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/Lectures/Lecture7.lean | coe_fib_eq | [202, 1] | [210, 9] | ring | n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ k * ϕ - ψ ^ k * ψ + (ϕ ^ k - ψ ^ k) = ϕ ^ k * (ϕ + 1) - ψ ^ k * (ψ + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n k : ℕ
ih1 : ↑(fib k) = (ϕ ^ k - ψ ^ k) / (ϕ - ψ)
ih2 : ↑(fib (k + 1)) = (ϕ ^ (k + 1) - ψ ^ (k + 1)) / (ϕ - ψ)
⊢ ϕ ^ k * ϕ - ψ ^ k * ψ + (ϕ ^ k - ψ ^ k) = ϕ ^ k * (ϕ + 1) - ψ ^ k * (ψ + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | fac_dvd_fac | [245, 1] | [245, 70] | sorry | n✝ n m : ℕ
h : n ≤ m
⊢ fac n ∣ fac m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n✝ n m : ℕ
h : n ≤ m
⊢ fac n ∣ fac m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | intro s hs | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
⊢ {s | IsOpen s} ⊆ {s | IsOpen s} | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ s ∈ {s | IsOpen s} | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
⊢ {s | IsOpen s} ⊆ {s | IsOpen s}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | simp | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ s ∈ {s | IsOpen s} | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ IsOpen s | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ s ∈ {s | IsOpen s}
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | simp at hs | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ IsOpen s | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : IsOpen s
⊢ IsOpen s | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : s ∈ {s | IsOpen s}
⊢ IsOpen s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | induction hs | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : IsOpen s
⊢ IsOpen s | case basic
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s s✝ : Set α
a✝ : s✝ ∈ g
⊢ IsOpen s✝
case univ
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
⊢ IsOpen Set.univ
case inter
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s s✝ t✝ : Set α
a✝¹ : GenerateOpen g s✝
a✝ : GenerateOpen g t✝
a_ih✝¹ : IsOpen s✝
a_ih✝ : IsOpen t✝
⊢ IsOpen (s✝ ∩ t✝)
case sUnion
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S✝ : Set (Set α)
a✝ : ∀ s ∈ S✝, GenerateOpen g s
a_ih✝ : ∀ s ∈ S✝, IsOpen s
⊢ IsOpen (⋃₀ S✝) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
hs : IsOpen s
⊢ IsOpen s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | case basic s hs =>
apply h
exact hs | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ IsOpen s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ IsOpen s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | case univ =>
simp | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
⊢ IsOpen Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
⊢ IsOpen Set.univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | case inter s s' _ _ ihs ihs' =>
apply IsOpen.inter
exact ihs
exact ihs' | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen (s ∩ s') | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen (s ∩ s')
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | case sUnion S _ hS =>
apply isOpen_sUnion
exact hS | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ IsOpen (⋃₀ S) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ IsOpen (⋃₀ S)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | apply h | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ IsOpen s | case a
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ s ∈ g | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ IsOpen s
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | exact hs | case a
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ s ∈ g | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s : Set α
hs : s ∈ g
⊢ s ∈ g
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | simp | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
⊢ IsOpen Set.univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
⊢ IsOpen Set.univ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | apply IsOpen.inter | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen (s ∩ s') | case h₁
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s
case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s' | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen (s ∩ s')
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | exact ihs | case h₁
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s
case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s' | case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s' | Please generate a tactic in lean4 to solve the state.
