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stringlengths 6
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https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | simp only [Set.mem_setOf_eq] at hel | x : βN0
A : Set βN0 := {b | plus (z, b) = b}
hzmem : z β A
n : βN0
hel : n β A
β’ plus (z, S n) = S n | x : βN0
A : Set βN0 := {b | plus (z, b) = b}
hzmem : z β A
n : βN0
hel : plus (z, n) = n
β’ plus (z, S n) = S n |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_plus_x_eq_eq | [300, 1] | [317, 12] | rw [splus, hel] | x : βN0
A : Set βN0 := {b | plus (z, b) = b}
hzmem : z β A
n : βN0
hel : plus (z, n) = n
β’ plus (z, S n) = S n | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | let A := {y : N0 | plus ((S x), y) = S (plus (x, y))} | x y : βN0
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | have hzmem : z β A := by
simp only [Set.mem_setOf_eq]
simp [zplus] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | have hind : (β n : N0, n β A β (S n) β A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
rw [splus, hel, βsplus] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | have heq := p3 A β¨hzmem, hindβ© | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp [A, Set.ext_iff] at heq | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x_1 : Ξ±),
(β (x_2 : x_1 β N0),
plus (S x, { val := x_1, property := (_ : x_1 β N0) }) =
S (plus (x, { val := x_1, property := (_ : x_1 β N0) }))) β
x_1 β N0
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | specialize heq y | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x_1 : Ξ±),
(β (x_2 : x_1 β N0),
plus (S x, { val := x_1, property := (_ : x_1 β N0) }) =
S (plus (x, { val := x_1, property := (_ : x_1 β N0) }))) β
x_1 β N0
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βy β N0),
plus (S x, { val := βy, property := (_ : βy β N0) }) = S (plus (x, { val := βy, property := (_ : βy β N0) }))) β
βy β N0
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βy β N0),
plus (S x, { val := βy, property := (_ : βy β N0) }) = S (plus (x, { val := βy, property := (_ : βy β N0) }))) β
βy β N0
β’ plus (S x, y) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : plus (S x, y) = S (plus (x, y))
β’ plus (S x, y) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | exact heq | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : plus (S x, y) = S (plus (x, y))
β’ plus (S x, y) = S (plus (x, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp only [Set.mem_setOf_eq] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
β’ z β A | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
β’ plus (S x, z) = S (plus (x, z)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp [zplus] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
β’ plus (S x, z) = S (plus (x, z)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | intros n hel | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
β’ β n β A, S n β A | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp only [Set.mem_setOf_eq] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ plus (S x, S n) = S (plus (x, S n)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | simp only [Set.mem_setOf_eq] at hel | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ plus (S x, S n) = S (plus (x, S n)) | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : plus (S x, n) = S (plus (x, n))
β’ plus (S x, S n) = S (plus (x, S n)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_eq_succ_plus | [321, 1] | [338, 12] | rw [splus, hel, βsplus] | x y : βN0
A : Set βN0 := {y | plus (S x, y) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : plus (S x, n) = S (plus (x, n))
β’ plus (S x, S n) = S (plus (x, S n)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | let A := {x : N0 | plus ((S y), x) = S (plus (x, y))} | x y : βN0
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | have hzmem : z β A := by
simp only [Set.mem_setOf_eq]
rw [zplus, zero_plus_x_eq_eq] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | have hind : (β n : N0, n β A β (S n) β A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
rw [splus, hel, succ_plus_eq_succ_plus] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | have heq := p3 A β¨hzmem, hindβ© | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | simp [A, Set.ext_iff] at heq | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x : Ξ±),
(β (x_1 : x β N0),
plus (S y, { val := x, property := (_ : x β N0) }) = S (plus ({ val := x, property := (_ : x β N0) }, y))) β
x β N0
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | specialize heq x | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x : Ξ±),
