A mathematical proof is any mathematical argument which demonstrates the truth of a mathematical statement. Informal proofs are typically rendered in natural language and are held true by consensus; formal proofs are typically rendered symbolically and can be checked mechanically. "Proofs" can be valid or invalid; only the former kind constitutes actual proof, whereas the latter kind usually refers to a flawed attempt at proof.
Questions tagged [proof]
828 questions
2
votes
1 answer
Proof of dynamic programming solution for Leetcode 818: Racecar
The problem is as follows:
Problem:
"Your car starts at position 0 and speed +1 on an infinite number line. Your car can go into negative positions. Your car drives automatically according to a sequence of instructions 'A' (accelerate) and 'R'…
punypaw
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2
votes
2 answers
How do I write ∀x ( P(x) and Q(x) ) in Coq?
I'm trying out Coq, but I'm not completely sure what I'm doing. Is:
Theorem new_theorem : forall x, P:Prop /\ Q:Prop
Equivalent to:
∀x ( P(x) and Q(x) )
Edit: I think they are.
Peter
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How do I formally prove the safety of a Rust crate with generics and trait bounds?
How to give formal proofs by hand? Is there a language specification, as formal as formal logic, that we can build our proof on?
For example, how do I know if this module panics?
mod my_mod {
use std::marker::PhantomData;
…
pepperjuice
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2
votes
1 answer
Modular arithmetic proofs in agda
I'm trying to prove (n : ℕ) → ∃[ m ] n * n ≡ 3 * m + 2 → ⊥. Typically I would prove this by rewriting it in terms of congruence, and then splitting on each case. There doesn't seem to be a modular arithmetic module in agda-stdlib. How should I…
Ace shinigami
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votes
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I'm trying to build a proof in Coq that two different permutation definitions are equivalent, but the non-inductive side is not working
The two definitions are these:
Inductive perm : list nat -> list nat -> Prop :=
| perm_eq: forall l1, perm l1 l1
| perm_swap: forall x y l1, perm (x :: y :: l1) (y :: x :: l1)
| perm_hd: forall x l1 l2, perm l1 l2 -> perm (x :: l1) (x :: l2)
|…
Andrey
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2
votes
1 answer
Proof of optimality of greedy algorithm for scheduling
Unable to come up with a formal proof of optimality for algorithm A for the given problem. Have convinced myself that it is possible to execute some optimal schedule O in increasing order of the events' deadline. But don't know how to formally prove…
Ajax
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votes
1 answer
Coq: Simpl in match pattern when having an inequality hypothesis
I have a definition involving match, similar like this:
Definition five (n: nat): bool :=
match n with
| 5 => true
| _ => false
end.
I try to proof something similar like this:
Theorem fiveT: forall (n: nat),
n <> 5 -> five n = false.
Proof.…
Leo G.
- 33
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votes
2 answers
Can you check for duplicates by taking the sum of the array and then the product of the array?
Let's say we have an array of size N with values from 1 to N inside it. We want to check if this array has any duplicates. My friend suggested two ways that I showed him were wrong:
Take the sum of the array and check it against the sum…
Majd Odeh
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votes
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Missing Order.Lattice function from Arend standard library added by Intellij IDEA plugin
I tried to prove some lemmas about Implicative Lattice and decided to use function meet-monotone from Order.Lattice.MeetSemilattice. I found this function in manually downloaded standard lib, but when I'm trying to use it in my code the syntax…
warmte
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- 2
2
votes
0 answers
Coinductive principle for streams
I am trying to prove the following principle for stream predicates (defined in the standard library).
From Coq Require Import Streams.
Lemma mystream_ind :
forall A (P : Stream A -> Prop),
(forall s, ForAll P (tl s) -> ForAll P s) ->
…
pjm
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votes
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Equivalence of Edit Distance and Alignment Distance
(from: https://math.mit.edu/classes/18.417/Slides/alignment.pdf)
The slide on the 11th page talks about how the Edit Distance and the Alignment Distance are equivalent.
I understand how to prove that the Edit Distance will always be less than or…
datoad
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votes
0 answers
Does this strict weak ordering have a name (spoilers for a specific coding puzzle)
There is a coding puzzle I have encountered on one of those sites (I don't recall if it was leetcode or something else) which goes as follows: Given a list of strings, return the lexicographically smallest concatenation that uses each of the strings…
Cereal
- 156
- 7
2
votes
2 answers
Proving A → ¬ (¬ A ∧ B) in Lean
I am having a hard time proving A → ¬ (¬ A ∧ B) with the Lean theorem prover. I set it up like this:
example : A → ¬ (¬ A ∧ B) :=
assume h1: ¬ (¬ A ∧ B),
assume h2: A,
assume h3: B,
show false, from sorry
I was unable to find examples to prove this…
Lina Nguyen
- 21
- 1
2
votes
2 answers
Convert a Bits8 to a `Subset Nat (`LT` 256)`
I have the following module:
module Nat256
import Data.DPair
import Data.Bits
public export
bits8ToNat256 : Bits8 -> Subset Nat (`LT` 256)
bits8ToNat256 i =
case i of
0 => 0
1 => 1
2 => 2
3 => 3
4 => 4
5 => 5
6 => 6
…
Janus Troelsen
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votes
2 answers
How to write a 'safe' head in coq?
I'm trying to do something in Coq similar to this liquid Haskell trick, which defines a partial function but proves it's actually total:
{-@ head :: {xs:[a] | len xs > 0} -> a @-}
head (x:xs) = x
Here is my first attempt, but Coq doesn't like it…
Felipe
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