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computation.lagda
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\begin{code}
{-# OPTIONS --rewriting #-}
open import Level using (Level ; 0ℓ ; Lift ; lift ; lower) renaming (suc to lsuc)
open import Agda.Builtin.Bool
open import Agda.Builtin.Equality
open import Agda.Builtin.Equality.Rewrite
open import Agda.Builtin.Sigma
open import Relation.Nullary
open import Relation.Unary using (Pred; Decidable)
open import Relation.Binary.PropositionalEquality hiding ([_]) -- using (sym ; subst ; _∎ ; _≡⟨_⟩_)
open ≡-Reasoning
open import Data.Product
open import Data.Sum
open import Data.Empty
open import Data.Maybe
open import Data.Unit using (⊤ ; tt)
open import Data.Nat using (ℕ ; _≟_ ; _<_ ; _≤_ ; _≥_ ; _≤?_ ; _<?_ ; suc ; _+_ ; pred)
open import Data.Nat.Properties
open import Agda.Builtin.String
open import Agda.Builtin.String.Properties
open import Data.List
open import Data.List.Properties
open import Data.List.Relation.Unary.Any
open import Data.List.Relation.Binary.Subset.Propositional
open import Data.List.Relation.Binary.Subset.Propositional.Properties
open import Data.List.Membership.Propositional
open import Data.List.Membership.DecSetoid(≡-decSetoid) using (_∈?_)
open import Data.List.Membership.Propositional.Properties
open import util
open import name
open import calculus
open import terms
open import world
open import choice
open import compatible
open import getChoice
open import choiceExt
open import newChoice
open import encode
module computation {L : Level} (W : PossibleWorlds {L})
(C : Choice) (M : Compatible W C) (G : GetChoice {L} W C M) (E : ChoiceExt {L} W C)
(N : NewChoice W C M G)
(EC : Encode)
where
open import worldDef(W)
open import choiceDef{L}(C)
open import getChoiceDef(W)(C)(M)(G)
open import choiceExtDef(W)(C)(M)(G)(E)
open import newChoiceDef(W)(C)(M)(G)(N)
open import encodeDef(EC)
\end{code}
We now define part of OpenTT's syntax and operational semantics.
\begin{code}
T→Name : Term → Name
T→Name t = fst (freshName (names t))
ret : (t : Term) (w : 𝕎·) → Maybe (Term × 𝕎·)
ret t w = just (t , w)
-- recursive call of WREC
WRECr : Term → Term → Term
WRECr r f = LAMBDA (WREC (APPLY (shiftUp 0 f) (VAR 0)) (shiftUp 3 r))
WRECsub : Term → Term → Term → Term
WRECsub a f r = sub (WRECr r f) (sub (shiftUp 0 f) (sub (shiftUp 0 (shiftUp 0 a)) r))
ENCr : Term → Term
ENCr t = NEGD (APPLY t (NUM (encode· (ENC t))))
NATRECr : ℕ → Term → Term → Term
NATRECr 0 b c = b
NATRECr (suc n) b c = APPLY2 c (NUM n) (NATREC (NUM n) b c)
SPREADsub : Term → Term → Term → Term
SPREADsub u v b = sub v (sub (shiftUp 0 u) b)
step : ∀ (T : Term) (w : 𝕎·) → Maybe (Term × 𝕎·)
-- VAR
step (VAR v) w = nothing
-- NAT
--step NAT = ret NAT
-- QNAT
step QNAT = ret QNAT
-- TNAT
--step TNAT = ret TNAT
-- LT
step (LT a b) = ret (LT a b)
-- QLT
step (QLT a b) = ret (QLT a b)
-- NUM
step (NUM n) = ret (NUM n)
-- IFLT
step (IFLT a b c d) w with is-NUM a
... | inj₁ (n , p) with is-NUM b
... | inj₁ (m , q) with n <? m
... | yes r = ret c w
... | no r = ret d w
step (IFLT a b c d) w | inj₁ (n , p) | inj₂ q with step b w
... | just (b' , w') = ret (IFLT a b' c d) w'
... | nothing = nothing
step (IFLT a b c d) w | inj₂ p with step a w
... | just (a' , w') = ret (IFLT a' b c d) w'
... | nothing = nothing
-- IFEQ
step (IFEQ a b c d) w with is-NUM a
... | inj₁ (n , p) with is-NUM b
... | inj₁ (m , q) with n ≟ m
... | yes r = ret c w
... | no r = ret d w
step (IFEQ a b c d) w | inj₁ (n , p) | inj₂ q with step b w
... | just (b' , w') = ret (IFEQ a b' c d) w'
... | nothing = nothing
step (IFEQ a b c d) w | inj₂ p with step a w
... | just (a' , w') = ret (IFEQ a' b c d) w'
... | nothing = nothing
-- SUC
step (SUC a) w with is-NUM a
... | inj₁ (n , p) = ret (NUM (suc n)) w
... | inj₂ p with step a w
... | just (a' , w') = ret (SUC a') w'
... | nothing = nothing
-- NATREC
step (NATREC a b c) w with is-NUM a
... | inj₁ (n , p) = ret (NATRECr n b c) w
... | inj₂ p with step a w
... | just (a' , w') = ret (NATREC a' b c) w'
... | nothing = nothing
-- PI
step (PI a b) = ret (PI a b)
-- LAMBDA
step (LAMBDA t) = ret (LAMBDA t)
-- APPLY
-- access the n^th choice in the history of choices made for "name"
step (APPLY f a) w with is-LAM f
... | inj₁ (t , p) = ret (sub a t) w
... | inj₂ x with is-CS f
... | inj₁ (name , p) with is-NUM a
