url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact Set.InjOn.mono (fun x hx β¦ (mem_Ioo.1 (mem_filter.1 hx).1).2.le) $
fun x hx y hy β¦ tsub_inj_right hx hy | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
hfim : image (fun x => n - x) (filter (fun x => x β B) (Ioo 0 n)) = filter (fun x => x β {n} - B) (Ioo 0 n)
β’ Set.InjOn (fun x => n - x) β(filter (fun x => x β B) (Ioo 0 n)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β hca, β hcb] at hc | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rwa [β Finset.card_pos] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hin : 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
β’ (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β filter_or, β tsub_zero n, β Nat.card_Ioo] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun a => a β A β¨ a β {n - 0} - B) (Ioo 0 (n - 0))).card β€ (Ioo 0 n).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact card_filter_le _ _ | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
β’ (filter (fun a => a β A β¨ a β {n - 0} - B) (Ioo 0 (n - 0))).card β€ (Ioo 0 n).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [card_union_add_card_inter, β lem1, countelements] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1) | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n)).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card =
(filter (fun x => x β A) (Icc 1 (n - 1))).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | congr | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ (filter (fun x => x β A) (Ioo 0 n)).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card =
(filter (fun x => x β A) (Icc 1 (n - 1))).card + (filter (fun x => x β {n} - B) (Ioo 0 n)).card | case e_a.e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ n = ((n - 1).add 0).succ |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | exact (tsub_add_cancel_of_le $ Nat.succ_le_iff.2 hn1).symm | case e_a.e_s.e_s.e_b
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ n = ((n - 1).add 0).succ | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β hui] at hc | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A (n - 1) + countelements B (n - 1)
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | by_contra! hip | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
β’ 0 < (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | have hnn : n β€ (n - 1) := le_trans hip0 hip1 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : n β€ n - 1
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β not_lt] at hnn | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : n β€ n - 1
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ False |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | apply hnn | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ False | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ n - 1 < n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [propext (Nat.lt_iff_le_pred hn1)] | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc :
n β€
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
hun : (filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hui :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card =
countelements A (n - 1) + countelements B (n - 1)
hun1 :
(filter (fun x => x β A) (Ioo 0 n) βͺ filter (fun x => x β {n} - B) (Ioo 0 n)).card +
(filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€
n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip0 : n β€ n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card
hip : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ 0
hip1 : n - 1 + (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => x β {n} - B) (Ioo 0 n)).card β€ n - 1
hnn : Β¬n - 1 < n
β’ n - 1 < n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | rw [β filter_and, β coe_nonempty, coe_filter, Set.setOf_and, Set.setOf_and, Set.setOf_mem_eq,
Set.inter_comm] at lem3 | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (filter (fun x => x β A) (Ioo 0 n) β© filter (fun x => β x_1 β B, n - x_1 = x) (Ioo 0 n)).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (A β© {a | β x β B, n - x = a} β© {a | a β Ioo 0 n}).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sumset_contains_n | [41, 1] | [98, 56] | convert lem3 using 3 <;> ext <;> simp | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hc : n β€ countelements A n + countelements B n
h : n β A + B
hnA : n β A
hnB : n β B
hca : countelements A (n - 1) = countelements A n
hcb : countelements B (n - 1) = countelements B n
hn1 : n > 0
lem1 : (filter (fun x => x β {n} - B) (Ioo 0 n)).card = countelements B (n - 1)
lem3 : (A β© {a | β x β B, n - x = a} β© {a | a β Ioo 0 n}).Nonempty
β’ (A β© ({n} - B) β© Set.Ioo 0 n).Nonempty | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | refine Set.eq_univ_of_forall $ fun n β¦ sumset_contains_n hA hB ?_ | A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
β’ A + B = Set.univ | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
β’ n β€ countelements A n + countelements B n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | obtain rfl | hn := eq_or_ne n 0 | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
β’ n β€ countelements A n + countelements B n | case inl
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
β’ 0 β€ countelements A 0 + countelements B 0
case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ n β€ countelements A n + countelements B n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | rw [β Nat.cast_le (Ξ± := β), β one_le_div (by positivity)] | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ n β€ countelements A n + countelements B n | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ 1 β€ β(countelements A n + countelements B n) / βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | calc
