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/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.ext | [43, 1] | [46, 12] | funext _ x | m : Type u β Type v
instβΒΉ : Monad m
n : Type u β Type w
instβ : Monad n
F G : MonadHom m n
h : β {Ξ± : Type u} (x : m Ξ±), app F x = app G x
β’ F.app = G.app | case h.h
m : Type u β Type v
instβΒΉ : Monad m
n : Type u β Type w
instβ : Monad n
F G : MonadHom m n
h : β {Ξ± : Type u} (x : m Ξ±), app F x = app G x
xβ : Type u
x : m xβ
β’ app F x = app G x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u β Type v
instβΒΉ : Monad m
n : Type u β Type w
instβ : Monad n
F G : MonadHom m n
h : β {Ξ± : Type u} (x : m Ξ±), app F x = app G x
β’ F.app = G.app
TACTIC:
|
https://github.com/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.ext | [43, 1] | [46, 12] | exact h x | case h.h
m : Type u β Type v
instβΒΉ : Monad m
n : Type u β Type w
instβ : Monad n
F G : MonadHom m n
h : β {Ξ± : Type u} (x : m Ξ±), app F x = app G x
xβ : Type u
x : m xβ
β’ app F x = app G x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.h
m : Type u β Type v
instβΒΉ : Monad m
n : Type u β Type w
instβ : Monad n
F G : MonadHom m n
h : β {Ξ± : Type u} (x : m Ξ±), app F x = app G x
xβ : Type u
x : m xβ
β’ app F x = app G x
TACTIC:
|
https://github.com/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.naturality | [49, 1] | [53, 24] | funext x | m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
β’ app F β Functor.map f = Functor.map f β app F | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (app F β Functor.map f) x = (Functor.map f β app F) x | Please generate a tactic in lean4 to solve the state.
STATE:
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
β’ app F β Functor.map f = Functor.map f β app F
TACTIC:
|
https://github.com/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.naturality | [49, 1] | [53, 24] | dsimp | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (app F β Functor.map f) x = (Functor.map f β app F) x | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ app F (f <$> x) = f <$> app F x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (app F β Functor.map f) x = (Functor.map f β app F) x
TACTIC:
|
https://github.com/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.naturality | [49, 1] | [53, 24] | conv =>
lhs; rw [map_eq_pure_bind, F.app_bind]; rhs; ext a; dsimp; rw [F.app_pure] | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ app F (f <$> x) = f <$> app F x | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (do
let a β app F x
pure (f a)) =
f <$> app F x | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ app F (f <$> x) = f <$> app F x
TACTIC:
|
https://github.com/Junology/dijkstra.git | 81543f27ea34811910c10871ecf7e6243f952b19 | Dijkstra/Control/Monad/Hom.lean | MonadHom.naturality | [49, 1] | [53, 24] | rw [map_eq_pure_bind] | case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (do
let a β app F x
pure (f a)) =
f <$> app F x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
m : Type u β Type v
instβΒ³ : Monad m
instβΒ² : LawfulMonad m
n : Type u β Type w
instβΒΉ : Monad n
instβ : LawfulMonad n
F : MonadHom m n
Ξ± Ξ² : Type u
f : Ξ± β Ξ²
x : m Ξ±
β’ (do
let a β app F x
pure (f a)) =
f <$> app F x
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/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | have h1 : β t n, n β€ f^[t] n :=
Nat.rec Nat.le_refl Ξ» t h1 n β¦ (h0 n).trans (h1 _) | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
β’ f = id | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
β’ f = id | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
β’ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | have h2 : StrictMono f :=
strictMono_nat_of_lt_succ Ξ» n β¦ (h1 _ (f n)).trans_lt (h n) | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
β’ f = id | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
β’ f = id | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
β’ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | refine funext Ξ» n β¦ (h0 n).antisymm' ?_ | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
β’ f = id | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f n β€ n | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
β’ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | rw [β Nat.lt_succ_iff, β h2.lt_iff_lt] | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f n β€ n | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f (f n) < f n.succ | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f n β€ n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | exact (h1 _ _).trans_lt (h n) | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f (f n) < f n.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
