Library iris.base_logic.upred

The basic definition of the uPred type, its metric and functor laws. You probably do not want to import this file. Instead, import base_logic.base_logic; that will also give you all the primitive and many derived laws for the logic. A good way of understanding this definition of the uPred OFE is to consider the OFE uPred0 of monotonous SProp predicates. That is, uPred0 is the OFE of non-expansive functions from M to SProp that are monotonous with respect to CMRA inclusion. This notion of monotonicity has to be stated in the SProp logic. Together with the usual closedness property of SProp, this gives exactly uPred_mono. Then, we quotient uPred0 *in the siProp logic* with respect to equivalence on valid elements of M. That is, we quotient with respect to the following *siProp* equivalence relation: P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1) When seen from the ambiant logic, obtaining this quotient requires definig both a custom Equiv and Dist. It is worth noting that this equivalence relation admits canonical representatives. More precisely, one can show that every equivalence class contains exactly one element P0 such that: ∀ x, (✓ x → P0(x)) → P0(x) (2) (Again, this assertion has to be understood in siProp). Intuitively, this says that P0 trivially holds whenever the resource is invalid. Starting from any element P, one can find this canonical representative by choosing: P0(x) := ✓ x → P(x) (3) Hence, as an alternative definition of uPred, we could use the set of canonical representatives (i.e., the subtype of monotonous siProp predicates that verify (2)). This alternative definition would save us from using a quotient. However, the definitions of the various connectives would get more complicated, because we have to make sure they all verify (2), which sometimes requires some adjustments. We would moreover need to prove one more property for every logical connective. Note that, somewhat curiously, uPred M is *not* in general a Camera, at least not if all propositions are considered "valid" Camera elements. It fails to satisfy the extension axiom. Here is the counterexample: We use M := (option Ex {A,B})^2 -- so we have pairs whose components are ε, A or B. Let
  P r n := (ownM (A,A) ∧ ▷ False) ∨ ownM (A,B) ∨ ownM (B,A) ∨ ownM (B,B)
         ↔ r = (A,A) ∧ n = 0 ∨
           r = (A,B) ∨
           r = (B,A) ∨
           r = (B,B)
 Q1 r n := ownM (A, ε) ∨ ownM (B, ε)
         ↔ (A, ε) ≼ r ∨ (B, ε) ≼ r
           ("Left component is not ε")
 Q2 r n := ownM (ε, A) ∨ ownM (ε, B)
         ↔ (ε, A) ≼ r ∨ (ε, B) ≼ r
           ("Right component is not ε")

These are all sufficiently closed and non-expansive and whatnot. We have P ≡{0}≡ Q1 × Q2. So assume extension holds, then we get Q1', Q2' such that
  P ≡ Q1' ∗ Q2'
 Q1 ≡{0}≡ Q1'
 Q2 ≡{0}≡ Q2'

Now comes the contradiction: We know that P (A,A) 1 does *not* hold, but I am going to show that (Q1' ∗ Q2') (A,A) 1 holds, which would be a contraction. To this end, I will show (a) Q1' (A,ε) 1 and (b) Q2' (ε,A) 1. The result (Q1' ∗ Q2') (A,A) follows from (A,ε) ⋅ (ε,A) = (A,A). (a) Proof of Q1' (A,ε) 1. We have P (A,B) 1, and thus Q1' r1 1 and Q2' r2 1 for some r1 ⋅ r2 = (A,B). There are four possible decompositions r1 ⋅ r2:

• (ε,ε) ⋅ (A,B): This would give us Q1' (ε,ε) 1, from which we obtain (through down-closure and the equality Q1 ≡{0}≡ Q1' above) that Q1 (ε,ε) 0. However, we know that's false.
• (A,B) ⋅ (ε,ε): Can be excluded for similar reasons (the second resource must not be ε in the 2nd component).
• (ε,B) ⋅ (A,ε): Can be excluded for similar reasons (the first resource must not be ε in the 1st component).
• (A,ε) ⋅ (ε,B): This gives us the desired Q1' (A,ε) 1.

(b) Proof of Q2' (ε,A) 1. We have P (B,A) 1, and thus Q1' r1 1 and Q2' r2 1 for some r1 ⋅ r2 = (B,A). There are again four possible decompositions, and like above we can exclude three of them. This leaves us with (B,ε) ⋅ (ε,A) and thus Q2' (ε,A) 1. This completes the proof.

When working in the model, it is convenient to be able to treat uPred as nat → M → Prop. But we only want to locally break the uPred abstraction this way.

functor

logical entailement

logical connectives

Global uPred-specific Notation

Primitive logical rules. These are not directly usable later because they do not refer to the BI connectives.

The notations below are implicitly local due to the section, so we do not mind the overlap with the general BI notations.

Entailment

Non-expansiveness and setoid morphisms

Introduction and elimination rules

BI connectives

Persistently

Plainly

Later

Internal equality

This is really just a special case of an entailment between two siProp, but we do not have the infrastructure to express the more general case. This temporary proof rule will be replaced by the proper one eventually.

Basic update modality