STATE:
case h₁
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s
case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | exact ihs' | case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h₂
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s✝ s s' : Set α
a✝¹ : GenerateOpen g s
a✝ : GenerateOpen g s'
ihs : IsOpen s
ihs' : IsOpen s'
⊢ IsOpen s'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | apply isOpen_sUnion | n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ IsOpen (⋃₀ S) | case h
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ ∀ t_1 ∈ S, IsOpen t_1 | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ IsOpen (⋃₀ S)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture7.lean | le_generateFrom_iff_subset_isOpen | [258, 1] | [276, 13] | exact hS | case h
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ ∀ t_1 ∈ S, IsOpen t_1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
n : ℕ
α : Type u_1
t : TopologicalSpace α
g : Set (Set α)
h : g ⊆ {s | IsOpen s}
s : Set α
S : Set (Set α)
a✝ : ∀ s ∈ S, GenerateOpen g s
hS : ∀ s ∈ S, IsOpen s
⊢ ∀ t_1 ∈ S, IsOpen t_1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | rcases n with _ | n | n : ℕ
⊢ 2 ^ (n - 1) ≤ fac n | case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero
case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
⊢ 2 ^ (n - 1) ≤ fac n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | induction' n with n ih | case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n) | case succ.zero
⊢ 2 ^ (Nat.succ 0 - 1) ≤ fac (Nat.succ 0)
case succ.succ
n : ℕ
ih : 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n)
⊢ 2 ^ (Nat.succ (n + 1) - 1) ≤ fac (Nat.succ (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : ℕ
⊢ 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | simp at * | case succ.succ
n : ℕ
ih : 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n)
⊢ 2 ^ (Nat.succ (n + 1) - 1) ≤ fac (Nat.succ (n + 1)) | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ^ (n + 1) ≤ fac (Nat.succ (n + 1)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 ^ (Nat.succ n - 1) ≤ fac (Nat.succ n)
⊢ 2 ^ (Nat.succ (n + 1) - 1) ≤ fac (Nat.succ (n + 1))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | rw [pow_succ, fac] | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ^ (n + 1) ≤ fac (Nat.succ (n + 1)) | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 * 2 ^ n ≤ (n + 1 + 1) * fac (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ^ (n + 1) ≤ fac (Nat.succ (n + 1))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | apply Nat.mul_le_mul _ ih | case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 * 2 ^ n ≤ (n + 1 + 1) * fac (n + 1) | n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ≤ n + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.succ
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 * 2 ^ n ≤ (n + 1 + 1) * fac (n + 1)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | repeat' apply Nat.succ_le_succ | n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ≤ n + 1 + 1 | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 0 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 2 ≤ n + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | apply zero_le | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 0 ≤ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a.a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 0 ≤ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | simp [fac] | case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ 2 ^ (Nat.zero - 1) ≤ fac Nat.zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | simp [fac] | case succ.zero
⊢ 2 ^ (Nat.succ 0 - 1) ≤ fac (Nat.succ 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.zero
⊢ 2 ^ (Nat.succ 0 - 1) ≤ fac (Nat.succ 0)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | pow_two_le_fac | [9, 1] | [18, 16] | apply Nat.succ_le_succ | case a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 1 ≤ n + 1 | case a.a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 0 ≤ n | Please generate a tactic in lean4 to solve the state.
STATE:
case a
n : ℕ
ih : 2 ^ n ≤ fac (Nat.succ n)
⊢ 1 ≤ n + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | symm | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) / 6 = ∑ i in range (n + 1), i ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ ∑ i in range (n + 1), i ^ 2 = n * (n + 1) * (2 * n + 1) / 6
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) / 6 = ∑ i in range (n + 1), i ^ 2 | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) / 6 = ∑ i in range (n + 1), i ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | induction' n with n ih | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2 | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) * (2 * 0 + 1) = 6 * ∑ i in range (0 + 1), i ^ 2
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = 6 * ∑ i in range (n + 1 + 1), i ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | rw [Finset.sum_range_succ, mul_add 6, ← ih] | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = 6 * ∑ i in range (n + 1 + 1), i ^ 2 | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = 6 * ∑ i in range (n + 1 + 1), i ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | ring | case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
ih : n * (n + 1) * (2 * n + 1) = 6 * ∑ i in range (n + 1), i ^ 2
⊢ (n + 1) * (n + 1 + 1) * (2 * (n + 1) + 1) = n * (n + 1) * (2 * n + 1) + 6 * (n + 1) ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | norm_num | α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 6 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n✝ n : ℕ
⊢ 0 < 6
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | sum_sqr | [27, 1] | [33, 7] | simp | case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) * (2 * 0 + 1) = 6 * ∑ i in range (0 + 1), i ^ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
α : Type u_1
s : Finset ℕ
f : ℕ → ℕ
n : ℕ
⊢ 0 * (0 + 1) * (2 * 0 + 1) = 6 * ∑ i in range (0 + 1), i ^ 2
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_add | [51, 1] | [54, 15] | induction' n with n ih | n : MyNat
⊢ add zero n = n | case zero
⊢ add zero zero = zero
case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : MyNat
⊢ add zero n = n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_add | [51, 1] | [54, 15] | rw [add, ih] | case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : MyNat
ih : add zero n = n
⊢ add zero (succ n) = succ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_add | [51, 1] | [54, 15] | rfl | case zero
⊢ add zero zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ add zero zero = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_add | [56, 1] | [60, 6] | induction' n with n ih | m n : MyNat
⊢ add (succ m) n = succ (add m n) | case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero)
