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/Basic.lean | List.Index.find_none | [91, 1] | [103, 32] | split at heq | case cons.h_2.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝ : p head = false
heq :
(match Index.find? fun i => p i.tail with
| some x_1 => some x_1.tail
| none => none) =
none
a✝ : xs.Index
⊢... | case cons.h_2.tail.h_1
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝¹ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
x✝ : xs.Index
heq✝ : (Index.find? fun i => p i.tail) = some x✝
heq : some x✝.tail = no... | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.h_2.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝ : p head = false
heq :
(match Index.find? fun i => p i.tail with
| some x_1 =>... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.find_none | [91, 1] | [103, 32] | contradiction | case cons.h_2.tail.h_1
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝¹ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
x✝ : xs.Index
heq✝ : (Index.find? fun i => p i.tail) = some x✝
heq : some x✝.tail = no... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.h_2.tail.h_1
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝¹ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
x✝ : xs.Index
heq✝ : (Ind... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.find_none | [91, 1] | [103, 32] | next h => rw [ih _ h] | case cons.h_2.tail.h_2
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝¹ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
heq✝ : (Index.find? fun i => p i.tail) = none
heq : none = none
⊢ p a✝.tail = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.h_2.tail.h_2
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝¹ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
heq✝ : (Index.find? fun i... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.find_none | [91, 1] | [103, 32] | rw [ih _ h] | α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
h : (Index.find? fun i => p i.tail) = none
heq : none = none
⊢ p a✝.tail = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ {p : xs.Index → Bool} (i : xs.Index), Index.find? p = none → p i = false
p : (x :: xs).Index → Bool
c✝ : Bool
heq✝ : p head = false
a✝ : xs.Index
opt✝ : Option xs.Index
h : (Index.find? fun i => p i.tail) = none
heq : ... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | unfold search | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
⊢ p (search h) | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
⊢ p
(match hi : Index.find? fun i => decide ((fun i => p i) i) with
| some i => i
| none => absurd h ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
⊢ p (search h)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | split | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
⊢ p
(match hi : Index.find? fun i => decide ((fun i => p i) i) with
| some i => i
| none => absurd h ⋯) | case h_1
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
i✝ : xs.Index
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ p i✝
case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
heq✝ : (Index.find? fun i => decide ((fun i =>... | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
⊢ p
(match hi : Index.find? fun i => decide ((fun i => p i) i) with
| some i => i
| none => absurd h ⋯)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | next h =>
apply of_decide_eq_true
exact find_some _ h | case h_1
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
i✝ : xs.Index
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ p i✝
case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
heq✝ : (Index.find? fun i => decide ((fun i =>... | case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ p (absurd h ⋯) | Please generate a tactic in lean4 to solve the state.
STATE:
case h_1
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
i✝ : xs.Index
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ p i✝
case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred ... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | next f =>
absurd h
intro ⟨j, hj⟩
have := find_none j f
rw [decide_eq_true hj] at this
contradiction | case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ p (absurd h ⋯) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h_2
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
heq✝ : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ p (absurd h ⋯)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | apply of_decide_eq_true | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h✝ : ∃ i, p i
i✝ : xs.Index
h : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ p i✝ | case a
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h✝ : ∃ i, p i
i✝ : xs.Index
h : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ decide (p i✝) = true | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h✝ : ∃ i, p i
i✝ : xs.Index
h : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ p i✝
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | exact find_some _ h | case a
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h✝ : ∃ i, p i
i✝ : xs.Index
h : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ decide (p i✝) = true | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h✝ : ∃ i, p i
i✝ : xs.Index
h : (Index.find? fun i => decide ((fun i => p i) i)) = some i✝
⊢ decide (p i✝) = true
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | absurd h | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ p (absurd h ⋯) | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ ¬∃ i, p i | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ p (absurd h ⋯)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | intro ⟨j, hj⟩ | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ ¬∃ i, p i | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
⊢ ¬∃ i, p i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | have := find_none j f | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
⊢ False | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : decide ((fun i => p i) j) = false
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | rw [decide_eq_true hj] at this | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : decide ((fun i => p i) j) = false
⊢ False | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : true = false
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : decide ((fun i => p i) j) = false
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_prop | [114, 1] | [125, 18] | contradiction | α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : true = false
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
inst✝ : DecidablePred p
h : ∃ i, p i
f : (Index.find? fun i => decide ((fun i => p i) i)) = none
j : xs.Index
hj : p j
this : true = false
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_eq | [127, 1] | [131, 6] | cases h | α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
ip : DecidablePred p
iq : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : p = q
⊢ search hp = search hq | case refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
iq : DecidablePred p
hq : ∃ j, p j
⊢ search hp = search hq | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
ip : DecidablePred p
iq : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : p = q
⊢ search hp = search hq
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_eq | [127, 1] | [131, 6] | cases Subsingleton.elim ip iq | case refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
iq : DecidablePred p
hq : ∃ j, p j
⊢ search hp = search hq | case refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp hq : ∃ j, p j
⊢ search hp = search hq | Please generate a tactic in lean4 to solve the state.
