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/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | exact? | aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
this : b * b < c * c
β’ b < c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
this : b * b < c * c
β’ b < c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | linarith | aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
thisβ : b * b < c * c
this : b < c
β’ c * c < (b + 1) * (b + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
thisβ : b * b < c * c
this : b < c
β’ c * c < (b + 1) * (b + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | exact? | aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
thisβΒΉ : b * b < c * c
thisβ : b < c
this : c * c < (b + 1) * (b + 1)
β’ c < b + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
aβ bβ : β
haβ : 0 < aβ
hb : 0 < bβ
a b : β
ha : 0 < a
hab : a β€ b
c : β
hc : b * b + a = c * c
thisβΒΉ : b * b < c * c
thisβ : b < c
this : c * c < (b + 1) * (b + 1)
β’ c < b + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | exact .inr (h1 ha h) | case inl
a b : β
ha : 0 < a
hb : 0 < b
h1 : β {a b : β}, 0 < a β a β€ b β Β¬IsSquare (b ^ 2 + a)
h : a β€ b
β’ Β¬IsSquare (a ^ 2 + b) β¨ Β¬IsSquare (b ^ 2 + a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
a b : β
ha : 0 < a
hb : 0 < b
h1 : β {a b : β}, 0 < a β a β€ b β Β¬IsSquare (b ^ 2 + a)
h : a β€ b
β’ Β¬IsSquare (a ^ 2 + b) β¨ Β¬IsSquare (b ^ 2 + a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Solutions/Solutions4.lean | exercise4_6 | [155, 1] | [167, 25] | exact .inl (h1 hb h) | case inr
a b : β
ha : 0 < a
hb : 0 < b
h1 : β {a b : β}, 0 < a β a β€ b β Β¬IsSquare (b ^ 2 + a)
h : b β€ a
β’ Β¬IsSquare (a ^ 2 + b) β¨ Β¬IsSquare (b ^ 2 + a) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
a b : β
ha : 0 < a
hb : 0 < b
h1 : β {a b : β}, 0 < a β a β€ b β Β¬IsSquare (b ^ 2 + a)
h : b β€ a
β’ Β¬IsSquare (a ^ 2 + b) β¨ Β¬IsSquare (b ^ 2 + a)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.le_def | [391, 1] | [391, 85] | simp | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ U β€ V β βU β βV | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ U β€ V β βU β βV
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.cl_le_iff | [426, 1] | [427, 37] | sorry | X : Type u_1
instβ : TopologicalSpace X
Uβ V U : RegularOpens X
C : Closeds X
β’ cl U β€ C β U β€ Closeds.int C | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V U : RegularOpens X
C : Closeds X
β’ cl U β€ C β U β€ Closeds.int C
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.cl_int | [429, 1] | [429, 48] | sorry | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ Closeds.int (cl U) = U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ Closeds.int (cl U) = U
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_inf | [445, 1] | [447, 14] | have : U β V = (U.cl β V.cl).int := rfl | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ β(U β V) = βU β© βV | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = βU β© βV | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ β(U β V) = βU β© βV
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_inf | [445, 1] | [447, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = βU β© βV | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = βU β© βV
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sup | [449, 1] | [451, 14] | have : U β V = (U.cl β V.cl).int := rfl | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ β(U β V) = interior (closure (βU βͺ βV)) | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = interior (closure (βU βͺ βV)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
β’ β(U β V) = interior (closure (βU βͺ βV))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sup | [449, 1] | [451, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = interior (closure (βU βͺ βV)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ Vβ U V : RegularOpens X
this : U β V = Closeds.int (cl U β cl V)
β’ β(U β V) = interior (closure (βU βͺ βV))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_top | [453, 1] | [455, 14] | have : (β€ : RegularOpens X) = (β€ : Closeds X).int := rfl | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ ββ€ = univ | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β€ = Closeds.int β€
β’ ββ€ = univ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ ββ€ = univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_top | [453, 1] | [455, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β€ = Closeds.int β€
β’ ββ€ = univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β€ = Closeds.int β€
β’ ββ€ = univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_bot | [457, 1] | [459, 14] | have : (β₯ : RegularOpens X) = (β₯ : Closeds X).int := rfl | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ ββ₯ = β
| X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β₯ = Closeds.int β₯
β’ ββ₯ = β
| Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
β’ ββ₯ = β
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_bot | [457, 1] | [459, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β₯ = Closeds.int β₯
β’ ββ₯ = β
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
this : β₯ = Closeds.int β₯
β’ ββ₯ = β
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sInf | [461, 1] | [465, 14] | have : sInf U = (sInf (cl '' U)).int := rfl | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
β’ β(sInf U) = interior (ββ ((fun u => closure βu) '' U)) | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
