M Derivation.agda => Derivation.agda +449 -24
@@ 1,19 1,29 @@
{-# OPTIONS --without-K #-}
-open import Level
+open import Level using (0ℓ)
open import Axiom.Extensionality.Propositional using (Extensionality)
+open import Axiom.ExcludedMiddle
module Derivation (funExt : Extensionality 0ℓ 0ℓ) where
-import Level
-open Level using (0ℓ)
-
-open import Signature
-open import PropositionalLogic funExt
+open import Data.Empty
+open import Data.Bool
+open import Data.Bool.Properties
+open import Data.Product
+open import Data.Sum
+open import Data.Unit using (⊤; tt)
+open import Relation.Binary.PropositionalEquality
+open ≡-Reasoning
+open import Relation.Nullary
open import Relation.Unary
hiding (_⊢_; _∈_)
renaming (｛_｝ to ⟦_⟧)
+open import Signature
+open import Structure funExt
+open import PropositionalLogic funExt
+open import Utils
+
Assumptions : (σ : Signature) → Set₂
Assumptions σ = Pred (LP σ) (Level.suc 0ℓ)
@@ 21,7 31,8 @@ private
variable
σ : Signature
ψ ϕ χ : LP σ
- Γ Δ : Assumptions σ
+ A : Structure σ
+ Γ Γ′ Δ Δ′ : Assumptions σ
infixr 3 _⊢_
@@ 45,25 56,439 @@ data _⊢_
: Δ ⊢ ψ
→ (∀ δ → δ ∈ Δ → Γ ⊢ δ)
→ Γ ⊢ ψ
+ ∧-intro
+ : Γ ⊢ ϕ
+ → Δ ⊢ ψ
+ → Γ ∪ Δ ⊢ ϕ ∧′ ψ
+ ∧-elim₁
+ : Γ ⊢ ϕ ∧′ ψ
+ → Γ ⊢ ϕ
+ ∧-elim₂
+ : Γ ⊢ ϕ ∧′ ψ
+ → Γ ⊢ ψ
→-intro
- : (ϕ ψ : LP σ)
- → (Γ ∪ ⟦ ϕ ⟧) ⊢ ψ
- → Γ ⊢ (ϕ →′ ψ)
+ : Γ ∪ ⟦ ϕ ⟧ ⊢ ψ
+ → Γ ⊢ ϕ →′ ψ
+ →-elim
+ : Γ ⊢ ϕ
+ → Δ ⊢ ϕ →′ ψ
+ → Γ ∪ Δ ⊢ ψ
+ ↔-intro
+ : Γ ⊢ ϕ →′ ψ
+ → Δ ⊢ ψ →′ ϕ
+ → Γ ∪ Δ ⊢ ϕ ↔′ ψ
+ ↔-elim₁
+ : Γ ⊢ ϕ ↔′ ψ
+ → Γ ⊢ ϕ →′ ψ
+ ↔-elim₂
+ : Γ ⊢ ϕ ↔′ ψ
+ → Γ ⊢ ψ →′ ϕ
¬-intro
- : (ϕ : LP σ)
- → (Γ ∪ ⟦ ϕ ⟧) ⊢ ⊥′
- → Γ ⊢ (¬′ ϕ)
+ : Γ ∪ ⟦ ϕ ⟧ ⊢ ⊥′
+ → Γ ⊢ ¬′ ϕ
+ ¬-elim
+ : Γ ⊢ ϕ
+ → Δ ⊢ ¬′ ϕ
+ → Γ ∪ Δ ⊢ ⊥′
RAA
- : (ϕ : LP σ)
- → (Γ ∪ ⟦ ¬′ ϕ ⟧) ⊢ ⊥′
- → Γ ⊢ ϕ
- →-elim
+ : Γ ∪ ⟦ ¬′ ϕ ⟧ ⊢ ⊥′
+ → Γ ⊢ ϕ
+ ∨-intro₁
: Γ ⊢ ϕ
- → Γ ⊢ (ϕ →′ ψ)
- → Γ ⊢ ψ
+ → Γ ⊢ ϕ ∨′ ψ
+ ∨-intro₂
+ : Γ ⊢ ψ
+ → Γ ⊢ ϕ ∨′ ψ
∨-elim
- : Γ ⊢ (ϕ ∨′ ψ)
- → (Γ ∪ ⟦ ϕ ⟧) ⊢ χ
- → (Γ ∪ ⟦ ψ ⟧) ⊢ χ
- → Γ ⊢ χ
- -- TODO: add the rest of the derivation rules
+ : Γ ∪ ⟦ ϕ ⟧ ⊢ χ
+ → Δ ∪ ⟦ ψ ⟧ ⊢ χ
+ → Γ ∪ Δ ∪ ⟦ ϕ ∨′ ψ ⟧ ⊢ χ
+
+_⊬_ : Assumptions σ → LP σ → Set₂
+Γ ⊬ ϕ = ¬ (Γ ⊢ ϕ)
+
+IsModel : Structure σ → Assumptions σ → Set₁
+IsModel A Γ = ∀ ϕ → ϕ ∈ Γ → A * ϕ ≡ true
+
+_⊨_ : Assumptions σ → LP σ → Set₁
+Γ ⊨ ϕ = ∀ A → IsModel A Γ → A * ϕ ≡ true