STATE:
case refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
iq : DecidablePred p
hq : ∃ j, p j
⊢ search hp = search hq
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_eq | [127, 1] | [131, 6] | cases Subsingleton.elim hp hq | case refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp hq : ∃ j, p j
⊢ search hp = search hq | case refl.refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
⊢ search hp = search hp | Please generate a tactic in lean4 to solve the state.
STATE:
case refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp hq : ∃ j, p j
⊢ search hp = search hq
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_eq | [127, 1] | [131, 6] | rfl | case refl.refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
⊢ search hp = search hp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refl.refl.refl
α✝ : Type u_1
xs : List α✝
p : xs.Index → Prop
ip : DecidablePred p
hp : ∃ i, p i
⊢ search hp = search hp
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_ext | [133, 1] | [137, 22] | intro h | α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
⊢ (∀ (i : xs.Index), p i ↔ q i) → search hp = search hq | α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ search hp = search hq | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
⊢ (∀ (i : xs.Index), p i ↔ q i) → search hp = search hq
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_ext | [133, 1] | [137, 22] | apply search_eq | α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ search hp = search hq | case h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ (fun i => p i) = fun i => q i | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ search hp = search hq
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_ext | [133, 1] | [137, 22] | funext i | case h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ (fun i => p i) = fun i => q i | case h.h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
i : xs.Index
⊢ p i = q i | Please generate a tactic in lean4 to solve the state.
STATE:
case h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
⊢ (fun i => p i) = fun i => q i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.search_ext | [133, 1] | [137, 22] | exact propext (h i) | case h.h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
i : xs.Index
⊢ p i = q i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
α✝ : Type u_1
xs : List α✝
p q : xs.Index → Prop
inst✝¹ : DecidablePred p
inst✝ : DecidablePred q
hp : ∃ i, p i
hq : ∃ j, q j
h : ∀ (i : xs.Index), p i ↔ q i
i : xs.Index
⊢ p i = q i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | funext _ _ i | ⊢ @Index.toNat = @toNatTR | case h.h.h
x✝¹ : Type u_1
x✝ : List x✝¹
i : x✝.Index
⊢ i.toNat = i.toNatTR | Please generate a tactic in lean4 to solve the state.