h0 : β (n : β), n β€ f n
h1 : β (t n : β), n β€ f^[t] n
h2 : StrictMono f
n : β
β’ f (f n) < f n.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | suffices β k n : β, k β€ n β k β€ f n from Ξ» n β¦ this n n n.le_refl | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
β’ β (n : β), n β€ f n | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
β’ β (k n : β), k β€ n β k β€ f n | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
β’ β (n : β), n β€ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | refine Nat.rec (Ξ» k _ β¦ (f k).zero_le) (Ξ» k h0 n h1 β¦ ?_) | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
β’ β (k n : β), k β€ n β k β€ f n | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n
β’ k.succ β€ f n | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
β’ β (k n : β), k β€ n β k β€ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | rcases n with _ | n | f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n
β’ k.succ β€ f n | case zero
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
h1 : k.succ β€ 0
β’ k.succ β€ f 0
case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k.succ β€ f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n
β’ k.succ β€ f n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | exact absurd k.succ_pos h1.not_lt | case zero
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
h1 : k.succ β€ 0
β’ k.succ β€ f 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
h1 : k.succ β€ 0
β’ k.succ β€ f 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | refine (h n).trans_le' ?_ | case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k.succ β€ f (n + 1) | case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k β€ f^[g n + 2] n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k.succ β€ f (n + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | generalize g n + 2 = t | case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k β€ f^[g n + 2] n | case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
t : β
β’ k β€ f^[t] n | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
β’ k β€ f^[g n + 2] n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.main_step | [28, 1] | [46, 32] | exact t.rec (Nat.succ_le_succ_iff.mp h1)
(Ξ» t h2 β¦ (h0 _ h2).trans_eq (f.iterate_succ_apply' _ _).symm) | case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
t : β
β’ k β€ f^[t] n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f g : β β β
h : β (k : β), f^[g k + 2] k < f (k + 1)
k : β
h0 : β (n : β), k β€ n β k β€ f n
n : β
h1 : k.succ β€ n + 1
t : β
β’ k β€ f^[t] n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | refine Iff.symm β¨Ξ» h k β¦ ?_, Ξ» h β¦ ?_β© | f g : β β β
β’ (β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)) β f = id β§ g = fun x => 0 | case refine_1
f g : β β β
h : f = id β§ g = fun x => 0
k : β
β’ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
case refine_2
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
β’ f = id β§ g = fun x => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
β’ (β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)) β f = id β§ g = fun x => 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | rcases h with β¨rfl, rflβ© | case refine_1
f g : β β β
h : f = id β§ g = fun x => 0
k : β
β’ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1) | case refine_1.intro
k : β
β’ id^[(fun x => 0) k + 2] k + ((fun x => 0)^[id k + 1] k + (fun x => 0) (k + 1)) + 1 = id (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f g : β β β
h : f = id β§ g = fun x => 0
k : β
β’ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | rw [iterate_id, iterate_succ_apply'] | case refine_1.intro
k : β
β’ id^[(fun x => 0) k + 2] k + ((fun x => 0)^[id k + 1] k + (fun x => 0) (k + 1)) + 1 = id (k + 1) | case refine_1.intro
k : β
β’ id k + (0 + (fun x => 0) (k + 1)) + 1 = id (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
k : β
β’ id^[(fun x => 0) k + 2] k + ((fun x => 0)^[id k + 1] k + (fun x => 0) (k + 1)) + 1 = id (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | rfl | case refine_1.intro
k : β
β’ id k + (0 + (fun x => 0) (k + 1)) + 1 = id (k + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
k : β
β’ id k + (0 + (fun x => 0) (k + 1)) + 1 = id (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | obtain rfl : f = id := by
refine main_step (g := g) (Ξ» k β¦ ?_)
rw [β h, Nat.lt_succ_iff]
exact Nat.le_add_right _ _ | case refine_2
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
β’ f = id β§ g = fun x => 0 | case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