(β (x_1 : x β N0),
plus (S y, { val := x, property := (_ : x β N0) }) = S (plus ({ val := x, property := (_ : x β N0) }, y))) β
x β N0
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βx β N0),
plus (S y, { val := βx, property := (_ : βx β N0) }) = S (plus ({ val := βx, property := (_ : βx β N0) }, y))) β
βx β N0
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βx β N0),
plus (S y, { val := βx, property := (_ : βx β N0) }) = S (plus ({ val := βx, property := (_ : βx β N0) }, y))) β
βx β N0
β’ plus (S y, x) = S (plus (x, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : plus (S y, x) = S (plus (x, y))
β’ plus (S y, x) = S (plus (x, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | exact heq | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
hind : β n β A, S n β A
heq : plus (S y, x) = S (plus (x, y))
β’ plus (S y, x) = S (plus (x, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | simp only [Set.mem_setOf_eq] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
β’ z β A | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
β’ plus (S y, z) = S (plus (z, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | rw [zplus, zero_plus_x_eq_eq] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
β’ plus (S y, z) = S (plus (z, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | intros n hel | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
β’ β n β A, S n β A | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | simp only [Set.mem_setOf_eq] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ plus (S y, S n) = S (plus (S n, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | simp only [Set.mem_setOf_eq] at hel | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : n β A
β’ plus (S y, S n) = S (plus (S n, y)) | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : plus (S y, n) = S (plus (n, y))
β’ plus (S y, S n) = S (plus (S n, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus | [340, 1] | [357, 12] | rw [splus, hel, succ_plus_eq_succ_plus] | x y : βN0
A : Set βN0 := {x | plus (S y, x) = S (plus (x, y))}
hzmem : z β A
n : βN0
hel : plus (S y, n) = S (plus (n, y))
β’ plus (S y, S n) = S (plus (S n, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | let A := {x : N0 | mul (z, x) = z} | x : βN0
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | have hzmem : z β A := by
simp only [Set.mem_setOf_eq]
rw [zmul] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | have hind : (β n : N0, n β A β (S n) β A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
rw [smul, hel, zero_plus_x_eq_eq] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | have heq := p3 A β¨hzmem, hindβ© | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | simp [A, Set.ext_iff] at heq | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : β (x : Ξ±), (β (x_1 : x β N0), mul (z, { val := x, property := (_ : x β N0) }) = z) β x β N0
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | specialize heq x | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : β (x : Ξ±), (β (x_1 : x β N0), mul (z, { val := x, property := (_ : x β N0) }) = z) β x β N0
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : (β (x_1 : βx β N0), mul (z, { val := βx, property := (_ : βx β N0) }) = z) β βx β N0
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : (β (x_1 : βx β N0), mul (z, { val := βx, property := (_ : βx β N0) }) = z) β βx β N0
β’ mul (z, x) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : mul (z, x) = z
β’ mul (z, x) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | exact heq | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
hind : β n β A, S n β A
heq : mul (z, x) = z
β’ mul (z, x) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | simp only [Set.mem_setOf_eq] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
β’ z β A | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
β’ mul (z, z) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | rw [zmul] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
β’ mul (z, z) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | intros n hel | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
β’ β n β A, S n β A | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | simp only [Set.mem_setOf_eq] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : n β A
β’ mul (z, S n) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | simp only [Set.mem_setOf_eq] at hel | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : n β A
β’ mul (z, S n) = z | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : mul (z, n) = z