STATE:
⊢ @Index.toNat = @toNatTR
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | induction i with simp_all only [Index.toNat, toNatTR, toNatTR.loop]
| tail => rw [lem] | case h.h.h
x✝¹ : Type u_1
x✝ : List x✝¹
i : x✝.Index
⊢ i.toNat = i.toNatTR | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h
x✝¹ : Type u_1
x✝ : List x✝¹
i : x✝.Index
⊢ i.toNat = i.toNatTR
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | rw [lem] | case h.h.h.tail
x✝² : Type u_1
x✝¹ xs✝ : List x✝²
x✝ : x✝²
a✝ : xs✝.Index
a_ih✝ : a✝.toNat = toNatTR.loop a✝ 0
⊢ toNatTR.loop a✝ 0 + 1 = toNatTR.loop a✝ (0 + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h.h.tail
x✝² : Type u_1
x✝¹ xs✝ : List x✝²
x✝ : x✝²
a✝ : xs✝.Index
a_ih✝ : a✝.toNat = toNatTR.loop a✝ 0
⊢ toNatTR.loop a✝ 0 + 1 = toNatTR.loop a✝ (0 + 1)
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | induction i generalizing n with
| head => rfl
| tail _ ih => simp [toNatTR.loop, ih] | α : Type ?u.14515
xs : List α
i : xs.Index
n : Nat
⊢ toNatTR.loop i (n + 1) = toNatTR.loop i n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type ?u.14515
xs : List α
i : xs.Index
n : Nat
⊢ toNatTR.loop i (n + 1) = toNatTR.loop i n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | rfl | case head
α : Type ?u.14515
xs : List α
x✝ : α
xs✝ : List α
n : Nat
⊢ toNatTR.loop head (n + 1) = toNatTR.loop head n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case head
α : Type ?u.14515
xs : List α
x✝ : α
xs✝ : List α
n : Nat
⊢ toNatTR.loop head (n + 1) = toNatTR.loop head n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_eq_toNatTR | [150, 10] | [158, 43] | simp [toNatTR.loop, ih] | case tail
α : Type ?u.14515
xs xs✝ : List α
x✝ : α
a✝ : xs✝.Index
ih : ∀ (n : Nat), toNatTR.loop a✝ (n + 1) = toNatTR.loop a✝ n + 1
n : Nat
⊢ toNatTR.loop a✝.tail (n + 1) = toNatTR.loop a✝.tail n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tail
α : Type ?u.14515
xs xs✝ : List α
x✝ : α
a✝ : xs✝.Index
ih : ∀ (n : Nat), toNatTR.loop a✝ (n + 1) = toNatTR.loop a✝ n + 1
n : Nat
⊢ toNatTR.loop a✝.tail (n + 1) = toNatTR.loop a✝.tail n + 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_lt_length | [160, 1] | [166, 44] | induction xs with
| nil => cases i
| cons x xs ih =>
cases i with
| head => exact Nat.zero_lt_succ ..
| tail => apply Nat.succ_lt_succ (ih _) | α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ i.toNat < xs.length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ i.toNat < xs.length
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_lt_length | [160, 1] | [166, 44] | cases i | case nil
α✝ : Type u_1
i : [].Index
⊢ i.toNat < [].length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α✝ : Type u_1
i : [].Index
⊢ i.toNat < [].length
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_lt_length | [160, 1] | [166, 44] | cases i with
| head => exact Nat.zero_lt_succ ..
| tail => apply Nat.succ_lt_succ (ih _) | case cons
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), i.toNat < xs.length
i : (x :: xs).Index
⊢ i.toNat < (x :: xs).length | 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.Index), i.toNat < xs.length
i : (x :: xs).Index
⊢ i.toNat < (x :: xs).length
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_lt_length | [160, 1] | [166, 44] | exact Nat.zero_lt_succ .. | case cons.head
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), i.toNat < xs.length
⊢ head.toNat < (x :: xs).length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.head
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), i.toNat < xs.length
⊢ head.toNat < (x :: xs).length
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_lt_length | [160, 1] | [166, 44] | apply Nat.succ_lt_succ (ih _) | case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), i.toNat < xs.length
a✝ : xs.Index
⊢ a✝.tail.toNat < (x :: xs).length | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), i.toNat < xs.length
a✝ : xs.Index
⊢ a✝.tail.toNat < (x :: xs).length
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | induction xs with
| nil => cases i
| cons x xs ih =>
cases i with
| head => rfl
| tail i =>
apply congrArg tail
exact ih .. | α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ Index.ofFin i.toFin = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α✝ : Type u_1
xs : List α✝
i : xs.Index
⊢ Index.ofFin i.toFin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | cases i | case nil
α✝ : Type u_1
i : [].Index
⊢ Index.ofFin i.toFin = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α✝ : Type u_1
i : [].Index
⊢ Index.ofFin i.toFin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | cases i with
| head => rfl
| tail i =>
apply congrArg tail
exact ih .. | case cons
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : (x :: xs).Index
