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 | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | cases i | case nil
α✝ : Type u_1
i : (i : [].Index) × i.val.Index
⊢ (join i).unjoin = i | case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).unjoin = ⟨fst✝, snd✝⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α✝ : Type u_1
i : (i : [].Index) × i.val.Index
⊢ (join i).unjoin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | contradiction | case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).unjoin = ⟨fst✝, snd✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).unjoin = ⟨fst✝, snd✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | match i with
| ⟨head, _⟩ => simp only [join, unjoin, unappend_append]
| ⟨tail _, _⟩ => simp only [join, unjoin, unappend_append]; rw [ih] | case cons
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
⊢ (join i).unjoin = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
⊢ (join i).unjoin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | simp only [join, unjoin, unappend_append] | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
snd✝ : head.val.Index
⊢ (join ⟨head, snd✝⟩).unjoin = ⟨head, snd✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
snd✝ : head.val.Index
⊢ (join ⟨head, snd✝⟩).unjoin = ⟨head, snd✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | simp only [join, unjoin, unappend_append] | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
a✝ : xss.Index
snd✝ : a✝.tail.val.Index
⊢ (join ⟨a✝.tail, snd✝⟩).unjoin = ⟨a✝.tail, snd✝⟩ | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
a✝ : xss.Index
snd✝ : a✝.tail.val.Index
⊢ ⟨(join ⟨a✝, snd✝⟩).unjoin.fst.tail, (join ⟨a✝, snd✝⟩).unjoin.snd⟩ = ⟨a✝.tail, snd✝⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
a✝ : xss.Index
snd✝ : a✝.tail.val.Index
⊢ (join ⟨a✝.tail, snd✝⟩).unjoin = ⟨a✝.tail, snd✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_join | [16, 1] | [22, 72] | rw [ih] | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
a✝ : xss.Index
snd✝ : a✝.tail.val.Index
⊢ ⟨(join ⟨a✝, snd✝⟩).unjoin.fst.tail, (join ⟨a✝, snd✝⟩).unjoin.snd⟩ = ⟨a✝.tail, snd✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).unjoin = i
i : (i : (xs :: xss).Index) × i.val.Index
a✝ : xss.Index
snd✝ : a✝.tail.val.Index
⊢ ⟨(join ⟨a✝, snd✝⟩).unjoin.fst.tail, (join ⟨a✝, snd✝⟩).unjoin.sn... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | induction xss with
| nil => contradiction
| cons xs xss ih =>
match h : unappend k with
| .inl j => rw [unappend_eq_iff_eq_append] at h; rw [h, unjoin, unappend_append, join]
| .inr k => rw [unappend_eq_iff_eq_append] at h; rw [h, unjoin, unappend_append, join, ih] | α : Type u_1
xss : List (List α)
k : xss.join.Index
⊢ join k.unjoin = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xss : List (List α)
k : xss.join.Index
⊢ join k.unjoin = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | contradiction | case nil
α : Type u_1
k : [].join.Index
⊢ join k.unjoin = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
k : [].join.Index
⊢ join k.unjoin = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | match h : unappend k with
| .inl j => rw [unappend_eq_iff_eq_append] at h; rw [h, unjoin, unappend_append, join]
| .inr k => rw [unappend_eq_iff_eq_append] at h; rw [h, unjoin, unappend_append, join, ih] | case cons
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
⊢ join k.unjoin = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
⊢ join k.unjoin = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | rw [unappend_eq_iff_eq_append] at h | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
j : xs.Index
h : k.unappend = Sum.inl j
⊢ join k.unjoin = k | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
j : xs.Index
h : k = append (Sum.inl j)
⊢ join k.unjoin = k | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
j : xs.Index
h : k.unappend = Sum.inl j
⊢ join k.unjoin = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | rw [h, unjoin, unappend_append, join] | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
j : xs.Index
h : k = append (Sum.inl j)
⊢ join k.unjoin = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k : (xs :: xss).join.Index
j : xs.Index
h : k = append (Sum.inl j)
⊢ join k.unjoin = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | rw [unappend_eq_iff_eq_append] at h | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k✝ : (xs :: xss).join.Index