β’ id = id β§ g = fun x => 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
β’ f = id β§ g = fun x => 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | refine β¨rfl, funext Ξ» n β¦ ?_β© | case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
β’ id = id β§ g = fun x => 0 | case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
n : β
β’ g n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
β’ id = id β§ g = fun x => 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | simp_rw [iterate_id, Function.id_def, Nat.succ_inj',
add_right_eq_self, add_eq_zero_iff] at h | case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
n : β
β’ g n = 0 | case refine_2
g : β β β
n : β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g n = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
g : β β β
h : β (k : β), id^[g k + 2] k + (g^[id k + 1] k + g (k + 1)) + 1 = id (k + 1)
n : β
β’ g n = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | rcases n with _ | n | case refine_2
g : β β β
n : β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g n = 0 | case refine_2.zero
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g 0 = 0
case refine_2.succ
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
n : β
β’ g (n + 1) = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
g : β β β
n : β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g n = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | exacts [(h 0).1, (h n).2] | case refine_2.zero
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g 0 = 0
case refine_2.succ
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
n : β
β’ g (n + 1) = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.zero
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
β’ g 0 = 0
case refine_2.succ
g : β β β
h : β (k : β), g^[k + 1] k = 0 β§ g (k + 1) = 0
n : β
β’ g (n + 1) = 0
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | refine main_step (g := g) (Ξ» k β¦ ?_) | f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
β’ f = id | f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k < f (k + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
β’ f = id
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | rw [β h, Nat.lt_succ_iff] | f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k < f (k + 1) | f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k β€ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k < f (k + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution | [49, 1] | [63, 30] | exact Nat.le_add_right _ _ | f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k β€ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β β β
h : β (k : β), f^[g k + 2] k + (g^[f k + 1] k + g (k + 1)) + 1 = f (k + 1)
k : β
β’ f^[g k + 2] k β€ f^[g k + 2] k + (g^[f k + 1] k + g (k + 1))
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.PNat_exists_Nat_conj | [74, 1] | [77, 53] | simp_rw [PNat.succPNat_natPred] | f : β+ β β+
n : β+
β’ f n = ((fun n => (f n.succPNat).natPred) n.natPred).succPNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β+ β β+
n : β+
β’ f n = ((fun n => (f n.succPNat).natPred) n.natPred).succPNat
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.PNat_eq_Nat_conj_iff | [79, 1] | [82, 80] | rw [h, PNat.succPNat_natPred, PNat.succPNat_natPred] | f : β+ β β+
g : β β β
h : g = fun n => (f n.succPNat).natPred
n : β+
β’ f n = (g n.natPred).succPNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β+ β β+
g : β β β
h : g = fun n => (f n.succPNat).natPred
n : β+
β’ f n = (g n.natPred).succPNat
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.PNat_Nat_conj_iterate | [84, 1] | [88, 52] | rw [iterate_succ_apply', iterate_succ_apply',
PNat_Nat_conj_iterate f m k] | f : β β β
m : β+
k : β
β’ (fun n => (f n.natPred).succPNat)^[k + 1] m = (f^[k + 1] m.natPred).succPNat | f : β β β
m : β+
k : β
β’ (f (f^[k] m.natPred).succPNat.natPred).succPNat = (f (f^[k] m.natPred)).succPNat | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
m : β+
k : β
β’ (fun n => (f n.natPred).succPNat)^[k + 1] m = (f^[k + 1] m.natPred).succPNat
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.PNat_Nat_conj_iterate | [84, 1] | [88, 52] | rfl | f : β β β
m : β+
k : β
β’ (f (f^[k] m.natPred).succPNat.natPred).succPNat = (f (f^[k] m.natPred)).succPNat | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
m : β+
k : β
β’ (f (f^[k] m.natPred).succPNat.natPred).succPNat = (f (f^[k] m.natPred)).succPNat
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | obtain β¨f, rflβ© := PNat_exists_Nat_conj f | f g : β+ β β+
β’ (β (n : β+), f^[β(g n) + 1] n + (g^[β(f n)] n + g (n + 1)) = f (n + 1) + 1) β f = id β§ g = fun x => 1 | case intro