⊢ Index.ofFin i.toFin = 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.Index), Index.ofFin i.toFin = i
i : (x :: xs).Index
⊢ Index.ofFin i.toFin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | rfl | case cons.head
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
⊢ Index.ofFin head.toFin = head | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.head
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
⊢ Index.ofFin head.toFin = head
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | apply congrArg tail | case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : xs.Index
⊢ Index.ofFin i.tail.toFin = i.tail | case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : xs.Index
⊢ Index.ofFin ⟨i.toNat, ⋯⟩ = i | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : xs.Index
⊢ Index.ofFin i.tail.toFin = i.tail
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.ofFin_toFin | [183, 1] | [191, 18] | exact ih .. | case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : xs.Index
⊢ Index.ofFin ⟨i.toNat, ⋯⟩ = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons.tail
α✝ : Type u_1
x : α✝
xs : List α✝
ih : ∀ (i : xs.Index), Index.ofFin i.toFin = i
i : xs.Index
⊢ Index.ofFin ⟨i.toNat, ⋯⟩ = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | induction xs with
| nil => cases i; contradiction
| cons x xs ih =>
match i with
| ⟨0,_⟩ => rfl
| ⟨i+1,h⟩ =>
apply congrArg Nat.succ
rw [ih] | α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).toNat = ↑i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).toNat = ↑i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | cases i | case nil
α : Type u_1
i : Fin [].length
⊢ (Index.ofFin i).toNat = ↑i | case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).toNat = ↑⟨val✝, isLt✝⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
i : Fin [].length
⊢ (Index.ofFin i).toNat = ↑i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | contradiction | case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).toNat = ↑⟨val✝, isLt✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).toNat = ↑⟨val✝, isLt✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | match i with
| ⟨0,_⟩ => rfl
| ⟨i+1,h⟩ =>
apply congrArg Nat.succ
rw [ih] | case cons
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i : Fin (x :: xs).length
⊢ (Index.ofFin i).toNat = ↑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 : Fin xs.length), (Index.ofFin i).toNat = ↑i
i : Fin (x :: xs).length
⊢ (Index.ofFin i).toNat = ↑i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | rfl | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i : Fin (x :: xs).length
isLt✝ : 0 < (x :: xs).length
⊢ (Index.ofFin ⟨0, isLt✝⟩).toNat = ↑⟨0, isLt✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i : Fin (x :: xs).length
isLt✝ : 0 < (x :: xs).length
⊢ (Index.ofFin ⟨0, isLt✝⟩).toNat = ↑⟨0, isLt✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | apply congrArg Nat.succ | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i✝ : Fin (x :: xs).length
i : Nat
h : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i + 1, h⟩).toNat = ↑⟨i + 1, h⟩ | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i✝ : Fin (x :: xs).length
i : Nat
h : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i, ⋯⟩).toNat = i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i✝ : Fin (x :: xs).length
i : Nat
h : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i + 1, h⟩).toNat = ↑⟨i + 1, h⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toNat_ofFin | [193, 1] | [201, 14] | rw [ih] | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i✝ : Fin (x :: xs).length
i : Nat
h : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i, ⋯⟩).toNat = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).toNat = ↑i
i✝ : Fin (x :: xs).length
i : Nat
h : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i, ⋯⟩).toNat = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toFin_ofFin | [203, 1] | [205, 20] | apply Fin.eq_of_val_eq | α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).toFin = i | case a
α : Type u_1
xs : List α
i : Fin xs.length
⊢ ↑(Index.ofFin i).toFin = ↑i | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).toFin = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.toFin_ofFin | [203, 1] | [205, 20] | apply toNat_ofFin | case a
α : Type u_1
xs : List α
i : Fin xs.length
⊢ ↑(Index.ofFin i).toFin = ↑i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
α : Type u_1
xs : List α
i : Fin xs.length
⊢ ↑(Index.ofFin i).toFin = ↑i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | induction xs with
| nil => cases i; contradiction
| cons x xs ih =>
match i with
| ⟨0, _⟩ => rfl
| ⟨i+1, _⟩ => simp [Index.ofFin, ih] | α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).val = xs.get i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