k : xss.join.Index
h : k✝.unappend = Sum.inr k
⊢ join k✝.unjoin = k✝ | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k✝ : (xs :: xss).join.Index
k : xss.join.Index
h : k✝ = append (Sum.inr k)
⊢ join k✝.unjoin = k✝ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k✝ : (xs :: xss).join.Index
k : xss.join.Index
h : k✝.unappend = Sum.inr k
⊢ join k✝.unjoin = k✝
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_unjoin | [24, 1] | [30, 95] | rw [h, unjoin, unappend_append, join, ih] | α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k✝ : (xs :: xss).join.Index
k : xss.join.Index
h : k✝ = append (Sum.inr k)
⊢ join k✝.unjoin = k✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
xss : List (List α)
ih : ∀ (k : xss.join.Index), join k.unjoin = k
k✝ : (xs :: xss).join.Index
k : xss.join.Index
h : k✝ = append (Sum.inr k)
⊢ join k✝.unjoin = k✝
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_eq_iff_eq_unjoin | [32, 1] | [35, 33] | constructor | α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ join i = k ↔ i = k.unjoin | case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ join i = k → i = k.unjoin
case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ i = k.unjoin → join i = k | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ join i = k ↔ i = k.unjoin
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_eq_iff_eq_unjoin | [32, 1] | [35, 33] | intro h | case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ join i = k → i = k.unjoin | case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : join i = k
⊢ i = k.unjoin | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ join i = k → i = k.unjoin
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_eq_iff_eq_unjoin | [32, 1] | [35, 33] | rw [←h, unjoin_join] | case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : join i = k
⊢ i = k.unjoin | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : join i = k
⊢ i = k.unjoin
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_eq_iff_eq_unjoin | [32, 1] | [35, 33] | intro h | case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ i = k.unjoin → join i = k | case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : i = k.unjoin
⊢ join i = k | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
⊢ i = k.unjoin → join i = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.join_eq_iff_eq_unjoin | [32, 1] | [35, 33] | rw [h, join_unjoin] | case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : i = k.unjoin
⊢ join i = k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
k : xss.join.Index
h : i = k.unjoin
⊢ join i = k
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_eq_iff_eq_join | [37, 1] | [40, 33] | constructor | α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k.unjoin = i ↔ k = join i | case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k.unjoin = i → k = join i
case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k = join i → k.unjoin = i | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k.unjoin = i ↔ k = join i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_eq_iff_eq_join | [37, 1] | [40, 33] | intro h | case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k.unjoin = i → k = join i | case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k.unjoin = i
⊢ k = join i | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k.unjoin = i → k = join i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_eq_iff_eq_join | [37, 1] | [40, 33] | rw [←h, join_unjoin] | case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k.unjoin = i
⊢ k = join i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k.unjoin = i
⊢ k = join i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_eq_iff_eq_join | [37, 1] | [40, 33] | intro h | case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k = join i → k.unjoin = i | case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k = join i
⊢ k.unjoin = i | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
⊢ k = join i → k.unjoin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.unjoin_eq_iff_eq_join | [37, 1] | [40, 33] | rw [h, unjoin_join] | case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k = join i
⊢ k.unjoin = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α✝ : Type u_1
xss : List (List α✝)
k : xss.join.Index