g : β+ β β+
f : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β(g n) + 1] n + (g^[β((fun n => (f n.natPred).succPNat) n)] n + g (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ g = fun x => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
f g : β+ β β+
β’ (β (n : β+), f^[β(g n) + 1] n + (g^[β(f n)] n + g (n + 1)) = f (n + 1) + 1) β f = id β§ g = fun x => 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | obtain β¨g, rflβ© := PNat_exists_Nat_conj g | case intro
g : β+ β β+
f : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β(g n) + 1] n + (g^[β((fun n => (f n.natPred).succPNat) n)] n + g (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ g = fun x => 1 | case intro.intro
f g : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n) + 1] n +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n)] n +
(fun n => (g n.natPred).succPNat) (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ (fun n => (g n.natPred).succPNat) = fun x => 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
g : β+ β β+
f : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β(g n) + 1] n + (g^[β((fun n => (f n.natPred).succPNat) n)] n + g (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ g = fun x => 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | rw [eq_comm, PNat_eq_Nat_conj_iff, eq_comm (b := Ξ» _ β¦ 1),
PNat_eq_Nat_conj_iff, PNat_to_Nat_prop, Iff.comm] | case intro.intro
f g : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n) + 1] n +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n)] n +
(fun n => (g n.natPred).succPNat) (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ (fun n => (g n.natPred).succPNat) = fun x => 1 | case intro.intro
f g : β β β
β’ ((f = fun n => (id n.succPNat).natPred) β§ g = fun n => PNat.natPred 1) β
β (n : β),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
β’ (β (n : β+),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n) + 1] n +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n)] n +
(fun n => (g n.natPred).succPNat) (n + 1)) =
(fun n => (f n.natPred).succPNat) (n + 1) + 1) β
(fun n => (f n.natPred).succPNat) = id β§ (fun n => (g n.natPred).succPNat) = fun x => 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | refine final_solution.symm.trans <| forall_congr' (Ξ» n β¦ ?_) | case intro.intro
f g : β β β
β’ ((f = fun n => (id n.succPNat).natPred) β§ g = fun n => PNat.natPred 1) β
β (n : β),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1 | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
β’ ((f = fun n => (id n.succPNat).natPred) β§ g = fun n => PNat.natPred 1) β
β (n : β),
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | rw [β PNat.coe_inj, PNat_Nat_conj_iterate, PNat_Nat_conj_iterate] | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1 | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
β((f^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat.natPred).succPNat +
((g^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat.natPred).succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1))) =
β((fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(fun n => (f n.natPred).succPNat)^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat +
((fun n => (g n.natPred).succPNat)^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1)) =
(fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | simp_rw [Nat.natPred_succPNat, PNat.add_coe, Nat.succPNat_coe, Nat.succ_add] | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
β((f^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat.natPred).succPNat +
((g^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat.natPred).succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1))) =
β((fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1) | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
β((f^[β((fun n => (g n.natPred).succPNat) n.succPNat) + 1] n.succPNat.natPred).succPNat +
((g^[β((fun n => (f n.natPred).succPNat) n.succPNat)] n.succPNat.natPred).succPNat +
(fun n => (g n.natPred).succPNat) (n.succPNat + 1))) =
β((fun n => (f n.natPred).succPNat) (n.succPNat + 1) + 1)
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | rw [β add_left_inj 2] | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 + 2 = f (n + 1) + 2 β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 = f (n + 1) β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2011/A4/A4.lean | IMOSL.IMO2011A4.final_solution_PNat | [91, 1] | [101, 29] | rfl | case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 + 2 = f (n + 1) + 2 β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f g : β β β
n : β
β’ f^[g n + 2] n + (g^[f n + 1] n + g (n + 1)) + 1 + 2 = f (n + 1) + 2 β
