Linear Logical Predicates for Strong Normalisation

(1) Let𝑃⊢_𝜂 𝑥:U^𝑓 𝐴,𝑄 ⊢_𝜂 𝑥:U^𝑓 𝐴and𝑆 ⊆ {𝑅| 𝑅⊢_𝜂 𝑦 :∧𝐴}. Then,

par{(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑃})| |(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑄})}

simulates

cut{cell𝑥(𝑦.𝑆) |𝑥| share𝑥{𝑃| |𝑄}}

(2) Let𝑃⊢_𝜂 𝑥:U^𝑒𝐴,𝑄⊢_𝜂 𝑥 :U^𝑓 𝐴and𝑆⊆ {𝑅 |𝑅⊢_𝜂 𝑦:∧𝐴}. Then,

par{(cut{empty𝑥(𝑦.𝑆) |𝑥| 𝑃})| |(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑄})}

simulates

cut{empty𝑥(𝑦.𝑆) |𝑥| share𝑥{𝑃 | |𝑄}}

Proof. Define

𝒮 ≜^𝒮1∪ 𝒮₂∪ 𝒮₃ where

𝒮₁ ≜ ^{{(𝑀, 𝑁) | ∃𝑃⊢}𝜂 𝑥 :U^𝑓 𝐴,∃𝑄⊢_𝜂 𝑥 :U^𝑓 𝐴. 𝑀≡_ccut{cell𝑥(𝑦.𝑆) |𝑥| share𝑥{𝑃 | |𝑄}}

and𝑁 ≡_cpar{(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑃})| |(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑄})}}

𝒮₂ ≜ ^{{(𝑀, 𝑁) | ∃𝑃⊢}𝜂 𝑥 :U^𝑒𝐴,∃𝑄⊢_𝜂 𝑥:U^𝑓 𝐴. 𝑀 ≡_ccut{empty𝑥(𝑦.𝑆) |𝑥| share𝑥 {𝑃 | |𝑄}}

and𝑁 ≡_cpar{(cut{empty𝑥(𝑦.𝑆) |𝑥| 𝑃})| |(cut{cell𝑥(𝑦.𝑆) |𝑥| 𝑄})}}

𝒮₃ ≜ ^{{(𝑀, 𝑁) | ∃𝑃⊢}𝜂 ∅;∅,∃𝒞∃𝒟. 𝑀≡_c 𝒞 ◦ 𝒟[𝑃] and𝑁 ≡_cpar{𝒞[𝑃]| |𝒟[𝑃]}}

We prove that𝒮is a simulation, by performing case analysis first on(𝑀, 𝑁) ∈ 𝒮and then on 𝑀 →_c 𝑀^′. Complete proof in AppendixE.

Lemma 21(SN: Basic Properties). The following properties hold (1) If𝑃is SN and𝑃≡_c𝑄, then𝑄is SN.

(2) If𝑃is SN and𝑃→_c𝑄, then𝑄is SN.

(3) Suppose𝑄is SN whenever𝑃→_c𝑄. Then,𝑃is SN.

(4) If𝑃and𝑄are SN, thenpar{𝑃| | 𝑄}is SN.

(5) If𝑄is SN and𝑄simulates𝑃, then𝑃is SN.

Proof. All properties are easy to establish, in particular we have the following: in (1) 𝑁(𝑃)=𝑁(𝑄), in (2)𝑁(𝑄)=𝑁(𝑃) −1, in (3)𝑁(𝑃)=(_max{𝑄 | 𝑃 →_c𝑄}) +1 and in (4) 𝑁(par{𝑃 | |𝑄})=𝑁(𝑃) +𝑁(𝑄).

We will now introduce the orthogonal, which will play a key role when defining logical predicates for strong normalisation. As we will see, each logical predicate is defined by taking the orthogonal of some set. In the following, we write𝑃_𝑥to emphasise that𝑥is the only free name of𝑃.

Definition 32(Orthogonal(−)^⊥). Let𝑆be a subset of processes𝑄_𝑥 with a single free name𝑥. We define the orthogonal of𝑆, written𝑆^⊥, by

𝑆^⊥ ≜^{{𝑃𝑥 | ∀𝑄𝑥 ∈ 𝑆.cut{𝑃𝑥 |𝑥| 𝑄𝑥}}is SN}

The orthogonal satisfies some well-known properties, as stated by the following lemma.

Lemma 22(Orthogonal: Basic Properties). The following properties hold (1) If𝑃 ∈𝑆^⊥and𝑃≡_c 𝑄, then𝑄 ∈𝑆^⊥.

(2) If𝑃 ∈𝑆^⊥and𝑃→_c𝑄,then𝑄 ∈𝑆^⊥. (3) If𝑆₁ ⊆𝑆₂, then𝑆^⊥

2 ⊆𝑆^⊥

1

(4) 𝑆⊆ 𝑆^⊥⊥. (5) 𝑆^⊥⊥⊥ =𝑆^⊥

(6) Let𝒮be a collection of sets. Then,(Ð_𝒮)⊥

=Ñ

𝑆∈𝒮𝑆^⊥.

(7) Let𝒮be a collection of sets𝑆s.t.𝑆=𝑆^⊥⊥, whenever𝑆∈ 𝒮. Then,(Ñ_𝒮)⊥⊥

=Ñ_𝒮. Proof. (1) Follows by Lemma21(1).

(2) Follows by Lemma21(2).

J𝑥 :𝑋

K𝜎 ≜ 𝜎(𝑋)[𝑥]

J𝑥 :1K𝜎 ≜ ^{{𝑃 | 𝑃≡}cclose𝑥and𝑃is SN}^⊥⊥

J𝑥 :𝐴⊗𝐵

K𝜎 ≜ ^{{𝑃 | ∃𝑃}1, 𝑃₂. 𝑃≡_csend𝑥(𝑦.𝑃₁);𝑃₂and𝑃₁ ∈ J𝑦 :𝐴

K𝜎and𝑃₂ ∈ J𝑥 :𝐵

K𝜎}^⊥⊥

J𝑥 :𝐴⊕𝐵

K𝜎 ≜ ^{{𝑃 | ∃𝑄.(𝑃 ≡}c𝑥.inl;𝑄and 𝑄∈ J𝑥 :𝐴

K𝜎) or (𝑃≡_c 𝑥.inr;𝑄and𝑄∈ J𝑥 :𝐵

K𝜎)}^⊥⊥

J𝑥 :!𝐴

K𝜎 ≜ ^{{𝑃 | ∃𝑄. 𝑃≡}c!𝑥(𝑦);𝑄and𝑄 ∈ J𝑦 :𝐴

K𝜎}^⊥⊥

J𝑥 :∃𝑋.𝐴

K ≜ ^{{𝑃 | ∃𝑄, 𝑆 ∈ ℛ[−}:𝐵]. 𝑃≡_csendty𝑥 𝐵;𝑄and𝑄∈ J𝑥 :𝐴

K_𝜎^{[𝑋↦→𝑆]}}^⊥⊥

J𝑥 :𝜇𝑋. 𝐴

K𝜎 ≜ ⁽Ñ_{{𝑆 ∈ ℛ[−}

:𝜇𝑋.𝐴] | unfold_𝜇 𝑥;J𝑥:𝐴

K𝜎[𝑋↦→𝑆] ⊆𝑆})^⊥⊥

J𝑥 :∧𝐴

K𝜎 ≜ ^{{𝑃 | ∃𝑄. 𝑃≡}caffine𝑥;𝑄and𝑄∈ J𝑥 :𝐴

K𝜎}^⊥⊥

J𝑥 :S^𝑓 𝐴

K𝜎 ≜ ^{{𝑃 | 𝑃≡}ccell𝑥(𝑦.

J𝑦:∧𝐴

K𝜎)and 𝑃is SN}^⊥⊥

J𝑥 :S^𝑒𝐴

K𝜎 ≜ ^{{𝑃 | 𝑃≡}cempty𝑥(𝑦.

J𝑦 :∧𝐴

K𝜎)and𝑃is SN}^⊥⊥

J𝑥 :𝐴

K𝜎 ≜ J𝑥 :𝐴 K

⊥𝜎

Figure 9.1: Logical predicateJ𝑥 :𝐴 K𝜎. (3) Suppose𝑃∈ 𝑆^⊥

2.

So let𝑄 ∈𝑆₁. Since𝑆₁⊆ 𝑆₂, then𝑄 ∈𝑆₂. Since𝑃∈ 𝑆^⊥

2, thencut{𝑃 |𝑥| 𝑄}is SN.

Thus,𝑃 ∈𝑆^⊥

1.

(4) Let𝑃 ∈ 𝑆. We want𝑃 ∈𝑆^⊥⊥. Take𝑄 ∈𝑆^⊥. It suffices to show thatcut{𝑃 |𝑥| 𝑄}is SN. It follows from𝑄 ∈𝑆^⊥and𝑃 ∈𝑆.