... | inj₁ (n , q) = Data.Maybe.map (λ t → t , w) (getT n name w)
... | inj₂ y with step a w
... | just (u , w') = ret (APPLY (CS name) u) w'
... | nothing = nothing
step (APPLY f a) w | inj₂ x {-- ¬LAM --} | inj₂ name {-- ¬SEQ --} with is-MSEQ f
... | inj₁ (s , r) = ret (MAPP s a) w
... | inj₂ r with step f w
... | just (g , w') = ret (APPLY g a) w'
... | nothing = nothing
{--step (APPLY (CS name) (NUM n)) w = Data.Maybe.map (λ t → t , w) (getT n name w)
step (APPLY (CS name) t) w with step t w
... | just (u , w') = ret (APPLY (CS name) u) w'
... | nothing = nothing
step (APPLY (LAMBDA t) u) w = ret (sub u t) w
step (APPLY f a) w with step f w
... | just (g , w') = ret (APPLY g a) w'
... | nothing = nothing--}
-- FRESH
-- This creates a new choice name and adds it to the current world with the default restriction
-- TODO: allow other restrictions
step (FRESH t) w = ret (shiftNameDown 0 (renn 0 (newChoiceT+ w t) t)) (startNewChoiceT Res⊤ w t)
-- LOAD
step (LOAD t) w = ret AX (startNewChoices Res⊤ w t)
-- CHOOSE
step (CHOOSE n t) w with is-NAME n
... | inj₁ (name , p) = ret AX (chooseT name w t)
... | inj₂ x with step n w
... | just (m , w') = ret (CHOOSE m t) w'
... | nothing = nothing
{--step (CHOOSE (CS name) t) w = ret AX w -- TODO: return a 𝕎· too
step (CHOOSE n t) w with step n w
... | just (m , w') = ret (CHOOSE m t) w'
... | nothing = nothing--}
-- IFC₀
{--step (IFC0 a b c) w with isValue⊎ a
... | inj₁ x with decT₀ a
... | inj₁ y = ret b w
... | inj₂ y = ret c w
step (IFC0 a b c) w | inj₂ x with step a w
... | just (a' , w') = ret (IFC0 a' b c) w'
... | nothing = nothing--}
-- FIX
step (FIX f) w with is-LAM f
... | inj₁ (t , p) = ret (sub (FIX (LAMBDA t)) t) w
... | inj₂ x with step f w
... | just (g , w') = ret (FIX g) w'
... | nothing = nothing
{--step (FIX (LAMBDA t)) w = ret (sub (FIX (LAMBDA t)) t) w
step (FIX f) w with step f w
... | just (g , w') = ret (FIX g) w'
... | nothing = nothing--}
-- LET
step (LET a f) w with isValue⊎ a
... | inj₁ x = ret (sub a f) w
... | inj₂ x with step a w
... | just (b , w') = ret (LET b f) w'
... | nothing = nothing
-- W
step (WT a f x) = ret (WT a f x)
-- SUP
step (SUP a b) = ret (SUP a b)
{---- DSUP
step (DSUP a b) w with is-SUP a
... | inj₁ (u , v , p) = ret (sub v (sub u b)) w
... | inj₂ x with step a w
... | just (t , w') = ret (DSUP t b) w'
... | nothing = nothing--}
-- WREC
step (WREC t r) w with is-SUP t
... | inj₁ (a , f , p) = ret (WRECsub a f r) w
... | inj₂ x with step t w
... | just (t' , w') = ret (WREC t' r) w'
... | nothing = nothing
-- M
step (MT a f x) = ret (MT a f x)
{---- MSUP
step (MSUP a b) = ret (MSUP a b)
-- DMSUP
step (DMSUP a b) w with is-MSUP a
... | inj₁ (u , v , p) = ret (sub v (sub u b)) w
... | inj₂ x with step a w
... | just (t , w') = ret (DMSUP t b) w'
... | nothing = nothing--}
-- SUM
step (SUM a b) = ret (SUM a b)
-- PAIR
step (PAIR a b) = ret (PAIR a b)
-- SPREAD
step (SPREAD a b) w with is-PAIR a
... | inj₁ (u , v , p) = ret (SPREADsub u v b) w
... | inj₂ x with step a w
... | just (t , w') = ret (SPREAD t b) w'
... | nothing = nothing
{--step (SPREAD (PAIR a b) c) w = ret (sub b (sub a c)) w
step (SPREAD a b) w with step a w
... | just (t , w') = ret (SPREAD t b) w'
... | nothing = nothing--}
-- SET
step (SET a b) = ret (SET a b)
-- ISECT
step (ISECT a b) = ret (ISECT a b)
-- TUNION
step (TUNION a b) = ret (TUNION a b)
-- UNION
step (UNION a b) = ret (UNION a b)
-- QTUNION
--step (QTUNION a b) = ret (QTUNION a b)
-- INL
step (INL a) = ret (INL a)
-- INR
step (INR a) = ret (INR a)
-- DECIDE
step (DECIDE a b c) w with is-INL a
... | inj₁ (t , p) = ret (sub t b) w
... | inj₂ x with is-INR a
... | inj₁ (t , p) = ret (sub t c) w
... | inj₂ y with step a w
... | just (t , w') = ret (DECIDE t b c) w'
... | nothing = nothing
{--step (DECIDE (INL a) b c) w = ret (sub a b) w
step (DECIDE (INR a) b c) w = ret (sub a c) w
step (DECIDE a b c) w with step a w
... | just (t , w') = ret (DECIDE t b c) w'
... | nothing = nothing--}
-- EQ
step (EQ a b c) = ret (EQ a b c)
-- EQB
--step (EQB a b c d) = ret (EQB a b c d)
-- AX
step AX = ret AX
-- FREE
step FREE = ret FREE
-- MSEQ
step (MSEQ f) = ret (MSEQ f)
-- APPLY
-- access the n^th choice in the history of choices made for "name"
step (MAPP s a) w with is-NUM a
... | inj₁ (n , q) = ret (NUM (s n)) w