β’ mul (z, S n) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | zero_mul_eq_zero | [360, 1] | [377, 12] | rw [smul, hel, zero_plus_x_eq_eq] | x : βN0
A : Set βN0 := {x | mul (z, x) = z}
hzmem : z β A
n : βN0
hel : mul (z, n) = z
β’ mul (z, S n) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | let A := {x : N0 | mul (S z, x) = x} | x : βN0
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | have hzmem : z β A := by
simp only [Set.mem_setOf_eq]
rw [zmul] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | have hind : (β n : N0, n β A β (S n) β A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
rw [smul, hel, splus, zplus] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | have heq := p3 A β¨hzmem, hindβ© | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | simp [A, Set.ext_iff] at heq | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x : Ξ±),
(β (x_1 : x β N0), mul (S z, { val := x, property := (_ : x β N0) }) = { val := x, property := (_ : x β N0) }) β
x β N0
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | specialize heq x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq :
β (x : Ξ±),
(β (x_1 : x β N0), mul (S z, { val := x, property := (_ : x β N0) }) = { val := x, property := (_ : x β N0) }) β
x β N0
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βx β N0), mul (S z, { val := βx, property := (_ : βx β N0) }) = { val := βx, property := (_ : βx β N0) }) β
βx β N0
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq :
(β (x_1 : βx β N0), mul (S z, { val := βx, property := (_ : βx β N0) }) = { val := βx, property := (_ : βx β N0) }) β
βx β N0
β’ mul (S z, x) = x | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq : mul (S z, x) = x
β’ mul (S z, x) = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | exact heq | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
hind : β n β A, S n β A
heq : mul (S z, x) = x
β’ mul (S z, x) = x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | simp only [Set.mem_setOf_eq] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
β’ z β A | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
β’ mul (S z, z) = z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | rw [zmul] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
β’ mul (S z, z) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | intros n hel | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
β’ β n β A, S n β A | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | simp only [Set.mem_setOf_eq] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : n β A
β’ mul (S z, S n) = S n |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | simp only [Set.mem_setOf_eq] at hel | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : n β A
β’ mul (S z, S n) = S n | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : mul (S z, n) = n
β’ mul (S z, S n) = S n |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self | [380, 1] | [397, 12] | rw [smul, hel, splus, zplus] | x : βN0
A : Set βN0 := {x | mul (S z, x) = x}
hzmem : z β A
n : βN0
hel : mul (S z, n) = n
β’ mul (S z, S n) = S n | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | let A := {x : N0 | motive x} | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | have hzmem : z β A := by
simp only [Set.mem_setOf_eq]
exact hz | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | have hind : (β n : N0, n β A β (S n) β A) := by
intros n hel
simp only [Set.mem_setOf_eq]
simp only [Set.mem_setOf_eq] at hel
specialize hs n
exact hs hel | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | have heq := p3 A β¨hzmem, hindβ© | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | simp [A, Set.ext_iff] at heq | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : Subtype.val '' A = N0
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : β (x : Ξ±), (β (x_1 : x β N0), motive { val := x, property := (_ : x β N0) }) β x β N0
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | specialize heq x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : β (x : Ξ±), (β (x_1 : x β N0), motive { val := x, property := (_ : x β N0) }) β x β N0
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : (β (x_1 : βx β N0), motive { val := βx, property := (_ : βx β N0) }) β βx β N0
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | simp only [Subtype.coe_eta, Subtype.coe_prop, exists_const, iff_true] at heq | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : (β (x_1 : βx β N0), motive { val := βx, property := (_ : βx β N0) }) β βx β N0
β’ motive x | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : motive x
β’ motive x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | exact heq | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