_ β€ _ := hAB
_ β€ _ := add_le_add (schnirelmannDensity_le_div hn) (schnirelmannDensity_le_div hn)
_ = _ := by push_cast; rw [add_div]; rfl | case inr
A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ 1 β€ β(countelements A n + countelements B n) / βn | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | exact countelements_nonneg A 0 | case inl
A B : Set β
n : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
β’ 0 β€ countelements A 0 + countelements B 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | positivity | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ 0 < βn | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | push_cast | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ β(filter (fun x => x β A) (Ioc 0 n)).card / βn + β(filter (fun x => x β B) (Ioc 0 n)).card / βn =
β(countelements A n + countelements B n) / βn | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ β(filter (fun x => x β A) (Ioc 0 n)).card / βn + β(filter (fun x => x β B) (Ioc 0 n)).card / βn =
(β(countelements A n) + β(countelements B n)) / βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | rw [add_div] | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ β(filter (fun x => x β A) (Ioc 0 n)).card / βn + β(filter (fun x => x β B) (Ioc 0 n)).card / βn =
(β(countelements A n) + β(countelements B n)) / βn | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ β(filter (fun x => x β A) (Ioc 0 n)).card / βn + β(filter (fun x => x β B) (Ioc 0 n)).card / βn =
β(countelements A n) / βn + β(countelements B n) / βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | sum_schnirelmannDensity_ge_one_sumset_nat | [100, 1] | [109, 45] | rfl | A B : Set β
nβ : β
hA : 0 β A
hB : 0 β B
hAB : 1 β€ schnirelmannDensity A + schnirelmannDensity B
n : β
hn : n β 0
β’ β(filter (fun x => x β A) (Ioc 0 n)).card / βn + β(filter (fun x => x β B) (Ioc 0 n)).card / βn =
β(countelements A n) / βn + β(countelements B n) / βn | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | set Ξ± := schnirelmannDensity A with halpha | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
β’ schnirelmannDensity A + schnirelmannDensity B - schnirelmannDensity A * schnirelmannDensity B β€
schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
β’ Ξ± + schnirelmannDensity B - Ξ± * schnirelmannDensity B β€ schnirelmannDensity (A + B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | set Ξ² := schnirelmannDensity B with hbeta | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
β’ Ξ± + schnirelmannDensity B - Ξ± * schnirelmannDensity B β€ schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ schnirelmannDensity (A + B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ² := by ring | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ schnirelmannDensity (A + B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [β dum] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ Ξ± + Ξ² - Ξ± * Ξ² β€ schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ Ξ± * (1 - Ξ²) + Ξ² β€ schnirelmannDensity (A + B) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rintro n | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | obtain rfl | n1 := n.eq_zero_or_pos | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inl
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ (Ξ± * (1 - Ξ²) + Ξ²) * β0 β€ β(countelements (A + B) 0)
case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | ring | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
β’ Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ² | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [schnirelmannDensity] | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ Ξ± * (1 - Ξ²) + Ξ² β€ schnirelmannDensity (A + B) | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β¨
n, β(filter (fun x => x β A + B) (Ioc 0 βn)).card / ββn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have : Nonempty {n // n β 0} := by
use 1
trivial | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β¨
n, β(filter (fun x => x β A + B) (Ioc 0 βn)).card / ββn | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β¨
n, β(filter (fun x => x β A + B) (Ioc 0 βn)).card / ββn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | apply le_ciInf | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β¨
n, β(filter (fun x => x β A + B) (Ioc 0 βn)).card / ββn | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
β’ β (x : { n // 0 < n }), Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | intro x | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
β’ β (x : { n // 0 < n }), Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hx : (x : β) β 0 := by aesop | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [le_div_iff] | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ Ξ± * (1 - Ξ²) + Ξ² β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card / ββx | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card
case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ 0 < ββx |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | use 1 | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ Nonempty { n // n β 0 } | case property
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ 1 β 0 |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | trivial | case property
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