β’ β(sInf U) = interior (ββ ((fun u => closure βu) '' U)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
β’ β(sInf U) = interior (ββ ((fun u => closure βu) '' U))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sInf | [461, 1] | [465, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
β’ β(sInf U) = interior (ββ ((fun u => closure βu) '' U)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sInf U = Closeds.int (sInf (cl '' U))
β’ β(sInf U) = interior (ββ ((fun u => closure βu) '' U))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.Closeds.coe_sSup | [467, 1] | [470, 19] | have : sSup C = Closeds.closure (sSup ((β) '' C)) := rfl | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
β’ β(sSup C) = closure (ββ (SetLike.coe '' C)) | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(sSup C) = closure (ββ (SetLike.coe '' C)) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
β’ β(sSup C) = closure (ββ (SetLike.coe '' C))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.Closeds.coe_sSup | [467, 1] | [470, 19] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(sSup C) = closure (ββ (SetLike.coe '' C)) | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(Closeds.closure (β x β C, βx)) = closure (β x β C, βx) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(sSup C) = closure (ββ (SetLike.coe '' C))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.Closeds.coe_sSup | [467, 1] | [470, 19] | rfl | X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(Closeds.closure (β x β C, βx)) = closure (β x β C, βx) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
U V : RegularOpens X
C : Set (Closeds X)
this : sSup C = Closeds.closure (sSup (SetLike.coe '' C))
β’ β(Closeds.closure (β x β C, βx)) = closure (β x β C, βx)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sSup | [472, 1] | [476, 14] | have : sSup U = (sSup (cl '' U)).int := rfl | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
β’ β(sSup U) = interior (closure (ββ ((fun u => closure βu) '' U))) | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
β’ β(sSup U) = interior (closure (ββ ((fun u => closure βu) '' U))) | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
β’ β(sSup U) = interior (closure (ββ ((fun u => closure βu) '' U)))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_sSup | [472, 1] | [476, 14] | simp [this] | X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
β’ β(sSup U) = interior (closure (ββ ((fun u => closure βu) '' U))) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V : RegularOpens X
U : Set (RegularOpens X)
this : sSup U = Closeds.int (sSup (cl '' U))
β’ β(sSup U) = interior (closure (ββ ((fun u => closure βu) '' U)))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture11.lean | RegularOpens.coe_compl | [490, 1] | [491, 79] | sorry | X : Type u_1
instβ : TopologicalSpace X
Uβ V U : RegularOpens X
β’ βUαΆ = interior (βU)αΆ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type u_1
instβ : TopologicalSpace X
Uβ V U : RegularOpens X
β’ βUαΆ = interior (βU)αΆ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | rw [add_pow] | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
β’ (x + y) ^ p = x ^ p + y ^ p | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
β’ (x + y) ^ p = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | have h2 : p.Prime := hp.out | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | have h3 : 0 < p := h2.pos | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | have h4 : range p = insert 0 (Ioo 0 p) | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | case h4
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ range p = insert 0 (Ioo 0 p)
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | have h5 : β i β Ioo 0 p, p β£ Nat.choose p i := by sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | have h6 : β i in Ioo 0 p, x ^ i * y ^ (p - i) * Nat.choose p i = 0 :=
calc
_ = β i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 := by sorry
_ = 0 := by sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
h6 : β i in Ioo 0 p, x ^ i * y ^ (p - i) * β(Nat.choose p i) = 0
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
h6 : β i in Ioo 0 p, x ^ i * y ^ (p - i) * β(Nat.choose p i) = 0
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
h6 : β i in Ioo 0 p, x ^ i * y ^ (p - i) * β(Nat.choose p i) = 0
β’ β m in range (p + 1), x ^ m * y ^ (p - m) * β(Nat.choose p m) = x ^ p + y ^ p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | ext (_|_) <;> simp [h3] | case h4
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ range p = insert 0 (Ioo 0 p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h4
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
β’ range p = insert 0 (Ioo 0 p)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
β’ β i β Ioo 0 p, p β£ Nat.choose p i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
β’ β i β Ioo 0 p, p β£ Nat.choose p i
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β i in Ioo 0 p, x ^ i * y ^ (p - i) * β(Nat.choose p i) = β i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β i in Ioo 0 p, x ^ i * y ^ (p - i) * β(Nat.choose p i) = β i in Ioo 0 p, x ^ i * y ^ (p - i) * 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_3 | [71, 1] | [82, 8] | sorry | p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
p : β
hp : Fact (Nat.Prime p)
K : Type u_1
instβΒΉ : Field K
instβ : CharP K p
x y : K
h2 : Nat.Prime p
h3 : 0 < p
h4 : range p = insert 0 (Ioo 0 p)
h5 : β i β Ioo 0 p, p β£ Nat.choose p i