i : (i : xss.Index) × i.val.Index
h : k = join i
⊢ k.unjoin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | induction xss with
| nil => cases i; contradiction
| cons xs xss ih =>
match i with
| ⟨head, j⟩ => rw [join, val_append_inl]
| ⟨tail i, j⟩ => rw [join, val_append_inr, ih] | α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
⊢ (join i).val = i.snd.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xss : List (List α✝)
i : (i : xss.Index) × i.val.Index
⊢ (join i).val = i.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | cases i | case nil
α✝ : Type u_1
i : (i : [].Index) × i.val.Index
⊢ (join i).val = i.snd.val | case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).val = ⟨fst✝, snd✝⟩.snd.val | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α✝ : Type u_1
i : (i : [].Index) × i.val.Index
⊢ (join i).val = i.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | contradiction | case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).val = ⟨fst✝, snd✝⟩.snd.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mk
α✝ : Type u_1
fst✝ : [].Index
snd✝ : fst✝.val.Index
⊢ (join ⟨fst✝, snd✝⟩).val = ⟨fst✝, snd✝⟩.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | match i with
| ⟨head, j⟩ => rw [join, val_append_inl]
| ⟨tail i, j⟩ => rw [join, val_append_inr, ih] | case cons
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i : (i : (xs :: xss).Index) × i.val.Index
⊢ (join i).val = i.snd.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i : (i : (xs :: xss).Index) × i.val.Index
⊢ (join i).val = i.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | rw [join, val_append_inl] | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i : (i : (xs :: xss).Index) × i.val.Index
j : head.val.Index
⊢ (join ⟨head, j⟩).val = ⟨head, j⟩.snd.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i : (i : (xs :: xss).Index) × i.val.Index
j : head.val.Index
⊢ (join ⟨head, j⟩).val = ⟨head, j⟩.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_join | [51, 1] | [57, 51] | rw [join, val_append_inr, ih] | α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i✝ : (i : (xs :: xss).Index) × i.val.Index
i : xss.Index
j : i.tail.val.Index
⊢ (join ⟨i.tail, j⟩).val = ⟨i.tail, j⟩.snd.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
xss : List (List α✝)
ih : ∀ (i : (i : xss.Index) × i.val.Index), (join i).val = i.snd.val
i✝ : (i : (xs :: xss).Index) × i.val.Index
i : xss.Index
j : i.tail.val.Index
⊢ (join ⟨i.tail, j⟩).val = ⟨i.tail, j⟩.snd.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Join.lean | List.Index.val_unjoin | [59, 1] | [60, 45] | rw [←join_unjoin k, val_join, unjoin_join] | α : Type u_1
xss : List (List α)
k : xss.join.Index
⊢ k.unjoin.snd.val = k.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xss : List (List α)
k : xss.join.Index
⊢ k.unjoin.snd.val = k.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Basic.lean | Int.succ_mk_zero | [17, 1] | [18, 57] | rw [Int.mk, Int.subNatNat, Nat.zero_sub, Nat.sub_zero] | m : Nat
⊢ m + 1 ⊖ 0 = ofNat (m + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Nat
⊢ m + 1 ⊖ 0 = ofNat (m + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Basic.lean | Int.succ_mk_succ | [20, 1] | [21, 82] | rw [Int.mk, Int.subNatNat, Int.subNatNat, Nat.succ_sub_succ, Nat.succ_sub_succ] | m n : Nat
⊢ m + 1 ⊖ n + 1 = m ⊖ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m n : Nat
⊢ m + 1 ⊖ n + 1 = m ⊖ n
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Int/Basic.lean | Int.mk_zero | [23, 9] | [24, 57] | rw [Int.mk, Int.subNatNat, Nat.zero_sub, Nat.sub_zero] | m : Nat
⊢ m ⊖ 0 = ofNat m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : Nat
⊢ m ⊖ 0 = ofNat m
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.val_iota | [27, 1] | [30, 50] | induction i with
| head => rfl
| tail i ih => rw [iota, val_tail, val_map, ih] | α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ i.iota.val = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ i.iota.val = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.val_iota | [27, 1] | [30, 50] | rfl | case head
α✝ : Type u_1
xs : List α✝
x✝ : α✝
xs✝ : List α✝
⊢ head.iota.val = head | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case head
α✝ : Type u_1
xs : List α✝
x✝ : α✝
xs✝ : List α✝
⊢ head.iota.val = head
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.val_iota | [27, 1] | [30, 50] | rw [iota, val_tail, val_map, ih] | case tail
