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stringlengths 6
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https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | ext x | β’ fn '' Set.Icc 3 5 = Set.Icc 12 28 | case h
x : β
β’ x β fn '' Set.Icc 3 5 β x β Set.Icc 12 28 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | simp only [ge_iff_le, not_le, gt_iff_lt, Set.mem_image, Set.mem_Icc, fn] | case h
x : β
β’ x β fn '' Set.Icc 3 5 β x β Set.Icc 12 28 | case h
x : β
β’ (β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x) β 12 β€ x β§ x β€ 28 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | constructor | case h
x : β
β’ (β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x) β 12 β€ x β§ x β€ 28 | case h.mp
x : β
β’ (β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x) β 12 β€ x β§ x β€ 28
case h.mpr
x : β
β’ 12 β€ x β§ x β€ 28 β β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | intro hex | case h.mp
x : β
β’ (β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x) β 12 β€ x β§ x β€ 28 | case h.mp
x : β
hex : β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x
β’ 12 β€ x β§ x β€ 28 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | rcases hex with β¨y, β¨β¨h3le, h5geβ© , hfyβ©β© | case h.mp
x : β
hex : β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x
β’ 12 β€ x β§ x β€ 28 | case h.mp.intro.intro.intro
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ 12 β€ x β§ x β€ 28 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | constructor | case h.mp.intro.intro.intro
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ 12 β€ x β§ x β€ 28 | case h.mp.intro.intro.intro.left
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ 12 β€ x
case h.mp.intro.intro.intro.right
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ x β€ 28 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.mp.intro.intro.intro.left
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ 12 β€ x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.mp.intro.intro.intro.right
x y : β
hfy : y ^ 2 + 3 = x
h3le : 3 β€ y
h5ge : y β€ 5
β’ x β€ 28 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | intro hlege | case h.mpr
x : β
β’ 12 β€ x β§ x β€ 28 β β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x | case h.mpr
x : β
hlege : 12 β€ x β§ x β€ 28
β’ β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | rcases hlege with β¨h12ge, h28leβ© | case h.mpr
x : β
hlege : 12 β€ x β§ x β€ 28
β’ β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x | case h.mpr.intro
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | use (Real.sqrt (x - 3)) | case h.mpr.intro
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ β x_1, (3 β€ x_1 β§ x_1 β€ 5) β§ x_1 ^ 2 + 3 = x | case h
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ (3 β€ Real.sqrt (x - 3) β§ Real.sqrt (x - 3) β€ 5) β§ Real.sqrt (x - 3) ^ 2 + 3 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | constructor | case h
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ (3 β€ Real.sqrt (x - 3) β§ Real.sqrt (x - 3) β€ 5) β§ Real.sqrt (x - 3) ^ 2 + 3 = x | case h.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 β€ Real.sqrt (x - 3) β§ Real.sqrt (x - 3) β€ 5
case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ Real.sqrt (x - 3) ^ 2 + 3 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | constructor | case h.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 β€ Real.sqrt (x - 3) β§ Real.sqrt (x - 3) β€ 5 | case h.left.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 β€ Real.sqrt (x - 3)
case h.left.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ Real.sqrt (x - 3) β€ 5 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | refine Real.le_sqrt_of_sq_le ?h.left.left.h | case h.left.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 β€ Real.sqrt (x - 3) | case h.left.left.h
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 ^ 2 β€ x - 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | linarith | case h.left.left.h
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 3 ^ 2 β€ x - 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | refine Real.sqrt_le_iff.mpr ?h.left.right.a | case h.left.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ Real.sqrt (x - 3) β€ 5 | case h.left.right.a
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ 5 β§ x - 3 β€ 5 ^ 2 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | constructor | case h.left.right.a
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ 5 β§ x - 3 β€ 5 ^ 2 | case h.left.right.a.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ 5
case h.left.right.a.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ x - 3 β€ 5 ^ 2 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.left.right.a.left
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ 5 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.left.right.a.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ x - 3 β€ 5 ^ 2 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | rw [Real.sq_sqrt] | case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ Real.sqrt (x - 3) ^ 2 + 3 = x | case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ x - 3 + 3 = x
