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/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | rw [(finChainFnOfgood h).mem_iterRangeCompl_iff, mem_range, not_iff_comm, not_lt] | case intro.a
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
β’ n β (finChainFnOfgood h).iterRangeCompl 4 β n β range (f^[4] 0) | case intro.a
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
β’ f^[4] 0 β€ n β n β Set.range f^[4] | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
β’ n β (finChainFnOfgood h).iterRangeCompl 4 β n β range (f^[4] 0)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | refine β¨Ξ» h1 β¦ β¨n - f^[4] 0, ?_β©, Ξ» β¨k, h1β© β¦ ?_β© | case intro.a
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
β’ f^[4] 0 β€ n β n β Set.range f^[4] | case intro.a.refine_1
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
h1 : f^[4] 0 β€ n
β’ f^[4] (n - f^[4] 0) = n
case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
β’ f^[4] 0 β€ n β n β Set.range f^[4]
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | rw [β card_range (f^[4] 0), β this,
(finChainFnOfgood h).iterRangeCompl_card, h0, card_singleton] | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
this : (finChainFnOfgood h).iterRangeCompl 4 = range (f^[4] 0)
β’ f^[4] 0 = 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
this : (finChainFnOfgood h).iterRangeCompl 4 = range (f^[4] 0)
β’ f^[4] 0 = 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | rw [iter_four_eq h, Nat.add_sub_of_le h1] | case intro.a.refine_1
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
h1 : f^[4] 0 β€ n
β’ f^[4] (n - f^[4] 0) = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a.refine_1
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
h1 : f^[4] 0 β€ n
β’ f^[4] (n - f^[4] 0) = n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | rw [β h1, iter_four_eq h k] | case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ n | case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ f^[4] 0 + k | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | exact Nat.le_add_right _ k | case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ f^[4] 0 + k | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.a.refine_2
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
n : β
xβ : n β Set.range f^[4]
k : β
h1 : f^[4] k = n
β’ f^[4] 0 β€ f^[4] 0 + k
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_eq_add_four | [116, 1] | [117, 58] | rw [iter_four_eq h, add_comm, iter_four_zero_eq_four h] | f : β β β
h : good f
n : β
β’ f^[4] n = n + 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
n : β
β’ f^[4] n = n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.map_add_four | [119, 1] | [121, 23] | have h1 := iter_four_eq_add_four h | f : β β β
h : good f
n : β
β’ f (n + 4) = f n + 4 | f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (n + 4) = f n + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
n : β
β’ f (n + 4) = f n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.map_add_four | [119, 1] | [121, 23] | rw [β h1, β h1] | f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (n + 4) = f n + 4 | f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (f^[4] n) = f^[4] (f n) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (n + 4) = f n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.map_add_four | [119, 1] | [121, 23] | rfl | f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (f^[4] n) = f^[4] (f n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
n : β
h1 : β (n : β), f^[4] n = n + 4
β’ f (f^[4] n) = f^[4] (f n)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rcases exists_rangeCompl_eq_singleton h with β¨a, h0β© | f : β β β
h : good f
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h1 := iterRangeCompl_three_subset h | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | let C := finChainFnOfgood h | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
C : FinChainFn f := finChainFnOfgood h
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [C.iterRangeCompl_succ, C.iterRangeCompl_succ, C.iterRangeCompl_one] at h1 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
C : FinChainFn f := finChainFnOfgood h
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
C.exactIterRange 2 βͺ (C.exactIterRange 1 βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
h1 : (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
C : FinChainFn f := finChainFnOfgood h
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | unfold FinChainFn.exactIterRange at h1 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
C.exactIterRange 2 βͺ (C.exactIterRange 1 βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
image f^[2] C.rangeCompl βͺ (image f^[1] C.rangeCompl βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
C.exactIterRange 2 βͺ (C.exactIterRange 1 βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h0, image_singleton, image_singleton, image_singleton, union_subset_iff,
