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lean4-htt/tests/lean/bool_simp.lean
Joe Hendrix 01104cc81e
chore: bool and prop lemmas for Mathlib compatibility and improved confluence (#3508) 
This adds a number of lemmas for simplification of `Bool` and `Prop`
terms. It pulls lemmas from Mathlib and adds additional lemmas where
confluence or consistency suggested they are needed.

It has been tested against Mathlib using some automated test
infrastructure.

That testing module is not yet included in this PR, but will be included
as part of this.

Note. There are currently some comments saying the origin of the simp
rule. These will be removed prior to merging, but are added to clarify
where the rule came from during review.

---------

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
2024-03-04 23:56:30 +00:00

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variable (p q : Prop)
variable (b c d : Bool)
variable (u v w : Prop) [Decidable u] [Decidable v] [Decidable w]
-- Specific regressions found when introducing Boolean normalization
#check_simp (b != !c) = false ~> b = !c
#check_simp ¬(u → v ∨ w) ~> u ∧ ¬v ∧ ¬w
#check_simp decide (u ∧ (v → False)) ~> decide u && !decide v
#check_simp decide (cond true b c = true) ~> b
#check_simp decide (ite u b c = true) ~> ite u b c
#check_simp true ≠ (b || c) ~> b = false ∧ c = false
#check_simp ¬((!b = false) ∧ (c = false)) ~> b = true → c = true
#check_simp (((!b) && c) ≠ false) ~> b = false ∧ c = true
#check_simp (cond b false c ≠ false) ~> b = false ∧ c
#check_simp (b && c) = false ~> b → c = false
#check_simp (b && c) ≠ false ~> b ∧ c
#check_simp decide (u → False) ~> !decide u
#check_simp decide (¬u) ~> !decide u
#check_simp (b = true) ≠ (c = false) ~> b = c
#check_simp (b != c) != (false != d) ~> b != (c != d)
#check_simp (b == false) ≠ (c != d) ~> b = (c != d)
#check_simp (b = true) ≠ (c = false) ~> b = c
#check_simp ¬b = !c ~> b = c
#check_simp (b == c) = false ~> ¬(b = c)
#check_simp (true ≠ if u then b else c) ~> (if u then b = false else c = false)
#check_simp (u ∧ v → False) ~> u → v → False
#check_simp (u = (v ≠ w)) ~> (u ↔ ¬(v ↔ w))
#check_simp ((b = false) = (c = false)) ~> (!b) = (!c)
#check_simp True ≠ (c = false) ~> c = true
#check_simp u ∧ u ∧ v ~> u ∧ v
#check_simp b || (b || c) ~> b || c
#check_simp ((b ≠ c) : Bool) ~> !(decide (b = c))
#check_simp ((u ≠ v) : Bool) ~> !((u : Bool) == (v : Bool))
#check_simp decide (u → False) ~> !(decide u)
#check_simp decide (¬u) ~> !u
-- Specific regressions done
-- Round trip isomorphisms
#check_simp (decide (b : Prop)) ~> b
#check_simp ((u : Bool) : Prop) ~> u
/- # not -/
variable [Decidable u]
-- Ground
#check_simp (¬True) ~> False
#check_simp (¬true) ~> False
#check_simp (!True) ~> false
#check_simp (!true) ~> false
#check_simp (¬False) ~> True
#check_simp (!False) ~> true
#check_simp (¬false) ~> True
#check_simp (!false) ~> true
/- # Coercions and not -/
#check_simp ¬p !~>
#check_simp !b !~>
#check_simp (¬u : Prop) !~>
#check_simp (¬u : Bool) ~> !u
#check_simp (!u : Prop) ~> ¬u
#check_simp (!u : Bool) !~>
#check_simp (¬b : Prop) ~> b = false
#check_simp (¬b : Bool) ~> !b
#check_simp (!b : Prop) ~> b = false
#check_simp (!b : Bool) !~>
#check_simp (¬¬u) ~> u
/- # and -/
-- Validate coercions
#check_simp (u ∧ v : Prop) !~>
#check_simp (u ∧ v : Bool) ~> u && v
#check_simp (u && v : Prop) ~> u ∧ v
#check_simp (u && v : Bool) !~>
#check_simp (b ∧ c : Prop) !~>
#check_simp (b ∧ c : Bool) ~> b && c
#check_simp (b && c : Prop) ~> b ∧ c
#check_simp (b && c : Bool) !~>
-- Partial evaluation
#check_simp (True ∧ v : Prop) ~> v
#check_simp (True ∧ v : Bool) ~> (v : Bool)
#check_simp (True && v : Prop) ~> v
#check_simp (True && v : Bool) ~> (v : Bool)
#check_simp (true ∧ c : Prop) ~> (c : Prop)
#check_simp (true ∧ c : Bool) ~> c
#check_simp (true && c : Prop) ~> (c : Prop)
#check_simp (true && c : Bool) ~> c
#check_simp (u ∧ True : Prop) ~> u