(5) From (2) and (3) follows𝑆^⊥⊥⊥ ⊆𝑆^⊥. From (3) follows𝑆^⊥ ⊆ (𝑆^⊥)^⊥⊥=𝑆^⊥⊥⊥. (6) We prove that (i)(Ð_𝒮)⊥ ⊆Ñ

𝑆∈𝒮𝑆^⊥and (ii)Ñ

𝑆∈𝒮𝑆^⊥⊆ (Ð_𝒮)⊥

. (ii) follows immediately by Def.32.

So let us consider (i).

Let𝑆∈ 𝒮. Applying (3) to𝑆⊆Ð_𝒮

yields(Ð_𝒮)⊥ ⊆𝑆^⊥. Then,(Ð_𝒮)⊥ ⊆Ñ

𝑆∈𝒮𝑆^⊥. (7) We have

(Ñ_𝒮)⊥⊥

= (Ñ

𝑆∈𝒮𝑆)^⊥⊥

= (Ñ

𝑆∈𝒮𝑆^⊥⊥)^⊥⊥ (𝑆=𝑆^⊥⊥, whenever𝑆 ∈ 𝒮)

= (Ð

𝑆∈𝒮𝑆^⊥)^⊥⊥⊥ (from (6))

= (Ð

𝑆∈𝒮𝑆^⊥)^⊥ (from (5))

= Ñ

𝑆∈𝒮𝑆^⊥⊥ (from (6))

= Ñ

𝑆∈𝒮𝑆 (𝑆=𝑆^⊥⊥, whenever𝑆 ∈ 𝒮)

Logical PredicatesJ𝑥: 𝐴 K𝜎

We will now introduce the logical predicatesJ𝑥:𝐴

K𝜎for strong normalisation. Since we are working with polymorphic and inductive types, the definition is parametric on a map 𝜎from type variables to reducibility candidates. So let us define reducibility candidates first.

Definition 33(Reducibility Candidates𝑅[𝑥 :𝐴]). Given a type𝐴and a name𝑥we define a reducibility candidate at𝑥 : 𝐴, denoted by𝑅[𝑥 : 𝐴]as a set of SN processes𝑃 ⊢ 𝑥 : 𝐴which is equal to its biorthogonal, i.e.𝑅[𝑥 :𝐴]=𝑅[𝑥 :𝐴]^⊥⊥.

We letℛ[− : 𝐴]be the set of all reducibility candidates 𝑅[𝑥 : 𝐴] for some name 𝑥. Reducibility candidates are ordered by set-inclusion⊆, the least candidate being∅^⊥⊥. Definition 34 (Logical Predicate J𝑥 : 𝐴

K𝜎). By induction on the type𝐴, we define the sets J𝑥 :𝐴

K𝜎an shown in Fig.9.1, such thatJ𝑥 :U𝑓 𝐴

K𝜎andJ𝑥 :U𝑒 𝐴

K𝜎areJ−:∧𝐴K-preserving on𝑥.The definition is direct for the positive types, the negative types are defined by the last clause, by orthogonality.

For the positive types 𝐴, the predicate J𝑥 : 𝐴

K𝜎 takes the biorthogonal of some base set 𝑆 of processes𝑃 that offer an action, further conditions then characterise the process constituents of the actions. In the base cases close 𝑥, cell 𝑥(𝑦.

J𝑦 : ∧𝐴

K𝜎)and empty 𝑥(𝑦.

J𝑦 :∧𝐴

K𝜎), where the action does not have any further process constituents, we simply require the action offering process to be SN.

The presence of duality give us some succinctness in the presentation of the logical predicates, since, for the negative types𝐴, the predicateJ𝑥:𝐴

K𝜎is simply defined as the biorthogonal of the logical predicate for its dual𝐴type. In fact, we can also establish this property for the positive types (Lemma23(4)), thereby lifting duality to the logical level using the orthogonal operation. As a pleasant consequence we conclude immediately that if𝑃

J𝑥:𝐴

K𝜎and𝑄 ∈ J𝑥:𝐴

K𝜎, then the resulting cut compositioncut{𝑃 |𝑥| 𝑄}is SN.

By exploiting the properties satisfied by the orthogonal (Lemma22) we obtain a strategy to establish the membership𝑃 ∈

J𝑥 :𝐴

K𝜎. For the positive types we haveJ𝑥 :𝐴

K𝜎=𝑆^⊥⊥, for some set𝑆. Since𝑆⊆𝑆^⊥⊥(Lemma22(4)), we can conclude that𝑃 ∈

J𝑥 :𝐴

K𝜎, provided we prove𝑃 ∈𝑆. On the other hand, for the negative types we haveJ𝑥 :𝐴

K𝜎=𝑆^⊥⊥⊥. But since𝑆^⊥⊥⊥ =𝑆^⊥(Lemma22(5)), it is equivalent to prove that for all𝑄∈ 𝑆,cut{𝑃 |𝑥| 𝑄}is SN. These strategies will be applied throughout the proof of the Fundamental Lemma27.

In all cases, with some exceptions, when definingJ𝑥:𝐴

K𝜎we simply propagate map 𝜎without modifications. The exceptions are the defining clauses corresponding to the existential∃𝑋.𝐴and the inductive types𝜇𝑋. 𝐴, in which we extend the map𝜎with an assignment for the type variable 𝑋. Furthermore, the definition of the predicate for a type variable J𝑥 : 𝑋

K𝜎 picks the corresponding reducibility candidate 𝜎(𝑋)= 𝑅[𝑦 : 𝐵], instantiated at name𝑥: {𝑥/𝑦}𝑅[𝑦:𝐵].

The definition ofJ𝑥 :𝜇𝑋. 𝐴

K𝜎relies on the constructionunfold_𝜇 𝑥;𝑆, that for any set 𝑆, is defined according to

unfold_𝜇 𝑥;𝑆≜ ^{{𝑃 | ∃𝑄. 𝑃≡}cunfold_𝜇 𝑥;𝑄and 𝑄∈ 𝑆}

Similarly, given a set𝑆, we defineunfold_𝜈 𝑥;𝐴by

unfold_𝜈 𝑥;𝑆≜ ^{{𝑃 | ∃𝑄. 𝑃≡}cunfold_𝜈 𝑥;𝑄and 𝑄 ∈𝑆}

The following lemma states some basic properties about the logical predicates.

Lemma 23(Logical Predicates: Basic Properties). The following properties hold (1) If𝑃 ∈

J𝑥:𝐴

K𝜎, then{𝑦/𝑥}𝑃 ∈ J𝑦 :𝐴

K𝜎. (2) If𝑃 ∈

J𝑥:𝐴

K𝜎and𝑃≡_c𝑄, then𝑄 ∈ J𝑥:𝐴

K𝜎. (3) If𝑃 ∈

J𝑥:𝐴

K𝜎and𝑃→_c𝑄, then𝑄∈ J𝑥:𝐴

K𝜎. (4) J𝑥:𝐴

K𝜎 =J𝑥 :𝐴 K

⊥𝜎. (5) J𝑥:{𝐵/𝑋}𝐴

K𝜎=J𝑥:𝐴 K𝜎[𝑋↦→

J𝑥:𝐵 K𝜎]. (6) J𝑥:𝐴

K𝜎[𝑋↦→𝑆^⊥]=J𝑥:{𝑋/𝑋}𝐴

K𝜎[𝑋↦→𝑆].

Proof. Property (1) is trivial. Properties (2) and (3) follows by Lemma22(1) and Lemma22(2), respectively. Property (4) follows directly by Def.34for half of the types. The remaining half follows by Lemma22(5). Properties (5) and (6) are straightforward by induction on 𝐴.

The logical predicates are preserved by name substitution, the congruence relation

≡_c and the reduction relation→_c(Lemma23(1)-(3)). Property Lemma23(4) relates the logical predicates of duality related types, using the orthogonal. Lemma23(5)-(6) relate type variable substitution with the parametric map 𝜎.

We use the interference-sensitive reference cells (Def.30) to define the logical predicates J𝑥:S^𝑓 𝐴

K𝜎andJ𝑐:S^𝑒 𝐴

K𝜎, for the state full and the state empty modalities, respectively.

This allows us to internalise state interference in the definition of the logical predicate itself and, as consequence, we can reason compositionally about state sharing as witnessed by the following lemma

Lemma 24. The following properties hold (1) If𝑃₁∈

J𝑐:U^𝑓 𝐴

K𝜎and𝑃₂∈

J𝑐:U^𝑓 𝐴

K𝜎, thenshare𝑐{𝑃₁ | |𝑃₂} ∈

J𝑐:U^𝑓 𝐴 K𝜎. (2) If𝑃₁∈

J𝑐:U^𝑒 𝐴

K𝜎and𝑃₂ ∈

J𝑐:U^𝑒 𝐴

K𝜎, thenshare𝑐{𝑃₁ | |𝑃₂} ∈

J𝑐:U^𝑒 𝐴 K𝜎.