union_subset_iff, f.iterate_succ_apply, f.iterate_one] at h1 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
image f^[2] C.rangeCompl βͺ (image f^[1] C.rangeCompl βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 : {f (f a)} β {0, (f 0).succ} βͺ {a.succ} β§ {f a} β {0, (f 0).succ} βͺ {a.succ} β§ {a} β {0, (f 0).succ} βͺ {a.succ}
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
image f^[2] C.rangeCompl βͺ (image f^[1] C.rangeCompl βͺ C.rangeCompl) β
{0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | iterate 3 rw [singleton_subset_iff, mem_union,
mem_singleton, mem_insert, mem_singleton] at h1 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 : {f (f a)} β {0, (f 0).succ} βͺ {a.succ} β§ {f a} β {0, (f 0).succ} βͺ {a.succ} β§ {a} β {0, (f 0).succ} βͺ {a.succ}
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 : {f (f a)} β {0, (f 0).succ} βͺ {a.succ} β§ {f a} β {0, (f 0).succ} βͺ {a.succ} β§ {a} β {0, (f 0).succ} βͺ {a.succ}
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h2 : β n, f (f n) β n := Ξ» n h2 β¦ by
apply absurd (iter_four_eq_add_four h n)
change f (f (f (f n))) β n + 4
rw [h2, h2, self_ne_add_right]
exact Nat.succ_ne_zero 3 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h3 : β n, f n β n := Ξ» n h3 β¦ h2 n <| Function.iterate_fixed h3 2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rcases h1 with β¨h1, h4, (rfl | rfl) | h5β© | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 β¨ f 0 = (f 0).succ) β¨ f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2
case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ) β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
case intro.intro.intro.inr
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h1 : (f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ
h4 : (f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ
h5 : a = a.succ
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [singleton_subset_iff, mem_union,
mem_singleton, mem_insert, mem_singleton] at h1 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ {a} β {0, (f 0).succ} βͺ {a.succ}
β’ f = Nat.succ β¨ f = answer2 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ {a} β {0, (f 0).succ} βͺ {a.succ}
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | apply absurd (iter_four_eq_add_four h n) | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ False | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ Β¬f^[4] n = n + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | change f (f (f (f n))) β n + 4 | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ Β¬f^[4] n = n + 4 | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ f (f (f (f n))) β n + 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ Β¬f^[4] n = n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h2, h2, self_ne_add_right] | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ f (f (f (f n))) β n + 4 | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ 4 β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ f (f (f (f n))) β n + 4
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact Nat.succ_ne_zero 3 | f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ 4 β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h1 :
((f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ) β§
((f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ) β§ ((a = 0 β¨ a = (f 0).succ) β¨ a = a.succ)
n : β
h2 : f (f n) = n
β’ 4 β 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_right (h3 0), or_iff_right (f 0).succ_ne_self.symm] at h4 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 β¨ f 0 = (f 0).succ) β¨ f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : (f 0 = 0 β¨ f 0 = (f 0).succ) β¨ f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_right (h2 0), h4, or_iff_left (h3 1)] at h1 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : (f (f 0) = 0 β¨ f (f 0) = (f 0).succ) β¨ f (f 0) = Nat.succ 0
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h5 : f (f (f 0)) = f 1 + 1 := h 0 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h6 : f (f (f (f 0))) = 4 := iter_four_zero_eq_four h | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5, h1] at h6 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f (f (f (f 0))) = 4
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, h1] at h5 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (f (f 0)) = f 1 + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | left | case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | refine funext (add_four_induction h4 h1 h5 h6 ?_) | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ β (n : β), f n = n.succ β f (n + 4) = (n + 4).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ f = Nat.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | intro n h7 | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ β (n : β), f n = n.succ β f (n + 4) = (n + 4).succ | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : β
h7 : f n = n.succ
β’ f (n + 4) = (n + 4).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