case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ x - 3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ x - 3 + 3 = x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | image_fn | [171, 1] | [195, 18] | nlinarith | case h.right
x : β
h12ge : 12 β€ x
h28le : x β€ 28
β’ 0 β€ x - 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | ext x | β’ fn β»ΒΉ' Set.Icc 12 19 = Set.Icc 3 4 βͺ Set.Icc (-4) (-3) | case h
x : β
β’ x β fn β»ΒΉ' Set.Icc 12 19 β x β Set.Icc 3 4 βͺ Set.Icc (-4) (-3) |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | simp only [ge_iff_le, not_le, gt_iff_lt, Set.mem_preimage, Set.mem_Icc, neg_le_neg_iff,
neg_lt_neg_iff, Set.mem_union, fn] | case h
x : β
β’ x β fn β»ΒΉ' Set.Icc 12 19 β x β Set.Icc 3 4 βͺ Set.Icc (-4) (-3) | case h
x : β
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 β 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | constructor | case h
x : β
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 β 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp
x : β
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 β 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3
case h.mpr
x : β
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 β 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | intro hlege | case h.mp
x : β
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 β 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp
x : β
hlege : 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | rcases hlege with β¨h12le, h19geβ© | case h.mp
x : β
hlege : 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 12 β€ x ^ 2 + 3
h19ge : x ^ 2 + 3 β€ 19
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | have heq12 : (12 : β) = (3^2) + 3 := by ring | case h.mp.intro
x : β
h12le : 12 β€ x ^ 2 + 3
h19ge : x ^ 2 + 3 β€ 19
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 12 β€ x ^ 2 + 3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | rw [heq12, add_le_add_iff_right, sq_le_sq, abs_of_pos (by simp), le_abs, le_neg] at h12le | case h.mp.intro
x : β
h12le : 12 β€ x ^ 2 + 3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | have heq19 : (19 : β) = (4^2) + 3 := by ring | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | have lt04 : (0 : β) < 4 := by simp | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | rw [heq19, add_le_add_iff_right, sq_le_sq, abs_of_pos lt04, abs_le] at h19ge | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | rcases h12le with (h3le | hm3ge) | case h.mp.intro
x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro.inl
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
h3le : 3 β€ x
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3
case h.mp.intro.inr
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
hm3ge : x β€ -3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | ring | x : β
h12le : 12 β€ x ^ 2 + 3
h19ge : x ^ 2 + 3 β€ 19
β’ 12 = 3 ^ 2 + 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | simp | x : β
h12le : |3| β€ |x|
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 0 < 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | ring | x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
β’ 19 = 4 ^ 2 + 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | simp | x : β
h12le : 3 β€ x β¨ x β€ -3
h19ge : x ^ 2 + 3 β€ 19
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
β’ 0 < 4 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | left | case h.mp.intro.inl
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
h3le : 3 β€ x
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro.inl.h
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
h3le : 3 β€ x
β’ 3 β€ x β§ x β€ 4 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | exact β¨h3le, h19ge.2β© | case h.mp.intro.inl.h
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
h3le : 3 β€ x
β’ 3 β€ x β§ x β€ 4 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | right | case h.mp.intro.inr
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
hm3ge : x β€ -3
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 | case h.mp.intro.inr.h
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
hm3ge : x β€ -3
β’ -4 β€ x β§ x β€ -3 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | exact β¨h19ge.1, hm3geβ© | case h.mp.intro.inr.h
x : β
h19ge : -4 β€ x β§ x β€ 4
heq12 : 12 = 3 ^ 2 + 3
heq19 : 19 = 4 ^ 2 + 3
lt04 : 0 < 4
hm3ge : x β€ -3
β’ -4 β€ x β§ x β€ -3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | intro hlege | case h.mpr
x : β
β’ 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3 β 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 | case h.mpr
x : β
hlege : 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | rcases hlege with (β¨h3le, h4geβ© | β¨hm4le, hm3geβ©) | case h.mpr
x : β
hlege : 3 β€ x β§ x β€ 4 β¨ -4 β€ x β§ x β€ -3
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 | case h.mpr.inl.intro
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19