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ add (succ m) n = succ (add m n)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_add | [56, 1] | [60, 6] | rw [add, ih] | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n)) | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ add (succ m) (succ n) = succ (add m (succ n))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_add | [56, 1] | [60, 6] | rfl | case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add (succ m) n = succ (add m n)
⊢ succ (succ (add m n)) = succ (add m (succ n))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_add | [56, 1] | [60, 6] | rfl | case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add (succ m) zero = succ (add m zero)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_comm | [62, 1] | [66, 25] | induction' n with n ih | m n : MyNat
⊢ add m n = add n m | case zero
m : MyNat
⊢ add m zero = add zero m
case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ add m n = add n m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_comm | [62, 1] | [66, 25] | rw [add, succ_add, ih] | case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : add m n = add n m
⊢ add m (succ n) = add (succ n) m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_comm | [62, 1] | [66, 25] | rw [zero_add] | case zero
m : MyNat
⊢ add m zero = add zero m | case zero
m : MyNat
⊢ add m zero = m | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add m zero = add zero m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_comm | [62, 1] | [66, 25] | rfl | case zero
m : MyNat
⊢ add m zero = m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ add m zero = m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_assoc | [68, 1] | [72, 6] | induction' k with k ih | m n k : MyNat
⊢ add (add m n) k = add m (add n k) | case zero
m n : MyNat
⊢ add (add m n) zero = add m (add n zero)
case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ add (add m n) (succ k) = add m (add n (succ k)) | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : MyNat
⊢ add (add m n) k = add m (add n k)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_assoc | [68, 1] | [72, 6] | rw [add, ih] | case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ add (add m n) (succ k) = add m (add n (succ k)) | case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ succ (add m (add n k)) = add m (add n (succ k)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ add (add m n) (succ k) = add m (add n (succ k))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_assoc | [68, 1] | [72, 6] | rfl | case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ succ (add m (add n k)) = add m (add n (succ k)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : MyNat
ih : add (add m n) k = add m (add n k)
⊢ succ (add m (add n k)) = add m (add n (succ k))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.add_assoc | [68, 1] | [72, 6] | rfl | case zero
m n : MyNat
⊢ add (add m n) zero = add m (add n zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m n : MyNat
⊢ add (add m n) zero = add m (add n zero)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_add | [74, 1] | [77, 36] | induction' k with k ih | m n k : MyNat
⊢ mul m (add n k) = add (mul m n) (mul m k) | case zero
m n : MyNat
⊢ mul m (add n zero) = add (mul m n) (mul m zero)
case succ
m n k : MyNat
ih : mul m (add n k) = add (mul m n) (mul m k)
⊢ mul m (add n (succ k)) = add (mul m n) (mul m (succ k)) | Please generate a tactic in lean4 to solve the state.
STATE:
m n k : MyNat
⊢ mul m (add n k) = add (mul m n) (mul m k)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_add | [74, 1] | [77, 36] | rw [add, mul, mul, ih, add_assoc] | case succ
m n k : MyNat
ih : mul m (add n k) = add (mul m n) (mul m k)
⊢ mul m (add n (succ k)) = add (mul m n) (mul m (succ k)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n k : MyNat
ih : mul m (add n k) = add (mul m n) (mul m k)
⊢ mul m (add n (succ k)) = add (mul m n) (mul m (succ k))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_add | [74, 1] | [77, 36] | rfl | case zero
m n : MyNat
⊢ mul m (add n zero) = add (mul m n) (mul m zero) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m n : MyNat
⊢ mul m (add n zero) = add (mul m n) (mul m zero)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_mul | [79, 1] | [83, 6] | induction' n with n ih | n : MyNat
⊢ mul zero n = zero | case zero
⊢ mul zero zero = zero
case succ
n : MyNat
ih : mul zero n = zero
⊢ mul zero (succ n) = zero | Please generate a tactic in lean4 to solve the state.
STATE:
n : MyNat
⊢ mul zero n = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_mul | [79, 1] | [83, 6] | rw [mul, ih] | case succ
n : MyNat
ih : mul zero n = zero
⊢ mul zero (succ n) = zero | case succ
n : MyNat
ih : mul zero n = zero
⊢ add zero zero = zero | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : MyNat
ih : mul zero n = zero
⊢ mul zero (succ n) = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_mul | [79, 1] | [83, 6] | rfl | case succ
n : MyNat
ih : mul zero n = zero
⊢ add zero zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
n : MyNat
ih : mul zero n = zero
⊢ add zero zero = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.zero_mul | [79, 1] | [83, 6] | rfl | case zero
⊢ mul zero zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
⊢ mul zero zero = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_mul | [85, 1] | [89, 6] | induction' n with n ih | m n : MyNat
⊢ mul (succ m) n = add (mul m n) n | case zero
m : MyNat
⊢ mul (succ m) zero = add (mul m zero) zero
case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ mul (succ m) (succ n) = add (mul m (succ n)) (succ n) | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ mul (succ m) n = add (mul m n) n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_mul | [85, 1] | [89, 6] | rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] | case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ mul (succ m) (succ n) = add (mul m (succ n)) (succ n) | case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ add (mul m n) (succ (add m n)) = add (mul m n) (add m (succ n)) | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ mul (succ m) (succ n) = add (mul m (succ n)) (succ n)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_mul | [85, 1] | [89, 6] | rfl | case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ add (mul m n) (succ (add m n)) = add (mul m n) (add m (succ n)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : mul (succ m) n = add (mul m n) n
⊢ add (mul m n) (succ (add m n)) = add (mul m n) (add m (succ n))
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.succ_mul | [85, 1] | [89, 6] | rfl | case zero
m : MyNat
⊢ mul (succ m) zero = add (mul m zero) zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ mul (succ m) zero = add (mul m zero) zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_comm | [91, 1] | [95, 25] | induction' n with n ih | m n : MyNat
⊢ mul m n = mul n m | case zero
m : MyNat
⊢ mul m zero = mul zero m
case succ
m n : MyNat
ih : mul m n = mul n m
⊢ mul m (succ n) = mul (succ n) m | Please generate a tactic in lean4 to solve the state.