β’ β (n : β), f n = n.succ β f (n + 4) = (n + 4).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [map_add_four h, h7] | case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : β
h7 : f n = n.succ
β’ f (n + 4) = (n + 4).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inl.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {0}
h1 : f (Nat.succ 0) = (Nat.succ 0).succ
h4 : f 0 = Nat.succ 0
h5 : f (Nat.succ 0).succ = (Nat.succ 0).succ + 1
h6 : f ((Nat.succ 0).succ + 1) = 4
n : β
h7 : f n = n.succ
β’ f (n + 4) = (n + 4).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_left (h3 _)] at h4 | case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ) β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : (f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ) β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [or_iff_left (h2 _)] at h1 | case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : (f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ) β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rcases h4 with h4 | h4 | case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0 β¨ f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, or_iff_left (h3 _)] at h1 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1 := h (f 0).succ | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | have h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5 := iter_four_eq_add_four h _ | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5, h1, zero_add] at h6 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f (f (f (f (f 0 + 1)))) = f 0 + 5
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, h1, zero_add] at h5 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f (f (f (f 0 + 1))) = f (f 0 + 2) + 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h5] at h1 h4 h6 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f 0).succ.succ = 0
h4 : f (f 0).succ = (f 0).succ.succ
h5 : f 0 = 1
h6 : f 1 = f 0 + 5
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | right | case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | refine funext (add_four_induction h5 h6 h4 h1 ?_) | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = answer2 | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ β (n : β), f n = answer2 n β f (n + 4) = answer2 (n + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | intro n h7 | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ β (n : β), f n = answer2 n β f (n + 4) = answer2 (n + 4) | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : β
h7 : f n = answer2 n
β’ f (n + 4) = answer2 (n + 4) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
β’ β (n : β), f n = answer2 n β f (n + 4) = answer2 (n + 4)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [map_add_four h, h7, answer2] | case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : β
h7 : f n = answer2 n
β’ f (n + 4) = answer2 (n + 4) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inr.h
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (Nat.succ 1).succ = 0
h4 : f (Nat.succ 1) = (Nat.succ 1).succ
h5 : f 0 = 1
h6 : f 1 = 1 + 5
n : β
h7 : f n = answer2 n
β’ f (n + 4) = answer2 (n + 4)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | rw [h4, or_iff_right (h3 0)] at h1 | case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2 | case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f (f (f 0).succ) = 0 β¨ f (f (f 0).succ) = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact absurd h1 (Nat.lt_succ.mpr (f 0).le_succ).ne | case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inl.inr.inl
f : β β β
h : good f
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h0 : (finChainFnOfgood h).rangeCompl = {(f 0).succ}
h1 : f 0 = (f 0).succ.succ
h4 : f (f 0).succ = 0
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_imp_succ_or_answer2 | [123, 1] | [168, 38] | exact absurd h5 a.lt_succ_self.ne | case intro.intro.intro.inr
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h1 : (f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ
h4 : (f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ
h5 : a = a.succ
β’ f = Nat.succ β¨ f = answer2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.inr
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
C : FinChainFn f := finChainFnOfgood h
h2 : β (n : β), f (f n) β n
h3 : β (n : β), f n β n
h1 : (f (f a) = 0 β¨ f (f a) = (f 0).succ) β¨ f (f a) = a.succ
h4 : (f a = 0 β¨ f a = (f 0).succ) β¨ f a = a.succ
h5 : a = a.succ
β’ f = Nat.succ β¨ f = answer2
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | by_contra h | a : β β β
ha : Antitone a
β’ β C N, β (n : β), a (n + N) = C | a : β β β
ha : Antitone a
h : Β¬β C N, β (n : β), a (n + N) = C
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β β β
ha : Antitone a
β’ β C N, β (n : β), a (n + N) = C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | simp only [not_exists, not_forall] at h | a : β β β
ha : Antitone a
h : Β¬β C N, β (n : β), a (n + N) = C