case h.mpr.inr.intro
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | constructor | case h.mpr.inl.intro
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 | case h.mpr.inl.intro.left
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ 12 β€ x ^ 2 + 3
case h.mpr.inl.intro.right
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ x ^ 2 + 3 β€ 19 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | nlinarith | case h.mpr.inl.intro.left
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ 12 β€ x ^ 2 + 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | nlinarith | case h.mpr.inl.intro.right
x : β
h3le : 3 β€ x
h4ge : x β€ 4
β’ x ^ 2 + 3 β€ 19 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | constructor | case h.mpr.inr.intro
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ 12 β€ x ^ 2 + 3 β§ x ^ 2 + 3 β€ 19 | case h.mpr.inr.intro.left
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ 12 β€ x ^ 2 + 3
case h.mpr.inr.intro.right
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ x ^ 2 + 3 β€ 19 |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | nlinarith | case h.mpr.inr.intro.left
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ 12 β€ x ^ 2 + 3 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | preimage_fn | [197, 1] | [223, 18] | nlinarith | case h.mpr.inr.intro.right
x : β
hm4le : -4 β€ x
hm3ge : x β€ -3
β’ x ^ 2 + 3 β€ 19 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | simp only [Set.subset_inter_iff, Set.image_subset_iff, Set.subset_def] | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
β’ f '' (X β© Y) β f '' X β© f '' Y | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
β’ β x β f '' (X β© Y), x β f '' X β© f '' Y |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | simp [Set.subset_def] | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
β’ β x β f '' (X β© Y), x β f '' X β© f '' Y | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
β’ β (x : Ξ²), β x_1 β X, x_1 β Y β f x_1 = x β (β x_2 β X, f x_2 = x) β§ β x_2 β Y, f x_2 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | intros a b bmemX bmemY img | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
β’ β (x : Ξ²), β x_1 β X, x_1 β Y β f x_1 = x β (β x_2 β X, f x_2 = x) β§ β x_2 β Y, f x_2 = x | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ (β x β X, f x = a) β§ β x β Y, f x = a |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | constructor | Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ (β x β X, f x = a) β§ β x β Y, f x = a | case left
Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ β x β X, f x = a
case right
Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ β x β Y, f x = a |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | use b | case left
Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ β x β X, f x = a | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_subset | [226, 1] | [238, 10] | use b | case right
Ξ± Ξ² : Type
X Y : Set Ξ±
f : Ξ± β Ξ²
a : Ξ²
b : Ξ±
bmemX : b β X
bmemY : b β Y
img : f b = a
β’ β x β Y, f x = a | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | use (β€) | β’ β Ξ± Ξ² f X Y, f '' (X β© Y) β f '' X β© f '' Y | case h
β’ β Ξ² f X Y, f '' (X β© Y) β f '' X β© f '' Y |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | use (β€) | case h
β’ β Ξ² f X Y, f '' (X β© Y) β f '' X β© f '' Y | case h
β’ β f X Y, f '' (X β© Y) β f '' X β© f '' Y |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | use (Ξ» _ => 0) | case h
β’ β f X Y, f '' (X β© Y) β f '' X β© f '' Y | case h
β’ β X Y, (fun x => 0) '' (X β© Y) β (fun x => 0) '' X β© (fun x => 0) '' Y |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | use ({1} : Set β€) | case h
β’ β X Y, (fun x => 0) '' (X β© Y) β (fun x => 0) '' X β© (fun x => 0) '' Y | case h
β’ β Y, (fun x => 0) '' ({1} β© Y) β (fun x => 0) '' {1} β© (fun x => 0) '' Y |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | use ({2} : Set β€) | case h
β’ β Y, (fun x => 0) '' ({1} β© Y) β (fun x => 0) '' {1} β© (fun x => 0) '' Y | case h
β’ (fun x => 0) '' ({1} β© {2}) β (fun x => 0) '' {1} β© (fun x => 0) '' {2} |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | simp only [Set.inter_singleton_nonempty, Set.image_singleton, Set.inter_self, ne_eq] | case h
β’ (fun x => 0) '' ({1} β© {2}) β (fun x => 0) '' {1} β© (fun x => 0) '' {2} | case h
β’ Β¬(fun x => 0) '' ({1} β© {2}) = {0} |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | intro heq | case h
β’ Β¬(fun x => 0) '' ({1} β© {2}) = {0} | case h
heq : (fun x => 0) '' ({1} β© {2}) = {0}
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | rw [Set.inter_def] at heq | case h
heq : (fun x => 0) '' ({1} β© {2}) = {0}
β’ False | case h
heq : (fun x => 0) '' {a | a β {1} β§ a β {2}} = {0}
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | simp at heq | case h
heq : (fun x => 0) '' {a | a β {1} β§ a β {2}} = {0}
β’ False | case h
heq : (fun x => 0) '' {a | a = 1 β§ a = 2} = {0}
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | have hf {a : β€} : (a = 1 β§ a = 2) = False := by