β’ β i in Ioo 0 p, x ^ i * y ^ (p - i) * 0 = 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Assignments/Assignment5.lean | exercise5_4 | [90, 1] | [94, 8] | sorry | R : Type u_1
M : Type u_2
M' : Type u_3
instββ΅ : Ring R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
instβΒ² : Nontrivial M
instβΒΉ : NoZeroSMulDivisors R M
instβ : Module R (M ββ[R] M)
h : β (r : R) (f : M ββ[R] M) (x : M), β(r β’ f) x = r β’ βf x
r s : R
β’ r * s = s * r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
M : Type u_2
M' : Type u_3
instββ΅ : Ring R
instββ΄ : AddCommGroup M
instβΒ³ : Module R M
instβΒ² : Nontrivial M
instβΒΉ : NoZeroSMulDivisors R M
instβ : Module R (M ββ[R] M)
h : β (r : R) (f : M ββ[R] M) (x : M), β(r β’ f) x = r β’ βf x
r s : R
β’ r * s = s * r
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | cases m | m : β
h0 : m β 0
h1 : m β 1
β’ 2 β€ m | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m β 0
h1 : m β 1
β’ 2 β€ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | contradiction | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | case succ m hm =>
cases m; contradiction
repeat' apply Nat.succ_le_succ
apply zero_le | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | cases m | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1 | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | contradiction | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | repeat' apply Nat.succ_le_succ | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | apply zero_le | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | apply Nat.succ_le_succ | case succ.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 1 β€ nβ + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 1 β€ nβ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | by_cases np : n.Prime | n : β
h : 2 β€ n
β’ β p, Nat.Prime p β§ p β£ n | case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : 2 β€ n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | induction' n using Nat.strong_induction_on with n ih | case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | rw [Nat.prime_def_lt] at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | push_neg at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | rcases np h with β¨m, mltn, mdvdn, mne1β© | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | have : m β 0 := by
intro mz
rw [mz, zero_dvd_iff] at mdvdn
linarith | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | have mgt2 : 2 β€ m := two_le this mne1 | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | by_cases mp : m.Prime | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n | case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | . rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ©
use p, pp
apply pdvd.trans mdvdn | case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | use n, np | case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | intro mz | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ m β 0 | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ m β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | rw [mz, zero_dvd_iff] at mdvdn | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | linarith | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | use m, mp | case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ© | case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | use p, pp | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n | case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [16, 1] | [32, 27] | apply pdvd.trans mdvdn | case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | intro n | β’ β (n : β), β p > n, Nat.Prime p | n : β
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β (n : β), β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | have : 2 β€ Nat.factorial (n + 1) + 1 := by
apply Nat.succ_le_succ
exact Nat.succ_le_of_lt (Nat.factorial_pos _) | n : β
β’ β p > n, Nat.Prime p | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | rcases exists_prime_factor this with β¨p, pp, pdvdβ© | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | refine' β¨p, _, ppβ© | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | by_contra ple | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | push_neg at ple | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | have : p β£ Nat.factorial (n + 1) := by
apply Nat.dvd_factorial
apply pp.pos
linarith | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False | case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | have : p β£ 1 := by
convert Nat.dvd_sub' pdvd this
simp | case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False | case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | have := Nat.le_of_dvd zero_lt_one this | case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False | case intro.intro
n : β
thisβΒ² : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβΒΉ : p β£ Nat.factorial (n + 1)
thisβ : p β£ 1
this : p β€ 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | linarith [pp.two_le] | case intro.intro
n : β
thisβΒ² : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβΒΉ : p β£ Nat.factorial (n + 1)
thisβ : p β£ 1
this : p β€ 1
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
thisβΒ² : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβΒΉ : p β£ Nat.factorial (n + 1)
thisβ : p β£ 1
this : p β€ 1
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | apply Nat.succ_le_succ | n : β
β’ 2 β€ Nat.factorial (n + 1) + 1 | case a
n : β