β’ False | a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β β β
ha : Antitone a
h : Β¬β C N, β (n : β), a (n + N) = C
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rcases h (a 0).succ with β¨N, hβ© | a : β β β
ha : Antitone a
h : β (C : β), β N, β (n : β), a (n + N) + C β€ a 0
β’ False | case intro
a : β β β
ha : Antitone a
hβ : β (C : β), β N, β (n : β), a (n + N) + C β€ a 0
N : β
h : β (n : β), a (n + N) + (a 0).succ β€ a 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
a : β β β
ha : Antitone a
h : β (C : β), β N, β (n : β), a (n + N) + C β€ a 0
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | exact (h 0).not_lt (Nat.lt_add_left _ (a 0).lt_succ_self) | case intro
a : β β β
ha : Antitone a
hβ : β (C : β), β N, β (n : β), a (n + N) + C β€ a 0
N : β
h : β (n : β), a (n + N) + (a 0).succ β€ a 0
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β β β
ha : Antitone a
hβ : β (C : β), β N, β (n : β), a (n + N) + C β€ a 0
N : β
h : β (n : β), a (n + N) + (a 0).succ β€ a 0
β’ False
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | induction' C with C hC | a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C : β
β’ β N, β (n : β), a (n + N) + C β€ a 0 | case zero
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
β’ β N, β (n : β), a (n + N) + 0 β€ a 0
case succ
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C : β
hC : β N, β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C : β
β’ β N, β (n : β), a (n + N) + C β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | exact β¨0, Ξ» n β¦ ha n.zero_leβ© | case zero
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
β’ β N, β (n : β), a (n + N) + 0 β€ a 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
β’ β N, β (n : β), a (n + N) + 0 β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rcases hC with β¨N, hCβ© | case succ
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C : β
hC : β N, β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C : β
hC : β N, β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rcases (hC 0).lt_or_eq with h0 | h0 | case succ.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
case succ.intro.inr
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | refine β¨N, Ξ» n β¦ h0.trans_le' (Nat.add_le_add_right (ha ?_) C)β© | case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
n : β
β’ 0 + N β€ n + N | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | exact Nat.add_le_add_right n.zero_le N | case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
n : β
β’ 0 + N β€ n + N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inl
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C < a 0
n : β
β’ 0 + N β€ n + N
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rcases h (a 0 - C) N with β¨K, h1β© | case succ.intro.inr
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a 0 - C
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rw [β h0, Nat.add_sub_cancel, Nat.zero_add] at h1 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a 0 - C
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a 0 - C
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | refine β¨K + N, Ξ» n β¦ Nat.add_one_le_iff.mpr ?_β© | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
β’ β N, β (n : β), a (n + N) + (C + 1) β€ a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | rw [β h0, Nat.zero_add] | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a 0 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a N + C | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | refine Nat.add_lt_add_right ((ha <| (K + N).le_add_left n).trans_lt ?_) C | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a N + C | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ a (K + N) < a N | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ (a (n + (K + N))).add C < a N + C
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/Extra/NatSequence/AntitoneConst.lean | IMOSL.Extra.NatSeq_antitone_imp_const | [19, 1] | [35, 60] | exact (ha (N.le_add_left K)).lt_of_ne h1 | case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ a (K + N) < a N | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.intro.inr.intro
a : β β β
ha : Antitone a
h : β (x x_1 : β), β x_2, Β¬a (x_2 + x_1) = x
C N : β
hC : β (n : β), a (n + N) + C β€ a 0
h0 : a (0 + N) + C = a 0
K : β
h1 : Β¬a (K + N) = a N
n : β
β’ a (K + N) < a N
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | by_cases h1 : c = 0 | c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
β’ 2 * (2 * c) β£ m - 3 | case pos
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : c = 0
β’ 2 * (2 * c) β£ m - 3
case neg
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
β’ 2 * (2 * c) β£ m - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
β’ 2 * (2 * c) β£ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rwa [h1, mul_zero, β h1] | case pos
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : c = 0
β’ 2 * (2 * c) β£ m - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : c = 0