hind : β n β A, S n β A
heq : motive x
β’ motive x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | simp only [Set.mem_setOf_eq] | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
β’ z β A | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
β’ motive z |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | exact hz | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
β’ motive z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | intros n hel | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
β’ β n β A, S n β A | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | simp only [Set.mem_setOf_eq] | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : n β A
β’ S n β A | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : n β A
β’ motive (S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | simp only [Set.mem_setOf_eq] at hel | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : n β A
β’ motive (S n) | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : motive n
β’ motive (S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | specialize hs n | motive : βN0 β Prop
hz : motive z
hs : β (n : βN0), motive n β motive (S n)
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : motive n
β’ motive (S n) | motive : βN0 β Prop
hz : motive z
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : motive n
hs : motive n β motive (S n)
β’ motive (S n) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | generic_recursor | [400, 1] | [421, 12] | exact hs hel | motive : βN0 β Prop
hz : motive z
x : βN0
A : Set βN0 := {x | motive x}
hzmem : z β A
n : βN0
hel : motive n
hs : motive n β motive (S n)
β’ motive (S n) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self' | [423, 1] | [429, 32] | apply generic_recursor | β’ β (x : βN0), mul (S z, x) = x | case hz
β’ mul (S z, z) = z
case hs
β’ β (n : βN0), mul (S z, n) = n β mul (S z, S n) = S n |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self' | [423, 1] | [429, 32] | rw [zmul] | case hz
β’ mul (S z, z) = z | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self' | [423, 1] | [429, 32] | intros x hi | case hs
β’ β (n : βN0), mul (S z, n) = n β mul (S z, S n) = S n | case hs
x : βN0
hi : mul (S z, x) = x
β’ mul (S z, S x) = S x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_zero_mul_eq_self' | [423, 1] | [429, 32] | rw [smul, hi, splus, zplus] | case hs
x : βN0
hi : mul (S z, x) = x
β’ mul (S z, S x) = S x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus' | [431, 1] | [438, 43] | apply generic_recursor | β’ β (x y : βN0), plus (S y, x) = S (plus (x, y)) | case hz
β’ β (y : βN0), plus (S y, z) = S (plus (z, y))
case hs
β’ β (n : βN0), (β (y : βN0), plus (S y, n) = S (plus (n, y))) β β (y : βN0), plus (S y, S n) = S (plus (S n, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus' | [431, 1] | [438, 43] | intro y | case hz
β’ β (y : βN0), plus (S y, z) = S (plus (z, y)) | case hz
y : βN0
β’ plus (S y, z) = S (plus (z, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus' | [431, 1] | [438, 43] | rw [zplus, zero_plus_x_eq_eq] | case hz
y : βN0
β’ plus (S y, z) = S (plus (z, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus' | [431, 1] | [438, 43] | intros n hi y | case hs
β’ β (n : βN0), (β (y : βN0), plus (S y, n) = S (plus (n, y))) β β (y : βN0), plus (S y, S n) = S (plus (S n, y)) | case hs
n : βN0
hi : β (y : βN0), plus (S y, n) = S (plus (n, y))
y : βN0
β’ plus (S y, S n) = S (plus (S n, y)) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex5.lean | succ_plus_n_eq_succ_n_plus' | [431, 1] | [438, 43] | rw [splus, hi, succ_plus_eq_succ_plus] | case hs
n : βN0
hi : β (y : βN0), plus (S y, n) = S (plus (n, y))
y : βN0
β’ plus (S y, S n) = S (plus (S n, y)) | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | induction' n with n ih | n : β
β’ (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3 | case zero
β’ (β x in Finset.range (Nat.zero + 1), x) ^ 2 = β x in Finset.range (Nat.zero + 1), x ^ 3
case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (β x in Finset.range (Nat.succ n + 1), x) ^ 2 = β x in Finset.range (Nat.succ n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | ring_nf | case zero
β’ (β x in Finset.range (Nat.zero + 1), x) ^ 2 = β x in Finset.range (Nat.zero + 1), x ^ 3 | case zero
β’ (β x in Finset.range (1 + Nat.zero), x) ^ 2 = β x in Finset.range (1 + Nat.zero), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | simp only [Nat.zero_eq, add_zero, Finset.range_one, Finset.sum_singleton, ne_eq,
OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow'] | case zero
