feat: Decidable as subtype of Bool

[Rob23oba]

This PR changes the definition of Decidable p to a structure containing a Bool and a proof of either p or ¬p.

This is basically the approach proposed by @kmill in #2038.

Due to bugs in the old compiler, this was previously not possible; however, now that the new compiler is enabled, this works perfectly fine.

Using Bool in the definition of Decidable has several advantages, in particular

• There are many more definitional equalities, e.g.

  variable (a b : Bool)
  
  #check (rfl : decide (a = true) = a)
  #check (rfl : decide (a = false) = !a)
  #check (rfl : decide (a = true ∧ b = true) = a && b)
  #check (rfl : decide (a = true ∨ b = true) = a || b)
  #check (rfl : decide (¬a) = !a)
  #check (rfl : decide (a = true ↔ b = true) = (a == b))

• The decide tactic no longer needs to carry proofs with it, improving performance for well-written Decidable instances.
• LawfulBEq and DecidableEq are now compatible: When using the DecidableEq instance provided by LawfulBEq, decide (a = b) is definitionally equivalent to a == b.
• Decidable no longer needs special casing in the compiler.

In order to take full advantage from these changes, it is recommended to use the decidable_of_bool and decidable_of_iff functions to construct Decidable instances.

This is a breaking change, but in part due to Decidable.isTrue and Decidable.isFalse remaining as match_patterns, surprisingly few (meta-)programs break.

[ghost]

Mathlib CI status (docs):

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[TwoFX]

Please also make a change to stdlib_flags.h so that CI runs the stage2 build.

[TwoFX]

I think it would be good to add a test for all the new defeqs that this change achieves, both as a regression test and a bit of documentation of intention.

[TwoFX]

Merging master will resolve the Lake test failures (this is due to a GH Actions quirk)

[TwoFX]

!radar

[leanprover-radar]

Benchmark results for 52a7c6a against 44a3664 are in. There are significant results. @TwoFX

• ✅ build//instructions: -112.6G (-0.97%)

Large changes (9✅, 2🟥)

• ✅ build/module/Std.Data.HashMap.RawLemmas//instructions: -10.1G (-9.83%)
• ✅ build/module/Std.Http.Data.Headers.Name//instructions: -7.0G (-36.70%)
• ✅ compiled/ilean_roundtrip//instructions: -325.1M (-1.45%)
• 🟥 compiled/phashmap//instructions: +72.8M (+0.84%)
• 🟥 elab/cbv_decide//instructions: +3.1G (+10.34%)
• ✅ elab/cbv_decide//task-clock: -357ms (-11.84%)
• ✅ elab/cbv_decide//wall-clock: -343ms (-11.40%)
• ✅ elab/cbv_divisors//instructions: -2.7G (-6.66%)
• ✅ elab/whnfMatcherImplicitTransparencyCaching//instructions: -2.8G (-10.23%)
• ✅ elab/whnfMatcherImplicitTransparencyCaching//task-clock: -219ms (-11.88%)
• ✅ elab/whnfMatcherImplicitTransparencyCaching//wall-clock: -135ms (-12.16%)

Medium changes (29✅, 5🟥)

• ✅ build/module/Init.Data.String.Basic//instructions: -2.8G (-11.23%) (reduced significance based on absolute threshold)
• ✅ build/module/Init.Data.Vector.Lemmas//instructions: -1.5G (-3.67%) (reduced significance based on absolute threshold)
• ✅ build/module/Lake.Build.Facets//instructions: -1.5G (-15.97%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Elab.StructInst//instructions: -1.0G (-3.29%)
• ✅ build/module/Lean.Elab.Tactic.Omega.Frontend//instructions: -1.2G (-5.11%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Elab.Tactic.VCGen.Solve//instructions: -2.1G (-9.19%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Sym.DSimp.EvalGround//instructions: -1.1G (-5.40%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Sym.Simp.EvalGround//instructions: -1.5G (-5.76%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.BVDecide.Normalize.Simproc//instructions: -1.5G (-4.55%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.FunInd//instructions: -1.0G (-2.67%)
• ✅ build/module/Lean.Meta.Tactic.Grind.AC.Eq//instructions: -1.6G (-5.81%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Arith.CommRing.EqCnstr//instructions: -3.1G (-7.96%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Arith.CommRing.Internalize//instructions: -1.4G (-14.08%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Arith.Cutsat.EqCnstr//instructions: -2.0G (-8.40%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Arith.Cutsat.Search//instructions: -2.9G (-12.28%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Arith.Linear.Search//instructions: -1.5G (-8.88%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Core//instructions: -1.1G (-8.37%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.EMatch//instructions: -1.2G (-3.29%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.Tactic.Grind.Internalize//instructions: -1.2G (-6.58%) (reduced significance based on absolute threshold)
• ✅ build/module/Lean.Meta.WrapInstance//instructions: -2.4G (-21.04%) (reduced significance based on absolute threshold)
• and 14 more

Small changes (365✅, 407🟥)

• 🟥 build/module/Init.Control.Lawful.Instances//instructions: +28.7M (+0.43%)
• 🟥 build/module/Init.Control.Lawful.MonadAttach//instructions: +5.6M (+1.15%)
• 🟥 build/module/Init.Control.Lawful.MonadLift.Instances//instructions: +8.6M (+0.75%)
• 🟥 build/module/Init.Control.Lawful.MonadLift//instructions: +6.1M (+1.25%)
• 🟥 build/module/Init.Control.StateRef//instructions: +8.2M (+0.91%)
• 🟥 build/module/Init.Core//instructions: +145.3M (+1.50%) (reduced significance based on absolute threshold)
• 🟥 build/module/Init.Data.Array.Attach//instructions: +32.1M (+0.31%)
• 🟥 build/module/Init.Data.Array.Basic//instructions: +25.1M (+0.23%)
• ✅ build/module/Init.Data.Array.Bootstrap//instructions: -39.9M (-1.71%)
• ✅ build/module/Init.Data.Array.Lemmas//instructions: -394.4M (-0.74%) (reduced significance based on absolute threshold)
• ✅ build/module/Init.Data.Array.Lex.Basic//instructions: -84.8M (-6.13%) (reduced significance based on absolute threshold)
• 🟥 build/module/Init.Data.Array.Lex//instructions: +6.2M (+1.25%)
• 🟥 build/module/Init.Data.Array.MapIdx//instructions: +33.4M (+0.38%)
• 🟥 build/module/Init.Data.Array.Monadic//instructions: +23.1M (+0.38%)
• 🟥 build/module/Init.Data.Array.QSort//instructions: +6.1M (+1.22%)
• 🟥 build/module/Init.Data.Array.Range//instructions: +19.8M (+0.50%)
• 🟥 build/module/Init.Data.Array.Sort//instructions: +6.8M (+1.30%)
• 🟥 build/module/Init.Data.Array//instructions: +6.4M (+1.17%)
• 🟥 build/module/Init.Data.BEq//instructions: +6.3M (+0.81%)
• ✅ build/module/Init.Data.BitVec.Bitblast//instructions: -419.5M (-0.84%) (reduced significance based on absolute threshold)
• and 751 more
• and 1 hidden

[TwoFX]

This is awesome, thank you for the hard work!