(f^[(g n + 1).succ] n + (g^[(f n).succ] n + (g (n.succPNat + 1).natPred).succ).succ).succ =
(f (n.succPNat + 1).natPred + β1).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_eq | [61, 1] | [64, 52] | rw [f.iterate_succ_apply, h, β h n, β f.iterate_succ_apply'] | f : β β β
h : good f
n : β
β’ f^[4] (n + 1) = f^[4] n + 1 | 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 + 1) = f^[4] n + 1
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_eq | [61, 1] | [64, 52] | rw [Nat.add_succ, h0, h1] | f : β β β
h : good f
h0 : β (n : β), f^[4] (n + 1) = f^[4] n + 1
n : β
h1 : f^[4] n = f^[4] 0 + n
β’ f^[4] n.succ = f^[4] 0 + n.succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
h0 : β (n : β), f^[4] (n + 1) = f^[4] n + 1
n : β
h1 : f^[4] n = f^[4] 0 + n
β’ f^[4] n.succ = f^[4] 0 + n.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.good_injective | [66, 1] | [68, 76] | rwa [iter_four_eq h, iter_four_eq h n, add_right_inj] at h0 | f : β β β
h : good f
m n : β
h0 : f^[4] m = f^[4] n
β’ m = n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
m n : β
h0 : f^[4] m = f^[4] n
β’ m = n
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | intro x h0 | f : β β β
h : good f
β’ (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
β’ (finChainFnOfgood h).iterRangeCompl 3 β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | rw [mem_union, mem_image, mem_insert, mem_singleton, or_assoc] | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x = 0 β¨ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x β {0, (f 0).succ} βͺ image Nat.succ (finChainFnOfgood h).rangeCompl
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | refine x.eq_zero_or_eq_succ_pred.imp id Ξ» h1 β¦ ?_ | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x = 0 β¨ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
h1 : x = x.pred.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
β’ x = 0 β¨ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | generalize x.pred = c at h1 | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
h1 : x = x.pred.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
c : β
h1 : x = c.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
h1 : x = x.pred.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | subst h1 | f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
c : β
h1 : x = c.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x | f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
β’ c.succ = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = c.succ | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
x : β
h0 : x β (finChainFnOfgood h).iterRangeCompl 3
c : β
h1 : x = c.succ
β’ x = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = x
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | refine (em <| c β Set.range f).imp (Ξ» h1 β¦ ?_) (Ξ» h1 β¦ β¨c, ?_, rflβ©) | f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
β’ c.succ = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = c.succ | case refine_1
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c.succ = (f 0).succ
case refine_2
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c β (finChainFnOfgood h).rangeCompl | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
β’ c.succ = (f 0).succ β¨ β a β (finChainFnOfgood h).rangeCompl, a.succ = c.succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | rcases h1 with β¨d, rflβ© | case refine_1
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c.succ = (f 0).succ | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
β’ (f d).succ = (f 0).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c.succ = (f 0).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | refine d.eq_zero_or_eq_succ_pred.elim (Ξ» h2 β¦ h2 βΈ rfl) (Ξ» h2 β¦ ?_) | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
β’ (f d).succ = (f 0).succ | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
β’ (f d).succ = (f 0).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | rw [(finChainFnOfgood h).mem_iterRangeCompl_iff] at h0 | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β (finChainFnOfgood h).iterRangeCompl 3
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | refine absurd β¨d.pred, ?_β© h0 | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ f^[3] d.pred = (f d).succ | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ (f d).succ = (f 0).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | rw [h, β d.pred.succ_eq_add_one, β h2] | case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ f^[3] d.pred = (f d).succ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1.intro
f : β β β
h : good f
d : β
