factorization at h | m n q p : ℕ
h : 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q
⊢ m = p ∧ n = q | m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ m = p ∧ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | rw [Nat.factorization_mul, Nat.factorization_mul] at h | m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ m = p ∧ n = q | m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p) + Nat.factorization (3 ^ q)
⊢ m = p ∧ n = q
case ha
m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 2 ^ p ≠ 0
case hb
m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 3 ^ q ≠ 0
case ha
m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 2 ^ m ≠ 0
case hb
m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 3 ^ n ≠ 0 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | all_goals positivity | case ha
m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 2 ^ p ≠ 0
case hb
m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 3 ^ q ≠ 0
case ha
m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 2 ^ m ≠ 0
case hb
m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 3 ^ n ≠ 0 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | simp_rw [Nat.factorization_pow] at h | m n q p : ℕ
h : Nat.factorization (2 ^ m) + Nat.factorization (3 ^ n) = Nat.factorization (2 ^ p) + Nat.factorization (3 ^ q)
⊢ m = p ∧ n = q | m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ m = p ∧ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | constructor | m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ m = p ∧ n = q | case left
m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ m = p
case right
m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | replace h := FunLike.congr_fun h 2 | case left
m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ m = p | case left
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 2 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 2
⊢ m = p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | have : ¬ 2 ∣ 3 := by norm_num | case left
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 2 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 2
⊢ m = p | case left
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 2 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 2
this : ¬2 ∣ 3
⊢ m = p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | simp_rw [Finsupp.add_apply, Finsupp.smul_apply, Nat.prime_two.factorization_self,
nsmul_one, Nat.factorization_eq_zero_of_not_dvd this, smul_zero, add_zero] at h | case left
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 2 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 2
this : ¬2 ∣ 3
⊢ m = p | case left
m n q p : ℕ
this : ¬2 ∣ 3
h : ↑m = ↑p
⊢ m = p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | exact h | case left
m n q p : ℕ
this : ¬2 ∣ 3
h : ↑m = ↑p
⊢ m = p | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | norm_num | m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 2 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 2
⊢ ¬2 ∣ 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | replace h := FunLike.congr_fun h 3 | case right
m n q p : ℕ
h : m • Nat.factorization 2 + n • Nat.factorization 3 = p • Nat.factorization 2 + q • Nat.factorization 3
⊢ n = q | case right
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
⊢ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | have : ¬ 3 ∣ 2 := Nat.not_dvd_of_pos_of_lt (by simp) (by simp) | case right
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
⊢ n = q | case right
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
this : ¬3 ∣ 2
⊢ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | simp_rw [Finsupp.add_apply, Finsupp.smul_apply, Nat.prime_three.factorization_self,
nsmul_one, Nat.factorization_eq_zero_of_not_dvd this, smul_zero, zero_add] at h | case right
m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
this : ¬3 ∣ 2
⊢ n = q | case right
m n q p : ℕ
this : ¬3 ∣ 2
h : ↑n = ↑q
⊢ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | exact h | case right
m n q p : ℕ
this : ¬3 ∣ 2
h : ↑n = ↑q
⊢ n = q | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | simp | m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
⊢ 0 < 2 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | simp | m n q p : ℕ
h : ↑(m • Nat.factorization 2 + n • Nat.factorization 3) 3 = ↑(p • Nat.factorization 2 + q • Nat.factorization 3) 3
⊢ 2 < 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | two_pow_three_pow_unique_factorization | [189, 1] | [205, 23] | positivity | case hb