β’ 1 β 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | aesop | Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
β’ ββx β 0 | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | specialize main x | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
main : (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(countelements (A + B) βx)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact main | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
main : (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(countelements (A + B) βx)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * ββx β€ β(filter (fun x => x β A + B) (Ioc 0 βx)).card | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | positivity | case H
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
main : β (n : β), (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n)
this : Nonempty { n // n β 0 }
x : { n // 0 < n }
hx : ββx β 0
β’ 0 < ββx | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | ring_nf | case inl
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ (Ξ± * (1 - Ξ²) + Ξ²) * β0 β€ β(countelements (A + B) 0) | case inl
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ 0 β€ β(countelements (A + B) 0) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | positivity | case inl
Aβ Bβ : Set β
n : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
β’ 0 β€ β(countelements (A + B) 0) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have claim : countelements A n + Ξ² * (n - countelements A n) β€
countelements (β a : A, {c β A + B | 0 < c - a β§ (c : β) β€ (next_elm A a n)}) n := by
have hcc (a : A) :
1 + countelements B (next_elm A a n - a - 1) β€
countelements {c β A + B | 0 < c - a β§ (c : β) β€ (next_elm A a n)} n := by
sorry
have hax (a x : A) (hh : a β x) :
{c β A + B | 0 < c - a β§ (c : β) β€ (next_elm A a n)} β©
{c β A + B | 0 < c - x β§ (c : β) β€ next_elm A x n} = β
:= by sorry
sorry | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have ht : countelements A n + Ξ² * (n - countelements A n) β€ countelements (A + B) n := by
apply le_trans claim _
norm_cast | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hc1 : countelements A n * (1 - Ξ²) + Ξ² * n =
countelements A n + Ξ² * (n - countelements A n) := by ring_nf | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hc3 : Ξ± * n * (1 - Ξ²) + Ξ² * n = (Ξ± * (1 - Ξ²) + Ξ²) * n := by ring_nf | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hc1] at hc2 | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hc3] at hc2 | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact le_trans hc2 ht | case inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc2 : (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements A n) + Ξ² * (βn - β(countelements A n))
hc3 : Ξ± * βn * (1 - Ξ²) + Ξ² * βn = (Ξ± * (1 - Ξ²) + Ξ²) * βn
β’ (Ξ± * (1 - Ξ²) + Ξ²) * βn β€ β(countelements (A + B) n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [tsub_pos_iff_lt, Set.sep_and, Set.iUnion_coe_set, Nat.lt_one_iff, coe_Icc, not_le,
Set.subset_inter_iff, Set.iUnion_subset_iff] | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n) | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ (β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β A + B) β§
β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β Set.Icc 1 n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | constructor | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ (β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β A + B) β§
β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β Set.Icc 1 n | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β A + B
case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β Set.Icc 1 n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | intro i hi x hx | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β Set.Icc 1 n | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β Set.Icc 1 n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [Set.mem_inter_iff, Set.mem_setOf_eq] at hx | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β Set.Icc 1 n | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β Set.Icc 1 n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [Set.mem_Icc] | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β Set.Icc 1 n | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ 1 β€ x β§ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | constructor | case right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ 1 β€ x β§ x β€ n | case right.left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ 1 β€ x
case right.right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | obtain β¨hx1, hx2, hx3β© := hx | case right.right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β€ n | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 : x β€ next_elm A β¨i, β―β© n
β’ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [next_elm] at hx3 | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 : x β€ next_elm A β¨i, β―β© n
β’ x β€ n | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 :
x β€
if h : (filter (fun x => x β A) (Ioc (ββ¨i, β―β©) n)).Nonempty then (filter (fun x => x β A) (Ioc (ββ¨i, β―β©) n)).min' h
else n
β’ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [mem_Ioc, and_imp, ne_eq, ite_not] at hx3 | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 :
x β€
if h : (filter (fun x => x β A) (Ioc (ββ¨i, β―β©) n)).Nonempty then (filter (fun x => x β A) (Ioc (ββ¨i, β―β©) n)).min' h
else n
β’ x β€ n | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 : x β€ if h : (filter (fun x => x β A) (Ioc i n)).Nonempty then (filter (fun x => x β A) (Ioc i n)).min' β― else n