... | inj₂ y with step a w
... | just (u , w') = ret (MAPP s u) w'
... | nothing = nothing
-- CS
step (CS name) = ret (CS name)
-- NAME
step (NAME name) = ret (NAME name)
-- TSQUASH
--step (TSQUASH a) = ret (TSQUASH a)
-- TTRUNC
--step (TTRUNC a) = ret (TTRUNC a)
-- NOWRITE
step NOWRITE = ret NOWRITE
-- NOREAD
step NOREAD = ret NOREAD
-- SUBSING
step (SUBSING a) = ret (SUBSING a)
-- PARTIAL
step (PARTIAL a) = ret (PARTIAL a)
-- FFDEFS
step (FFDEFS a b) = ret (FFDEFS a b)
-- PURE
step PURE = ret PURE
-- NOSEQ
step NOSEQ = ret NOSEQ
-- NOENC
step NOENC = ret NOENC
-- TERM
step (TERM t) = ret (TERM t)
-- ENC
step (ENC t) = ret (ENCr t) -- ret (NUM (Term→ℕ t))
-- UNIV
step (UNIV u) = ret (UNIV u)
-- LIFT
step (LIFT t) = ret (LIFT t)
-- LOWER
step (LOWER t) = ret (LOWER t)
-- LOWER
step (SHRINK t) = ret (SHRINK t)
steps : (n : ℕ) (tw : Term × 𝕎·) → Term × 𝕎·
steps 0 (t , w) = (t , w)
steps (suc n) (t , w) with step t w
... | just (u , w') = steps n (u , w')
... | nothing = (t , w)
stepsT : (n : ℕ) (t : Term) (w : 𝕎·) → Term
stepsT n t w = fst (steps n (t , w))
_⇓_at_ : ∀ (T T' : Term) (w : 𝕎·) → Set
T ⇓ T' at w = Σ ℕ (λ n → stepsT n T w ≡ T')
infix 30 _⇓_at_
_⇓_from_to_ : ∀ (T T' : Term) (w w' : 𝕎·) → Set(L)
T ⇓ T' from w to w' = Σ ℕ (λ n → steps n (T , w) ≡ (T' , w'))
infix 30 _⇓_from_to_
_⇓!_at_ : ∀ (T T' : Term) (w : 𝕎·) → Set(L)
T ⇓! T' at w = T ⇓ T' from w to w
infix 30 _⇓!_at_
-- T computes to T' in all extensions of w
_⇛_at_ : (T T' : Term) (w : 𝕎·) → Set(lsuc(L))
T ⇛ T' at w = ∀𝕎 w (λ w' _ → Lift {0ℓ} (lsuc(L)) (T ⇓ T' at w'))
infix 30 _⇛_at_
-- as opposed to the one above, this one does not allow the computation to change the world
_⇛!_at_ : (T T' : Term) (w : 𝕎·) → Set(lsuc(L))
T ⇛! T' at w = ∀𝕎 w (λ w' _ → Lift {L} (lsuc(L)) (T ⇓! T' at w'))
infix 30 _⇛!_at_
⇓-refl : (T : Term) (w : 𝕎·) → T ⇓ T at w
⇓-refl T w = (0 , refl)
-- values compute to themselves
stepVal : (a : Term) (w : 𝕎·) → isValue a → step a w ≡ ret a w
--stepVal NAT w v = refl
stepVal QNAT w v = refl
--stepVal TNAT w v = refl
stepVal (LT a b) w v = refl
stepVal (QLT a b) w v = refl
stepVal (NUM x) w v = refl
stepVal (PI a a₁) w v = refl
stepVal (LAMBDA a) w v = refl
stepVal (WT a a₁ a₂) w v = refl
stepVal (SUP a a₁) w v = refl
stepVal (MT a a₁ a₂) w v = refl
--stepVal (MSUP a a₁) w v = refl
stepVal (SUM a a₁) w v = refl
stepVal (PAIR a a₁) w v = refl
stepVal (SET a a₁) w v = refl
stepVal (ISECT a a₁) w v = refl
stepVal (TUNION a a₁) w v = refl
stepVal (UNION a a₁) w v = refl
--stepVal (QTUNION a a₁) w v = refl
stepVal (INL a) w v = refl
stepVal (INR a) w v = refl
stepVal (EQ a a₁ a₂) w v = refl
--stepVal (EQB a a₁ a₂ a₃) w v = refl
stepVal AX w v = refl
stepVal FREE w v = refl
stepVal (MSEQ x) w v = refl
stepVal (CS x) w v = refl
stepVal (NAME x) w v = refl
--stepVal (TSQUASH a) w v = refl
--stepVal (TTRUNC a) w v = refl
stepVal NOWRITE w v = refl
stepVal NOREAD w v = refl
stepVal (SUBSING a) w v = refl
stepVal (PARTIAL a) w v = refl
stepVal (FFDEFS a a₁) w v = refl
stepVal PURE w v = refl
stepVal NOSEQ w v = refl
stepVal NOENC w v = refl
stepVal (TERM a) w v = refl
stepVal (UNIV x) w v = refl
stepVal (LIFT x) w v = refl
stepVal (LOWER a) w v = refl
stepVal (SHRINK a) w v = refl
stepsVal : (a : Term) (w : 𝕎·) (n : ℕ) → isValue a → steps n (a , w) ≡ (a , w)
stepsVal a w 0 v = refl
stepsVal a w (suc n) v rewrite stepVal a w v = stepsVal a w n v
compVal : (a b : Term) (w : 𝕎·) → a ⇓ b at w → isValue a → a ≡ b
compVal a b w (n , c) v rewrite stepsVal a w n v = c
postulate
-- Howe's computational equivalence relation
_∼_at_ : (T T' : Term) (w : 𝕎·) → Set
-- ∼ is an equivalence relation
∼-refl : {a : Term} {w : 𝕎·} → a ∼ a at w
∼-sym : {a b : Term} {w : 𝕎·} → a ∼ b at w → b ∼ a at w
∼-trans : {a b c : Term} {w : 𝕎·} → a ∼ b at w → b ∼ c at w → a ∼ c at w
-- includes ⇓
⇓→∼ : {a b : Term} {w : 𝕎·} → a ⇓ b at w → a ∼ b at w
-- states that the argument does not contain any definition or choice sequence
nodefs : Term → Set
infix 30 _∼_at_
-- T computationally equivalent to T' in all extensions of w
_≈_at_ : (T T' : Term) (w : 𝕎·) → Set(lsuc(L))
T ≈ T' at w = ∀𝕎 w (λ w' _ → Lift {0ℓ} (lsuc(L)) (T ∼ T' at w'))
infix 30 _≈_at_
≈-refl : {a : Term} {w : 𝕎·} → a ≈ a at w
≈-refl {a} {w} w1 e1 = lift ∼-refl
≈-sym : {a b : Term} {w : 𝕎·} → a ≈ b at w → b ≈ a at w