#check_simp (u ∧ True : Bool) ~> (u : Bool)
#check_simp (u && True : Prop) ~> u
#check_simp (u && True : Bool) ~> (u : Bool)
#check_simp (b ∧ true : Prop) ~> (b : Prop)
#check_simp (b ∧ true : Bool) ~> b
#check_simp (b && true : Prop) ~> (b : Prop)
#check_simp (b && true : Bool) ~> b
#check_simp (False ∧ v : Prop) ~> False
#check_simp (False ∧ v : Bool) ~> false
#check_simp (False && v : Prop) ~> False
#check_simp (False && v : Bool) ~> false
#check_simp (false ∧ c : Prop) ~> False
#check_simp (false ∧ c : Bool) ~> false
#check_simp (false && c : Prop) ~> False
#check_simp (false && c : Bool) ~> false
#check_simp (u ∧ False : Prop) ~> False
#check_simp (u ∧ False : Bool) ~> false
#check_simp (u && False : Prop) ~> False
#check_simp (u && False : Bool) ~> false
#check_simp (b ∧ false : Prop) ~> False
#check_simp (b ∧ false : Bool) ~> false
#check_simp (b && false : Prop) ~> False
#check_simp (b && false : Bool) ~> false
-- Idempotence
#check_simp (u ∧ u) ~> u
#check_simp (u && u) ~> (u : Bool)
#check_simp (b ∧ b) ~> (b : Prop)
#check_simp (b && b) ~> b
-- Cancelation
#check_simp (u ∧ ¬u) ~> False
#check_simp (¬u ∧ u) ~> False
#check_simp (b && ¬b) ~> false
#check_simp (¬b && b) ~> false
-- Check we swap operators, but do apply deMorgan etc
#check_simp ¬(u ∧ v) ~> u → ¬v
#check_simp decide (¬(u ∧ v)) ~> !u || !v
#check_simp !(u ∧ v) ~> !u || !v
#check_simp ¬(b ∧ c) ~> b → c = false
#check_simp !(b ∧ c) ~> !b || !c
#check_simp ¬(u && v) ~> u → ¬ v
#check_simp ¬(b && c) ~> b = true → c = false
#check_simp !(u && v) ~> !u || !v
#check_simp !(b && c) ~> !b || !c
#check_simp ¬u ∧ ¬v !~>
#check_simp ¬b ∧ ¬c ~> ((b = false) ∧ (c = false))
#check_simp ¬u && ¬v ~> (!u && !v)
#check_simp ¬b && ¬c ~> (!b && !c)
-- Some ternary test cases
#check_simp (u ∧ (v ∧ w) : Prop) !~>
#check_simp (u ∧ (v ∧ w) : Bool) ~> (u && (v && w))
#check_simp ((u ∧ v) ∧ w : Prop) !~>
#check_simp ((u ∧ v) ∧ w : Bool) ~> ((u && v) && w)
#check_simp (b && (c && d) : Prop) ~> (b ∧ c ∧ d)
#check_simp (b && (c && d) : Bool) !~>
#check_simp ((b && c) && d : Prop) ~> ((b ∧ c) ∧ d)
#check_simp ((b && c) && d : Bool) !~>
/- # or -/
-- Validate coercions
#check_simp p ∨ q !~>
#check_simp q ∨ p !~>
#check_simp (u ∨ v : Prop) !~>
#check_simp (u ∨ v : Bool) ~> u || v
#check_simp (u || v : Prop) ~> u ∨ v
#check_simp (u || v : Bool) !~>
#check_simp (b ∨ c : Prop) !~>
#check_simp (b ∨ c : Bool) ~> b || c
#check_simp (b || c : Prop) ~> b ∨ c
#check_simp (b || c : Bool) !~>
-- Partial evaluation
#check_simp (True ∨ v : Prop) ~> True
#check_simp (True ∨ v : Bool) ~> true
#check_simp (True || v : Prop) ~> True
#check_simp (True || v : Bool) ~> true
#check_simp (true ∨ c : Prop) ~> True
#check_simp (true ∨ c : Bool) ~> true
#check_simp (true || c : Prop) ~> True
#check_simp (true || c : Bool) ~> true
#check_simp (u ∨ True : Prop) ~> True
#check_simp (u ∨ True : Bool) ~> true
#check_simp (u || True : Prop) ~> True
#check_simp (u || True : Bool) ~> true
#check_simp (b ∨ true : Prop) ~> True
#check_simp (b ∨ true : Bool) ~> true
#check_simp (b || true : Prop) ~> True
#check_simp (b || true : Bool) ~> true
#check_simp (False ∨ v : Prop) ~> v
#check_simp (False ∨ v : Bool) ~> (v : Bool)
#check_simp (False || v : Prop) ~> v
#check_simp (False || v : Bool) ~> (v : Bool)
#check_simp (false ∨ c : Prop) ~> (c : Prop)
#check_simp (false ∨ c : Bool) ~> c
#check_simp (false || c : Prop) ~> (c : Prop)
#check_simp (false || c : Bool) ~> c
#check_simp (u ∨ False : Prop) ~> u
#check_simp (u ∨ False : Bool) ~> (u : Bool)
#check_simp (u || False : Prop) ~> u
#check_simp (u || False : Bool) ~> (u : Bool)
#check_simp (b ∨ false : Prop) ~> (b : Prop)
#check_simp (b ∨ false : Bool) ~> b
#check_simp (b || false : Prop) ~> (b : Prop)
#check_simp (b || false : Bool) ~> b
-- Idempotence
#check_simp (u ∨ u) ~> u
#check_simp (u || u) ~> (u : Bool)
#check_simp (b ∨ b) ~> (b : Prop)
#check_simp (b || b) ~> b
-- Complement
-- Note. We may want to revisit this.