xs : List α
i : Fin xs.length
⊢ (Index.ofFin i).val = xs.get i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | cases i | case nil
α : Type u_1
i : Fin [].length
⊢ (Index.ofFin i).val = [].get i | case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).val = [].get ⟨val✝, isLt✝⟩ | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
i : Fin [].length
⊢ (Index.ofFin i).val = [].get i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | contradiction | case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).val = [].get ⟨val✝, isLt✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil.mk
α : Type u_1
val✝ : Nat
isLt✝ : val✝ < [].length
⊢ (Index.ofFin ⟨val✝, isLt✝⟩).val = [].get ⟨val✝, isLt✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | match i with
| ⟨0, _⟩ => rfl
| ⟨i+1, _⟩ => simp [Index.ofFin, ih] | case cons
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).val = xs.get i
i : Fin (x :: xs).length
⊢ (Index.ofFin i).val = (x :: xs).get 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 : Fin xs.length), (Index.ofFin i).val = xs.get i
i : Fin (x :: xs).length
⊢ (Index.ofFin i).val = (x :: xs).get i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | rfl | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).val = xs.get i
i : Fin (x :: xs).length
isLt✝ : 0 < (x :: xs).length
⊢ (Index.ofFin ⟨0, isLt✝⟩).val = (x :: xs).get ⟨0, isLt✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).val = xs.get i
i : Fin (x :: xs).length
isLt✝ : 0 < (x :: xs).length
⊢ (Index.ofFin ⟨0, isLt✝⟩).val = (x :: xs).get ⟨0, isLt✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Basic.lean | List.Index.val_ofFin_eq_get | [215, 1] | [221, 41] | simp [Index.ofFin, ih] | α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).val = xs.get i
i✝ : Fin (x :: xs).length
i : Nat
isLt✝ : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i + 1, isLt✝⟩).val = (x :: xs).get ⟨i + 1, isLt✝⟩ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
x : α
xs : List α
ih : ∀ (i : Fin xs.length), (Index.ofFin i).val = xs.get i
i✝ : Fin (x :: xs).length
i : Nat
isLt✝ : i + 1 < (x :: xs).length
⊢ (Index.ofFin ⟨i + 1, isLt✝⟩).val = (x :: xs).get ⟨i + 1, isLt✝⟩
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_map | [23, 1] | [26, 40] | induction i with
| head => rfl
| tail i ih => exact congrArg tail ih | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
⊢ unmap f (map f i) = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
⊢ unmap f (map f i) = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_map | [23, 1] | [26, 40] | rfl | case head
α : Type u_1
β : Type u_2
f : α → β
xs : List α
x✝ : α
xs✝ : List α
⊢ unmap f (map f head) = head | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case head
α : Type u_1
β : Type u_2
f : α → β
xs : List α
x✝ : α
xs✝ : List α
⊢ unmap f (map f head) = head
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_map | [23, 1] | [26, 40] | exact congrArg tail ih | case tail
α : Type u_1
β : Type u_2
f : α → β
xs xs✝ : List α
x✝ : α
i : xs✝.Index
ih : unmap f (map f i) = i
⊢ unmap f (map f i.tail) = i.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tail
α : Type u_1
β : Type u_2
f : α → β
xs xs✝ : List α
x✝ : α
i : xs✝.Index
ih : unmap f (map f i) = i
⊢ unmap f (map f i.tail) = i.tail
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_unmap | [28, 1] | [34, 43] | induction xs with
| nil => contradiction
| cons x xs ih =>
match i with
| head => rfl
| tail i => exact congrArg tail (ih i) | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
⊢ map f (unmap f i) = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
⊢ map f (unmap f i) = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_unmap | [28, 1] | [34, 43] | contradiction | case nil
α : Type u_1
β : Type u_2
f : α → β
i : (List.map f []).Index
⊢ map f (unmap f i) = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case nil
α : Type u_1
β : Type u_2
f : α → β
i : (List.map f []).Index
⊢ map f (unmap f i) = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_unmap | [28, 1] | [34, 43] | match i with
| head => rfl
| tail i => exact congrArg tail (ih i) | case cons
α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i : (List.map f (x :: xs)).Index
⊢ map f (unmap f i) = i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case cons
α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i : (List.map f (x :: xs)).Index
⊢ map f (unmap f i) = i
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_unmap | [28, 1] | [34, 43] | rfl | α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i : (List.map f (x :: xs)).Index
⊢ map f (unmap f head) = head | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i : (List.map f (x :: xs)).Index