+
+_⊭_ : Assumptions σ → LP σ → Set₁
+Γ ⊭ ϕ = ¬ (Γ ⊨ ϕ)
+
+Soundness : Set₂
+Soundness =
+ ∀ {σ} {Γ : Assumptions σ} ψ
+ → Γ ⊢ ψ → Γ ⊨ ψ
+
+Adequacy : Set₂
+Adequacy =
+ ∀ {σ} {Γ : Assumptions σ} ψ
+ → Γ ⊨ ψ → Γ ⊢ ψ
+
+Completeness : Set₂
+Completeness = Soundness × Adequacy
+
+IsConsistent : Assumptions σ → Set₂
+IsConsistent Γ = Γ ⊬ ⊥′
+
+--has-model→consistent : IsModel A Γ → IsConsistent Γ
+--has-model→consistent = {!!}
+
+model⊆ : IsModel A Δ → Γ ⊆ Δ → IsModel A Γ
+model⊆ = {!!}
+
+model∪₁ : IsModel A (Γ ∪ Δ) → IsModel A Γ
+model∪₁ = {!!}
+
+model∪₂ : IsModel A (Γ ∪ Δ) → IsModel A Δ
+model∪₂ = {!!}
+
+model∪ : IsModel A Γ → IsModel A Δ → IsModel A (Γ ∪ Δ)
+model∪ = {!!}
+
+model-singl : IsModel A ⟦ ψ ⟧ → A * ψ ≡ true
+model-singl {ψ = ψ} model = model ψ refl
+
+model-singl-inv : A * ψ ≡ true → IsModel A ⟦ ψ ⟧
+model-singl-inv prf ψ refl = prf
+
+∨-helper : ∀ {a b} → a ∨ b ≡ true → (a ≡ true) ⊎ (b ≡ true)
+∨-helper {false} {b} prf = inj₂ prf
+∨-helper {true} {b} prf = inj₁ prf
+
+model∨ : A * (ϕ ∨′ ψ) ≡ true → ((A * ϕ ≡ true) ⊎ (A * ψ ≡ true))
+model∨ {A = A} {ϕ = ϕ} {ψ = ψ} prf = ∨-helper lemma
+ where
+ lemma : (A * ϕ) ∨ (A * ψ) ≡ true
+ lemma = prf
+
+∧-helper₁ : ∀ {a b} → a ∧ b ≡ true → a ≡ true
+∧-helper₁ {a = true} refl = refl
+
+∧-helper₂ : ∀ {a b} → a ∧ b ≡ true → b ≡ true
+∧-helper₂ {a = true} refl = refl
+
+⇒b-cancel-left : ∀ {a} → false ⇒b a ≡ true
+⇒b-cancel-left {false} = refl
+⇒b-cancel-left {true} = refl
+
+⇒b-left-identity : ∀ {a} → true ⇒b a ≡ a
+⇒b-left-identity {false} = refl
+⇒b-left-identity {true} = refl
+
+sound : Soundness
+sound ψ (weakening subset D) A model = sound ψ D A (model⊆ model subset)
+sound ψ (axiom ψ) A model = model ψ refl
+sound ψ (transitive D x) A model = {!!}
+sound (ϕ ∧′ ψ) (∧-intro D₁ D₂) A model =
+ let
+ ind-hyp₁ = sound ϕ D₁ A (model∪₁ model)
+ ind-hyp₂ = sound ψ D₂ A (model∪₂ model)
+ in
+ begin
+ (A * ϕ) ∧ (A * ψ)
+ ≡⟨ cong₂ _∧_ ind-hyp₁ ind-hyp₂ ⟩
+ true ∧ true
+ ≡⟨⟩
+ true
+ ∎
+sound ψ (∧-elim₁ D) A model =
+ let ind-hyp = sound _ D A model
+ in ∧-helper₁ ind-hyp
+sound ψ (∧-elim₂ D) A model =
+ let ind-hyp = sound _ D A model
+ in ∧-helper₂ ind-hyp
+sound (ϕ →′ ψ) (→-intro D) A model with A * ϕ ≟ true
+... | yes prf =
+ let
+ ind-hyp = sound ψ D A (model∪ model (model-singl-inv prf))