≈-sym {a} {b} {w} h w1 e1 = lift (∼-sym (lower (h w1 e1)))
≈-trans : {a b c : Term} {w : 𝕎·} → a ≈ b at w → b ≈ c at w → a ≈ c at w
≈-trans {a} {b} {c} {w} h q w1 e1 = lift (∼-trans (lower (h w1 e1)) (lower (q w1 e1)))
≈-∼ : {a b : Term} {w : 𝕎·} → a ≈ b at w → a ∼ b at w
≈-∼ {a} {b} {w} h = lower (h w (⊑-refl· w))
compAllRefl : (T : Term) (w : 𝕎·) → T ⇛ T at w
compAllRefl T w = λ w' e → lift (⇓-refl T w')
compAllVal : {a b : Term} {w : 𝕎·} → a ⇛ b at w → isValue a → a ≡ b
compAllVal {a} {b} {w} c i = let c' = c _ (⊑-refl· w) in compVal _ _ _ (lower c') i
-- t1 and t2 compute to the same number and stay the same number in all extensions
strongMonEq : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
strongMonEq w t1 t2 = Σ ℕ (λ n → t1 ⇛ (NUM n) at w × t2 ⇛ (NUM n) at w)
⇛!sameℕ : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
⇛!sameℕ w t1 t2 = Σ ℕ (λ n → t1 ⇛! (NUM n) at w × t2 ⇛! (NUM n) at w)
#⇛!sameℕ : (w : 𝕎·) (t1 t2 : CTerm) → Set(lsuc(L))
#⇛!sameℕ w t1 t2 = ⇛!sameℕ w ⌜ t1 ⌝ ⌜ t2 ⌝
⇓sameℕ : (w : 𝕎·) (t1 t2 : Term) → Set
⇓sameℕ w t1 t2 = Σ ℕ (λ n → t1 ⇓ (NUM n) at w × t2 ⇓ (NUM n) at w)
-- t1 and t2 compute to the same number but that number can change over time
weakMonEq : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
weakMonEq w t1 t2 = ∀𝕎 w (λ w' _ → Lift {0ℓ} (lsuc(L)) (⇓sameℕ w' t1 t2))
⇓∼ℕ : (w : 𝕎·) (t1 t2 : Term) → Set(L)
⇓∼ℕ w t1 t2 = Σ ℕ (λ n → Σ 𝕎· (λ w' → t1 ⇓ (NUM n) from w to w' × t2 ⇓ (NUM n) from w to w'))
∀𝕎-⇓∼ℕ : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
∀𝕎-⇓∼ℕ w t1 t2 = ∀𝕎 w (λ w' _ → Lift {L} (lsuc(L)) (⇓∼ℕ w' t1 t2))
weakℕ : (w : 𝕎·) (t : Term) → Set(lsuc(L))
weakℕ w t = ∀𝕎 w (λ w' _ → Lift {0ℓ} (lsuc(L)) (Σ ℕ (λ n → t ⇓ NUM n at w')))
weakℕM : (w : 𝕎·) (f : 𝕎· → Maybe Term) → Set(lsuc(L))
weakℕM w f = ∀𝕎 w (λ w' _ → Lift {0ℓ} (lsuc(L)) (Σ Term (λ t → f w' ≡ just t × Σ ℕ (λ n → t ⇓ NUM n at w'))))
⇛to-same-CS : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
⇛to-same-CS w t1 t2 = Σ Name (λ n → t1 ⇛ (CS n) at w × t2 ⇛ (CS n) at w)
<NUM-pair : (w : 𝕎·) (t1 t2 : Term) → Set
<NUM-pair w t1 t2 = Σ ℕ (λ n → Σ ℕ (λ m → t1 ⇓ (NUM n) at w × t2 ⇓ (NUM m) at w × n < m))
lift-<NUM-pair : (w : 𝕎·) (t1 t2 : Term) → Set(lsuc(L))
lift-<NUM-pair w t1 t2 = Lift {0ℓ} (lsuc(L)) (<NUM-pair w t1 t2)
⇛-mon : {a b : Term} {w2 w1 : 𝕎·}
→ w1 ⊑· w2
→ a ⇛ b at w1
→ a ⇛ b at w2
⇛-mon {a} {b} {w2} {w1} ext c w' e' = c w' (⊑-trans· ext e')
maybeStep : (tw : Term × 𝕎·) → Term × 𝕎·
maybeStep (t , w) with step t w
... | just x = x
... | nothing = t , w
stepsR : (n : ℕ) (tw : Term × 𝕎·) → Term × 𝕎·
stepsR 0 (t , w) = t , w
stepsR (suc n) (t , w) = maybeStep (stepsR n (t , w))
step⊎ : (t : Term) (w : 𝕎·) → (Σ Term (λ u → Σ 𝕎· (λ w' → step t w ≡ just (u , w')))) ⊎ step t w ≡ nothing
step⊎ t w with step t w
... | just (u , w') = inj₁ (u , w' , refl)
... | nothing = inj₂ refl
steps≡ : (n : ℕ) (t : Term × 𝕎·) → steps (suc n) t ≡ maybeStep (steps n t)
steps≡ 0 (t , w) with step t w
... | just u = refl
... | nothing = refl
steps≡ (suc n) (t , w) with step⊎ t w
... | inj₁ (u , w' , p) rewrite p | steps≡ n (u , w') = refl
... | inj₂ p rewrite p | p = refl
steps≡stepsR : (n : ℕ) (t : Term × 𝕎·) → steps n t ≡ stepsR n t
steps≡stepsR 0 t = refl
steps≡stepsR (suc n) t rewrite sym (steps≡stepsR n t) | steps≡ n t = refl
step-APPLY-CS : (t : Term) (w : 𝕎·) (k : ℕ) (name : Name)
→ getT k name w ≡ just t
→ steps 1 (APPLY (CS name) (NUM k) , w) ≡ (t , w)
step-APPLY-CS t w k name gc rewrite gc = refl
step-APPLY-CS-¬NUM : (name : Name) (a b : Term) (w w' : 𝕎·)
→ ((n : ℕ) → ¬ a ≡ NUM n)
→ step a w ≡ just (b , w')
→ step (APPLY (CS name) a) w ≡ ret (APPLY (CS name) b) w'
--step-APPLY-CS-¬NUM name NAT b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name QNAT b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name TNAT b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (LT a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (QLT a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (NUM x) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = ⊥-elim (c x refl)