β’ 2 * (2 * c) β£ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rcases h with β¨d, hβ© | case neg
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
β’ 2 * (2 * c) β£ m - 3 | case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * (2 * c) β£ m - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
c m : β€
h : 2 * c β£ m - 3
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
β’ 2 * (2 * c) β£ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rw [h, mul_comm] | case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * (2 * c) β£ m - 3 | case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * c * 2 β£ 2 * c * d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * (2 * c) β£ m - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | apply mul_dvd_mul_left | case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * c * 2 β£ 2 * c * d | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 β£ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 * c * 2 β£ 2 * c * d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | have X : (2 : β€) β 0 := two_ne_zero | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 β£ d | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
β’ 2 β£ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
β’ 2 β£ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | have X0 : (3 / 2 : β€) = 1 := rfl | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
β’ 2 β£ d | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
β’ 2 β£ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | rw [f, eq_add_of_sub_eq h, add_mul, add_sub_assoc, mul_assoc, β mul_sub_one,
dvd_add_right β¨_, rflβ©, mul_assoc, add_comm, Int.add_mul_ediv_left _ _ X,
X0, add_sub_cancel_left, β two_add_one_eq_three, add_one_mul (Ξ± := β€),
β mul_assoc, dvd_add_right β¨_, rflβ©, mul_comm] at h0 | case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d | case neg.intro.h
c m : β€
h1 : Β¬c = 0
d : β€
h0 : c * 2 β£ c * d
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : β€
h0 : 2 * c β£ f m - 3
h1 : Β¬c = 0
d : β€
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.main_claim | [28, 1] | [41, 44] | exact Int.dvd_of_mul_dvd_mul_left h1 h0 | case neg.intro.h
c m : β€
h1 : Β¬c = 0
d : β€
h0 : c * 2 β£ c * d
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.h
c m : β€
h1 : Β¬c = 0
d : β€
h0 : c * 2 β£ c * d
h : m - 3 = 2 * c * d
X : 2 β 0
X0 : 3 / 2 = 1
β’ 2 β£ d
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [iff_not_comm, not_exists] | M : β€
β’ (β k, 2 β£ f^[k] M) β M β 3 | M : β€
β’ M = 3 β β (x : β), Β¬2 β£ f^[x] M | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
β’ (β k, 2 β£ f^[k] M) β M β 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine β¨Ξ» h n β¦ ?_, Ξ» h β¦ ?_β© | M : β€
β’ M = 3 β β (x : β), Β¬2 β£ f^[x] M | case refine_1
M : β€
h : M = 3
n : β
β’ Β¬2 β£ f^[n] M
case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ M = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
β’ M = 3 β β (x : β), Β¬2 β£ f^[x] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | suffices h0 : β n k, 2 ^ (n + 1) β£ f^[k] M - 3 by
let K := (M - 3).natAbs
refine eq_of_sub_eq_zero <| Int.eq_zero_of_abs_lt_dvd (h0 K 0) <| ?_
rw [β Int.natCast_natAbs, β Nat.cast_ofNat (n := 2), β Int.natCast_pow]
exact Int.ofNat_lt.mpr (K.lt_succ_self.trans K.succ.lt_two_pow) | case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ M = 3 | case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine Nat.rec (Ξ» k β¦ ?_) (Ξ» n h0 k β¦ ?_) | case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3 | case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ 2 ^ (Nat.zero + 1) β£ f^[k] M - 3
case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n.succ + 1) β£ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
β’ β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | have h0 : f 3 = 3 := rfl | case refine_1
M : β€
h : M = 3
n : β
β’ Β¬2 β£ f^[n] M | case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ f^[n] M | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : β€
h : M = 3
n : β
β’ Β¬2 β£ f^[n] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [h, Function.iterate_fixed h0, β two_add_one_eq_three] | case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ f^[n] M | case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ 2 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ f^[n] M
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact Int.two_not_dvd_two_mul_add_one 1 | case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ 2 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
M : β€
h : M = 3
n : β
h0 : f 3 = 3
β’ Β¬2 β£ 2 + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | let K := (M - 3).natAbs | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
β’ M = 3 | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ M = 3 | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