α✝ : Type u_1
xs xs✝ : List α✝
x✝ : α✝
i : xs✝.Index
ih : i.iota.val = i
⊢ i.tail.iota.val = i.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tail
α✝ : Type u_1
xs xs✝ : List α✝
x✝ : α✝
i : xs✝.Index
ih : i.iota.val = i
⊢ i.tail.iota.val = i.tail
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.iota_val | [32, 1] | [38, 98] | induction xs with
| nil => contradiction
| cons x xs ih =>
match i with
| head => rfl
| tail i => rw [←map_unmap Index.tail i, val_tail, val_unmap Index.tail, iota, ih, map_unmap] | α : Type u_1
xs : List α
i : xs.indexIota.Index
⊢ i.val.iota = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
i : xs.indexIota.Index
⊢ i.val.iota = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.iota_val | [32, 1] | [38, 98] | contradiction | case nil
α : Type u_1
i : [].indexIota.Index
⊢ i.val.iota = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
i : [].indexIota.Index
⊢ i.val.iota = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.iota_val | [32, 1] | [38, 98] | match i with
| head => rfl
| tail i => rw [←map_unmap Index.tail i, val_tail, val_unmap Index.tail, iota, ih, map_unmap] | case cons
α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i : (x :: xs).indexIota.Index
⊢ i.val.iota = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i : (x :: xs).indexIota.Index
⊢ i.val.iota = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.iota_val | [32, 1] | [38, 98] | rfl | α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i : (x :: xs).indexIota.Index
⊢ head.val.iota = head | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i : (x :: xs).indexIota.Index
⊢ head.val.iota = head
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Iota.lean | List.Index.iota_val | [32, 1] | [38, 98] | rw [←map_unmap Index.tail i, val_tail, val_unmap Index.tail, iota, ih, map_unmap] | α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i✝ : (x :: xs).indexIota.Index
i : (List.map tail xs.indexIota).Index
⊢ i.tail.val.iota = i.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : xs.indexIota.Index), i.val.iota = i
i✝ : (x :: xs).indexIota.Index
i : (List.map tail xs.indexIota).Index
⊢ i.tail.val.iota = i.tail
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_ne_one | [35, 11] | [36, 66] | by_toNat | x : Pos
⊢ x + 1 ≠ 1 | x : Pos
⊢ x.toNat + 1 ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ x + 1 ≠ 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_ne_one | [35, 11] | [36, 66] | intro h | x : Pos
⊢ x.toNat + 1 ≠ 1 | x : Pos
h : x.toNat + 1 = 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ x.toNat + 1 ≠ 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_ne_one | [35, 11] | [36, 66] | absurd (Nat.succ.inj h) | x : Pos
h : x.toNat + 1 = 1
⊢ False | x : Pos
h : x.toNat + 1 = 1
⊢ ¬x.toNat = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
h : x.toNat + 1 = 1
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_ne_one | [35, 11] | [36, 66] | exact Nat.ne_zero _ | x : Pos
h : x.toNat + 1 = 1
⊢ ¬x.toNat = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
h : x.toNat + 1 = 1
⊢ ¬x.toNat = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.one_add_eq_add_one | [38, 19] | [39, 32] | by_toNat | x : Pos
⊢ 1 + x = x + 1 | x : Pos
⊢ 1 + x.toNat = x.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ 1 + x = x + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.one_add_eq_add_one | [38, 19] | [39, 32] | exact Nat.one_add _ | x : Pos
⊢ 1 + x.toNat = x.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ 1 + x.toNat = x.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_succ | [41, 19] | [42, 34] | by_toNat | x y : Pos
⊢ x + (y + 1) = x + y + 1 | x y : Pos
⊢ x.toNat + (y.toNat + 1) = x.toNat + y.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + (y + 1) = x + y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_succ | [41, 19] | [42, 34] | exact Nat.add_succ .. | x y : Pos
⊢ x.toNat + (y.toNat + 1) = x.toNat + y.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x.toNat + (y.toNat + 1) = x.toNat + y.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_add | [44, 19] | [45, 34] | by_toNat | x y : Pos
⊢ x + 1 + y = x + y + 1 | x y : Pos
⊢ x.toNat + 1 + y.toNat = x.toNat + y.toNat + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + 1 + y = x + y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_add | [44, 19] | [45, 34] | exact Nat.succ_add .. | x y : Pos