step-APPLY-CS-¬NUM name (IFLT a a₁ a₂ a₃) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (IFEQ a a₁ a₂ a₃) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (SUC a) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (NATREC a a₁ a₂) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (PI a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (LAMBDA a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (APPLY a a₁) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (FIX a) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (LET a a₁) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (WT a a₁ a₂) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (SUP a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (MT a a₁ a₂) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (MSUP a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (SUM a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (PAIR a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (SET a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (ISECT a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (TUNION a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (UNION a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (QTUNION a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (INL a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (INR a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (EQ a a₁ a₂) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (EQB a a₁ a₂ a₃) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name AX b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name FREE b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (MSEQ x) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (CS x) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (NAME x) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (FRESH a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (LOAD a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (TSQUASH a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (TTRUNC a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name NOWRITE b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name NOREAD b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (SUBSING a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (PARTIAL a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (FFDEFS a a₁) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name PURE b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name NOSEQ b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name NOENC b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (TERM a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (UNIV x) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (LIFT a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (LOWER a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (SHRINK a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
step-APPLY-CS-¬NUM name (DECIDE a x y) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (SPREAD a x) b w w' c s rewrite s = refl
--step-APPLY-CS-¬NUM name (DSUP a x) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (WREC a x) b w w' c s rewrite s = refl
--step-APPLY-CS-¬NUM name (DMSUP a x) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (CHOOSE a a₁) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (MAPP x a) b w w' c s rewrite s = refl
step-APPLY-CS-¬NUM name (ENC a) b w w' c s rewrite sym (pair-inj₁ (just-inj s)) | sym (pair-inj₂ (just-inj s)) = refl
--step-APPLY-CS-¬NUM name (IFC0 a a₁ a₂) b w w' c s rewrite s = refl
Σ-steps-APPLY-CS≤ : (n : ℕ) (a b : Term) (w w' : 𝕎·) (name : Name)
→ steps n (a , w) ≡ (b , w')
→ Σ ℕ (λ m → m ≤ n × steps m (APPLY (CS name) a , w) ≡ (APPLY (CS name) b , w'))
Σ-steps-APPLY-CS≤ 0 a b w w' name h rewrite pair-inj₁ h | pair-inj₂ h = (0 , ≤-refl , refl)
Σ-steps-APPLY-CS≤ (suc n) a b w w' name h with step⊎ a w
... | inj₁ (u , w'' , p) rewrite p with is-NUM a
... | inj₁ (k , q) rewrite q | sym (pair-inj₁ (just-inj p)) | sym (pair-inj₂ (just-inj p)) | stepsVal (NUM k) w n tt | sym (pair-inj₁ h) | sym (pair-inj₂ h) = (0 , _≤_.z≤n , refl)
... | inj₂ q = (suc m , _≤_.s≤s l , g)
where
ms : Σ ℕ (λ m → m ≤ n × steps m (APPLY (CS name) u , w'') ≡ (APPLY (CS name) b , w'))
ms = Σ-steps-APPLY-CS≤ n u b w'' w' name h
m : ℕ
m = proj₁ ms
l : m ≤ n
l = proj₁ (proj₂ ms)
s : steps m (APPLY (CS name) u , w'') ≡ (APPLY (CS name) b , w')
s = proj₂ (proj₂ ms)
g : steps (suc m) (APPLY (CS name) a , w) ≡ (APPLY (CS name) b , w')
g rewrite step-APPLY-CS-¬NUM name a u w w'' q p = s
Σ-steps-APPLY-CS≤ (suc n) a b w w' name h | inj₂ p rewrite p | pair-inj₁ h | pair-inj₂ h = (0 , _≤_.z≤n , refl)