Proof. (1) By Def.32and Lemma22(5) we haveJ𝑐:U^𝑓 𝐴

K=𝑆^⊥, where 𝑆={𝑄 | 𝑄≡_ccell𝑐(𝑎.

J𝑎:∧𝐴 K)_𝜎}.

Let𝑄≡_ccell𝑐(𝑎.

J𝑎:∧𝐴 K)_𝜎.

We need to prove thatcut{𝑄 |𝑐| share𝑐{𝑃₁| |𝑃₂}}is SN.

By Lemma20(1) we conclude thatcut{𝑄 |𝑐| share𝑐{𝑃₁| |𝑃₂}}is simulated by par{(cut{𝑄 |𝑐| 𝑃₁})| |(cut{𝑄 |𝑐| 𝑃₂})}

By hypothesis,𝑃₁∈

J𝑐:U^𝑓 𝐴

K𝜎, hencecut{𝑄 |𝑐| 𝑃₁}is SN.

By hypothesis,𝑃₂∈

J𝑐:U^𝑓 𝐴

K𝜎, hencecut{𝑄 |𝑐| 𝑃₂}is SN.

Then,par{(cut{𝑄 |𝑐| 𝑃₁})| |(cut{𝑄 |𝑐| 𝑃₂})}is SN (Lemma21(4)).

Therefore,cut{𝑄 |𝑐| share𝑐{𝑃₁| |𝑃₂}}is SN (Lemma21(5)).

By hypothesis, for any 𝑦, both𝑃₁ and𝑃₂ areJ𝑦 : ∧𝐴

K-preserving on 𝑐. Applying Lemma19, we conclude thatshare𝑐{𝑃₁ | |𝑃₂}is alsoJ𝑦 :∧𝐴K-preserving on𝑐. (2) Similarly to (1), by applying the simulation Lemma20(2).

We will now state some properties concerning the logical predicate for inductive types.

But first, let us introduce the following definition.

Definition 35(𝜙^𝐴(𝑆)). Suppose that𝑋occurs positively on𝐴. Define 𝜙^𝐴(𝑆)≜^unfold𝜇𝑥;J𝑥 :𝐴

K_𝜎^{[𝑋↦→𝑆]}

J𝑥 :𝜇𝑋. 𝐴

K𝜎is defined as the biorthogonal of the intersection of all𝜙^𝐴-closed sets𝑆, i.e. sets𝑆s.t. 𝜙^𝐴(𝑆) ⊆𝑆. Since𝜙^𝐴is monotonic (Lemma25(1)), Knaster-Tarski theorem implies thatJ𝑥:𝜇𝑋. 𝐴

K𝜎is the least fixed point of𝜙^𝐴(Lemma25(2)). Symmetrically, we can obtain a greatest fixed point characterisation forJ𝑥 :𝜈𝑋. 𝐴

K𝜎(Lemma25(3)). Applying Kleene’s fixed point theorem we explicitly construct the fixed point of𝜙^𝐴(Lemma25(4)).

Lemma 25. The following properties hold

(1) The map𝜙^𝐴is monotonic, i.e. 𝜙^𝐴(𝑆₁) ⊆𝜙^𝐴(𝑆₂), whenever𝑆₁ ⊆𝑆₂. (2) J𝑥:𝜇𝑋. 𝐴

K𝜎is the least fixed point of𝜙^𝐴. (3) Let𝜓^𝐴(𝑆)≜ 𝜙{𝑋/𝑋}𝐴(𝑆^⊥)^⊥. Then,J𝑥 :𝜈𝑋. 𝐴

K𝜎is the greatest fixed point of𝜓^𝐴. (4) J𝑥:𝜇𝑋. 𝐴

K𝜎 =Ð

𝑛∈N𝜙^𝑛𝐴(∅^⊥⊥). (5) unfold_𝜈𝑥;J𝑥 :{𝜈𝑋. 𝐴/𝑋}𝐴

K𝜎 ⊆

J𝑥 :𝜈𝑋. 𝐴 K𝜎.

Proof. (1) We prove hypothesis (H1) if𝑆₁ ⊆ 𝑆₂, thenJ𝑥 : 𝐴

K𝜎[𝑋↦→𝑆₁] ⊆

J𝑥 : 𝐴

K𝜎[𝑋↦→𝑆₂], which implies (1).

The proof of (H1) is by induction on𝐴, we handle some representative cases.

Case: 𝐴=𝑌.

There are two cases to consider, depending on whether (i)𝑌≠𝑋or (ii)𝑌=𝑋. If (i), thenJ𝑥:𝑌

K_𝜎^{[𝑋↦→𝑆}1] =𝜎(𝑌)=J𝑥 :𝑌

K_𝜎^{[𝑋↦→𝑆}2]. If (ii), thenJ𝑥:𝑋

K_𝜎^{[𝑋↦→𝑆}1] =𝑆₁ ⊆𝑆₂=J𝑥 :𝑋

K_𝜎^{[𝑋↦→𝑆}2]. In either case (i)-(ii),J𝑥:𝑌

K_𝜎^{[𝑋↦→𝑆}1] ⊆ J𝑥 :𝑌

K_𝜎^{[𝑋↦→𝑆}2]. Case: 𝐴=1.

We haveJ𝑥 :1K_𝜎^{[𝑋↦→𝑆}1]=J𝑥:1K_𝜎^{[𝑋↦→𝑆}2]. Case: 𝐴=𝐴₁⊗𝐴₂.

By Def.34,

J𝑥:𝐴₁⊗𝐴₂

K𝜎[𝑋↦→𝑆]= 𝑓(𝑆)^⊥⊥

where

𝑓(𝑆)≜ ^{{𝑃 | ∃𝑃}1, 𝑃₂. 𝑃≈send𝑥(𝑦.𝑃₁);𝑃₂ and𝑃₁∈

J𝑦:𝐴₁

K𝜎[𝑋↦→𝑆]and𝑃₂∈

J𝑥 :𝐴₂

K𝜎[𝑋↦→𝑆]} Suppose that𝑆₁ ⊆𝑆₂. I.h. applied to𝐴₁and𝐴₂yields 𝑓(𝑆₁) ⊆ 𝑓(𝑆₂). Lemma22(3) applied twice to 𝑓(𝑆₁) ⊆ 𝑓(𝑆₂)yields

J𝑥 :𝐴₁⊗𝐴₂

K𝜎[𝑋↦→𝑆₁]= 𝑓(𝑆₁)^⊥⊥ ⊆ 𝑓(𝑆₂)^⊥⊥ =J𝑥 :𝐴₁⊗𝐴₂

K𝜎[𝑋↦→𝑆₂]

Case: 𝐴=𝜇𝑌. 𝐵. By Def.34

J𝑥 :𝜇𝑌. 𝐵

K𝜎[𝑋↦→𝑆]=(Ù

𝑓(𝑆))^⊥⊥

where

𝑓(𝑆)≜^{{𝑇∈ ℛ[−}:𝜇𝑌.𝐵] |unfold_𝜇 𝑥;J𝑥:𝐵

K𝜎[𝑋↦→𝑆,𝑌↦→𝑇] ⊆𝑇}

Suppose𝑆₁⊆ 𝑆₂. Let𝑇 ∈ 𝑓(𝑆₂). Then,unfold_𝜇 𝑥;J𝑥:𝐵

K𝜎[𝑋↦→𝑆₂,𝑌↦→𝑇] ⊆𝑇. I.h. applied to 𝐵 yields unfold_𝜇 𝑥;J𝑥 : 𝐵

K𝜎[𝑋↦→𝑆₁,𝑌↦→𝑇] ⊆ unfold_𝜇 𝑥;J𝑥 : 𝐵K_𝜎^{[𝑋↦→𝑆}2,𝑌↦→𝑇].

By transitivity of⊆,unfold_𝜇 𝑥;J𝑥 :𝐵

K_𝜎^{[𝑋↦→𝑆}1,𝑌↦→𝑇] ⊆𝑇. Hence,𝑇 ∈ 𝑓(𝑆₁).

This establishes 𝑓(𝑆₂) ⊆ 𝑓(𝑆₁). Then,Ñ_𝑓(𝑆

1) ⊆Ñ_𝑓(𝑆

2).

Lemma22(3) applied twice toÑ_𝑓(𝑆

1) ⊆Ñ_𝑓(𝑆

2)yields J𝑥 :𝜇𝑌. 𝐵

K𝜎[𝑋↦→𝑆₁]=(Ù

𝑓(𝑆₁))^⊥⊥ ⊆ (Ù

𝑓(𝑆₂))^⊥⊥=J𝑥:𝜇𝑌. 𝐵

K𝜎[𝑋↦→𝑆₂]

Case: 𝐴=S^𝑓 𝐵. By Def.34

J𝑥:S^𝑓 𝐵

K𝜎[𝑋↦→𝑆]= 𝑓(𝑆)^⊥⊥

where

𝑓(𝑆)≜ ^{{𝑃 |𝑃 ≡}ccell𝑥(𝑦.