-- Decidable excluded middle currently does not simplify.
#check_simp ( u ∨ ¬u) !~>
#check_simp (¬u ∨ u) !~>
#check_simp ( b || ¬b) ~> true
#check_simp (¬b || b) ~> true
-- Check we swap operators, but do apply deMorgan etc
#check_simp ¬(u ∨ v) ~> ¬u ∧ ¬v
#check_simp !(u ∨ v) ~> !u && !v
#check_simp ¬(b ∨ c) ~> b = false ∧ c =false
#check_simp !(b ∨ c) ~> !b && !c
#check_simp ¬(u || v) ~> ¬u ∧ ¬v
#check_simp ¬(b || c) ~> b = false ∧ c = false
#check_simp !(u || v) ~> !u && !v
#check_simp !(b || c) ~> !b && !c
#check_simp ¬u ∨ ¬v !~>
#check_simp (¬b) ∨ (¬c) ~> b = false ∨ c = false
#check_simp ¬u || ¬v ~> (!u || !v)
#check_simp ¬b || ¬c ~> (!b || !c)
-- Some ternary test cases
#check_simp (u ∨ (v ∨ w) : Prop) !~>
#check_simp (u ∨ (v ∨ w) : Bool) ~> (u || (v || w))
#check_simp ((u ∨ v) ∨ w : Prop) !~>
#check_simp ((u ∨ v) ∨ w : Bool) ~> ((u || v) || w)
#check_simp (b || (c || d) : Prop) ~> (b ∨ c ∨ d)
#check_simp (b || (c || d) : Bool) !~>
#check_simp ((b || c) || d : Prop) ~> ((b ∨ c) ∨ d)
#check_simp ((b || c) || d : Bool) !~>
/- # and/or -/
-- We don't currently do automatic simplification across and/or/not
-- This tests for non-unexpected reductions.
#check_simp p ∧ (p ∨ q) !~>
#check_simp (p ∨ q) ∧ p !~>
#check_simp u ∧ (v ∨ w) !~>
#check_simp u ∨ (v ∧ w) !~>
#check_simp (v ∨ w) ∧ u !~>
#check_simp (v ∧ w) ∨ u !~>
#check_simp b && (c || d) !~>
#check_simp b || (c && d) !~>
#check_simp (c || d) && b !~>
#check_simp (c && d) || b !~>
/- # implication -/
#check_simp (b → c) !~>
#check_simp (u → v) !~>
#check_simp p → q !~>
#check_simp decide (u → ¬v) ~> !u || !v
/- # iff -/
#check_simp (u = v : Prop) ~> u ↔ v
#check_simp (u = v : Bool) ~> u == v
#check_simp (u ↔ v : Prop) !~>
#check_simp (u ↔ v : Bool) ~> u == v
#check_simp (u == v : Prop) ~> u ↔ v
#check_simp (u == v : Bool) !~>
#check_simp (b = c : Prop) !~>
#check_simp (b = c : Bool) !~>
#check_simp (b ↔ c : Prop) ~> b = c
#check_simp (b ↔ c : Bool) ~> decide (b = c)
#check_simp (b == c : Prop) ~> b = c
#check_simp (b == c : Bool) !~>
-- Partial evaluation
#check_simp (True = v : Prop) ~> v
#check_simp (True = v : Bool) ~> (v : Bool)
#check_simp (True ↔ v : Prop) ~> v
#check_simp (True ↔ v : Bool) ~> (v : Bool)
#check_simp (True == v : Prop) ~> v
#check_simp (True == v : Bool) ~> (v : Bool)
#check_simp (true = c : Prop) ~> c = true
#check_simp (true = c : Bool) ~> c
#check_simp (true ↔ c : Prop) ~> c = true
#check_simp (true ↔ c : Bool) ~> c
#check_simp (true == c : Prop) ~> (c : Prop)
#check_simp (true == c : Bool) ~> c
#check_simp (v = True : Prop) ~> v
#check_simp (v = True : Bool) ~> (v : Bool)
#check_simp (v ↔ True : Prop) ~> v
#check_simp (v ↔ True : Bool) ~> (v : Bool)
#check_simp (v == True : Prop) ~> v