⊢ x.toNat + 1 + y.toNat = x.toNat + y.toNat + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x.toNat + 1 + y.toNat = x.toNat + y.toNat + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_assoc | [47, 11] | [48, 35] | by_toNat | x y z : Pos
⊢ x + y + z = x + (y + z) | x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + (y.toNat + z.toNat) | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + y + z = x + (y + z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_assoc | [47, 11] | [48, 35] | exact Nat.add_assoc .. | x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + (y.toNat + z.toNat) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + (y.toNat + z.toNat)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_comm | [50, 11] | [51, 34] | by_toNat | x y : Pos
⊢ x + y = y + x | x y : Pos
⊢ x.toNat + y.toNat = y.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x + y = y + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_comm | [50, 11] | [51, 34] | exact Nat.add_comm .. | x y : Pos
⊢ x.toNat + y.toNat = y.toNat + x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x.toNat + y.toNat = y.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_left_comm | [53, 11] | [54, 39] | by_toNat | x y z : Pos
⊢ x + (y + z) = y + (x + z) | x y z : Pos
⊢ x.toNat + (y.toNat + z.toNat) = y.toNat + (x.toNat + z.toNat) | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + (y + z) = y + (x + z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_left_comm | [53, 11] | [54, 39] | exact Nat.add_left_comm .. | x y z : Pos
⊢ x.toNat + (y.toNat + z.toNat) = y.toNat + (x.toNat + z.toNat) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat + (y.toNat + z.toNat) = y.toNat + (x.toNat + z.toNat)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_right_comm | [56, 11] | [57, 40] | by_toNat | x y z : Pos
⊢ x + y + z = x + z + y | x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + z.toNat + y.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + y + z = x + z + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_right_comm | [56, 11] | [57, 40] | exact Nat.add_right_comm .. | x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + z.toNat + y.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat + y.toNat + z.toNat = x.toNat + z.toNat + y.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_left_cancel | [59, 11] | [60, 38] | by_toNat | x y z : Pos
⊢ x + y = x + z → y = z | x y z : Pos
⊢ x.toNat + y.toNat = x.toNat + z.toNat → y.toNat = z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + y = x + z → y = z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_left_cancel | [59, 11] | [60, 38] | exact Nat.add_left_cancel | x y z : Pos
⊢ x.toNat + y.toNat = x.toNat + z.toNat → y.toNat = z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat + y.toNat = x.toNat + z.toNat → y.toNat = z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_right_cancel | [62, 11] | [63, 39] | by_toNat | x y z : Pos
⊢ x + y = z + y → x = z | x y z : Pos
⊢ x.toNat + y.toNat = z.toNat + y.toNat → x.toNat = z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x + y = z + y → x = z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.add_right_cancel | [62, 11] | [63, 39] | exact Nat.add_right_cancel | x y z : Pos
⊢ x.toNat + y.toNat = z.toNat + y.toNat → x.toNat = z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat + y.toNat = z.toNat + y.toNat → x.toNat = z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | induction x with
| one =>
apply Ne.symm
exact Pos.succ_ne_one y
| succ x H =>
intro h
rw [Pos.add_succ] at h
apply H
exact Pos.add_right_cancel h | x y : Pos
⊢ x ≠ y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≠ y + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | apply Ne.symm | case one
y : Pos
⊢ 1 ≠ y + 1 | case one.h
y : Pos
⊢ y + 1 ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one
y : Pos
⊢ 1 ≠ y + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | exact Pos.succ_ne_one y | case one.h
y : Pos
⊢ y + 1 ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one.h
y : Pos
⊢ y + 1 ≠ 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | intro h | case succ
y x : Pos
H : x ≠ y + x
⊢ x + 1 ≠ y + (x + 1) | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + (x + 1)
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ y + x