J𝑦 :∧𝐴

K𝜎[𝑋↦→𝑆])}

Suppose𝑆₁⊆ 𝑆₂. We prove that 𝑓(𝑆₂)^⊥⊆ 𝑓(𝑆₁)^⊥.

Let 𝑄 ∈ 𝑓(𝑆₂)^⊥. In order to show that 𝑄 ∈ 𝑓(𝑆₁)^⊥ we must show that cut{𝑃 |𝑥| 𝑄}is SN, when𝑃 ∈ 𝑓(𝑆₁).

We prove by induction on𝑁(𝑃)+𝑁(𝑄)that all the reductionscut{𝑃 |𝑥| 𝑄} →𝑅 are SN.

We handle only the interesting reduction, which corresponds to a cell-take interaction on session𝑥. Then

cut{𝑃 |𝑥| 𝑄} ≡_ccut{cell𝑥(𝑦.

J𝑦:∧𝐴

K𝜎[𝑋↦→𝑆₁]) |𝑥| take𝑥(𝑦);𝑄^′}

→_ccut{empty𝑥(𝑦.

J𝑦:∧𝐴

K_𝜎^{[𝑋↦→𝑆}1]) |𝑥| (cut{𝑃^′ |𝑦| 𝑄^′})}=𝑅 where𝑃 ≡_ccell𝑥(𝑦.

J𝑦:∧𝐴

K𝜎[𝑋↦→𝑆₁]),𝑄≡_ctake𝑥(𝑦);𝑄^′and𝑃^′is some element inJ𝑦 :∧𝐴

K𝜎[𝑋↦→𝑆₁]. By hypothesis,𝑄∈ 𝑓(𝑆₂)^⊥, hence cut{cell𝑥(𝑦.

J𝑦 :∧𝐴

K𝜎[𝑋↦→𝑆₂]) |𝑥| take𝑥(𝑦);𝑄^′} is SN.

Then, all the reductions ofcut{cell𝑥(𝑦.

J𝑦 :∧𝐴

K_𝜎^{[𝑋↦→𝑆}2]) |𝑥| take𝑥(𝑦);𝑄^′}are SN, in particular the following reduction can be obtained, since𝑃^′ ∈𝑆₁ ⊆𝑆₂:

cut{cell𝑥(𝑦.

J𝑦:∧𝐴

K𝜎[𝑋↦→𝑆₂]) |𝑥| take𝑥(𝑦);𝑄^′}

→cut{empty𝑥(𝑦.

J𝑦:∧𝐴

K_𝜎^{[𝑋↦→𝑆}2]) |𝑥| (cut{𝑃^′ |𝑦| 𝑄^′})}

(2) By Def.34

J𝑥 :𝜇𝑋. 𝐴

K𝜎 = (Ñ_{{𝑆∈ ℛ[−}

:𝜇𝑋.𝐴] | 𝜙^𝐴(𝑆) ⊆𝑆})^⊥⊥

Since a reducibility candidate is equal to its biorthogonal (Def.33), we can write J𝑥 :𝜇𝑋. 𝐴

K𝜎in the alternative form (Lemma22(7)) J𝑥 :𝜇𝑋. 𝐴

K𝜎=Ù

{𝑆 ∈ ℛ[−:𝜇𝑋.𝐴] | 𝜙^𝐴(𝑆) ⊆𝑆}

i.e.J𝑥 :𝜇𝑋. 𝐴

K𝜎is the intersection of all𝜙^𝐴-closed sets inℛ[−:𝜇𝑋.𝐴]. We now prove the following propositions

(i) J𝑥:𝜇𝑋. 𝐴

K𝜎is𝜙^𝐴-closed, i.e. 𝜙^𝐴(

J𝑥:𝜇𝑋. 𝐴 K𝜎) ⊆

J𝑥:𝜇𝑋. 𝐴 K𝜎. Let𝑆∈ ℛ[−:𝜇𝑋.𝐴]be a𝜙^𝐴-closed set.

By definition, we have (a)𝜙^𝐴(𝑆) ⊆𝑆and (b)J𝑥 :𝜇𝑋. 𝐴 K𝜎 ⊆𝑆.

Monotonicity of𝜙^𝐴(1) applied to (b) yields𝜙^𝐴(

J𝑥 :𝜇𝑋. 𝐴

K𝜎 ⊆𝜙^𝐴(𝑆). Hence, transitivity and (a) implies𝜙^𝐴(

J𝑥 :𝜇𝑋. 𝐴

K𝜎) ⊆𝑆. SinceJ𝑥 :𝜇𝑋. 𝐴

K𝜎is the intersection of all𝜙^𝐴-closed sets inℛ[−:𝜇𝑋.𝐴], then 𝜙^𝐴(

J𝑥:𝜇𝑋. 𝐴 K𝜎) ⊆

J𝑥 :𝜇𝑋. 𝐴 K𝜎. (ii) J𝑥 :𝜇𝑋. 𝐴

K𝜎 ⊆𝜙^𝐴(

J𝑥:𝜇𝑋. 𝐴 K𝜎).

Monotonicity of 𝜙^𝐴 (1) applied to (i) yields 𝜙^𝐴(𝜙^𝐴(

J𝑥 : 𝜇𝑋. 𝐴

K𝜎) ⊆ 𝜙^𝐴( J𝑥 : 𝜇𝑋. 𝐴

K𝜎), i.e. 𝜙^𝐴(

J𝑥 :𝜇𝑋. 𝐴

K𝜎)is𝜙^𝐴-closed.

SinceJ𝑥 :𝜇𝑋. 𝐴

K𝜎is the intersection of all𝜙^𝐴-closed sets inℛ[−:𝜇𝑋.𝐴], then J𝑥 :𝜇𝑋. 𝐴

K𝜎 ⊆𝜙^𝐴(

J𝑥:𝜇𝑋. 𝐴 K𝜎).

Propositions (i) and (ii) imply thatJ𝑥:𝜇𝑋. 𝐴

K𝜎is a fixed point of𝜙^𝐴.

Let 𝑆 ∈ ℛ[−: 𝜇𝑋.𝐴]be any fixed point of𝜙^𝐴. Then, in particular,𝑆is 𝜙^𝐴-closed, henceJ𝑥 :𝜇𝑋. 𝐴

K𝜎 ⊆𝜙^𝐴. Therefore,J𝑥 :𝜇𝑋. 𝐴

K𝜎is the least fixed point of𝜙^𝐴. (3) We need to prove the following propositions

(i) J𝑥 :𝜈𝑋. 𝑋𝐴

K𝜎is a fixed point of𝜓^𝐴. By (b),J𝑥:𝜇𝑋.{𝑋/𝑋}𝐴

K𝜎is a fixed point of𝜙{𝑋/𝑋}𝐴

𝜙{𝑋/𝑋}𝐴(

J𝑥:𝜇𝑋.{𝑋/𝑋}𝐴

K𝜎)=J𝑥 :𝜇𝑋.{𝑋/𝑋}𝐴 K𝜎

hence, applying the orthogonal to both sides of the equation yields 𝜙{𝑋/𝑋}𝐴(

J𝑥 :𝜇𝑋.{𝑋/𝑋}𝐴

K𝜎)^⊥ =J𝑥 :𝜇𝑋.{𝑋/𝑋}𝐴 K

⊥ 𝜎

SinceJ𝑥 : 𝜇𝑋. {𝑋/𝑋}𝐴 K

⊥𝜎 =J𝑥 :𝜈𝑋. 𝑋𝐴

K𝜎(Lemma23(4)) we can rewrite the equation in the equivalent form

𝜙{𝑋/𝑋}𝐴(

J𝑥 :𝜈𝑋. 𝑋𝐴 K

⊥𝜎)^⊥=J𝑥:𝜈𝑋. 𝑋𝐴 K𝜎

Then,J𝑥 :𝜈𝑋. 𝑋𝐴

K𝜎is a fixed point of𝜓^𝐴. (ii) If𝑆is a fixed point of𝜓^𝐴, then𝑆⊆

J𝑥 :𝜈𝑋. 𝑋𝐴 K𝜎. Suppose that𝑆is a fixed point of𝜓^𝐴, i.e.