h0 : (f d).succ β Set.range f^[3]
h2 : d = d.pred.succ
β’ f^[3] d.pred = (f d).succ
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iterRangeCompl_three_subset | [78, 1] | [91, 50] | rwa [(finChainFnOfgood h).mem_rangeCompl_iff] | case refine_2
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c β (finChainFnOfgood h).rangeCompl | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
h : good f
c : β
h0 : c.succ β (finChainFnOfgood h).iterRangeCompl 3
h1 : c β Set.range f
β’ c β (finChainFnOfgood h).rangeCompl
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | have h0 := (card_le_card <| iterRangeCompl_three_subset h).trans <|
(card_union_le _ _).trans <| Nat.add_le_add
(card_pair (f 0).succ_ne_zero.symm).le card_image_le | f : β β β
h : good f
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | f : β β β
h : good f
h0 : ((finChainFnOfgood h).iterRangeCompl 3).card β€ 2 + (finChainFnOfgood h).rangeCompl.card
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | rw [(finChainFnOfgood h).iterRangeCompl_card, Nat.succ_mul,
Nat.add_le_add_iff_right, mul_comm, β Nat.le_div_iff_mul_le (Nat.succ_pos 1)] at h0 | f : β β β
h : good f
h0 : ((finChainFnOfgood h).iterRangeCompl 3).card β€ 2 + (finChainFnOfgood h).rangeCompl.card
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | f : β β β
h : good f
h0 : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
h0 : ((finChainFnOfgood h).iterRangeCompl 3).card β€ 2 + (finChainFnOfgood h).rangeCompl.card
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | refine h0.eq_or_lt.elim card_eq_one.mp (Ξ» h0 β¦ ?_) | f : β β β
h : good f
h0 : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : (finChainFnOfgood h).rangeCompl.card < 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
h0 : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | rw [Nat.lt_one_iff, card_eq_zero, β (finChainFnOfgood h).surjective_iff] at h0 | f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : (finChainFnOfgood h).rangeCompl.card < 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : Function.Surjective f
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : (finChainFnOfgood h).rangeCompl.card < 2 / Nat.succ 1
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | obtain β¨a, h0β© := h0.iterate 3 0 | f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : Function.Surjective f
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | case intro
f : β β β
h : good f
h0βΒΉ : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0β : Function.Surjective f
a : β
h0 : f^[3] a = 0
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
h0β : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0 : Function.Surjective f
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.exists_rangeCompl_eq_singleton | [93, 1] | [103, 61] | exact absurd ((h a).symm.trans h0) (f a.succ).succ_ne_zero | case intro
f : β β β
h : good f
h0βΒΉ : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0β : Function.Surjective f
a : β
h0 : f^[3] a = 0
β’ β a, (finChainFnOfgood h).rangeCompl = {a} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
h : good f
h0βΒΉ : (finChainFnOfgood h).rangeCompl.card β€ 2 / Nat.succ 1
h0β : Function.Surjective f
a : β
h0 : f^[3] a = 0
β’ β a, (finChainFnOfgood h).rangeCompl = {a}
TACTIC:
|
https://github.com/mortarsanjaya/IMOSLLean4.git | be127d301e366822fbeeeda49d9fd5b998fb4eb5 | IMOSLLean4/IMO2013/A5/A5.lean | IMOSL.IMO2013A5.iter_four_zero_eq_four | [105, 1] | [114, 60] | obtain β¨a, h0β© := exists_rangeCompl_eq_singleton h | f : β β β
h : good f
β’ f^[4] 0 = 4 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ f^[4] 0 = 4 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
h : good f
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] | suffices (finChainFnOfgood h).iterRangeCompl 4 = range (f^[4] 0) by
rw [β card_range (f^[4] 0), β this,
(finChainFnOfgood h).iterRangeCompl_card, h0, card_singleton] | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ f^[4] 0 = 4 | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ (finChainFnOfgood h).iterRangeCompl 4 = range (f^[4] 0) | 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^[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] | ext n | case intro
f : β β β
h : good f
a : β
h0 : (finChainFnOfgood h).rangeCompl = {a}
β’ (finChainFnOfgood h).iterRangeCompl 4 = range (f^[4] 0) | 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) | 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’ (finChainFnOfgood h).iterRangeCompl 4 = 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] | 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:
|
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