β’ 1 β€ Nat.factorial (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ 2 β€ Nat.factorial (n + 1) + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | exact Nat.succ_le_of_lt (Nat.factorial_pos _) | case a
n : β
β’ 1 β€ Nat.factorial (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : β
β’ 1 β€ Nat.factorial (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | apply Nat.dvd_factorial | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β£ Nat.factorial (n + 1) | case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ 0 < p
case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β£ Nat.factorial (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | apply pp.pos | case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ 0 < p
case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1 | case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ 0 < p
case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | linarith | case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β€ n + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | convert Nat.dvd_sub' pdvd this | n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ p β£ 1 | case h.e'_4
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ 1 = Nat.factorial (n + 1) + 1 - Nat.factorial (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ p β£ 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [34, 1] | [53, 23] | simp | case h.e'_4
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ 1 = Nat.factorial (n + 1) + 1 - Nat.factorial (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ 1 = Nat.factorial (n + 1) + 1 - Nat.factorial (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [82, 1] | [87, 13] | cases prime_q.eq_one_or_self_of_dvd _ h | p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
β’ p = q | case inl
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = 1
β’ p = q
case inr
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = q
β’ p = q | Please generate a tactic in lean4 to solve the state.
STATE:
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
β’ p = q
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [82, 1] | [87, 13] | assumption | case inr
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = q
β’ p = q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = q
β’ p = q
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | Nat.Prime.eq_of_dvd_of_prime | [82, 1] | [87, 13] | linarith [prime_p.two_le] | case inl
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = 1
β’ p = q | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
p q : β
prime_p : Nat.Prime p
prime_q : Nat.Prime q
h : p β£ q
hβ : p = 1
β’ p = q
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | intro hβ hβ | s : Finset β
p : β
prime_p : Nat.Prime p
β’ (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s | s : Finset β
p : β
prime_p : Nat.Prime p
hβ : β n β s, Nat.Prime n
hβ : p β£ β n in s, n
β’ p β s | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
p : β
prime_p : Nat.Prime p
β’ (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | induction' s using Finset.induction_on with a s ans ih | s : Finset β
p : β
prime_p : Nat.Prime p
hβ : β n β s, Nat.Prime n
hβ : p β£ β n in s, n
β’ p β s | case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p β£ β n in β
, n
β’ p β β
case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : β n β insert a s, Nat.Prime n
hβ : p β£ β n in insert a s, n
β’ p β insert a s | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
p : β
prime_p : Nat.Prime p
hβ : β n β s, Nat.Prime n
hβ : p β£ β n in s, n
β’ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | simp [Finset.prod_insert ans, prime_p.dvd_mul] at hβ hβ | case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : β n β insert a s, Nat.Prime n
hβ : p β£ β n in insert a s, n
β’ p β insert a s | case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p β insert a s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : β n β insert a s, Nat.Prime n
hβ : p β£ β n in insert a s, n
β’ p β insert a s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | rw [mem_insert] | case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p β insert a s | case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p = a β¨ p β s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p β insert a s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | rcases hβ with hβ | hβ | case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p = a β¨ p β s | case insert.inl
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a β¨ p β s
case insert.inr
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p = a β¨ p β s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a β¨ p β£ β n in s, n
β’ p = a β¨ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | right | case insert.inr
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p = a β¨ p β s | case insert.inr.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p β s | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.inr
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p = a β¨ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | exact ih hβ.2 hβ | case insert.inr.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p β s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.inr.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ β n in s, n
β’ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | simp at hβ | case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p β£ β n in β
, n
β’ p β β
| case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p = 1
β’ p β β
| Please generate a tactic in lean4 to solve the state.