β’ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | split_ifs at hx3 with h | case right.right.intro.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
hx3 : x β€ if h : (filter (fun x => x β A) (Ioc i n)).Nonempty then (filter (fun x => x β A) (Ioc i n)).min' β― else n
β’ x β€ n | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
h : (filter (fun x => x β A) (Ioc i n)).Nonempty
hx3 : x β€ (filter (fun x => x β A) (Ioc i n)).min' β―
β’ x β€ n
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
h : Β¬(filter (fun x => x β A) (Ioc i n)).Nonempty
hx3 : x β€ n
β’ x β€ n |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | intro i hi x hx | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
β’ β (i : β) (i_1 : i β A), {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n} β A + B | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [Set.mem_inter_iff] at hx | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β© {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β A + B | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β§ x β {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [Set.mem_setOf_eq] at hx | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : x β {x | x β A + B β§ i < x} β§ x β {x | x β A + B β§ x β€ next_elm A β¨i, β―β© n}
β’ x β A + B | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact hx.1.1 | case left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ x β A + B | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rcases i.eq_zero_or_pos with i0 | i1 | case right.left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
β’ 1 β€ x | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ 1 β€ x
case right.left.inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i1 : i > 0
β’ 1 β€ x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [Nat.succ_le] | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ 1 β€ x | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ 0 < x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [β i0] | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ 0 < x | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ i < x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact hx.1.2 | case right.left.inl
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i0 : i = 0
β’ i < x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [Nat.succ_le] | case right.left.inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i1 : i > 0
β’ 1 β€ x | case right.left.inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i1 : i > 0
β’ 0 < x |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | apply lt_trans i1 hx.1.2 | case right.left.inr
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx : (x β A + B β§ i < x) β§ x β A + B β§ x β€ next_elm A β¨i, β―β© n
i1 : i > 0
β’ 0 < x | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact hx3.trans (mem_Ioc.1 (mem_filter.1 $ min'_mem _ _).1).2 | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
h : (filter (fun x => x β A) (Ioc i n)).Nonempty
hx3 : x β€ (filter (fun x => x β A) (Ioc i n)).min' β―
β’ x β€ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simpa using hx3 | case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
i : β
hi : i β A
x : β
hx1 : x β A + B β§ i < x
hx2 : x β A + B
h : Β¬(filter (fun x => x β A) (Ioc i n)).Nonempty
hx3 : x β€ n
β’ x β€ n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [countelements, countelements] | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
β’ countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
β’ (filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n)).card β€
(filter (fun x => x β A + B) (Icc 1 n)).card |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | apply card_le_card | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
β’ (filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n)).card β€
(filter (fun x => x β A + B) (Icc 1 n)).card | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
β’ filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n) β
filter (fun x => x β A + B) (Icc 1 n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | intro y | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
β’ filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n) β
filter (fun x => x β A + B) (Icc 1 n) | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n) β
y β filter (fun x => x β A + B) (Icc 1 n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | repeat rw [mem_filter] | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β filter (fun x => x β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) (Icc 1 n) β
y β filter (fun x => x β A + B) (Icc 1 n) | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β y β Icc 1 n β§ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | intro hy | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β y β Icc 1 n β§ y β A + B | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β Icc 1 n β§ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | constructor | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β Icc 1 n β§ y β A + B | case a.left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β Icc 1 n
case a.right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [mem_filter] | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β y β filter (fun x => x β A + B) (Icc 1 n) | case a
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
β’ y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β y β Icc 1 n β§ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact hy.1 | case a.left