simp_all only [eq_iff_iff, iff_false, not_and, OfNat.one_ne_ofNat, not_false_eq_true, implies_true] | case h
heq : (fun x => 0) '' {a | a = 1 β§ a = 2} = {0}
β’ False | case h
heq : (fun x => 0) '' {a | a = 1 β§ a = 2} = {0}
hf : β {a : β€}, (a = 1 β§ a = 2) = False
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | simp_rw [hf] at heq | case h
heq : (fun x => 0) '' {a | a = 1 β§ a = 2} = {0}
hf : β {a : β€}, (a = 1 β§ a = 2) = False
β’ False | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : (fun x => 0) '' {a | False} = {0}
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | simp at heq | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : (fun x => 0) '' {a | False} = {0}
β’ False | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : β
= {0}
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | symm at heq | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : β
= {0}
β’ False | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : {0} = β
β’ False |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | exact Set.singleton_ne_empty 0 heq | case h
hf : β {a : β€}, (a = 1 β§ a = 2) = False
heq : {0} = β
β’ False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | inter_image_fun_inter_not_eq | [241, 1] | [263, 37] | simp_all only [eq_iff_iff, iff_false, not_and, OfNat.one_ne_ofNat, not_false_eq_true, implies_true] | heq : (fun x => 0) '' {a | a = 1 β§ a = 2} = {0}
a : β€
β’ (a = 1 β§ a = 2) = False | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | rw [Function.Bijective, Function.Injective, Function.Surjective] | β’ Function.Bijective bij_fun | β’ (β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ) β§ β (b : { x // x > Real.sqrt 2 }), β a, bij_fun a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | simp only [Subtype.forall, ge_iff_le] | β’ (β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ) β§ β (b : { x // x > Real.sqrt 2 }), β a, bij_fun a = b | β’ (β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ) β§
β (a : β) (b : a > Real.sqrt 2), β a_1, bij_fun a_1 = { val := a, property := b } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | constructor | β’ (β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ) β§
β (a : β) (b : a > Real.sqrt 2), β a_1, bij_fun a_1 = { val := a, property := b } | case left
β’ β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ
case right
β’ β (a : β) (b : a > Real.sqrt 2), β a_1, bij_fun a_1 = { val := a, property := b } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | intros a b heq | case left
β’ β β¦aβ aβ : ββ¦, bij_fun aβ = bij_fun aβ β aβ = aβ | case left
a b : β
heq : bij_fun a = bij_fun b
β’ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | simp_rw [bij_fun] at heq | case left
a b : β
heq : bij_fun a = bij_fun b
β’ a = b | case left
a b : β
heq :
{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } =
{ val := Real.exp b + Real.sqrt 2, property := (_ : Real.exp b + Real.sqrt 2 > Real.sqrt 2) }
β’ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | apply_fun(Ξ» e => Real.log (e - Real.sqrt 2)) at heq | case left
a b : β
heq :
{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } =
{ val := Real.exp b + Real.sqrt 2, property := (_ : Real.exp b + Real.sqrt 2 > Real.sqrt 2) }
β’ a = b | case left
a b : β
heq :
Real.log
(β{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } - Real.sqrt 2) =
Real.log
(β{ val := Real.exp b + Real.sqrt 2, property := (_ : Real.exp b + Real.sqrt 2 > Real.sqrt 2) } - Real.sqrt 2)
β’ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | simp only [add_sub_cancel, Real.log_exp] at heq | case left
a b : β
heq :
Real.log
(β{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } - Real.sqrt 2) =
Real.log
(β{ val := Real.exp b + Real.sqrt 2, property := (_ : Real.exp b + Real.sqrt 2 > Real.sqrt 2) } - Real.sqrt 2)
β’ a = b | case left
a b : β
heq : a = b
β’ a = b |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | exact heq | case left
a b : β
heq : a = b
β’ a = b | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | intros x hle | case right
β’ β (a : β) (b : a > Real.sqrt 2), β a_1, bij_fun a_1 = { val := a, property := b } | case right
x : β
hle : x > Real.sqrt 2
β’ β a, bij_fun a = { val := x, property := hle } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | simp_rw [bij_fun] | case right
x : β
hle : x > Real.sqrt 2
β’ β a, bij_fun a = { val := x, property := hle } | case right
x : β
hle : x > Real.sqrt 2
β’ β a,
{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } =
{ val := x, property := hle } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | use (Real.log (x - Real.sqrt 2)) | case right
x : β
hle : x > Real.sqrt 2
β’ β a,
{ val := Real.exp a + Real.sqrt 2, property := (_ : Real.exp a + Real.sqrt 2 > Real.sqrt 2) } =
{ val := x, property := hle } | case h
x : β
hle : x > Real.sqrt 2
β’ { val := Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2,
property := (_ : Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2 > Real.sqrt 2) } =
{ val := x, property := hle } |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | simp only [Subtype.mk.injEq] | case h