stepsT→steps : {n : ℕ} {t u : Term} {w : 𝕎·}
→ stepsT n t w ≡ u
→ steps n (t , w) ≡ (u , snd (steps n (t , w)))
stepsT→steps {n} {t} {u} {w} h rewrite sym h | sym (pair-eta (steps n (t , w))) = refl
steps→stepsT : {n : ℕ} {t u : Term} {w : 𝕎·}
→ steps n (t , w) ≡ (u , snd (steps n (t , w)))
→ stepsT n t w ≡ u
steps→stepsT {n} {t} {u} {w} h rewrite h = refl
steps→stepsT' : {n : ℕ} {t u : Term} {w w' : 𝕎·}
→ steps n (t , w) ≡ (u , w')
→ stepsT n t w ≡ u
steps→stepsT' {n} {t} {u} {w} {w'} h rewrite h = refl
step-steps-trans : {w w' w'' : 𝕎·} {a b c : Term} {n : ℕ}
→ step a w ≡ just (b , w')
→ steps n (b , w') ≡ (c , w'')
→ steps (suc n) (a , w) ≡ (c , w'')
step-steps-trans {w} {w'} {w''} {a} {b} {c} {n} c₁ c₂ rewrite c₁ = c₂
step-stepsT-trans : {w w' : 𝕎·} {a b c : Term} {n : ℕ}
→ step a w ≡ just (b , w')
→ stepsT n b w' ≡ c
→ stepsT (suc n) a w ≡ c
step-stepsT-trans {w} {w'} {a} {b} {c} {n} c₁ c₂ rewrite c₁ = c₂
step-⇓-trans : {w w' : 𝕎·} {a b c : Term} → step a w ≡ just (b , w') → b ⇓ c at w' → a ⇓ c at w
step-⇓-trans {w} {w'} {a} {b} {c} c₁ (n , c₂) =
suc n ,
step-stepsT-trans {w} {w'} {a} {b} {c} {n} c₁ c₂
steps-⇓-trans : {w w' : 𝕎·} {a b c : Term} (n : ℕ) → steps n (a , w) ≡ (b , w') → b ⇓ c at w' → a ⇓ c at w
steps-⇓-trans {w} {w'} {a} {b} {c} 0 c₁ c₂ rewrite pair-inj₁ c₁ | pair-inj₂ c₁ = c₂
steps-⇓-trans {w} {w'} {a} {b} {c} (suc n) c₁ c₂ with step⊎ a w
... | inj₁ (u , w'' , p) rewrite p = step-⇓-trans p (steps-⇓-trans n c₁ c₂)
... | inj₂ p rewrite p | pair-inj₁ c₁ | pair-inj₂ c₁ = c₂
_#⇓_at_ : (T T' : CTerm) (w : 𝕎·) → Set
T #⇓ T' at w = ⌜ T ⌝ ⇓ ⌜ T' ⌝ at w
infix 30 _#⇓_at_
_#⇓!_at_ : (T T' : CTerm) (w : 𝕎·) → Set(L)
T #⇓! T' at w = ⌜ T ⌝ ⇓! ⌜ T' ⌝ at w
infix 30 _#⇓!_at_
_#⇓_from_to_ : (T T' : CTerm) (w1 w2 : 𝕎·) → Set(L)
T #⇓ T' from w1 to w2 = ⌜ T ⌝ ⇓ ⌜ T' ⌝ from w1 to w2
infix 30 _#⇓_from_to_
_#⇛_at_ : (T T' : CTerm) (w : 𝕎·) → Set(lsuc(L))
T #⇛ T' at w = ⌜ T ⌝ ⇛ ⌜ T' ⌝ at w
infix 30 _#⇛_at_
_#⇛!_at_ : (T T' : CTerm) (w : 𝕎·) → Set(lsuc(L))
T #⇛! T' at w = ⌜ T ⌝ ⇛! ⌜ T' ⌝ at w
infix 30 _#⇛!_at_
#compAllRefl : (T : CTerm) (w : 𝕎·) → T #⇛ T at w
#compAllRefl (ct T cT) w i = compAllRefl T w i
#compAllVal : {a b : CTerm} {w : 𝕎·} → a #⇛ b at w → #isValue a → a ≡ b
#compAllVal {ct a ca} {ct b cb} {w} c i = CTerm≡ (compAllVal c i)
#strongMonEq : (w : 𝕎·) (t1 t2 : CTerm) → Set(lsuc(L))
#strongMonEq w t1 t2 = strongMonEq w ⌜ t1 ⌝ ⌜ t2 ⌝
#∀𝕎-⇓∼ℕ : (w : 𝕎·) (t1 t2 : CTerm) → Set(lsuc(L))
#∀𝕎-⇓∼ℕ w t1 t2 = ∀𝕎-⇓∼ℕ w ⌜ t1 ⌝ ⌜ t2 ⌝
#weakMonEq : (w : 𝕎·) (t1 t2 : CTerm) → Set(lsuc(L))
#weakMonEq w t1 t2 = weakMonEq w ⌜ t1 ⌝ ⌜ t2 ⌝
#weakMonEq→ : {w : 𝕎·} {a b : CTerm}
→ #weakMonEq w a b
→ Σ ℕ (λ n → a #⇓ #NUM n at w × b #⇓ #NUM n at w)
#weakMonEq→ {w} {a} {B} h = lower (h w (⊑-refl· w))
#weakMonEq-#NUM : (w : 𝕎·) (k : ℕ) → #weakMonEq w (#NUM k) (#NUM k)
#weakMonEq-#NUM w k w' e' = lift (k , ⇓-refl (NUM k) w' , ⇓-refl (NUM k) w')
#strongMonEq-#NUM : (w : 𝕎·) (k : ℕ) → #strongMonEq w (#NUM k) (#NUM k)
#strongMonEq-#NUM w k = k , compAllRefl (NUM k) w , compAllRefl (NUM k) w
strongMonEq-refl : {w : 𝕎·} {a b : Term}
→ strongMonEq w a b
→ strongMonEq w a a
strongMonEq-refl {w} {a} {b} (n , c₁ , c₂) = n , c₁ , c₁
strongMonEq-refl-rev : {w : 𝕎·} {a b : Term}
→ strongMonEq w a b
→ strongMonEq w b b
strongMonEq-refl-rev {w} {a} {b} (n , c₁ , c₂) = n , c₂ , c₂
weakMonEq-refl : {w : 𝕎·} {a b : Term}
→ weakMonEq w a b
→ weakMonEq w a a
weakMonEq-refl {w} {a} {b} wm w1 e1 = lift (fst z , fst (snd z) , fst (snd z))
where
z : Σ ℕ (λ n → a ⇓ NUM n at w1 × b ⇓ NUM n at w1)
z = lower (wm w1 e1)
weakMonEq-refl-rev : {w : 𝕎·} {a b : Term}
→ weakMonEq w a b
→ weakMonEq w b b
weakMonEq-refl-rev {w} {a} {b} wm w1 e1 = lift (fst z , snd (snd z) , snd (snd z))
where
z : Σ ℕ (λ n → a ⇓ NUM n at w1 × b ⇓ NUM n at w1)
z = lower (wm w1 e1)
strongMonEq-sym : {w : 𝕎·} {a b : Term}
→ strongMonEq w a b
→ strongMonEq w b a
strongMonEq-sym {w} {a} {b} (n , c₁ , c₂) = n , c₂ , c₁
⇓-APPLY-CS : (w : 𝕎·) (a b : Term) (name : Name)
→ a ⇓ b at w
→ (APPLY (CS name) a) ⇓ (APPLY (CS name) b) at w
⇓-APPLY-CS w a b name (n , c) = fst c' , steps→stepsT' {fst c'} (snd (snd c'))
where
c' : Σ ℕ (λ m → m ≤ n × steps m (APPLY (CS name) a , w) ≡ (APPLY (CS name) b , snd (steps n (a , w))))
c' = Σ-steps-APPLY-CS≤ n a b w (snd (steps n (a , w))) name (stepsT→steps {n} {a} {b} {w} c)