𝜓^𝐴(𝑆)=𝜙{𝑋/𝑋}𝐴(𝑆^⊥)^⊥=𝑆

Applying the orthogonal to both sides of the equation yields 𝜙{𝑋/𝑋}𝐴(𝑆^⊥)^⊥⊥=𝑆^⊥

Since𝜙{𝑋/𝑋}𝐴(𝑆^⊥) ⊆𝜙{𝑋/𝑋}𝐴(𝑆^⊥)^⊥⊥(Lemma22(4)), then 𝜙{𝑋/𝑋}𝐴(𝑆^⊥) ⊆𝑆^⊥

i.e.𝑆^⊥is a𝜙{𝑋/𝑋}𝐴-closed set.

Then, by Def.34

J𝑥:𝜇𝑋.{𝑋/𝑋}𝐴

K𝜎⊆𝑆^⊥

Applying the orthogonal to the inequation (Lemma22(3)) yields 𝑆^⊥⊥ ⊆

J𝑥:𝜇𝑋.{𝑋/𝑋}𝐴 K

⊥ 𝜎

Since𝑆⊆𝑆^⊥⊥(Lemma22(2)), we obtain 𝑆⊆

J𝑥:𝜇𝑋.{𝑋/𝑋}𝐴 K

⊥𝜎

Finally, sinceJ𝑥:𝜇𝑋.{𝑋/𝑋}𝐴 K

⊥𝜎 =J𝑥 :𝜈𝑋. 𝐴

K𝜎(Lemma23(4)) we have 𝑆 ⊆

J𝑥 :𝜈𝑋. 𝐴 K𝜎

(4) We prove thatÐ

𝑛∈N𝜙^𝑛𝐴(∅^⊥⊥)is the least fixed point of𝜙^𝐴. By (b) it follows thatJ𝑥 :𝜇𝑋. 𝐴

K𝜎=Ð

𝑛∈N𝜙^𝑛𝐴(∅^⊥⊥). We need to prove the following propositions (i) Ð

𝑛∈N𝜙^𝑛𝐴(∅^⊥⊥)is a fixed point of𝜙^𝐴. We have

𝜙^𝐴(Ø

𝑛∈N

𝜙𝐴^𝑛(∅^⊥⊥))=Ø

𝑛>0

𝜙^𝑛𝐴(∅^⊥⊥)

Since 𝜙⁰𝐴(∅^⊥⊥) = ∅^⊥⊥ is the least reducibility candidate, we have𝜙⁰𝐴(∅^⊥⊥) ⊆ 𝜙^𝑛𝐴(∅^⊥⊥), for any𝑛 >0.

Then

Ø

𝑛>0

𝜙^𝑛𝐴(∅^⊥⊥)=Ø

𝑛∈N

𝜙^𝑛𝐴(∅^⊥⊥) Therefore

𝜙^𝐴(Ø

𝑛∈N

𝜙^𝑛𝐴(∅^⊥⊥))= Ø

𝑛∈N

𝜙^𝑛𝐴(∅^⊥⊥) (ii) If𝑆is fixed point of𝜙^𝐴, thenÐ

𝑛∈N𝜙^𝑛𝐴(∅^⊥⊥) ⊆𝑆.

We that𝜙^𝑛𝐴(∅^⊥⊥) ⊆𝑆, for all𝑛 ∈N. The proof is by induction on𝑛. Case: 𝑛 =0.

Since𝜙⁰𝐴(∅^⊥⊥)=∅^⊥⊥is the least reducibility candidate,𝜙⁰𝐴(∅^⊥⊥) ⊆𝑆. Case: 𝑛 =𝑚+1.

By i.h. we have

𝜙^𝑚𝐴(∅^⊥⊥) ⊆𝑆 Monotonicity of𝜙^𝐴(1) implies

𝜙^𝑚+𝐴 ¹(∅^⊥⊥)) ⊆ 𝜙(𝑆) Since𝜙(𝑆)=𝑆, then

𝜙𝐴^𝑚+1(∅^⊥⊥)) ⊆ 𝑆

(5) Let𝑃≡_cunfold_𝜈 𝑥;𝑃^′, where𝑃^′∈

J𝑥 :{𝜈𝑋. 𝐴/𝑋}𝐴 K𝜎. Let𝐵≜ ^{{𝑋/𝑋}𝐴}, hence𝜈𝑋. 𝐴 =𝜇𝑋. 𝐵.

We prove thatcut{𝑃 |𝑥| 𝑄} is SN, for all𝑄 ∈

J𝑥 : 𝜇𝑋. 𝐵

K𝜎, by analysing all the possible reductions ofcut{𝑃 |𝑥| 𝑄}and concluding that all of them are SN.

The critical reduction is the unfold-unfold interaction on session𝑥, in which case 𝑄 ≡_cunfold_𝜈𝑥;𝑄^′, andcut{𝑃 |𝑥| 𝑄} →_ccut{𝑃^′ |𝑥| 𝑄^′}.

By (2) we conclude that𝑄^′∈

J𝑥 :{𝜇𝑋. 𝐵/𝑋}𝐵 K𝜎.

Since{𝜈𝑋. 𝐴/𝑋}𝐴 ={𝜇𝑋. 𝐵/𝑋}𝐵, we conclude thatcut{𝑃^′ |𝑥| 𝑄^′}is SN.

Extended Logical Predicate and Fundamental Lemma The logical predicates J𝑥 : 𝐴

K𝜎 introduced previously apply to processes that have a single free name𝑥. Now, we will extend the logical predicate to general typed processes 𝑃∈

J⊢_𝜂 Δ;ΓKby composing it alongΔandΓwith processes from the basic logical predicates (Def. 34) and by replacing the free process variables by elements of the appropriate reducibility candidate, according to the following definition.

Definition 36(Extended Logical PredicateJ⊢_𝜂 Δ;ΓK𝜎). We defineℒ

J⊢_𝜂 Δ;ΓK𝜎inductively on Δ,Γand𝜂as the set of processes𝑃⊢_𝜂 Δ;Γs.t.

𝑃∈ ℒ

J⊢_𝜂 ∅;∅

K𝜎iff𝑃is SN. 𝑃∈ ℒ

J⊢_𝜂 Δ, 𝑥:𝐴;ΓK𝜎iff ∀𝑄 ∈ J𝑥 :𝐴

K𝜎.cut{𝑄 |𝑥:𝐴| 𝑃} ∈ ℒ

J⊢_∅ Δ;ΓK𝜎. 𝑃∈ ℒ

J⊢_𝜂 Δ;Γ, 𝑥:𝐴

K𝜎iff∀𝑄 ∈ J𝑦 :𝐴

K𝜎.cut!{𝑦.𝑄|𝑥 :𝐴|𝑃} ∈ ℒ

J⊢_∅ Δ;ΓK𝜎. 𝑃∈ ℒ

J⊢

𝜂,𝑋(𝑥,𝑦)↦→® Δ^′,𝑥:𝑌;ΓΔ;ΓK𝜎iff∀𝑄 ∈𝜎(𝑌).{𝑄/𝑋}𝑃 ∈ ℒ

J⊢_𝜂Δ;ΓK𝜎. The base caseℒ

J⊢_𝜂 Δ;ΓK𝜎corresponds to the set of closed well-typed SN processes.

We will now introduce some auxiliary definitions which allows us to give a more succinct definition ofℒ

J⊢_∅ ∅;∅

K𝜎. We define the setJΔK𝜎of linear logical contexts atΔis inductively by

J∅

K𝜎 ≜ ^{−} JΔ, 𝑥:𝐴

K𝜎 ≜ ^{{cut{𝑃 |𝑥} :𝐴| 𝒞} | 𝑃∈ J𝑥:𝐴

K𝜎and𝒞 ∈ JΔK𝜎} Similarly, we define the setJΓK

!𝜎of unrestricted logical contexts atΓinductively by J∅

K

!𝜎 ≜^{−} JΓ, 𝑦:𝐴 K

!𝜎 ≜^{cut!{𝑦.𝑃|𝑥:𝐴|𝒞} | 𝑃 ∈ J𝑦 :𝐴

K𝜎and𝒞 ∈ JΓK

!𝜎} We extend𝑁 from processes to contexts𝒞 ∈

JΔK𝜎by𝑁(−)=0 and𝑁(cut{𝑃 |𝑥| 𝒞^′})= 𝑁(𝑃) +𝑁(𝒞^′).

Given a map

𝜂=𝑋₁( ®𝑥₁) ↦→Δ₁;Γ₁, . . . , 𝑋𝑛( ®𝑥𝑛) ↦→Δ𝑛;Γ𝑛

we defineJ𝜂K_𝜎as the set of all substitution maps𝜂^′s.t.

