Research proof index

Illustration that equiv_implies_canonical_eq does not hold for all functions f -- in particular, does not hold for Prod.fst. (One canonicalize produces [(1, false)], and the other produces [(1, true)].) -/ example : xs = [(1, false), (1, true)] → ys = [(1, true), (1, false)] → xs ≡ ys ∧ (canonicalize Prod.fst xs) ≠ (canonicalize Prod.fst ys) :...

our own variant of map_congr, for mapM -/ public theorem mapM_congr [Monad m] [LawfulMonad m] {f g : α → m β} : ∀ {l : List α}, (∀ x ∈ l, f x = g x) → mapM f l = mapM g l := by intro l rw [← mapM'_eq_mapM, ← mapM'_eq_mapM] exact mapM'_congr public theorem to_option_distr_mapM' {α β ε} {l : List α} {f: α → Except ε β} : (List.mapM' f l).toOption = (List.ma...

our own variant of map_congr, for mapM -/ public theorem mapM_congr [Monad m] [LawfulMonad m] {f g : α → m β} : ∀ {l : List α}, (∀ x ∈ l, f x = g x) → mapM f l = mapM g l := by intro l rw [← mapM'_eq_mapM, ← mapM'_eq_mapM] exact mapM'_congr public theorem to_option_distr_mapM' {α β ε} {l : List α} {f: α → Except ε β} : (List.mapM' f l).toOption = (List.ma...

Applying mapM on the RHS list of Forall₂ leads to new Forall₂ relation if certain conditions are met.

Note the converse is not true: counterexample R is =, xs is [1], ys is [1, 2]

kind of a transitivity property, if you squint