⊢ map f (unmap f head) = head
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_unmap | [28, 1] | [34, 43] | exact congrArg tail (ih i) | α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i✝ : (List.map f (x :: xs)).Index
i : (List.map f xs).Index
⊢ map f (unmap f i.tail) = i.tail | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
x : α
xs : List α
ih : ∀ (i : (List.map f xs).Index), map f (unmap f i) = i
i✝ : (List.map f (x :: xs)).Index
i : (List.map f xs).Index
⊢ map f (unmap f i.tail) = i.tail
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_eq_iff_eq_unmap | [36, 1] | [39, 31] | constructor | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ map f i = j ↔ i = unmap f j | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ map f i = j → i = unmap f j
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ i = unmap f j → map f i = j | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ map f i = j ↔ i = unmap f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_eq_iff_eq_unmap | [36, 1] | [39, 31] | intro h | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ map f i = j → i = unmap f j | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : map f i = j
⊢ i = unmap f j | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ map f i = j → i = unmap f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_eq_iff_eq_unmap | [36, 1] | [39, 31] | rw [←h, unmap_map] | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : map f i = j
⊢ i = unmap f j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : map f i = j
⊢ i = unmap f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_eq_iff_eq_unmap | [36, 1] | [39, 31] | intro h | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ i = unmap f j → map f i = j | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : i = unmap f j
⊢ map f i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
⊢ i = unmap f j → map f i = j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.map_eq_iff_eq_unmap | [36, 1] | [39, 31] | rw [h, map_unmap] | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : i = unmap f j
⊢ map f i = j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
j : (List.map f xs).Index
h : i = unmap f j
⊢ map f i = j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_eq_iff_eq_map | [41, 1] | [44, 31] | constructor | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ unmap f i = j ↔ i = map f j | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ unmap f i = j → i = map f j
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ i = map f j → unmap f i = j | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ unmap f i = j ↔ i = map f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_eq_iff_eq_map | [41, 1] | [44, 31] | intro h | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ unmap f i = j → i = map f j | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : unmap f i = j
⊢ i = map f j | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ unmap f i = j → i = map f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_eq_iff_eq_map | [41, 1] | [44, 31] | rw [←h, map_unmap] | case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : unmap f i = j
⊢ i = map f j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : unmap f i = j
⊢ i = map f j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_eq_iff_eq_map | [41, 1] | [44, 31] | intro h | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ i = map f j → unmap f i = j | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : i = map f j
⊢ unmap f i = j | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
⊢ i = map f j → unmap f i = j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.unmap_eq_iff_eq_map | [41, 1] | [44, 31] | rw [h, unmap_map] | case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : i = map f j
⊢ unmap f i = j | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
j : xs.Index
h : i = map f j
⊢ unmap f i = j
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.val_map | [55, 1] | [58, 26] | induction i with
| head => rfl
| tail _ ih => exact ih | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
⊢ (map f i).val = f i.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : xs.Index
⊢ (map f i).val = f i.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.val_map | [55, 1] | [58, 26] | rfl | case head
α : Type u_1
β : Type u_2
f : α → β
xs : List α
x✝ : α
xs✝ : List α
⊢ (map f head).val = f head.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case head
α : Type u_1
β : Type u_2
f : α → β
xs : List α
x✝ : α
xs✝ : List α
⊢ (map f head).val = f head.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.val_map | [55, 1] | [58, 26] | exact ih | case tail
α : Type u_1
β : Type u_2
f : α → β