𝜂^′=𝑋₁( ®𝑥₁) ↦→𝑄₁, . . . , 𝑋𝑛( ®𝑥𝑛) ↦→𝑄𝑛

where𝑄₁ ∈ ℒ

J⊢_∅ Δ₁;Γ₁K𝜎, . . . , 𝑄𝑛 ∈ ℒ

J⊢_∅Δ𝑛;Γ𝑛

K𝜎. Then, Def.36is equivalent to the following

𝑃∈ ℒ

J⊢_𝜂 Δ;ΓK𝜎 iff ∀𝜂^′∈ J𝜂K𝜎∀𝒞 ∈

JΔK𝜎∀𝒟 ∈ JΓK

!𝜎.𝜂^′(𝒞 ◦ 𝒟[𝑃])is SN. where we denote by 𝜂^′(𝑃) the process obtained by substituting the variables in 𝑃 by processes according to𝜂^′.

The following property establishes an equivalence between the extended logical pred-icate and the basic logical predpred-icates of Def. 34. In one direction it establishes that if 𝑃 ∈ ℒ

J⊢_∅ Δ, 𝑥 : 𝐴;ΓK𝜎, then we can cut the process along Δand Γand prove that the resulting cut composition is an element ofJ𝑥:𝐴

K𝜎. Lemma 26. The following two propositions

(1) 𝑃 ∈ ℒ

J⊢_𝜂 Δ, 𝑥:𝐴;ΓK𝜎. (2) For all𝒞 ∈

JΔK𝜎and𝒟 ∈ JΓK

!𝜎,𝒞 ◦ 𝒟[𝑃] ∈ J𝑥:𝐴

K𝜎. are equivalent.

Proof. By Lemma23(4).

Lemma26gives us a degree of freedom in the sense that we can choose an arbitrary typed channel 𝑥 : 𝐴from a nonempty linear typing contextΔ of a typed process𝑃 ⊢_𝜂 Δ;Γ and cut the remaining linear context. We conclude this section with the proof of the Fundamental Lemma 27, from which strong normalisation (Theorem 5) follows immediately.

Lemma 27(Fundamental Lemma). If𝑃⊢_𝜂 Δ;Γ, then𝑃 ∈ ℒ

J⊢_𝜂 Δ;ΓK𝜎.

Proof. By induction on the structure of a typing derivation for𝑃 ⊢_𝜂 Δ;Γ. Cases [Tcut], [Tfwd], [Tcut!] follow immediately becauseJ𝑥 :𝐴

K=J𝑥 :𝐴 K

⊥. Case [T0] follows because 0is SN and case [Tmix] follows becausepar{𝑃| |𝑄}is SN whenever𝑃and𝑄are SN. For the positive types𝐴, the logical predicateJ𝑥:𝐴

K𝜎is defined as the biorthogonal of some set𝑆, hence for the typing rules that introduce a positive type𝐴the strategy is to show that the introduced action𝑃lies in𝑆 ⊆𝑆^⊥⊥. For the negative types𝐴:J𝑥 :𝐴

K𝜎 =𝑆^⊥⊥⊥ =𝑆^⊥, hence the strategy for the typing rules that that introduce an action𝑄that types with a negative type𝑥 :𝐴is to show thatcut{𝑃 |𝑥 :𝐴| 𝑄}is SN, for all𝑃 ∈𝑆. Particularly, for rule [Tcorec], where 𝐴 = 𝜇𝑋. 𝐵, we proceed by induction on the depth 𝑛of unfolding, since 𝑆Ð

𝑛∈N𝜙^𝑛𝐵(∅^⊥⊥). Cases [Tcell] and [Tempty] follow by applying the simulations Lemma18(1)-(2). Cases [Tsh], [TshL], [TshR] follows after applying thedecomposition of the share as a mixas given by Lemma20(1)-(2). We illustrate the proof with some cases. In the cases in which the recursive map𝜂that annotates the typing judgments𝑃⊢_𝜂 Δ;Γplays no role and is essentially propagated from the conclusion to the premises of the typing

rule we omit it, working as if the process𝑃 did not have any free recursion variable𝑋. Similarly for the map 𝜎 which annotates the logical predicates J𝑥 : 𝐴

K𝜎. The complete proof can be found in AppendixE.

Case [Tcut]:

𝑃₁⊢Δ₁, 𝑥:𝐴;Γ 𝑃₂ ⊢Δ₂, 𝑥:𝐴;Γ cut{𝑃₁ |𝑥| 𝑃₂} ⊢Δ₁,Δ₂;Γ Let𝒞₁ ∈

JΔ₁K,𝒞₂∈

JΔ₂Kand𝒟 ∈ JΓK

!. We have

𝒞₁◦ 𝒞₂◦ 𝒟[cut{𝑃₁ |𝑥| 𝑃₂}] ≡_ccut{(𝒞₁◦ 𝒟[𝑃₁]) |𝑥| (𝒞₂◦ 𝒟[𝑃₂])}

I.h. and Lemma26applied to𝑃₁ ⊢Δ₁, 𝑥:𝐴;Γyields𝒞₁◦ 𝒟[𝑃₁] ∈

J𝑥:𝐴K.

I.h. and Lemma26applied to𝑃₂ ⊢Δ₂, 𝑥:𝐴;Γyields𝒞₂◦ 𝒟[𝑃₂] ∈ J𝑥:𝐴

K.

By applying Lemma23(4) we conclude thatcut{(𝒞₁◦ 𝒟[𝑃₁]) |𝑥| (𝒞₂◦ 𝒟[𝑃₂])}is SN.

Hence,𝒞 ◦ 𝒟[cut{𝑃₁ |𝑥| 𝑃₂}]is SN.

Case: [Tcorec]

{𝑥/𝑧}{ ®𝑦/ ®𝑤}𝑃^′⊢_𝜂′ Δ^′, 𝑥:𝐴;Γ 𝜂^′=𝜂, 𝑌(𝑥,𝑦) ↦→® Δ^′, 𝑥:𝑋;Γ corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦] ⊢® _𝜂 Δ^′, 𝑥 :𝜈𝑋. 𝐴;Γ

Let𝜌∈J𝜂K𝜎,𝒞 ∈

JΔ^′K𝜎and𝒟 ∈ JΓK

!𝜎.

We prove that𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])] ∈®

J𝑥 :𝜈𝑋. 𝐴 K𝜎. By Lemma26, this implies thatcorec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦] ∈ ℒ®

J⊢_𝜂 Δ^′, 𝑥:𝜈𝑋. 𝐴;ΓK𝜎. By Lemma25(5), we have

J𝑥 :𝜈𝑋. 𝐴

K𝜎 =Ù

𝑛∈N

𝜙^𝑛_{{𝑋/𝑋}𝐴}(∅^⊥⊥)^⊥ where𝜙{𝑋/𝑋}𝐴(𝑆)≜^unfold𝜇 𝑥;J𝑥:{𝑋/𝑋}𝐴

K𝜎[𝑋↦→𝑆]. We prove (H1):

∀𝑛 ∈N^{, ∀}𝜌∈J𝜂K𝜎, ∀𝒞 ∈

JΔ^′K𝜎, ∀𝒟 ∈ JΓK

!𝜎.

𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])] ∈® 𝜙^𝑛_{{𝑋/𝑋}𝐴}(∅^⊥⊥)^⊥ Proof of (H1) is by induction on𝑛 ∈N:

Case: 𝑛 =0.

Follows because𝒞◦𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])] ∈ ∅® ^⊥and since𝜙⁰_{{𝑋/𝑋}𝐴}(∅^⊥⊥)^⊥=

∅^⊥⊥⊥ =∅^⊥(Lemma22(5)).

Case: 𝑛 =𝑚+1.

Let𝑄∈ 𝜙_{{𝑋/𝑋}𝐴}^𝑚+1 (∅^⊥⊥).

Then𝑄≡_cunfold_𝜇 𝑥;𝑄^′, where𝑄^′ ∈

J𝑥:{𝑋/𝑋}𝐴

K_𝜎^[𝑋↦→_𝜓^𝑚_𝐴^(∅⊥⊥)]. We prove (H2)

cut{𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])]® |𝑥| 𝑄}is SN by induction on𝑁(𝒞) +𝑁(𝜌) +𝑁(𝑄).

Suppose thatcut{𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′ [𝑥,𝑦])]® |𝑥| 𝑄} →_c 𝑅. There are two cases to consider:

Case: (i)𝑅is obtained by an internal reduction of either𝒞,𝜌or𝑄. Case: (ii)𝑅is obtained by an interaction on session𝑥.

Case (i) follows by inner inductive hypothesis (H2).

So let us consider case (ii). Then

cut{𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])]® |𝑥| 𝑄}

≡_ccut{𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])]® |𝑥| unfold_𝜇 𝑥;𝑄^′}

→_ccut{𝒞 ◦ 𝒟[𝜌({𝑥/𝑧}{ ®𝑦/ ®𝑤}{corec𝑌(𝑧,𝑤)® ;𝑃^′/𝑌}𝑃^′)] |𝑥| 𝑄^′}

=cut{𝒞 ◦ 𝒟[𝜌^′({𝑥/𝑧}{ ®𝑦/ ®𝑤}𝑃^′)] |𝑥| 𝑄^′}=𝑅 where𝜌^′ =𝜌, 𝑌(𝑥,𝑦) ↦→® 𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′).