STATE:
m n : MyNat
⊢ mul m n = mul n m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_comm | [91, 1] | [95, 25] | rw [mul, ih, succ_mul] | case succ
m n : MyNat
ih : mul m n = mul n m
⊢ mul m (succ n) = mul (succ n) m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
m n : MyNat
ih : mul m n = mul n m
⊢ mul m (succ n) = mul (succ n) m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_comm | [91, 1] | [95, 25] | rw [zero_mul] | case zero
m : MyNat
⊢ mul m zero = mul zero m | case zero
m : MyNat
⊢ mul m zero = zero | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ mul m zero = mul zero m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/solutions/Solutions_S02_Induction_and_Recursion.lean | MyNat.mul_comm | [91, 1] | [95, 25] | rfl | case zero
m : MyNat
⊢ mul m zero = zero | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
m : MyNat
⊢ mul m zero = zero
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rcases le_or_gt 0 x with h | h | x✝ y x : ℝ
⊢ x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ x ≤ |x|
case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | . rw [abs_of_neg h]
linarith | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rw [abs_of_nonneg h] | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x| | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ -x | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.le_abs_self | [12, 1] | [16, 13] | linarith | case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ -x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y x : ℝ
h : 0 > x
⊢ x ≤ -x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rcases le_or_gt 0 x with h | h | x✝ y x : ℝ
⊢ -x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ |x|
case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y x : ℝ
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | . rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rw [abs_of_nonneg h] | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ |x| | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ x | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | linarith | case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y x : ℝ
h : 0 ≤ x
⊢ -x ≤ x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.neg_le_abs_self | [18, 1] | [22, 22] | rw [abs_of_neg h] | case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y x : ℝ
h : 0 > x
⊢ -x ≤ |x|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rcases le_or_gt 0 (x + y) with h | h | x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y|
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
x✝ y✝ x y : ℝ
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | . rw [abs_of_neg h]
linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rw [abs_of_nonneg h] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y| | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | linarith [le_abs_self x, le_abs_self y] | case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x✝ y✝ x y : ℝ
h : 0 ≤ x + y
⊢ x + y ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | rw [abs_of_neg h] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y| | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ |x + y| ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_add | [24, 1] | [29, 52] | linarith [neg_le_abs_self x, neg_le_abs_self y] | case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x✝ y✝ x y : ℝ
h : 0 > x + y
⊢ -(x + y) ≤ |x| + |y|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases le_or_gt 0 y with h | h | x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y
case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : ℝ
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rw [abs_of_neg h] | case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > y
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | constructor | case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y
case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : ℝ
h : 0 > y
⊢ x < -y ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rw [abs_of_nonneg h] | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ y
⊢ x < |y| ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | constructor | case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y
case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : ℝ
h : 0 ≤ y
⊢ x < y ↔ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ y
⊢ x < y → x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | left | case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.h
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ y
⊢ x < y ∨ x < -y → x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases h' with h' | h' | case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : ℝ
h : 0 ≤ y
h' : x < y ∨ x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | . linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inl
x y : ℝ
h : 0 ≤ y
h' : x < y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | linarith | case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.inr
x y : ℝ
h : 0 ≤ y
h' : x < -y
⊢ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > y
⊢ x < -y → x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | right | case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < y ∨ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.h
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | intro h' | case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > y
⊢ x < y ∨ x < -y → x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | rcases h' with h' | h' | case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y | case inr.mpr.inl
x y : ℝ
h : 0 > y
h' : x < y
⊢ x < -y
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : ℝ
h : 0 > y
h' : x < y ∨ x < -y
⊢ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | . exact h' | case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inr
x y : ℝ
h : 0 > y
h' : x < -y
⊢ x < -y
TACTIC:
|
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