β’ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine eq_of_sub_eq_zero <| Int.eq_zero_of_abs_lt_dvd (h0 K 0) <| ?_ | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ M = 3 | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ |M - 3| < 2 ^ (K + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ M = 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [β Int.natCast_natAbs, β Nat.cast_ofNat (n := 2), β Int.natCast_pow] | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ |M - 3| < 2 ^ (K + 1) | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ β(M - 3).natAbs < β(OfNat.ofNat 2 ^ (K + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ |M - 3| < 2 ^ (K + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact Int.ofNat_lt.mpr (K.lt_succ_self.trans K.succ.lt_two_pow) | M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ β(M - 3).natAbs < β(OfNat.ofNat 2 ^ (K + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
h0 : β (n k : β), 2 ^ (n + 1) β£ f^[k] M - 3
K : β := (M - 3).natAbs
β’ β(M - 3).natAbs < β(OfNat.ofNat 2 ^ (K + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [Int.dvd_iff_emod_eq_zero, Nat.zero_add, pow_one,
β Int.even_iff, Int.even_sub', Int.odd_iff_not_even,
Int.even_iff, β Int.dvd_iff_emod_eq_zero] | case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ 2 ^ (Nat.zero + 1) β£ f^[k] M - 3 | case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ Β¬2 β£ f^[k] M β Odd 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ 2 ^ (Nat.zero + 1) β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact iff_of_true (h k) (Int.odd_iff.mpr rfl) | case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ Β¬2 β£ f^[k] M β Odd 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
k : β
β’ Β¬2 β£ f^[k] M β Odd 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [pow_succ', pow_succ'] | case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n.succ + 1) β£ f^[k] M - 3 | case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * (2 * 2 ^ n) β£ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n.succ + 1) β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | refine main_claim ?_ ?_ | case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * (2 * 2 ^ n) β£ f^[k] M - 3 | case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f^[k] M - 3
case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f (f^[k] M) - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * (2 * 2 ^ n) β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [β pow_succ'] | case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f^[k] M - 3 | case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact h0 k | case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k] M - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_1
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | rw [β pow_succ', β f.iterate_succ_apply'] | case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f (f^[k] M) - 3 | case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k.succ] M - 3 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 * 2 ^ n β£ f (f^[k] M) - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2015/N1/N1.lean | IMOSL.IMO2015N1.final_solution | [44, 1] | [66, 66] | exact h0 (k + 1) | case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k.succ] M - 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.refine_2.refine_2
M : β€
h : β (x : β), Β¬2 β£ f^[x] M
n : β
h0 : β (k : β), 2 ^ (n + 1) β£ f^[k] M - 3
k : β
β’ 2 ^ (n + 1) β£ f^[k.succ] M - 3
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.Even_iff_bodd | [28, 1] | [30, 59] | rw [Nat.even_iff, Nat.mod_two_of_bodd, Bool.cond_eq_ite,
Nat.one_ne_zero.ite_eq_right_iff, Bool.bool_iff_false] | k : β
β’ Even k β k.bodd = false | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
k : β
β’ Even k β k.bodd = false
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.xor_eq_false_iff_eq | [32, 1] | [33, 55] | rw [β Bool.bool_iff_false, Bool.xor_iff_ne, not_not] | a b : Bool
β’ xor a b = false β a = b | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b : Bool
β’ xor a b = false β a = b
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | let f : β β Fin N β Bool := Ξ» a k β¦ (Ξ© (a.succ + k)).bodd | N : β
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | have h : Β¬f.Injective := not_injective_infinite_finite f | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬Function.Injective f
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2009/N2/N2.lean | IMOSL.IMO2009N2.final_solution_part1 | [36, 1] | [48, 28] | rw [Function.Injective] at h | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬Function.Injective f
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬β β¦aβ aβ : ββ¦, f aβ = f aβ β aβ = aβ
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k))) | Please generate a tactic in lean4 to solve the state.
STATE:
N : β
f : β β Fin N β Bool := fun a k => (Ξ© (a.succ + βk)).bodd
h : Β¬Function.Injective f
β’ β a b, a β b β§ β k < N, Even (Ξ© ((a + k) * (b + k)))
TACTIC:
|
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