m n q p : ℕ
h : Nat.factorization (2 ^ m * 3 ^ n) = Nat.factorization (2 ^ p * 3 ^ q)
⊢ 3 ^ n ≠ 0 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | let f : (𝒫 (Set.univ : Set (ℕ × ℕ))) → (𝒫 (Set.univ : Set ℕ)) := by
intro a_set_of_nxn
exact {
val := {2^x.1 * 3^x.2 | x ∈ a_set_of_nxn.val}
property := by simp only [Set.powerset_univ, Set.mem_univ]
} | ⊢ ∃ fg, Function.Bijective fg | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
⊢ ∃ fg, Function.Bijective fg |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | let g : (𝒫 (Set.univ : Set ℕ)) → (𝒫 (Set.univ : Set (ℕ × ℕ))) := by
intro a_set_of_n
have a_set_of_nxn : Set (ℕ × ℕ) := a_set_of_n.val ×ˢ a_set_of_n.val
exact { val := a_set_of_nxn, property := by simp } | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
⊢ ∃ fg, Function.Bijective fg | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
⊢ ∃ fg, Function.Bijective fg |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact Function.Embedding.schroeder_bernstein hf hg | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
hg : Function.Injective g
⊢ ∃ fg, Function.Bijective fg | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intro a_set_of_nxn | ⊢ ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) | a_set_of_nxn : ↑(𝒫 Set.univ)
⊢ ↑(𝒫 Set.univ) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact {
val := {2^x.1 * 3^x.2 | x ∈ a_set_of_nxn.val}
property := by simp only [Set.powerset_univ, Set.mem_univ]
} | a_set_of_nxn : ↑(𝒫 Set.univ)
⊢ ↑(𝒫 Set.univ) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [Set.powerset_univ, Set.mem_univ] | a_set_of_nxn : ↑(𝒫 Set.univ)
⊢ {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rintro ⟨p, _⟩ ⟨q, _⟩ heq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
⊢ Function.Injective f | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : f { val := p, property := property✝¹ } = f { val := q, property := property✝ }
⊢ { val := p, property := property✝¹ } = { val := q, property := property✝ } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [f, Prod.exists, Subtype.mk.injEq, Set.ext_iff, Set.mem_setOf_eq] at heq | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : f { val := p, property := property✝¹ } = f { val := q, property := property✝ }
⊢ { val := p, property := property✝¹ } = { val := q, property := property✝ } | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
⊢ { val := p, property := property✝¹ } = { val := q, property := property✝ } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rw [Subtype.mk.injEq, Set.ext_iff] | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
⊢ { val := p, property := property✝¹ } = { val := q, property := property✝ } | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
⊢ ∀ (x : ℕ × ℕ), x ∈ p ↔ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intro x | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
⊢ ∀ (x : ℕ × ℕ), x ∈ p ↔ x ∈ q | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
x : ℕ × ℕ
⊢ x ∈ p ↔ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | let uniq := 2^x.1 * 3^x.2 | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
x : ℕ × ℕ
⊢ x ∈ p ↔ x ∈ q | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ p ↔ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | specialize heq uniq | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
heq : ∀ (x : ℕ), (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = x) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = x
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ p ↔ x ∈ q | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
heq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p ↔ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rcases heq with ⟨pimpq, qimpp⟩ | case mk.mk
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
heq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) ↔ ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p ↔ x ∈ q | case mk.mk.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p ↔ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | constructor | case mk.mk.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p ↔ x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p → x ∈ q
case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ q → x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intro hmemp | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p → x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | have ⟨a, ⟨b, ⟨hmemq, heqfac⟩⟩⟩
:= pimpq ⟨x.1, ⟨x.2, by simp only [Prod.mk.eta,and_true]; exact hmemp⟩⟩ | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