β’ (β x in Finset.range (1 + Nat.zero), x) ^ 2 = β x in Finset.range (1 + Nat.zero), x ^ 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | rw [Finset.range_succ,
Finset.sum_insert Finset.not_mem_range_self,
Finset.sum_insert Finset.not_mem_range_self] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (β x in Finset.range (Nat.succ n + 1), x) ^ 2 = β x in Finset.range (Nat.succ n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (n + 1 + β x in Finset.range (n + 1), x) ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | rw [add_sq] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (n + 1 + β x in Finset.range (n + 1), x) ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | let S := β x in Finset.range (n + 1), x | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | have hS : S = β x in Finset.range (n + 1), x := rfl | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | nth_rewrite 2 [βhS] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + (β x in Finset.range (n + 1), x) ^ 2 =
(n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | have bigger (n : β) : 1 β€ n + 1 := by
simp only [le_add_iff_nonneg_left, zero_le] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | rw [Finset.sum_range_id] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (n + 1) ^ 2 + 2 * (n + 1) * β x in Finset.range (n + 1), x + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | zify [bigger] | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) + S ^ 2 = (n + 1) ^ 3 + β x in Finset.range (n + 1), x ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | have hring (n : β€):
(n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2)
= (n + 1) ^ 3 := by
have htwodvd : 2 β£ ((n + 1) * (n + 1 - 1)) := by
rw [add_sub_cancel, mul_comm]
exact even_iff_two_dvd.1 (Int.even_mul_succ_self n)
qify [htwodvd, bigger]
push_cast
ring_nf | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
hring : β (n : β€), (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | zify at ih | case succ
n : β
ih : (β x in Finset.range (n + 1), x) ^ 2 = β x in Finset.range (n + 1), x ^ 3
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
hring : β (n : β€), (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 | case succ
n : β
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
hring : β (n : β€), (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3
ih : (β x in Finset.range (n + 1), βx) ^ 2 = β x in Finset.range (n + 1), βx ^ 3
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | rw [hring n, ih] | case succ
n : β
S : β := β x in Finset.range (n + 1), x
hS : S = β x in Finset.range (n + 1), x
bigger : β (n : β), 1 β€ n + 1
hring : β (n : β€), (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3
ih : (β x in Finset.range (n + 1), βx) ^ 2 = β x in Finset.range (n + 1), βx ^ 3
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) + (β x in Finset.range (n + 1), βx) ^ 2 =
(βn + 1) ^ 3 + β x in Finset.range (n + 1), βx ^ 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | simp only [le_add_iff_nonneg_left, zero_le] | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
n : β
β’ 1 β€ n + 1 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | have htwodvd : 2 β£ ((n + 1) * (n + 1 - 1)) := by
rw [add_sub_cancel, mul_comm]
exact even_iff_two_dvd.1 (Int.even_mul_succ_self n) | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
bigger : β (n : β), 1 β€ n + 1
n : β€
β’ (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3 | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
bigger : β (n : β), 1 β€ n + 1
n : β€
htwodvd : 2 β£ (n + 1) * (n + 1 - 1)
β’ (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | qify [htwodvd, bigger] | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
bigger : β (n : β), 1 β€ n + 1
n : β€
htwodvd : 2 β£ (n + 1) * (n + 1 - 1)
β’ (n + 1) ^ 2 + 2 * (n + 1) * ((n + 1) * (n + 1 - 1) / 2) = (n + 1) ^ 3 | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
bigger : β (n : β), 1 β€ n + 1
n : β€
htwodvd : 2 β£ (n + 1) * (n + 1 - 1)
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) = (βn + 1) ^ 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex3.lean | sum_squared_eq_sum_cubes | [7, 1] | [42, 21] | ring_nf | nβ : β
ih : (β x in Finset.range (nβ + 1), x) ^ 2 = β x in Finset.range (nβ + 1), x ^ 3
S : β := β x in Finset.range (nβ + 1), x
hS : S = β x in Finset.range (nβ + 1), x
bigger : β (n : β), 1 β€ n + 1
n : β€
htwodvd : 2 β£ (n + 1) * (n + 1 - 1)
β’ (βn + 1) ^ 2 + 2 * (βn + 1) * ((βn + 1) * (βn + 1 - 1) / 2) = (βn + 1) ^ 3 | no goals |
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