I.h. (H1) applied to𝑚yields

∀𝒞 ∈JΔ^′K, ∀𝒟 ∈ JΓK

!.

𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])] ∈® 𝜙^𝑚_{{𝑋/𝑋}𝐴}(∅^⊥⊥)^⊥ Hence, by Lemma26, we obtain

𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦]) ∈ ℒ®

J⊢_∅ Δ^′, 𝑥:𝑋;ΓK𝜎[𝑋↦→𝜓^𝑚_{{𝑋/𝑋}𝐴}(∅^⊥⊥)^⊥]

Therefore,𝜌^′ ∈J𝜂^′K𝜎.

Applying i.h. (outer i.h., fundamental lemma) to{𝑥/𝑧}{ ®𝑦/ ®𝑤}𝑃^′⊢_𝜂′ Δ^′, 𝑥:𝐴;Γ and Lemma26yields𝒞 ◦ 𝒟[𝜌^′({𝑥/𝑧}{ ®𝑦/ ®𝑤}𝑃^′)] ∈

J𝑥:𝐴

K𝜎[𝑋↦→𝜓^𝑚𝐴(∅^⊥⊥)^⊥]. Lemma23(6) implies𝒞 ◦ 𝒟[𝜌^′({𝑥/𝑧}{ ®𝑦/ ®𝑤}𝑃^′)] ∈

J𝑥:{𝑋/𝑋}𝐴

K𝜎[𝑋↦→𝜓^𝑚𝐴(∅^⊥⊥)]. By hypothesis, 𝑄^′ ∈

J𝑥 : {𝑋/𝑋}𝐴

K_𝜎^[𝑋↦→_𝜓^𝑚_𝐴^(∅⊥⊥)], hence by Lemma 23(3) we obtain that𝑅is SN.

In either case (i)-(ii),𝑅is SN.

By applying Lemma21(3) we conclude thatcut{𝒞◦𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])]® |𝑥| 𝑄}

is SN.

Therefore,𝒞 ◦ 𝒟[𝜌(corec𝑌(𝑧,𝑤)® ;𝑃^′[𝑥,𝑦])] ∈® 𝜙^𝑚+_{{𝑋/𝑋}𝐴}¹ (∅^⊥⊥)^⊥.

Case [Tsh]:

𝑃₁⊢_𝜂 Δ₁, 𝑐:U^𝑓 𝐴;Γ 𝑃₂ ⊢_𝜂Δ₂, 𝑐 :U^𝑓 𝐴;Γ share𝑐{𝑃 | |𝑄} ⊢_𝜂 Δ₁,Δ₂, 𝑐 :U^𝑓 𝐴;Γ Let𝒞₁ ∈

JΔ₁K,𝒞₂∈

JΔ₂Kand𝒟 ∈ JΓK

!. We have

𝒞₁◦ 𝒞₂◦ 𝒟[share𝑐{𝑃₁ | |𝑃₂}]

≡_cshare𝑐{𝒞₁◦ 𝒟[𝑃₁]| |𝒞₂◦ 𝒟[𝑃₂]}

I.h. and Lemma26applied to𝑃₁ ⊢_𝜂Δ₁, 𝑐 :U𝑓 𝐴;Γyields𝒞₁◦ 𝒟[𝑃₁] ∈

J𝑐:U𝑓 𝐴 K. I.h. and Lemma26applied to𝑃₂ ⊢_𝜂Δ₂, 𝑐 :U^𝑓 𝐴;Γyields𝒞₂◦ 𝒟[𝑃₂] ∈

J𝑐:U^𝑓 𝐴K.

By applying Lemma24(1) we conclude that𝒞₁◦𝒞₂◦𝒟[share𝑐{𝑃₁| |𝑃₂}] ∈

J𝑐:U^𝑓 𝐴K.

By Lemma26,share𝑐{𝑃₁ | |𝑃₂} ∈ ℒ

J⊢_𝜂 Δ₁,Δ₂, 𝑐:U^𝑓 𝐴;ΓK.

Cases: [TshL], [TshR]. Similarly to case [Tsh], by applying Lemma24(2).

Case: [Tcell]

𝑃^′⊢_𝜂 Δ^′, 𝑎:∧𝐴;Γ cell𝑐(𝑎.𝑃^′) ⊢_𝜂 Δ^′, 𝑐 :S^𝑓 𝐴;Γ Let𝒞 ∈

JΔ^′K,𝒟 ∈ JΓK

! and𝑄∈

J𝑐:U^𝑓 𝐴K.

I.h. and Lemma26applied to𝑃^′⊢_𝜂 Δ^′, 𝑎:∧𝐴;Γyields𝒞 ◦ 𝒟[𝑃^′] ∈

J𝑎 :∧𝐴K.

Since𝑄∈

J𝑐:U^𝑓 𝐴K, then𝑄isJ𝑎:∧𝐴K-preserving.

Hence, by Lemma18(1),cut{cell𝑐(𝑎.𝒞◦𝒟[𝑃^′]) |𝑐| 𝑄}is simulated bycut{cell𝑐(𝑎.

J𝑎:

∧𝐴K) |𝑐| 𝑄}. Since 𝑄 ∈

J𝑐 : U^𝑓 𝐴

K = 𝑆^⊥ where 𝑆 = {𝑅 | 𝑅 ≡_c cell 𝑐(𝑎.

J𝑎 : ∧𝐴

K)}, then cut{cell𝑐(𝑎.

J𝑎:∧𝐴

K) |𝑐| 𝑄}is SN.

Hence,cut{𝒞 ◦ 𝒟[cell𝑐(𝑎.𝑃^′)] |𝑐| 𝑄}is SN.

Then,cell𝑐(𝑎.𝑃^′) ∈ ℒ

J⊢_𝜂 Δ^′, 𝑐 :S^𝑓 𝐴;ΓK.

Case: [Tempty]

empty𝑐⊢_𝜂 𝑐:S^𝑒 𝐴;Γ Let𝒟 ∈

JΓK

!and𝑄 ∈

J𝑐:U^𝑒𝐴K.

Since𝑄∈

J𝑐:U^𝑒 𝐴K, then𝑄isJ𝑎 :∧𝐴K-preserving.

Hence, by Lemma 18(2), cut {empty 𝑐 |𝑐| 𝑄} is simulated by cut {empty 𝑐(

J𝑎 :

∧𝐴K.)|𝑐| 𝑄}. Since 𝑄 ∈

J𝑐 : U^𝑒 𝐴

K = 𝑆^⊥ where 𝑆 = {𝑅 | 𝑅 ≡_c empty 𝑐(

J𝑎 : ∧𝐴

K.}), then cut{empty𝑐(

J𝑎:∧𝐴

K.)|𝑐| 𝑄}is SN.

Hence,cut{𝒟[empty𝑐] |𝑐| 𝑄}is SN.

Then,empty𝑐 ∈ ℒ

J⊢_𝜂 𝑐:S^𝑒𝐴;ΓK.

Case: [Trelease]

release𝑐⊢_𝜂 𝑐:U^𝑓 𝐴;Γ By Def.32and Lemma22(5) we haveJ𝑥 :U^𝑓 𝐴

K=𝑆^⊥, where 𝑆={𝑄 | 𝑄≡_ccell𝑐(𝑎.

J𝑎:∧𝐴 K)}.

Let𝒟 ∈ JΓK

!and𝑄 ∈𝑆. Then,𝑄≡_ccell𝑐(𝑎.

J𝑎 :∧𝐴 K).

We prove that (H)cut{𝑄 |𝑐| 𝒟[release𝑐]}is SN, by induction on𝑁(𝑄). Suppose thatcut{𝑄 |𝑐| 𝒟[release𝑐]} →_c𝑅. There are two cases to consider:

Case: (i)𝑅is obtained by an internal reduction of either𝑄. Case: (ii)𝑅is obtained by an interaction on cut session𝑐. Case (i) follows by inner inductive hypothesis (H).

So let us consider case (ii). Then

cut{𝑄 |𝑐| 𝒟[release𝑐]} ≡_c𝒟[cut{cell𝑐(𝑎.

J𝑎 :∧𝐴

K) |𝑐| release𝑐}]→−^∗ _c𝒟[0]=𝑅 In either case (i)-(ii),𝑅is SN.

By applying Lemma21(3) we conclude thatcut{𝑄 |𝑐| 𝒟[release𝑐]}is SN.

Furthermore,release𝑐is vacuouslyJ𝑦 :∧𝐴K-preserving, for any𝑦. Therefore,𝒟[release𝑐] ∈

J𝑥:U^𝑓 𝐴 K. By Lemma26,release𝑐 ∈ ℒ

J⊢_𝜂 𝑎 :U^𝑓 𝐴;ΓK.