x : β
hle : x > Real.sqrt 2
β’ { val := Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2,
property := (_ : Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2 > Real.sqrt 2) } =
{ val := x, property := hle } | case h
x : β
hle : x > Real.sqrt 2
β’ Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | rw [Real.exp_log (by linarith)] | case h
x : β
hle : x > Real.sqrt 2
β’ Real.exp (Real.log (x - Real.sqrt 2)) + Real.sqrt 2 = x | case h
x : β
hle : x > Real.sqrt 2
β’ x - Real.sqrt 2 + Real.sqrt 2 = x |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | ring_nf | case h
x : β
hle : x > Real.sqrt 2
β’ x - Real.sqrt 2 + Real.sqrt 2 = x | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | bijective_fn | [274, 1] | [290, 12] | linarith | x : β
hle : x > Real.sqrt 2
β’ 0 < x - Real.sqrt 2 | no goals |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | intro hfac | m n q p : β
hβ : p β€ m
hβ : q β€ n
β’ 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q β m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | have hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n := by
exact dvd_mul_of_dvd_right (pow_dvd_pow 3 hβ) (2 ^ m) | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [βNat.div_eq_iff_eq_mul_left (by simp) hdvd3,
Nat.mul_div_assoc (2^m)
(by rw [pow_dvd_pow_iff (by simp) (by simp)]; exact hβ),
Nat.pow_div hβ (by simp)] at hfac | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ n = 2 ^ p * 3 ^ q
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ (n - q) = 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [mul_comm, βone_mul (2^p)] at hfac | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 2 ^ m * 3 ^ (n - q) = 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ m = 1 * 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | have hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m := by
exact dvd_mul_of_dvd_right (pow_dvd_pow 2 hβ) (3 ^ (n - q)) | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ m = 1 * 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ m = 1 * 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [βNat.div_eq_iff_eq_mul_left (by simp) hdvd2,
Nat.mul_div_assoc (3^(n-q))
(by rw [pow_dvd_pow_iff (by simp) (by simp)]; exact hβ),
Nat.pow_div hβ (by simp)] at hfac | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ m = 1 * 2 ^ p
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ (m - p) = 1
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [mul_eq_one_iff' (Nat.one_le_pow' (n - q) 2) (Nat.one_le_two_pow (m - p))] at hfac | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : 3 ^ (n - q) * 2 ^ (m - p) = 1
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : (2 + 1) ^ (n - q) = 1 β§ 2 ^ (m - p) = 1
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rcases hfac with β¨h3, h2β© | m n q p : β
hβ : p β€ m
hβ : q β€ n
hfac : (2 + 1) ^ (n - q) = 1 β§ 2 ^ (m - p) = 1
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
h3 : (2 + 1) ^ (n - q) = 1
h2 : 2 ^ (m - p) = 1
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | ring_nf at * | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 3 ^ (n - q) * 2 ^ m
h3 : (2 + 1) ^ (n - q) = 1
h2 : 2 ^ (m - p) = 1
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h2 : 2 ^ (m - p) = 1
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | apply_fun (Ξ» e => Nat.log 2 e) at h2 | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h2 : 2 ^ (m - p) = 1
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : Nat.log 2 (2 ^ (m - p)) = Nat.log 2 1
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [Nat.log_pow (by simp)] at h2 | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : Nat.log 2 (2 ^ (m - p)) = Nat.log 2 1
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m - p = Nat.log 2 1
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | simp at h2 | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m - p = Nat.log 2 1
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | have heqβ : m = p := by
linarith | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
heqβ : m = p
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | apply_fun (Ξ» e => Nat.log 3 e) at h3 | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
h3 : 3 ^ (n - q) = 1
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
heqβ : m = p
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
heqβ : m = p
h3 : Nat.log 3 (3 ^ (n - q)) = Nat.log 3 1
β’ m = p β§ n = q |
https://github.com/aronerben/lean4-playground.git | 5efced915ecee24cd723d28d00aa63f9c7ea0a9c | meetings/ex4.lean | two_pow_three_pow_unique_factorization' | [295, 1] | [327, 21] | rw [Nat.log_pow (by simp)] at h3 | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
heqβ : m = p
h3 : Nat.log 3 (3 ^ (n - q)) = Nat.log 3 1
β’ m = p β§ n = q | case intro
m n q p : β
hβ : p β€ m
hβ : q β€ n
hdvd3 : 3 ^ q β£ 2 ^ m * 3 ^ n
hdvd2 : 2 ^ p β£ 2 ^ m * 3 ^ (n - q)
h2 : m β€ p
heqβ : m = p
h3 : n - q = Nat.log 3 1
β’ m = p β§ n = q |
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