#check_simp (v == True : Bool) ~> (v : Bool)
#check_simp (c = true : Prop) !~>
#check_simp (c = true : Bool) ~> c
#check_simp (c ↔ true : Prop) ~> c = true
#check_simp (c ↔ true : Bool) ~> c
#check_simp (c == true : Prop) ~> c = true
#check_simp (c == true : Bool) ~> c
#check_simp (True = v : Prop) ~> v
#check_simp (True = v : Bool) ~> (v : Bool)
#check_simp (True ↔ v : Prop) ~> v
#check_simp (True ↔ v : Bool) ~> (v : Bool)
#check_simp (True == v : Prop) ~> v
#check_simp (True == v : Bool) ~> (v : Bool)
#check_simp (true = c : Prop) ~> c = true
#check_simp (true = c : Bool) ~> c
#check_simp (true ↔ c : Prop) ~> c = true
#check_simp (true ↔ c : Bool) ~> c
#check_simp (true == c : Prop) ~> (c : Prop)
#check_simp (true == c : Bool) ~> c
#check_simp (v = False : Prop) ~> ¬v
#check_simp (v = False : Bool) ~> !v
#check_simp (v ↔ False : Prop) ~> ¬v
#check_simp (v ↔ False : Bool) ~> !v
#check_simp (v == False : Prop) ~> ¬v
#check_simp (v == False : Bool) ~> !v
#check_simp (c = false : Prop) !~>
#check_simp (c = false : Bool) ~> !c
#check_simp (c ↔ false : Prop) ~> c = false
#check_simp (c ↔ false : Bool) ~> !c
#check_simp (c == false : Prop) ~> c = false
#check_simp (c == false : Bool) ~> !c
#check_simp (False = v : Prop) ~> ¬v
#check_simp (False = v : Bool) ~> !v
#check_simp (False ↔ v : Prop) ~> ¬v
#check_simp (False ↔ v : Bool) ~> !v
#check_simp (False == v : Prop) ~> ¬v
#check_simp (False == v : Bool) ~> !v
#check_simp (false = c : Prop) ~> c = false
#check_simp (false = c : Bool) ~> !c
#check_simp (false ↔ c : Prop) ~> c = false
#check_simp (false ↔ c : Bool) ~> !c
#check_simp (false == c : Prop) ~> c = false
#check_simp (false == c : Bool) ~> !c
-- Ternary (expand these)
#check_simp (u == (v = w)) ~> u == (v == w)
#check_simp (u == (v == w)) !~>
/- # bne -/
#check_simp p ≠ q ~> ¬(p ↔ q)
#check_simp (b != c : Bool) !~>
#check_simp ¬(p = q) ~> ¬(p ↔ q)
#check_simp b ≠ c ~> b ≠ c
#check_simp ¬(b = c) !~>
#check_simp ¬(b ↔ c) ~> ¬(b = c)
#check_simp (b != c : Prop) ~> b ≠ c
#check_simp u ≠ v ~> ¬(u ↔ v)
#check_simp ¬(u = v) ~> ¬(u ↔ v)
#check_simp ¬(u ↔ v) !~>
#check_simp ((u:Bool) != v : Bool) !~>
#check_simp ((u:Bool) != v : Prop) ~> ¬(u ↔ v)
/- # equality and and/or interactions -/
#check_simp (u == (v ∨ w)) ~> u == (v || w)
#check_simp (u == (v || w)) !~>
#check_simp ((u ∧ v) == w) ~> (u && v) == w
/- # ite/cond -/
#check_simp if b then c else d !~>
#check_simp if b then p else q !~>
#check_simp if u then p else q !~>
#check_simp if u then b else c !~>
#check_simp if u then u else q ~> ¬u → q
#check_simp if u then q else u ~> u ∧ q
#check_simp if u then q else q ~> q
#check_simp cond b c d !~>
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