Documentation

Cslib.Languages.CombinatoryLogic.Evaluation

Evaluation results #

This file formalises evaluation and normal forms of SKI terms.

Main definitions #

Main results #

References #

This file draws heavily from https://gist.github.com/b-mehta/e412c837818223b8f16ca0b4aa19b166.

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def Cslib.SKI.RedexFree :
SKI → Prop

The predicate that a term has no reducible sub-terms.

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    def Cslib.SKI.evalStep (x : SKI) :
    PLift x.RedexFree ⊕ SKI

    One-step evaluation as a function: either it returns a term that has been reduced by one step, or a proof that the term is redex free. Uses normal-order reduction.

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      theorem Cslib.SKI.evalStep_right_correct (x y : SKI) :
      x.evalStep = Sum.inr y → x.Red y

      The normal-order reduction implemented by evalStep indeed computes a one-step reduction.

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      theorem Cslib.SKI.redexFree_of_normal_red {x : SKI} (h : Relation.Normal Red x) :
      x.RedexFree
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      theorem Cslib.SKI.RedexFree.normal_red {x : SKI} (hx : x.RedexFree) :
      Relation.Normal Red x
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      theorem Cslib.SKI.redexFree_iff {x : SKI} :
      x.RedexFree ↔ Relation.Normal Red x

      A term is redex free iff it has no one-step reductions.

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      theorem Cslib.SKI.redexFree_iff_evalStep {x : SKI} :
      x.RedexFree ↔ x.evalStep.isLeft = true
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      @[implicit_reducible]
      instance Cslib.SKI.instDecidablePredRedexFree :
      DecidablePred RedexFree
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      theorem Cslib.SKI.redexFree_iff_mred_eq {x : SKI} :
      x.RedexFree ↔ ∀ (y : SKI), Relation.ReflTransGen Red x y ↔ x = y

      A term is redex free iff its only many-step reduction is itself.

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      theorem Cslib.SKI.mJoin_red_redexFree {x y : SKI} (hy : y.RedexFree) (h : Relation.MJoin Red x y) :
      Relation.ReflTransGen Red x y

      If a term has a common reduct with a normal term, it in fact reduces to that term.

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      theorem Cslib.SKI.confluent_redexFree {x y z : SKI} (hxy : Relation.ReflTransGen Red x y) (hxz : Relation.ReflTransGen Red x z) (hz : z.RedexFree) :
      Relation.ReflTransGen Red y z

      If x reduces to both y and z, and z is not reducible, then y reduces to z.

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      theorem Cslib.SKI.unique_normal_form {x y z : SKI} (hxy : Relation.ReflTransGen Red x y) (hxz : Relation.ReflTransGen Red x z) (hy : y.RedexFree) (hz : z.RedexFree) :
      y = z

      If x reduces to both y and z, and both y and z are in normal form, then they are equal.

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      theorem Cslib.SKI.eq_of_mJoin_red_redexFree {x y : SKI} (h : Relation.MJoin Red x y) (hx : x.RedexFree) (hy : y.RedexFree) :
      x = y

      If x and y are normal and have a common reduct, then they are equal.

      Injectivity for datatypes #

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      theorem Cslib.SKI.sk_nequiv :
      ¬Relation.MJoin Red S K
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      theorem Cslib.SKI.isBool_injective (x y : SKI) (u v : Bool) (hx : IsBool u x) (hy : IsBool v y) (hxy : Relation.MJoin Red x y) :
      u = v

      Injectivity for booleans.

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      theorem Cslib.SKI.TF_nequiv :
      ¬Relation.MJoin Red TT FF
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      def Cslib.SKI.churchK :
      ℕ → SKI

      A specialisation of Church : Nat → SKI.

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        theorem Cslib.SKI.churchK_church (n : ℕ) :
        churchK n = Church n K K
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        theorem Cslib.SKI.churchK_redexFree (n : ℕ) :
        (churchK n).RedexFree
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        @[simp]
        theorem Cslib.SKI.churchK_size (n : ℕ) :
        (churchK n).size = n + 1
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        theorem Cslib.SKI.churchK_injective :
        Function.Injective churchK
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        theorem Cslib.SKI.isChurch_injective (x y : SKI) (n m : ℕ) (hx : IsChurch n x) (hy : IsChurch m y) (hxy : Relation.MJoin Red x y) :
        n = m

        Injectivity for Church numerals

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        theorem Cslib.SKI.rice {P : SKI} (hP : ∀ (x : SKI), Relation.ReflTransGen Red (P.app x) TT ∨ Relation.ReflTransGen Red (P.app x) FF) (hxt : ∃ (x : SKI), Relation.ReflTransGen Red (P.app x) TT) (hxf : ∃ (x : SKI), Relation.ReflTransGen Red (P.app x) FF) :
        False

        Rice's theorem: no SKI term is a non-trivial predicate.

        More specifically, say a term P is a predicate if, for every term x, P · x reduces to either TT or FF. A predicate P is trivial if either it always reduces to true, or always to false. This version of Rice's theorem derives a contradiction from the existence of a predicate P and the existence of terms x for which P · x is true (P · x ↠ TT) and for which P · x is false (P · x ↠ FF).

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        theorem Cslib.SKI.rice' {P : SKI} (hP : ∀ (x : SKI), Relation.ReflTransGen Red (P.app x) TT ∨ Relation.ReflTransGen Red (P.app x) FF) :
        (∀ (x : SKI), Relation.ReflTransGen Red (P.app x) TT) ∨ ∀ (x : SKI), Relation.ReflTransGen Red (P.app x) FF

        Rice's theorem: any SKI predicate is trivial.

        This version of Rice's theorem proves (classically) that any SKI predicate P either is constantly true (ie P · x ↠ TT for every x) or is constantly false (P · x ↠ FF for every x).