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
|
|---|---|---|---|---|---|---|---|---|
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | apply Int.gcd_dvd_right | a b c x y : ℤ
h : a * x + b * y = c
⊢ ↑(Int.gcd a b) ∣ b | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_08/Diaphontine.lean | eqn_solvable_divides | [50, 1] | [62, 32] | apply Int.dvd_mul_right | a b c x y : ℤ
h : a * x + b * y = c
⊢ b ∣ b * y | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | match m, n, hyp with
| m, 0, _ => simp [minus]
| 0, _ +1, pf => contradiction
| m + 1, n + 1 , pf =>
simp [minus]
rw [← add_assoc]
simp [minus_add_cancel] | m n : ℕ
hyp : n ≤ m
⊢ minus m n hyp + n = m | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | simp [minus] | m✝ n : ℕ
hyp : n ≤ m✝
m : ℕ
x✝ : 0 ≤ m
⊢ minus m 0 x✝ + 0 = m | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | contradiction | m n : ℕ
hyp : n ≤ m
n✝ : ℕ
pf : n✝ + 1 ≤ 0
⊢ minus 0 (n✝ + 1) pf + (n✝ + 1) = 0 | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | simp [minus] | m✝ n✝ : ℕ
hyp : n✝ ≤ m✝
m n : ℕ
pf : n + 1 ≤ m + 1
⊢ minus (m + 1) (n + 1) pf + (n + 1) = m + 1 | m✝ n✝ : ℕ
hyp : n✝ ≤ m✝
m n : ℕ
pf : n + 1 ≤ m + 1
⊢ minus m n ⋯ + (n + 1) = m + 1 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | rw [← add_assoc] | m✝ n✝ : ℕ
hyp : n✝ ≤ m✝
m n : ℕ
pf : n + 1 ≤ m + 1
⊢ minus m n ⋯ + (n + 1) = m + 1 | m✝ n✝ : ℕ
hyp : n✝ ≤ m✝
m n : ℕ
pf : n + 1 ≤ m + 1
⊢ minus m n ⋯ + n + 1 = m + 1 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMinus.lean | Nat.minus_add_cancel | [42, 1] | [51, 28] | simp [minus_add_cancel] | m✝ n✝ : ℕ
hyp : n✝ ≤ m✝
m n : ℕ
pf : n + 1 ≤ m + 1
⊢ minus m n ⋯ + n + 1 = m + 1 | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_11/NatLe.lean | four_le_seven | [106, 1] | [115, 20] | apply Nat.le_step | ⊢ 4 ≤ 7 | case h
⊢ 4 ≤ 6 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_11/NatLe.lean | four_le_seven | [106, 1] | [115, 20] | apply Nat.le_step | case h
⊢ 4 ≤ 6 | case h.h
⊢ 4 ≤ 5 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_11/NatLe.lean | four_le_seven | [106, 1] | [115, 20] | apply Nat.le_step | case h.h
⊢ 4 ≤ 5 | case h.h.h
⊢ 4 ≤ 4 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_11/NatLe.lean | four_le_seven | [106, 1] | [115, 20] | apply Nat.le_refl | case h.h.h
⊢ 4 ≤ 4 | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_11/NatLe.lean | four_le_ten | [149, 1] | [150, 12] | decide | ⊢ 4 ≤ 10 | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.disagree_eq_maybe | [86, 1] | [89, 36] | cases ans <;> simp [disagree] | ans : Answer
⊢ disagree ans = ans → ans = Answer.maybe | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | intro h₁ h₂ | a b c : Answer
⊢ a ≤ b → b ≤ c → a ≤ c | a b c : Answer
h₁ : a ≤ b
h₂ : b ≤ c
⊢ a ≤ c |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | cases c₁:h₁ | a b c : Answer
h₁ : a ≤ b
h₂ : b ≤ c
⊢ a ≤ c | case no_le
b c : Answer
h₂ : b ≤ c
h₁ : Answer.no ≤ b
c₁ : h₁ = ⋯
⊢ Answer.no ≤ c
case le_yes
a c : Answer
h₁ : a ≤ Answer.yes
h₂ : Answer.yes ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c
case refl
a c : Answer
h₁ : a ≤ a
h₂ : a ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | case no_le =>
apply Answer.le.no_le | b c : Answer
h₂ : b ≤ c
h₁ : Answer.no ≤ b
c₁ : h₁ = ⋯
⊢ Answer.no ≤ c | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | case le_yes =>
cases c₂:h₂
case le_yes => apply Answer.le.le_yes
case refl => assumption | a c : Answer
h₁ : a ≤ Answer.yes
h₂ : Answer.yes ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | case refl => assumption | a c : Answer
h₁ : a ≤ a
h₂ : a ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | apply Answer.le.no_le | b c : Answer
h₂ : b ≤ c
h₁ : Answer.no ≤ b
c₁ : h₁ = ⋯
⊢ Answer.no ≤ c | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | cases c₂:h₂ | a c : Answer
h₁ : a ≤ Answer.yes
h₂ : Answer.yes ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c | case le_yes
a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes
case refl
a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | case le_yes => apply Answer.le.le_yes | a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | case refl => assumption | a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | apply Answer.le.le_yes | a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | assumption | a : Answer
h₁ : a ≤ Answer.yes
c₁ : h₁ = ⋯
h₂ : Answer.yes ≤ Answer.yes
c₂ : h₂ = ⋯
⊢ a ≤ Answer.yes | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_25/Answer.lean | Waffle.le_trans | [279, 1] | [290, 28] | assumption | a c : Answer
h₁ : a ≤ a
h₂ : a ≤ c
c₁ : h₁ = ⋯
⊢ a ≤ c | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | simp [List.remove] | case nil
α : Type u
inst✝ : DecidableEq α
a : α
⊢ List.length (List.remove a []) ≤ List.length [] | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | simp [List.remove] | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