⊢ x + 1 ≠ y + (x + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | rw [Pos.add_succ] at h | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + (x + 1)
⊢ False | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + (x + 1)
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | apply H | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ False | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ x = y + x | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_left | [65, 11] | [74, 33] | exact Pos.add_right_cancel h | case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ x = y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ y + x
h : x + 1 = y + x + 1
⊢ x = y + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | induction x with
| one =>
apply Ne.symm
rw [Pos.one_add_eq_add_one]
exact Pos.succ_ne_one y
| succ x H =>
intro h
rw [Pos.succ_add] at h
apply H
exact Pos.add_right_cancel h | x y : Pos
⊢ x ≠ x + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x ≠ x + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | apply Ne.symm | case one
y : Pos
⊢ 1 ≠ 1 + y | case one.h
y : Pos
⊢ 1 + y ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one
y : Pos
⊢ 1 ≠ 1 + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | rw [Pos.one_add_eq_add_one] | case one.h
y : Pos
⊢ 1 + y ≠ 1 | case one.h
y : Pos
⊢ y + 1 ≠ 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case one.h
y : Pos
⊢ 1 + y ≠ 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | exact Pos.succ_ne_one y | case one.h
y : Pos
⊢ y + 1 ≠ 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case one.h
y : Pos
⊢ y + 1 ≠ 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | intro h | case succ
y x : Pos
H : x ≠ x + y
⊢ x + 1 ≠ x + 1 + y | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + 1 + y
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ x + y
⊢ x + 1 ≠ x + 1 + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | rw [Pos.succ_add] at h | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + 1 + y
⊢ False | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + 1 + y
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | apply H | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ False | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ x = x + y | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.ne_add_self_right | [76, 11] | [86, 33] | exact Pos.add_right_cancel h | case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ x = x + y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
y x : Pos
H : x ≠ x + y
h : x + 1 = x + y + 1
⊢ x = x + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.one_mul | [88, 19] | [89, 32] | by_toNat | x : Pos
⊢ 1 * x = x | x : Pos
⊢ 1 * x.toNat = x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ 1 * x = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.one_mul | [88, 19] | [89, 32] | exact Nat.one_mul _ | x : Pos
⊢ 1 * x.toNat = x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ 1 * x.toNat = x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_one | [91, 19] | [92, 32] | by_toNat | x : Pos
⊢ x * 1 = x | x : Pos
⊢ x.toNat * 1 = x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ x * 1 = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_one | [91, 19] | [92, 32] | exact Nat.mul_one _ | x : Pos
⊢ x.toNat * 1 = x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Pos
⊢ x.toNat * 1 = x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_mul | [94, 11] | [95, 34] | by_toNat | x y : Pos
⊢ (x + 1) * y = x * y + y | x y : Pos
⊢ (x.toNat + 1) * y.toNat = x.toNat * y.toNat + y.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ (x + 1) * y = x * y + y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.succ_mul | [94, 11] | [95, 34] | exact Nat.succ_mul .. | x y : Pos
⊢ (x.toNat + 1) * y.toNat = x.toNat * y.toNat + y.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ (x.toNat + 1) * y.toNat = x.toNat * y.toNat + y.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_succ | [97, 11] | [98, 34] | by_toNat | x y : Pos
⊢ x * (y + 1) = x * y + x | x y : Pos
⊢ x.toNat * (y.toNat + 1) = x.toNat * y.toNat + x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x * (y + 1) = x * y + x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_succ | [97, 11] | [98, 34] | exact Nat.mul_succ .. | x y : Pos