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β Icc 1 n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | obtain β¨hy1, hy2β© := hy | case a.right
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy : y β Icc 1 n β§ y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β A + B | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hs : y β (A + B) β© (Icc 1 n) := by aesop | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β A + B | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
hs : y β (A + B) β© β(Icc 1 n)
β’ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [Set.mem_inter_iff] at hs | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
hs : y β (A + B) β© β(Icc 1 n)
β’ y β A + B | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
hs : y β A + B β§ y β β(Icc 1 n)
β’ y β A + B |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | exact hs.1 | case a.right.intro
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
hs : y β A + B β§ y β β(Icc 1 n)
β’ y β A + B | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | aesop | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
y : β
hy1 : y β Icc 1 n
hy2 : y β β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}
β’ y β (A + B) β© β(Icc 1 n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hcc (a : A) :
1 + countelements B (next_elm A a n - a - 1) β€
countelements {c β A + B | 0 < c - a β§ (c : β) β€ (next_elm A a n)} n := by
sorry | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
hcc :
β (a : βA),
1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | have hax (a x : A) (hh : a β x) :
{c β A + B | 0 < c - a β§ (c : β) β€ (next_elm A a n)} β©
{c β A + B | 0 < c - x β§ (c : β) β€ next_elm A x n} = β
:= by sorry | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
hcc :
β (a : βA),
1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
hcc :
β (a : βA),
1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n
hax :
β (a x : βA),
a β x β {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β© {c | c β A + B β§ 0 < c - βx β§ c β€ next_elm A x n} = β
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | sorry | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
hcc :
β (a : βA),
1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n
hax :
β (a x : βA),
a β x β {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β© {c | c β A + B β§ 0 < c - βx β§ c β€ next_elm A x n} = β
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | sorry | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
a : βA
β’ 1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | sorry | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
hcc :
β (a : βA),
1 + countelements B (next_elm A a n - βa - 1) β€ countelements {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} n
a x : βA
hh : a β x
β’ {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β© {c | c β A + B β§ 0 < c - βx β§ c β€ next_elm A x n} = β
| no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | apply le_trans claim _ | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
β’ β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n) | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
β’ β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) β€ β(countelements (A + B) n) |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | norm_cast | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
β’ β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n) β€ β(countelements (A + B) n) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | ring_nf | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
β’ β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n)) | no goals |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [halpha] | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
β’ Ξ± * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | by_cases hbo : Ξ² = 1 | Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Ξ² = 1
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn
case neg
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : ¬β = 1
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | rw [hbo] | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Ξ² = 1
β’ schnirelmannDensity A * βn * (1 - Ξ²) + Ξ² * βn β€ β(countelements A n) * (1 - Ξ²) + Ξ² * βn | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Ξ² = 1
β’ schnirelmannDensity A * βn * (1 - 1) + 1 * βn β€ β(countelements A n) * (1 - 1) + 1 * βn |
https://github.com/YaelDillies/LeanCamCombi.git | 034199694e3b91536d03bc4a8b0cdbd659cdf50f | LeanCamCombi/Mathlib/Combinatorics/Schnirelmann.lean | le_schnirelmannDensity_add | [115, 1] | [216, 26] | simp only [sub_self, mul_zero, one_mul, zero_add, le_refl] | case pos
Aβ Bβ : Set β
nβ : β
A B : Set β
hA : 0 β A
hB : 0 β B
Ξ± : β := schnirelmannDensity A
halpha : Ξ± = schnirelmannDensity A
Ξ² : β := schnirelmannDensity B
hbeta : Ξ² = schnirelmannDensity B
dum : Ξ± * (1 - Ξ²) + Ξ² = Ξ± + Ξ² - Ξ± * Ξ²
n : β
n1 : n > 0
lem : β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n} β (A + B) β© β(Icc 1 n)
aux : countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n β€ countelements (A + B) n
claim :
β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€
β(countelements (β a, {c | c β A + B β§ 0 < c - βa β§ c β€ next_elm A a n}) n)
ht : β(countelements A n) + Ξ² * (βn - β(countelements A n)) β€ β(countelements (A + B) n)
hc1 : β(countelements A n) * (1 - Ξ²) + Ξ² * βn = β(countelements A n) + Ξ² * (βn - β(countelements A n))
hbo : Ξ² = 1
β’ schnirelmannDensity A * βn * (1 - 1) + 1 * βn β€ β(countelements A n) * (1 - 1) + 1 * βn | no goals |
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