⊢ List.length (List.remove a (h' :: t)) ≤ List.length (h' :: t) | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
⊢ List.length (if a = h' then List.remove a t else h' :: List.remove a t) ≤ Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | split | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
⊢ List.length (if a = h' then List.remove a t else h' :: List.remove a t) ≤ Nat.succ (List.length t) | case cons.inl
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ Nat.succ (List.length t)
case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ List.length (h' :: List.remove a t) ≤ Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | apply Nat.le_step | case cons.inl
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ Nat.succ (List.length t) | case cons.inl.h
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ List.length t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | assumption | case cons.inl.h
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ List.length t | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | rw [List.length_cons] | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ List.length (h' :: List.remove a t) ≤ Nat.succ (List.length t) | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ Nat.succ (List.length (List.remove a t)) ≤ Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | apply Nat.succ_le_succ | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ Nat.succ (List.length (List.remove a t)) ≤ Nat.succ (List.length t) | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ List.length (List.remove a t) ≤ List.length t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_length_le | [17, 1] | [28, 17] | exact ih | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : List.length (List.remove a t) ≤ List.length t
h✝ : ¬a = h'
⊢ List.length (List.remove a t) ≤ List.length t | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | contradiction | case nil
α : Type u
inst✝ : DecidableEq α
a : α
hyp : a ∈ []
⊢ List.length (List.remove a []) < List.length [] | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | simp [List.remove] | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
⊢ List.length (List.remove a (h' :: t)) < List.length (h' :: t) | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
⊢ List.length (if a = h' then List.remove a t else h' :: List.remove a t) < Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | split | case cons
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
⊢ List.length (if a = h' then List.remove a t else h' :: List.remove a t) < Nat.succ (List.length t) | case cons.inl
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : a = h'
⊢ List.length (List.remove a t) < Nat.succ (List.length t)
case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ List.length (h' :: List.remove a t) < Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | apply Nat.lt_succ_of_le | case cons.inl
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : a = h'
⊢ List.length (List.remove a t) < Nat.succ (List.length t) | case cons.inl.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ List.length t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | apply remove_length_le | case cons.inl.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : a = h'
⊢ List.length (List.remove a t) ≤ List.length t | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | rw [List.length_cons] | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ List.length (h' :: List.remove a t) < Nat.succ (List.length t) | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ Nat.succ (List.length (List.remove a t)) < Nat.succ (List.length t) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | apply Nat.succ_lt_succ | case cons.inr
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ Nat.succ (List.length (List.remove a t)) < Nat.succ (List.length t) | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ List.length (List.remove a t) < List.length t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | have in_tail: a ∈ t := by
have : ¬ a = h' := by assumption
simp [List.mem_cons, this] at hyp
assumption | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ List.length (List.remove a t) < List.length t | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
in_tail : a ∈ t
⊢ List.length (List.remove a t) < List.length t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | exact ih in_tail | case cons.inr.a
α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
in_tail : a ∈ t
⊢ List.length (List.remove a t) < List.length t | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | have : ¬ a = h' := by assumption | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ a ∈ t | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ this : ¬a = h'