STATE:
case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p β£ β n in β
, n
β’ p β β
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | linarith [prime_p.two_le] | case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p = 1
β’ p β β
| no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case empty
p : β
prime_p : Nat.Prime p
hβ : β n β β
, Nat.Prime n
hβ : p = 1
β’ p β β
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | left | case insert.inl
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a β¨ p β s | case insert.inl.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.inl
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a β¨ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.mem_of_dvd_prod_primes | [89, 1] | [101, 19] | exact prime_p.eq_of_dvd_of_prime hβ.1 hβ | case insert.inl.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case insert.inl.h
p : β
prime_p : Nat.Prime p
a : β
s : Finset β
ans : a β s
ih : (β n β s, Nat.Prime n) β p β£ β n in s, n β p β s
hβ : Nat.Prime a β§ β a β s, Nat.Prime a
hβ : p β£ a
β’ p = a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | intro s | β’ β (s : Finset β), β p, Nat.Prime p β§ p β s | s : Finset β
β’ β p, Nat.Prime p β§ p β s | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β (s : Finset β), β p, Nat.Prime p β§ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | by_contra h | s : Finset β
β’ β p, Nat.Prime p β§ p β s | s : Finset β
h : Β¬β p, Nat.Prime p β§ p β s
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
β’ β p, Nat.Prime p β§ p β s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | push_neg at h | s : Finset β
h : Β¬β p, Nat.Prime p β§ p β s
β’ False | s : Finset β
h : β (p : β), Nat.Prime p β p β s
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
h : Β¬β p, Nat.Prime p β§ p β s
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | set s' := s.filter Nat.Prime with s'_def | s : Finset β
h : β (p : β), Nat.Prime p β p β s
β’ False | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
h : β (p : β), Nat.Prime p β p β s
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | have mem_s' : β {n : β}, n β s' β n.Prime := by
intro n
simp [s'_def]
apply h | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ False | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | have : 2 β€ (β i in s', i) + 1 := by
apply Nat.succ_le_succ
apply Nat.succ_le_of_lt
apply Finset.prod_pos
intro n ns'
apply (mem_s'.mp ns').pos | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
β’ False | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | rcases exists_prime_factor this with β¨p, pp, pdvdβ© | s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
β’ False | case intro.intro
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
p : β
pp : Nat.Prime p
pdvd : p β£ β i in s', i + 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite' | [103, 1] | [128, 23] | have : p β£ β i in s', i := by
apply dvd_prod_of_mem
rw [mem_s']
apply pp | case intro.intro
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
p : β
pp : Nat.Prime p
pdvd : p β£ β i in s', i + 1
β’ False | case intro.intro
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
thisβ : 2 β€ β i in s', i + 1
p : β
pp : Nat.Prime p
pdvd : p β£ β i in s', i + 1
this : p β£ β i in s', i
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
s : Finset β
h : β (p : β), Nat.Prime p β p β s
s' : Finset β := filter Nat.Prime s
s'_def : s' = filter Nat.Prime s
mem_s' : β {n : β}, n β s' β Nat.Prime n
this : 2 β€ β i in s', i + 1
p : β
pp : Nat.Prime p
pdvd : p β£ β i in s', i + 1
β’ False
TACTIC:
|
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