xs xs✝ : List α
x✝ : α
a✝ : xs✝.Index
ih : (map f a✝).val = f a✝.val
⊢ (map f a✝.tail).val = f a✝.tail.val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case tail
α : Type u_1
β : Type u_2
f : α → β
xs xs✝ : List α
x✝ : α
a✝ : xs✝.Index
ih : (map f a✝).val = f a✝.val
⊢ (map f a✝.tail).val = f a✝.tail.val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Index/Map.lean | List.Index.val_unmap | [60, 1] | [61, 42] | rw [←map_unmap f i, val_map, unmap_map] | α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
⊢ i.val = f (unmap f i).val | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
β : Type u_2
f : α → β
xs : List α
i : (List.map f xs).Index
⊢ i.val = f (unmap f i).val
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_div_two | [5, 9] | [5, 63] | omega | x : Nat
⊢ 2 * x / 2 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ 2 * x / 2 = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_div_two | [7, 9] | [7, 67] | omega | x : Nat
⊢ (2 * x + 1) / 2 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ (2 * x + 1) / 2 = x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_mod_two | [9, 9] | [9, 63] | omega | x : Nat
⊢ 2 * x % 2 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ 2 * x % 2 = 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit1_mod_two | [11, 9] | [11, 67] | omega | x : Nat
⊢ (2 * x + 1) % 2 = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x : Nat
⊢ (2 * x + 1) % 2 = 1
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.zero_bitwise | [41, 9] | [42, 21] | rw [bitwise] | f : Bool → Bool → Bool
y : Nat
⊢ bitwise f 0 y = bif f false true then y else 0 | f : Bool → Bool → Bool
y : Nat
⊢ (if 0 = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then 0 else 0
else
let n' := 0 / 2;
let m' := y / 2;
let b₁ := 0 % 2 = 1;
let b₂ := y % 2 = 1;
let r := bitwise f n' m';
if f (de... | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
y : Nat
⊢ bitwise f 0 y = bif f false true then y else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.zero_bitwise | [41, 9] | [42, 21] | simp | f : Bool → Bool → Bool
y : Nat
⊢ (if 0 = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then 0 else 0
else
let n' := 0 / 2;
let m' := y / 2;
let b₁ := 0 % 2 = 1;
let b₂ := y % 2 = 1;
let r := bitwise f n' m';
if f (de... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
y : Nat
⊢ (if 0 = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then 0 else 0
else
let n' := 0 / 2;
let m' := y / 2;
let b₁ := 0 % 2 = 1;
let b₂ := ... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_zero | [44, 9] | [45, 35] | rw [bitwise] | f : Bool → Bool → Bool
x : Nat
⊢ bitwise f x 0 = bif f true false then x else 0 | f : Bool → Bool → Bool
x : Nat
⊢ (if x = 0 then if f false true = true then 0 else 0
else
if 0 = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := 0 / 2;
let b₁ := x % 2 = 1;
let b₂ := 0 % 2 = 1;
let r := bitwise f n' m';
if f (de... | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x : Nat
⊢ bitwise f x 0 = bif f true false then x else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_zero | [44, 9] | [45, 35] | split <;> simp [*] | f : Bool → Bool → Bool
x : Nat
⊢ (if x = 0 then if f false true = true then 0 else 0
else
if 0 = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := 0 / 2;
let b₁ := x % 2 = 1;
let b₂ := 0 % 2 = 1;
let r := bitwise f n' m';
if f (de... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x : Nat
⊢ (if x = 0 then if f false true = true then 0 else 0
else
if 0 = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := 0 / 2;
let b₁ := x % 2 = 1;
let b₂ := ... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.pos_bitwise_pos | [47, 1] | [51, 34] | rw [bitwise] | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ bitwise f x y = 2 * bitwise f (x / 2) (y / 2) + bif f (decide (x % 2 = 1)) (decide (y % 2 = 1)) then 1 else 0 | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if x = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := y / 2;
let b₁ := x % 2 = 1;
let b₂ := y % 2 = 1;
let r := bitwise f... | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ bitwise f x y = 2 * bitwise f (x / 2) (y / 2) + bif f (decide (x % 2 = 1)) (decide (y % 2 = 1)) then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.pos_bitwise_pos | [47, 1] | [51, 34] | simp [hx, hy] | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if x = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := y / 2;
let b₁ := x % 2 = 1;
let b₂ := y % 2 = 1;
let r := bitwise f... | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if f (decide (x % 2 = 1)) (decide (y % 2 = 1)) = true then bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2) + 1
else bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2)) =