⊢ x.toNat * (y.toNat + 1) = x.toNat * y.toNat + x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x.toNat * (y.toNat + 1) = x.toNat * y.toNat + x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_assoc | [100, 11] | [101, 35] | by_toNat | x y z : Pos
⊢ x * y * z = x * (y * z) | x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * (y.toNat * z.toNat) | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * y * z = x * (y * z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_assoc | [100, 11] | [101, 35] | exact Nat.mul_assoc .. | x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * (y.toNat * z.toNat) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * (y.toNat * z.toNat)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_comm | [103, 11] | [104, 34] | by_toNat | x y : Pos
⊢ x * y = y * x | x y : Pos
⊢ x.toNat * y.toNat = y.toNat * x.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x * y = y * x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_comm | [103, 11] | [104, 34] | exact Nat.mul_comm .. | x y : Pos
⊢ x.toNat * y.toNat = y.toNat * x.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Pos
⊢ x.toNat * y.toNat = y.toNat * x.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_left_comm | [106, 11] | [107, 39] | by_toNat | x y z : Pos
⊢ x * (y * z) = y * (x * z) | x y z : Pos
⊢ x.toNat * (y.toNat * z.toNat) = y.toNat * (x.toNat * z.toNat) | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * (y * z) = y * (x * z)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_left_comm | [106, 11] | [107, 39] | exact Nat.mul_left_comm .. | x y z : Pos
⊢ x.toNat * (y.toNat * z.toNat) = y.toNat * (x.toNat * z.toNat) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat * (y.toNat * z.toNat) = y.toNat * (x.toNat * z.toNat)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_right_comm | [109, 11] | [110, 40] | by_toNat | x y z : Pos
⊢ x * y * z = x * z * y | x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * z.toNat * y.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * y * z = x * z * y
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_right_comm | [109, 11] | [110, 40] | exact Nat.mul_right_comm .. | x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * z.toNat * y.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat * y.toNat * z.toNat = x.toNat * z.toNat * y.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.left_distrib | [112, 11] | [113, 38] | by_toNat | x y z : Pos
⊢ x * (y + z) = x * y + x * z | x y z : Pos
⊢ x.toNat * (y.toNat + z.toNat) = x.toNat * y.toNat + x.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * (y + z) = x * y + x * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.left_distrib | [112, 11] | [113, 38] | exact Nat.left_distrib .. | x y z : Pos
⊢ x.toNat * (y.toNat + z.toNat) = x.toNat * y.toNat + x.toNat * z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x.toNat * (y.toNat + z.toNat) = x.toNat * y.toNat + x.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.right_distrib | [115, 11] | [116, 39] | by_toNat | x y z : Pos
⊢ (x + y) * z = x * z + y * z | x y z : Pos
⊢ (x.toNat + y.toNat) * z.toNat = x.toNat * z.toNat + y.toNat * z.toNat | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ (x + y) * z = x * z + y * z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.right_distrib | [115, 11] | [116, 39] | exact Nat.right_distrib .. | x y z : Pos
⊢ (x.toNat + y.toNat) * z.toNat = x.toNat * z.toNat + y.toNat * z.toNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ (x.toNat + y.toNat) * z.toNat = x.toNat * z.toNat + y.toNat * z.toNat
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_left_cancel | [118, 11] | [132, 36] | intro h | x y z : Pos
⊢ x * y = x * z → y = z | x y z : Pos
h : x * y = x * z
⊢ y = z | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
⊢ x * y = x * z → y = z
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Pos/Lemmas.lean | Pos.mul_left_cancel | [118, 11] | [132, 36] | induction y, z with
| one_one => rfl
| one_succ z _ =>
rw [Pos.mul_one, Pos.mul_succ] at h
absurd h
apply Pos.ne_add_self_left
| succ_one y _ =>
rw [Pos.mul_one, Pos.mul_succ] at h
absurd h.symm
apply Pos.ne_add_self_left
| succ_succ y z H =>
rw [Pos.mul_succ, Pos.mul_succ] at h
rw [H (Pos.add_right_can... | x y z : Pos
h : x * y = x * z
⊢ y = z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y z : Pos
h : x * y = x * z
⊢ y = z
TACTIC:
|
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