⊢ x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = uniq
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp_rw [uniq] at heqfac | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = uniq
⊢ x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | have ⟨heq1, heq2⟩ := two_pow_three_pow_unique_factorization heqfac | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rw [heq1, heq2] at hmemq | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (a, b) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ q | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (x.1, x.2) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact hmemq | case mk.mk.intro.mp
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
a b : ℕ
hmemq : (x.1, x.2) ∈ q
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ q | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [Prod.mk.eta,and_true] | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
⊢ (x.1, x.2) ∈ p ∧ 2 ^ x.1 * 3 ^ x.2 = uniq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact hmemp | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemp : x ∈ p
⊢ x ∈ p | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intro hmemq | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
⊢ x ∈ q → x ∈ p | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | have ⟨a, ⟨b, ⟨hmemp, heqfac⟩⟩⟩
:= qimpp ⟨x.1, ⟨x.2, by simp only [Prod.mk.eta,and_true]; exact hmemq⟩⟩ | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
⊢ x ∈ p | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp_rw [uniq] at heqfac | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = uniq
⊢ x ∈ p | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | have ⟨heq1, heq2⟩ := two_pow_three_pow_unique_factorization heqfac | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
⊢ x ∈ p | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rw [heq1, heq2] at hmemp | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (a, b) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ p | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (x.1, x.2) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ p |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact hmemp | case mk.mk.intro.mpr
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
a b : ℕ
hmemp : (x.1, x.2) ∈ p
heqfac : 2 ^ a * 3 ^ b = 2 ^ x.1 * 3 ^ x.2
heq1 : a = x.1
heq2 : b = x.2
⊢ x ∈ p | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [Prod.mk.eta,and_true] | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
⊢ (x.1, x.2) ∈ q ∧ 2 ^ x.1 * 3 ^ x.2 = uniq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
⊢ x ∈ q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact hmemq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
p : Set (ℕ × ℕ)
property✝¹ : p ∈ 𝒫 Set.univ
q : Set (ℕ × ℕ)
property✝ : q ∈ 𝒫 Set.univ
x : ℕ × ℕ
uniq : ℕ := 2 ^ x.1 * 3 ^ x.2
pimpq : (∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq
qimpp : (∃ a b, (a, b) ∈ q ∧ 2 ^ a * 3 ^ b = uniq) → ∃ a b, (a, b) ∈ p ∧ 2 ^ a * 3 ^ b = uniq
hmemq : x ∈ q
⊢ x ∈ q | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intro a_set_of_n | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
⊢ ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
a_set_of_n : ↑(𝒫 Set.univ)
⊢ ↑(𝒫 Set.univ) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | have a_set_of_nxn : Set (ℕ × ℕ) := a_set_of_n.val ×ˢ a_set_of_n.val | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
a_set_of_n : ↑(𝒫 Set.univ)
⊢ ↑(𝒫 Set.univ) | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
a_set_of_n : ↑(𝒫 Set.univ)
a_set_of_nxn : Set (ℕ × ℕ)
⊢ ↑(𝒫 Set.univ) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact { val := a_set_of_nxn, property := by simp } | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
a_set_of_n : ↑(𝒫 Set.univ)
a_set_of_nxn : Set (ℕ × ℕ)
⊢ ↑(𝒫 Set.univ) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
a_set_of_n : ↑(𝒫 Set.univ)
a_set_of_nxn : Set (ℕ × ℕ)
⊢ a_set_of_nxn ∈ 𝒫 Set.univ | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | intros a b heq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
⊢ Function.Injective g | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
heq : g a = g b
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [g, Subtype.mk.injEq, Set.prod_eq_prod_iff] at heq | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
heq : g a = g b
⊢ a = b | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
heq : ↑a = ↑b ∧ ↑a = ↑b ∨ (↑a = ∅ ∨ ↑a = ∅) ∧ (↑b = ∅ ∨ ↑b = ∅)