⊢ a ∈ t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | simp [List.mem_cons, this] at hyp | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ this : ¬a = h'
⊢ a ∈ t | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
h✝ this : ¬a = h'
hyp : a ∈ t
⊢ a ∈ t |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | assumption | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
h✝ this : ¬a = h'
hyp : a ∈ t
⊢ a ∈ t | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_24/Sampling.lean | remove_mem_length | [32, 1] | [47, 25] | assumption | α : Type u
inst✝ : DecidableEq α
a h' : α
t : List α
ih : a ∈ t → List.length (List.remove a t) < List.length t
hyp : a ∈ h' :: t
h✝ : ¬a = h'
⊢ ¬a = h' | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Labs/Lab02/AnswerLE.lean | Waffle.Answer.eq_of_le_le | [5, 1] | [6, 37] | sorry | a b : Answer
⊢ a ≤ b → b ≤ a → a = b | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | by_cases c:tail = [] | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ smallest (head :: tail) hyp ∈ head :: tail | case pos
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : tail = []
⊢ smallest (head :: tail) hyp ∈ head :: tail
case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
⊢ smallest (head :: tail) hyp ∈ head :: tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | simp [smallest, c] | case pos
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : tail = []
⊢ smallest (head :: tail) hyp ∈ head :: tail | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | simp [smallest, c] | case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
⊢ smallest (head :: tail) hyp ∈ head :: tail | case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | by_cases c':head ≤ smallest tail (by simp [c]) | case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail | case pos
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail
case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | simp [c] | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
⊢ tail ≠ [] | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | left | case pos
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail | case pos.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | assumption | case pos.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | right | case neg
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ head ≤ smallest tail ⋯ ∨ Nat.min head (smallest tail ⋯) ∈ tail | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ Nat.min head (smallest tail ⋯) ∈ tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | have lem :
Nat.min head (smallest tail (by simp [c])) = smallest tail (by simp [c]) := by
apply Nat.min_eq_right
apply Nat.le_of_not_le
assumption | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ Nat.min head (smallest tail ⋯) ∈ tail | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
lem : Nat.min head (smallest tail ⋯) = smallest tail ⋯
⊢ Nat.min head (smallest tail ⋯) ∈ tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | rw [lem] | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
lem : Nat.min head (smallest tail ⋯) = smallest tail ⋯
⊢ Nat.min head (smallest tail ⋯) ∈ tail | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
lem : Nat.min head (smallest tail ⋯) = smallest tail ⋯
⊢ smallest tail ⋯ ∈ tail |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | simp [smallest_in_list] | case neg.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
lem : Nat.min head (smallest tail ⋯) = smallest tail ⋯
⊢ smallest tail ⋯ ∈ tail | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | simp [c] | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ tail ≠ [] | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | apply Nat.min_eq_right | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ Nat.min head (smallest tail ⋯) = smallest tail ⋯ | case h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ smallest tail ⋯ ≤ head |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | apply Nat.le_of_not_le | case h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ smallest tail ⋯ ≤ head | case h.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ ¬head ≤ smallest tail ⋯ |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_in_list | [30, 1] | [48, 34] | assumption | case h.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
c : ¬tail = []
c' : ¬head ≤ smallest tail ⋯
⊢ ¬head ≤ smallest tail ⋯ | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | simp [smallest, hyp] | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ ∀ m ∈ head :: tail, smallest (head :: tail) hyp ≤ m | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ head ∧
∀ a ∈ tail, (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | apply And.intro | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ head ∧