map-pair-𝕎⊑ : (w w' : 𝕎·) (m : Maybe Term) (a : Term)
→ Data.Maybe.map (λ t → t , w) m ≡ just (a , w')
→ w ⊑· w'
map-pair-𝕎⊑ w w' (just x) a h rewrite sym (pair-inj₁ (just-inj h)) | sym (pair-inj₂ (just-inj h)) = ⊑-refl· _
step⊑ : {w w' : 𝕎·} {a b : Term} → step a w ≡ just (b , w') → w ⊑· w'
--step⊑ {w} {w'} {NAT} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {QNAT} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
--step⊑ {w} {w'} {TNAT} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {LT a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {QLT a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NUM x} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {IFLT x y t u} {b} comp with is-NUM x
... | inj₁ (n , p) with is-NUM y
... | inj₁ (m , q) with n <? m
... | yes r rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | no r rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {IFLT x y t u} {b} comp | inj₁ (n , p) | inj₂ q with step⊎ y w
... | inj₁ (y' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {y} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {IFLT x y t u} {b} comp | inj₂ p with step⊎ x w
... | inj₁ (x' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {x} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {IFEQ x y t u} {b} comp with is-NUM x
... | inj₁ (n , p) with is-NUM y
... | inj₁ (m , q) with n ≟ m
... | yes r rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | no r rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {IFEQ x y t u} {b} comp | inj₁ (n , p) | inj₂ q with step⊎ y w
... | inj₁ (y' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {y} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {IFEQ x y t u} {b} comp | inj₂ p with step⊎ x w
... | inj₁ (x' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {x} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {SUC x} {b} comp with is-NUM x
... | inj₁ (n , p) rewrite p | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ x w
... | inj₁ (x' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {x} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {NATREC x x₁ x₂} {b} comp with is-NUM x
... | inj₁ (0 , p) rewrite p | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₁ (suc n , p) rewrite p | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ x w
... | inj₁ (x' , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {x} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {PI a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {LAMBDA a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {APPLY a a₁} {b} comp with is-LAM a
... | inj₁ (t , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ x with is-CS a
... | inj₁ (name , p) with is-NUM a₁
... | inj₁ (n , q) = map-pair-𝕎⊑ w w' (getT n name w) b comp
... | inj₂ y with step⊎ a₁ w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a₁} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {APPLY a a₁} {b} comp | inj₂ x | inj₂ y with is-MSEQ a
... | inj₁ (s , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ r with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {MAPP s a} {b} comp with is-NUM a
... | inj₁ (n , q) rewrite sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ y with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {FIX a} {b} comp with is-LAM a
... | inj₁ (t , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {LET a f} {b} comp with isValue⊎ a
... | inj₁ x rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ x with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {WT a a₁ a₂} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {SUP a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
{--step⊑ {w} {w'} {DSUP a a₁} {b} comp with is-SUP a
... | inj₁ (u , v , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))--}
step⊑ {w} {w'} {WREC a a₁} {b} comp with is-SUP a
... | inj₁ (u , v , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {MT a a₁ a₂} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
{--step⊑ {w} {w'} {MSUP a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {DMSUP a a₁} {b} comp with is-MSUP a
... | inj₁ (u , v , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))--}
step⊑ {w} {w'} {SUM a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {PAIR a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {SPREAD a a₁} {b} comp with is-PAIR a