2 * bitwise f (x / 2) (y / 2) + bif f (decide (x % 2 = 1)) (decide (y % 2 =... | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if x = 0 then if f false true = true then y else 0
else
if y = 0 then if f true false = true then x else 0
else
let n' := x / 2;
let m' := y / 2;
let b₁ := x % 2... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.pos_bitwise_pos | [47, 1] | [51, 34] | split <;> simp [*, Nat.two_mul] | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if f (decide (x % 2 = 1)) (decide (y % 2 = 1)) = true then bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2) + 1
else bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2)) =
2 * bitwise f (x / 2) (y / 2) + bif f (decide (x % 2 = 1)) (decide (y % 2 =... | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ (if f (decide (x % 2 = 1)) (decide (y % 2 = 1)) = true then bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2) + 1
else bitwise f (x / 2) (y / 2) + bitwise f (x / 2) (y / 2)) =
2 * bitwise f ... |
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_flip | [53, 1] | [59, 68] | if x = 0 then
simp [*, flip]
else if y = 0 then
simp [*, flip]
else
simp [*, pos_bitwise_pos, flip, bitwise_flip f (x / 2) (y / 2)] | f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f x y = bitwise (flip f) y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
⊢ bitwise f x y = bitwise (flip f) y x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_flip | [53, 1] | [59, 68] | simp [*, flip] | f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ bitwise f x y = bitwise (flip f) y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : x = 0
⊢ bitwise f x y = bitwise (flip f) y x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_flip | [53, 1] | [59, 68] | if y = 0 then
simp [*, flip]
else
simp [*, pos_bitwise_pos, flip, bitwise_flip f (x / 2) (y / 2)] | f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ bitwise f x y = bitwise (flip f) y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝ : ¬x = 0
⊢ bitwise f x y = bitwise (flip f) y x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_flip | [53, 1] | [59, 68] | simp [*, flip] | f : Bool → Bool → Bool
x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ bitwise f x y = bitwise (flip f) y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝¹ : ¬x = 0
h✝ : y = 0
⊢ bitwise f x y = bitwise (flip f) y x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bitwise_flip | [53, 1] | [59, 68] | simp [*, pos_bitwise_pos, flip, bitwise_flip f (x / 2) (y / 2)] | f : Bool → Bool → Bool
x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ bitwise f x y = bitwise (flip f) y x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : Bool → Bool → Bool
x y : Nat
h✝¹ : ¬x = 0
h✝ : ¬y = 0
⊢ bitwise f x y = bitwise (flip f) y x
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | have hx2 : 2 * x ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0 | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | have hy2 : 2 * y ≠ 0 := by intro h; simp [Nat.mul_eq_zero] at h; contradiction | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0 | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
hy2 : 2 * y ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0 | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | simp [pos_bitwise_pos _ hx2 hy2] | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
hy2 : 2 * y ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
hy2 : 2 * y ≠ 0
⊢ bitwise f (2 * x) (2 * y) = 2 * bitwise f x y + bif f false false then 1 else 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | intro h | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ 2 * x ≠ 0 | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : 2 * x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
⊢ 2 * x ≠ 0
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | simp [Nat.mul_eq_zero] at h | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : 2 * x = 0
⊢ False | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : x = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : 2 * x = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | contradiction | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : x = 0
⊢ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
h : x = 0
⊢ False
TACTIC:
|
https://github.com/fgdorais/extra4.git | eb1c6c30f5790bed1bb4b01534d271a2490d9730 | Extra/Nat/Bitwise.lean | Nat.bit0_bitwise_bit0 | [61, 1] | [65, 35] | intro h | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
⊢ 2 * y ≠ 0 | x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
h : 2 * y = 0
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
x y : Nat
f : Bool → Bool → Bool
hx : x ≠ 0
hy : y ≠ 0
hx2 : 2 * x ≠ 0
⊢ 2 * y ≠ 0
TACTIC:
|
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