∀ a ∈ tail, (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a | case left
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ head
case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ ∀ a ∈ tail, (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | split <;> simp | case left
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ head | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | intro a hyp' | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
⊢ ∀ a ∈ tail, (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | have c'' : tail ≠ [] := by
intro contra
rw [contra] at hyp'
contradiction | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | simp [c''] | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ (if c : tail = [] then head else Nat.min head (smallest tail c)) ≤ a | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ head ≤ a ∨ smallest tail ⋯ ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | right | case right
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ head ≤ a ∨ smallest tail ⋯ ≤ a | case right.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ smallest tail ⋯ ≤ a |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | exact smallest_le tail c'' a hyp' | case right.h
l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
c'' : tail ≠ []
⊢ smallest tail ⋯ ≤ a | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | intro contra | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
⊢ tail ≠ [] | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
contra : tail = []
⊢ False |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | rw [contra] at hyp' | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ tail
contra : tail = []
⊢ False | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ []
contra : tail = []
⊢ False |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_01_20/NatMin.lean | smallest_le | [62, 1] | [75, 40] | contradiction | l : List ℕ
head : ℕ
tail : List ℕ
hyp : head :: tail ≠ []
a : ℕ
hyp' : a ∈ []
contra : tail = []
⊢ False | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Labs/Lab02/SelectionSort.lean | remove_length_le | [40, 1] | [41, 8] | sorry | a : ℕ
l : List ℕ
⊢ List.length (List.remove a l) ≤ List.length l | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Labs/Lab02/SelectionSort.lean | remove_mem_length | [45, 1] | [46, 8] | sorry | a : ℕ
l : List ℕ
hyp : a ∈ l
⊢ List.length (List.remove a l) < List.length l | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_10/FormalCalculus.lean | integral_point | [24, 1] | [29, 15] | unfold integral | f : ℝ → ℝ
int : Integrable f
a : ℝ
⊢ integral f a a = 0 | f : ℝ → ℝ
int : Integrable f
a : ℝ
⊢ Integrable.integral f a a = 0 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_10/FormalCalculus.lean | integral_point | [24, 1] | [29, 15] | have l := int.interval_union a a a | f : ℝ → ℝ
int : Integrable f
a : ℝ
⊢ Integrable.integral f a a = 0 | f : ℝ → ℝ
int : Integrable f
a : ℝ
l : Integrable.integral f a a = Integrable.integral f a a + Integrable.integral f a a
⊢ Integrable.integral f a a = 0 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_10/FormalCalculus.lean | integral_point | [24, 1] | [29, 15] | simp at l | f : ℝ → ℝ
int : Integrable f
a : ℝ
l : Integrable.integral f a a = Integrable.integral f a a + Integrable.integral f a a
⊢ Integrable.integral f a a = 0 | f : ℝ → ℝ
int : Integrable f
a : ℝ
l : Integrable.integral f a a = 0
⊢ Integrable.integral f a a = 0 |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Lec_03_10/FormalCalculus.lean | integral_point | [24, 1] | [29, 15] | assumption | f : ℝ → ℝ
int : Integrable f
a : ℝ
l : Integrable.integral f a a = 0
⊢ Integrable.integral f a a = 0 | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Extras/SumToN.lean | sum_formula | [7, 1] | [16, 13] | induction n with
| zero => rfl
| succ n ih =>
simp [sumTo] simp [left_distrib, ih]
linarith | n : ℕ
⊢ 2 * sumTo n = n * (n + 1) | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Extras/SumToN.lean | sum_formula | [7, 1] | [16, 13] | rfl | case zero
⊢ 2 * sumTo Nat.zero = Nat.zero * (Nat.zero + 1) | no goals |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Extras/SumToN.lean | sum_formula | [7, 1] | [16, 13] | simp [sumTo] | case succ
n : ℕ
ih : 2 * sumTo n = n * (n + 1)
⊢ 2 * sumTo (Nat.succ n) = Nat.succ n * (Nat.succ n + 1) | case succ
n : ℕ
ih : 2 * sumTo n = n * (n + 1)
⊢ 2 * (n + 1 + sumTo n) = Nat.succ n * (Nat.succ n + 1) |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Extras/SumToN.lean | sum_formula | [7, 1] | [16, 13] | simp [left_distrib, ih] | case succ
n : ℕ
ih : 2 * sumTo n = n * (n + 1)
⊢ 2 * (n + 1 + sumTo n) = Nat.succ n * (Nat.succ n + 1) | case succ
n : ℕ
ih : 2 * sumTo n = n * (n + 1)
⊢ 2 * n + 2 + (n * n + n) = Nat.succ n * Nat.succ n + Nat.succ n |