... | inj₁ (u , v , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ p with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {SET a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {ISECT a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {TUNION a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {UNION a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
--step⊑ {w} {w'} {QTUNION a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {INL a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {INR a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {DECIDE a a₁ a₂} {b} comp with is-INL a
... | inj₁ (t , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ x with is-INR a
... | inj₁ (t , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ y with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
step⊑ {w} {w'} {EQ a a₁ a₂} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
--step⊑ {w} {w'} {EQB a a₁ a₂ a₃} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {AX} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {FREE} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {MSEQ x} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {CS x} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NAME x} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {CHOOSE a a₁} {b} comp with is-NAME a
... | inj₁ (name , p) rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = choose⊑· name w (T→ℂ· a₁)
... | inj₂ x with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))
{--step⊑ {w} {w'} {IFC0 a a₁ a₂} {b} comp with isValue⊎ a
... | inj₁ x with decT₀ a
... | inj₁ y rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
... | inj₂ y rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {IFC0 a a₁ a₂} {b} comp | inj₂ y with step⊎ a w
... | inj₁ (u , w'' , z) rewrite z | sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = step⊑ {_} {_} {a} z
... | inj₂ z rewrite z = ⊥-elim (¬just≡nothing (sym comp))--}
step⊑ {w} {w'} {FRESH a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = startNewChoiceT⊏ Res⊤ w a
step⊑ {w} {w'} {LOAD a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = startNewChoices⊑ Res⊤ w a
--step⊑ {w} {w'} {TSQUASH a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
--step⊑ {w} {w'} {TTRUNC a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NOWRITE} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NOREAD} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {SUBSING a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {PARTIAL a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {FFDEFS a a₁} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {PURE} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NOSEQ} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {NOENC} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {TERM a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {ENC a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {UNIV x} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {LIFT a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {LOWER a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
step⊑ {w} {w'} {SHRINK a} {b} comp rewrite sym (pair-inj₁ (just-inj comp)) | sym (pair-inj₂ (just-inj comp)) = ⊑-refl· _
→𝕎 : {a b : Term} {w : 𝕎·} (c : a ⇓ b at w) → 𝕎·
→𝕎 {a} {b} {w} (n , c) = snd (steps n (a , w))
steps⊑ : (w : 𝕎·) (n : ℕ) (t : Term) → w ⊑· snd (steps n (t , w))
steps⊑ w 0 t = ⊑-refl· _
steps⊑ w (suc n) t with step⊎ t w
... | inj₁ (u , w' , z) rewrite z = ⊑-trans· (step⊑ {_} {_} {t} z) (steps⊑ w' n u)
... | inj₂ z rewrite z = ⊑-refl· _
⇓-trans₀ : {w : 𝕎·} {a b c : Term} (comp : a ⇓ b at w) → b ⇓ c at (→𝕎 comp) → a ⇓ c at w
⇓-trans₀ {w} {a} {b} {c} (n , c₁) c₂ = steps-⇓-trans n (stepsT→steps {n} c₁) c₂
⇓-trans₁ : {w w' : 𝕎·} {a b c : Term} → a ⇓ b from w to w' → b ⇓ c at w' → a ⇓ c at w
⇓-trans₁ {w} {w'} {a} {b} {c} (n , c₁) c₂ = steps-⇓-trans n c₁ c₂
⇓-trans : {w : 𝕎·} {a b c : Term} → a ⇓ b at w → b ⇛ c at w → a ⇓ c at w
⇓-trans {w} {a} {b} {c} (n , c₁) c₂ = steps-⇓-trans n (stepsT→steps {n} c₁) (lower (c₂ (snd (steps n (a , w))) (steps⊑ w n a)))
Σ-steps-APPLY-CS : (n : ℕ) (a t : Term) (w w' : 𝕎·) (k : ℕ) (name : Name)
→ steps n (a , w) ≡ (NUM k , w')
→ getT k name w' ≡ just t
→ Σ ℕ (λ m → steps m (APPLY (CS name) a , w) ≡ (t , w'))