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rcases heq with (coe_eq | ⟨ha, hb⟩) | f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
heq : ↑a = ↑b ∧ ↑a = ↑b ∨ (↑a = ∅ ∨ ↑a = ∅) ∧ (↑b = ∅ ∨ ↑b = ∅)
⊢ a = b | case inl
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
coe_eq : ↑a = ↑b ∧ ↑a = ↑b
⊢ a = b
case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ∅ ∨ ↑a = ∅
hb : ↑b = ∅ ∨ ↑b = ∅
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact SetCoe.ext coe_eq.1 | case inl
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
coe_eq : ↑a = ↑b ∧ ↑a = ↑b
⊢ a = b | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [or_self] at ha | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ∅ ∨ ↑a = ∅
hb : ↑b = ∅ ∨ ↑b = ∅
⊢ a = b | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
hb : ↑b = ∅ ∨ ↑b = ∅
ha : ↑a = ∅
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | simp only [or_self] at hb | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
hb : ↑b = ∅ ∨ ↑b = ∅
ha : ↑a = ∅
⊢ a = b | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ∅
hb : ↑b = ∅
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | rw [←hb] at ha | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ∅
hb : ↑b = ∅
⊢ a = b | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ↑b
hb : ↑b = ∅
⊢ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | power_nat_nat_card_eq_power_nat_csb | [207, 1] | [254, 53] | exact SetCoe.ext ha | case inr.intro
f : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_nxn =>
{ val := {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x},
property := (_ : {x | ∃ x_1 ∈ ↑a_set_of_nxn, 2 ^ x_1.1 * 3 ^ x_1.2 = x} ∈ 𝒫 Set.univ) }
hf : Function.Injective f
g : ↑(𝒫 Set.univ) → ↑(𝒫 Set.univ) :=
fun a_set_of_n =>
let_fun a_set_of_nxn := ↑a_set_of_n ×ˢ ↑a_set_of_n;
{ val := a_set_of_nxn, property := (_ : a_set_of_nxn ∈ 𝒫 Set.univ) }
a b : ↑(𝒫 Set.univ)
ha : ↑a = ↑b
hb : ↑b = ∅
⊢ a = b | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | intro hne | n : ↑N0
⊢ n ≠ z → ∃ m, n = S m | n : ↑N0
hne : n ≠ z
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | let A := {n : N0 | n ≠ z → ∃ m : N0, n = S m} | n : ↑N0
hne : n ≠ z
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | have hzmem : z ∈ A := by
simp only [ne_eq, Subtype.exists, Set.mem_setOf_eq, not_true_eq_false, IsEmpty.forall_iff] | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | have hind : (∀ n : N0, n ∈ A → (S n) ∈ A) := by
intros n _
simp only [ne_eq, Subtype.exists, Set.mem_setOf_eq]
intro _
use n
simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const] | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | have heq := p3 A ⟨hzmem, hind⟩ | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | simp [A, Set.ext_iff] at heq | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
∀ (x : α),
(∃ (x_1 : x ∈ N0),
¬{ val := x, property := (_ : x ∈ N0) } = z →
∃ a, ∃ (b : a ∈ N0), { val := x, property := (_ : x ∈ N0) } = S { val := a, property := b }) ↔
x ∈ N0
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | specialize heq n | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
∀ (x : α),
(∃ (x_1 : x ∈ N0),
¬{ val := x, property := (_ : x ∈ N0) } = z →
∃ a, ∃ (b : a ∈ N0), { val := x, property := (_ : x ∈ N0) } = S { val := a, property := b }) ↔
x ∈ N0
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
(∃ (x : ↑n ∈ N0),
¬{ val := ↑n, property := (_ : ↑n ∈ N0) } = z →
∃ a, ∃ (b : a ∈ N0), { val := ↑n, property := (_ : ↑n ∈ N0) } = S { val := a, property := b }) ↔
↑n ∈ N0
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
(∃ (x : ↑n ∈ N0),
¬{ val := ↑n, property := (_ : ↑n ∈ N0) } = z →
∃ a, ∃ (b : a ∈ N0), { val := ↑n, property := (_ : ↑n ∈ N0) } = S { val := a, property := b }) ↔
↑n ∈ N0
⊢ ∃ m, n = S m | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : ¬n = z → ∃ a, ∃ (h : a ∈ N0), n = S { val := a, property := (_ : a ∈ N0) }
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | rcases heq hne with ⟨a, ⟨h, heq⟩⟩ | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : ¬n = z → ∃ a, ∃ (h : a ∈ N0), n = S { val := a, property := (_ : a ∈ N0) }
⊢ ∃ m, n = S m | case intro.intro
n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq✝ : ¬n = z → ∃ a, ∃ (h : a ∈ N0), n = S { val := a, property := (_ : a ∈ N0) }
a : α
h : a ∈ N0
heq : n = S { val := a, property := (_ : a ∈ N0) }
⊢ ∃ m, n = S m |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | use { val := a, property := h } | case intro.intro
n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq✝ : ¬n = z → ∃ a, ∃ (h : a ∈ N0), n = S { val := a, property := (_ : a ∈ N0) }