https://github.com/siddhartha-gadgil/proofs-and-programs-2023.git | 92fd2d8464d8687b5b0bcd7b17612440c304d19c | PnP2023/Extras/SumToN.lean | sum_formula | [7, 1] | [16, 13] | linarith | case succ
n : ℕ
ih : 2 * sumTo n = n * (n + 1)
⊢ 2 * n + 2 + (n * n + n) = Nat.succ n * Nat.succ n + Nat.succ n | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | repeat (first | apply IsQuizSchema.consQuiz (by decide)
| apply IsQuizSchema.consNonQuiz (by decide)
| apply IsQuizSchema.nil) | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema (schema gradebook) | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | (first | apply IsQuizSchema.consQuiz (by decide)
| apply IsQuizSchema.consNonQuiz (by decide)
| apply IsQuizSchema.nil) | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [] | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | first | apply IsQuizSchema.consQuiz (by decide)
| apply IsQuizSchema.consNonQuiz (by decide)
| apply IsQuizSchema.nil | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [] | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | apply IsQuizSchema.consQuiz (by decide) | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [("final", Nat)] | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | decide | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ String.startsWith "quiz4" "quiz" = true | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | apply IsQuizSchema.consNonQuiz (by decide) | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [("final", Nat)] | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [] |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | decide | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ String.startsWith "final" "quiz" = false | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/ExamplePrograms.lean | gradebook_schema_is_quiz_schema | [140, 1] | [143, 42] | apply IsQuizSchema.nil | η : Type u_η
inst✝ : DecidableEq η
sch : Schema
⊢ IsQuizSchema [] | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Demo.lean | nrows_groupBySubtractive | [199, 1] | [205, 30] | have h_gbs := groupBySubtractive_spec7 t c hc | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
⊢ nrows (groupBySubtractive t c) ≤ nrows t | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ nrows (groupBySubtractive t c) ≤ nrows t |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Demo.lean | nrows_groupBySubtractive | [199, 1] | [205, 30] | rw [h_gbs] | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ nrows (groupBySubtractive t c) ≤ nrows t | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ List.length (List.unique (getColumn2 t c)) ≤ nrows t |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Demo.lean | nrows_groupBySubtractive | [199, 1] | [205, 30] | rw [←getColumn2_spec2 t c hc] | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ List.length (List.unique (getColumn2 t c)) ≤ nrows t | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ List.length (List.unique (getColumn2 t c)) ≤ List.length (getColumn2 t c) |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Demo.lean | nrows_groupBySubtractive | [199, 1] | [205, 30] | apply List.length_unique_le | η : Type u_1
τ : Type u_2
inst✝¹ : DecidableEq η
inst✝ : DecidableEq τ
schema : Schema
t : Table schema
c : η
hc : Schema.HasCol (c, τ) schema
h_gbs : nrows (groupBySubtractive t c) = List.length (List.unique (getColumn2 t c))
⊢ List.length (List.unique (getColumn2 t c)) ≤ List.length (getColumn2 t c) | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Schema.lean | Schema.lookup_of_colImpliesName | [353, 1] | [361, 33] | rw [colImpliesName_eq_1 (sch' := ss) (hdr := (nm, τ)),
lookup_eq_1] | η : Type u_η
dec_η : DecidableEq η
schema : Schema
nm : η
τ : Type u_1
ss : List Header
⊢ lookup ((nm, τ) :: ss) { fst := nm, snd := colImpliesName HasCol.hd } = (nm, τ) | no goals |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Schema.lean | Schema.lookup_of_colImpliesName | [353, 1] | [361, 33] | rw [colImpliesName_eq_2, lookup_eq_2] | η : Type u_η
dec_η : DecidableEq η
schema : Schema
nm : η
τ : Type u_1
head✝ : Header
ss : List Header
h : HasCol (nm, τ) ss
⊢ lookup (head✝ :: ss) { fst := nm, snd := colImpliesName (HasCol.tl h) } = (nm, τ) | η : Type u_η
dec_η : DecidableEq η
schema : Schema
nm : η
τ : Type u_1
head✝ : Header
ss : List Header
h : HasCol (nm, τ) ss
⊢ lookup ss { fst := nm, snd := colImpliesName h } = (nm, τ) |
https://github.com/jrr6/lean-tables.git | 7dfa8308e13cb7b15296cc63fa2cbd26c0d0f712 | Table/Schema.lean | Schema.lookup_of_colImpliesName | [353, 1] | [361, 33] | apply lookup_of_colImpliesName | η : Type u_η
dec_η : DecidableEq η
schema : Schema
nm : η
τ : Type u_1
head✝ : Header
ss : List Header
h : HasCol (nm, τ) ss
⊢ lookup ss { fst := nm, snd := colImpliesName h } = (nm, τ) | no goals |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.