Case: [Ttake]

𝑃^′ ⊢_𝜂Δ^′, 𝑎:∨𝐴, 𝑐 :U^𝑒 𝐴;Γ take𝑐(𝑎);𝑃^′ ⊢_𝜂 Δ^′, 𝑐 :U^𝑓 𝐴;Γ By Def.32and Lemma22(5) we haveJ𝑐:U^𝑓 𝐴

K=𝑆^⊥, where 𝑆={𝑄 | 𝑄≡_ccell𝑐(𝑎.

J𝑎:∧𝐴 K)}.

Let𝒞 ∈

JΔ^′Kand𝒟 ∈ JΓK

!and𝑄 ∈𝑆. Then,𝑄≡_ccell𝑐(𝑎.

J𝑎 :∧𝐴 K).

We prove that (H)cut{𝑄 |𝑐| 𝒞 ◦𝒟[take𝑐(𝑎);𝑃^′]}is SN, by induction on𝑁(𝑄)+𝑁(𝒞). Suppose thatcut{𝑄 |𝑐| 𝒞 ◦ 𝒟[take𝑐(𝑎);𝑃^′]} →_c𝑅. There are two cases to consider:

Case: (i)𝑅is obtained by an internal reduction of either𝑄or𝒞. Case: (ii)𝑅is obtained by an interaction on cut session𝑐.

Case (i) follows by inner inductive hypothesis (H). So let us consider case (ii). Then cut{𝑄 |𝑐| 𝒞 ◦ 𝒟[take𝑐(𝑎);𝑃^′]}

≡_ccut{cell𝑐(𝑎.

J𝑎 :∧𝐴

K) |𝑐| 𝒞 ◦ 𝒟[take𝑐(𝑎);𝑃^′]}

→_ccut{cell𝑐(𝑎.

J𝑎:∧𝐴

K) |𝑐| (cut{𝑄^′ |𝑎| 𝒞 ◦ 𝒟[𝑃^′]})} =𝑅 where𝑄^′∈

J𝑎:∧𝐴K.

By Def.32,J𝑐:S^𝑓 𝐴

K=𝑆^⊥⊥.

By Lemma22(4),𝑆 ⊆𝑆^⊥⊥, hencecell𝑐(𝑎.

J𝑎 :∧𝐴 K) ∈

J𝑐:S^𝑓 𝐴K.

Applying i.h. to𝑃^′ ⊢_𝜂 Δ^′, 𝑎:∨𝐴, 𝑐 :U^𝑒 𝐴;Γyields𝑅is SN.

In either case (i)-(ii),𝑅is SN.

By applying Lemma21(3) we conclude thatcut{𝑄 |𝑐| 𝒞 ◦ 𝒟[take𝑐(𝑎);𝑃^′]}is SN.

Now, we prove that 𝒞 ◦ 𝒟[take 𝑐(𝑎);𝑃^′] is J𝑎 : ∧𝐴

K𝜎-preserving, for any 𝑎. Let 𝑅 ∈

J𝑎 : ∧𝐴K. Applying i.h. to 𝑃^′ ⊢_𝜂 Δ^′, 𝑎 : ∨𝐴, 𝑐 : U^𝑒 𝐴;Γ we conclude that cut{𝑅 |𝑎| 𝒞 ◦ 𝒟[𝑃^′]} ∈

J𝑐:U^𝑒 𝐴

K, which implies thatcut{𝑅 |𝑎| 𝒞 ◦ 𝒟[𝑃^′]} ∈ J𝑐: U^𝑒 𝐴

Kand hencecut{𝑅 |𝑎| 𝒞 ◦ 𝒟[𝑃^′]}isJ𝑎:∧𝐴K-preserving.

Therefore,𝒞 ◦ 𝒟[take𝑐(𝑎);𝑃^′] ∈

J𝑐:U^𝑓 𝐴K.

By Lemma26,take𝑐(𝑎);𝑃^′ ∈ ℒ

J⊢_𝜂 Δ^′, 𝑐 :U^𝑓 𝐴;ΓK.

Case: [Tput]

𝑃₁ ⊢_𝜂Δ₁, 𝑎:∧𝐴;Γ 𝑃₂⊢_𝜂 Δ₂, 𝑐:U^𝑓 𝐴;Γ put𝑐(𝑎.𝑃₁);𝑃₂ ⊢_𝜂 Δ₁,Δ₂, 𝑐:U^𝑒 𝐴;Γ By Def.32and Lemma22(5) we haveJ𝑐:U^𝑒 𝐴

K=𝑆^⊥, where 𝑆 ={𝑄 | 𝑄≡_cempty𝑐(

J𝑎:∧𝐴 K.}).

Let𝒞₁ ∈

JΔ₁K,𝒞₂ ∈

JΔ₂Kand𝒟 ∈ JΓK

!and𝑄 ∈𝑆. Then,𝑄≡_cempty𝑐(

J𝑎 :∧𝐴 K.).

We prove that (H)cut{𝑄 |𝑐| 𝒞₁◦ 𝒞₂◦ 𝒟[put 𝑐(𝑎.𝑃₁);𝑃₂]} is SN, by induction on 𝑁(𝑄) +𝑁(𝒞₁) +𝑁(𝒞₂).

Suppose thatcut{𝑄 |𝑐| 𝒞₁◦ 𝒞₂◦ 𝒟[put𝑐(𝑎.𝑃₁);𝑃₂]} →_c𝑅. There are two cases to consider:

Case: (i)𝑅is obtained by an internal reduction of either𝑄,𝒞₁or𝒞₂. Case: (ii)𝑅is obtained by an interaction on cut session𝑐.

Case (i) follows by inner inductive hypothesis (H).So let us consider case (ii). Then cut{𝑄 |𝑐| 𝒞₁◦ 𝒞₂◦ 𝒟[put𝑐(𝑎.𝑃₁);𝑃₂]}

≡_ccut{empty𝑐(

J𝑎 :∧𝐴

K.)|𝑐| 𝒞₁◦ 𝒞₂◦ 𝒟[put𝑐(𝑎.𝑃₁);𝑃₂]}

≡_ccut{empty𝑐(

J𝑎 :∧𝐴

K.)|𝑐| put𝑐(𝑎.𝒞₁◦ 𝒟[𝑃₁]);𝒞₂◦ 𝒟[𝑃₂]}

→_ccut{cell𝑐(𝑎.

J𝑎:∧𝐴

K) |𝑐| 𝒞₂◦ 𝒟[𝑃₂]} =𝑅(*)

I.h. applied to𝑃₁ ⊢_𝜂 Δ₁, 𝑎:∧𝐴;Γyields𝒞₁◦ 𝒟[𝑃₁] ∈

J𝑎 :∧𝐴

K, hence reduction step (*).

By Def.32,J𝑐:S^𝑓 𝐴

K=𝑆^⊥⊥.

By Lemma22(4),𝑆 ⊆𝑆^⊥⊥, hencecell𝑐(𝑎.

J𝑎 :∧𝐴 K) ∈

J𝑐:S^𝑓 𝐴 K. Applying i.h. to𝑃₂⊢_𝜂 Δ₂, 𝑐 :U𝑓 𝐴;Γyields𝑅is SN.

In either case (i)-(ii),𝑅is SN.

By applying Lemma21(3) we conclude thatcut{𝑄 |𝑐| 𝒞₁◦ 𝒞₂◦ 𝒟[put𝑐(𝑎.𝑃₁);𝑃₂]}

is SN.

Now, we prove that𝒞₁◦ 𝒞₂◦ 𝒟[put 𝑐(𝑎.𝑃₁);𝑃₂]isJ𝑎 : ∧𝐴

K-preserving, for any 𝑎. Applying i.h. to𝑃₁⊢_𝜂 Δ₁, 𝑎:∧𝐴;Γwe conclude that𝒞₁◦ 𝒟[𝑃₁] ∈

J𝑎 :∧𝐴K. Applying i.h. to𝑃₂⊢_𝜂 Δ₂, 𝑐:U^𝑓 𝐴;Γwe conclude that𝒞₂◦ 𝒟[𝑃₂] ∈

J𝑐:U^𝑓 𝐴K, which implies that𝒞₂◦ 𝒟[𝑃₂]isJ𝑎:∧𝐴

K-preserving Therefore,𝒞₁◦ 𝒞₂◦ 𝒟[put𝑐(𝑎.𝑃₁);𝑃₂] ∈

J𝑐:U^𝑒 𝐴 K. By Lemma26,put𝑐(𝑎.𝑃₁);𝑃₂∈ ℒ

J⊢_𝜂 Δ₁,Δ₂, 𝑐 :U^𝑒 𝐴;ΓK.

Theorem 5(Strong Normalisation). If𝑃 ⊢_∅ ∅;∅, then𝑃is SN.

Proof. Immediately by Lemma27.

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