a : α
h : a ∈ N0
heq : n = S { val := a, property := (_ : a ∈ N0) }
⊢ ∃ m, n = S m | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | simp only [ne_eq, Subtype.exists, Set.mem_setOf_eq, not_true_eq_false, IsEmpty.forall_iff] | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
⊢ z ∈ A | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | intros n _ | n : ↑N0
hne : n ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
⊢ ∀ n ∈ A, S n ∈ A | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝ : n ∈ A
⊢ S n ∈ A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | simp only [ne_eq, Subtype.exists, Set.mem_setOf_eq] | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝ : n ∈ A
⊢ S n ∈ A | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝ : n ∈ A
⊢ ¬S n = z → ∃ a, ∃ (b : a ∈ N0), S n = S { val := a, property := b } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | intro _ | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝ : n ∈ A
⊢ ¬S n = z → ∃ a, ∃ (b : a ∈ N0), S n = S { val := a, property := b } | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝¹ : n ∈ A
a✝ : ¬S n = z
⊢ ∃ a, ∃ (b : a ∈ N0), S n = S { val := a, property := b } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | use n | n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝¹ : n ∈ A
a✝ : ¬S n = z
⊢ ∃ a, ∃ (b : a ∈ N0), S n = S { val := a, property := b } | case h
n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝¹ : n ∈ A
a✝ : ¬S n = z
⊢ ∃ (b : ↑n ∈ N0), S n = S { val := ↑n, property := b } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | every_nonzero_nat_successor | [270, 1] | [289, 34] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const] | case h
n✝ : ↑N0
hne : n✝ ≠ z
A : Set ↑N0 := {n | n ≠ z → ∃ m, n = S m}
hzmem : z ∈ A
n : ↑N0
a✝¹ : n ∈ A
a✝ : ¬S n = z
⊢ ∃ (b : ↑n ∈ N0), S n = S { val := ↑n, property := b } | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | let A := {b : N0 | plus (z, b) = b} | x : ↑N0
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | have hzmem : z ∈ A := by
simp only [Set.mem_setOf_eq]
exact zplus z | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | have hind : (∀ n : N0, n ∈ A → (S n) ∈ A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
rw [splus, hel] | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | have heq := p3 A ⟨hzmem, hind⟩ | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | simp [A, Set.ext_iff] at heq | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : Subtype.val '' A = N0
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
∀ (x : α),
(∃ (x_1 : x ∈ N0), plus (z, { val := x, property := (_ : x ∈ N0) }) = { val := x, property := (_ : x ∈ N0) }) ↔
x ∈ N0
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | specialize heq x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
∀ (x : α),
(∃ (x_1 : x ∈ N0), plus (z, { val := x, property := (_ : x ∈ N0) }) = { val := x, property := (_ : x ∈ N0) }) ↔
x ∈ N0
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
(∃ (x_1 : ↑x ∈ N0), plus (z, { val := ↑x, property := (_ : ↑x ∈ N0) }) = { val := ↑x, property := (_ : ↑x ∈ N0) }) ↔
↑x ∈ N0
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq :
(∃ (x_1 : ↑x ∈ N0), plus (z, { val := ↑x, property := (_ : ↑x ∈ N0) }) = { val := ↑x, property := (_ : ↑x ∈ N0) }) ↔
↑x ∈ N0
⊢ plus (z, x) = x | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : plus (z, x) = x
⊢ plus (z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | exact heq | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
hind : ∀ n ∈ A, S n ∈ A
heq : plus (z, x) = x
⊢ plus (z, x) = x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | simp only [Set.mem_setOf_eq] | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
⊢ z ∈ A | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
⊢ plus (z, z) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | exact zplus z | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
⊢ plus (z, z) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | intros n hel | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
⊢ ∀ n ∈ A, S n ∈ A | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
n : ↑N0
hel : n ∈ A
⊢ S n ∈ A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | simp only [Set.mem_setOf_eq] | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
n : ↑N0
hel : n ∈ A
⊢ S n ∈ A | x : ↑N0
A : Set ↑N0 := {b | plus (z, b) = b}
hzmem : z ∈ A
n : ↑N0
hel : n ∈ A
⊢ plus (z, S n) = S n |
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