+ in
+ begin
+ (A * ϕ) ⇒b (A * ψ)
+ ≡⟨ cong₂ (_⇒b_) prf ind-hyp ⟩
+ true ⇒b true
+ ≡⟨⟩
+ true
+ ∎
+... | no prf =
+ begin
+ (A * ϕ) ⇒b (A * ψ)
+ ≡⟨ cong (_⇒b (A * ψ)) (¬t→f prf) ⟩
+ false ⇒b (A * ψ)
+ ≡⟨ ⇒b-cancel-left ⟩
+ true
+ ∎
+sound ψ (→-elim {ϕ = ϕ} D₁ D₂) A model =
+ let
+ ind-hyp₁ : A * ϕ ≡ true
+ ind-hyp₁ = sound ϕ D₁ A (λ ϕ z → model ϕ (inj₁ z))
+
+ ind-hyp₂ : A * (ϕ →′ ψ) ≡ true
+ ind-hyp₂ = sound (ϕ →′ ψ) D₂ A (λ ϕ z → model ϕ (inj₂ z))
+
+ in
+ begin
+ A * ψ
+ ≡⟨ sym ⇒b-left-identity ⟩
+ true ⇒b (A * ψ)
+ ≡⟨ cong (_⇒b (A * ψ)) (sym ind-hyp₁) ⟩
+ (A * ϕ) ⇒b (A * ψ)
+ ≡⟨ ind-hyp₂ ⟩
+ true
+ ∎
+sound (ϕ ↔′ ψ) (↔-intro D₁ D₂) A model = {!!}
+sound (_ →′ _) (↔-elim₁ D) A x₁ = {!!}
+sound (_ →′ _) (↔-elim₂ D) A x₁ = {!!}
+sound (¬′ _) (¬-intro D) A x₁ = {!!}
+sound ⊥′ (¬-elim D D₁) A x₁ = {!!}
+sound ψ (RAA D) A x₁ = {!!}
+sound (ϕ ∨′ ψ) (∨-intro₁ D) A model =
+ let ind-hyp = sound ϕ D A model
+ in
+ begin
+ (A * ϕ) ∨ (A * ψ)
+ ≡⟨ cong (_∨ (A * ψ)) ind-hyp ⟩
+ true ∨ (A * ψ)
+ ≡⟨⟩
+ true
+ ∎
+sound (ϕ ∨′ ψ) (∨-intro₂ D) A model =
+ let ind-hyp = sound ψ D A model
+ in
+ begin
+ (A * ϕ) ∨ (A * ψ)
+ ≡⟨ cong ((A * ϕ) ∨_) ind-hyp ⟩
+ (A * ϕ) ∨ true
+ ≡⟨ ∨-comm (A * ϕ) true ⟩
+ true ∨ (A * ϕ)
+ ≡⟨⟩
+ true
+ ∎
+sound χ (∨-elim {Γ = Γ}{ϕ = ϕ}{Δ = Δ}{ψ = ψ} D₁ D₂) A model
+ with
+ model∨ {A = A} {ϕ = ϕ} {ψ = ψ}
+ (model-singl (model∪₂ (model∪₂ model)))
+... | inj₁ prf =
+ let
+ model′ : IsModel A (Γ ∪ ⟦ ϕ ⟧)
+ model′ = model∪ (model∪₁ model) (model-singl-inv prf)
+
+ ind-hyp : A * χ ≡ true
+ ind-hyp = sound χ D₁ A model′
+
+ in ind-hyp
+... | inj₂ prf =
+ let
+ model′ : IsModel A (Δ ∪ ⟦ ψ ⟧)
+ model′ = model∪ (model∪₁ (model∪₂ model)) (model-singl-inv prf)
+
+ ind-hyp : A * χ ≡ true
+ ind-hyp = sound χ D₂ A model′
+
+ in ind-hyp
+
+{-
+Lemma 3.10.3
+-}
+consistent→model→adequacy
+ : (∀ (Γ : Assumptions σ) → IsConsistent Γ → ∃[ A ] IsModel A Γ)
+ → Adequacy
+consistent→model→adequacy = {!!}
+
+{-
+Definition 3.10.4
+-}
+record IsHintikka (Γ : Assumptions σ) : Set₁ where
+ field
+ and : (ϕ ∧′ ψ) ∈ Γ → ϕ ∈ Γ × ψ ∈ Γ
+ or : (¬′ (ϕ ∧′ ψ)) ∈ Γ → ¬′ ϕ ∈ Γ ⊎ ¬′ ψ ∈ Γ
+ neg : (¬′ ¬′ ϕ) ∈ Γ → ϕ ∈ Γ
+ bot : ⊥′ ∉ Γ
+ vars : ∀ p prf → ¬ (var p prf ∈ Γ × ¬′ var p prf ∈ Γ)
+
+bool-DeMorgan-∧ : ∀ {a b} → not (a ∧ b) ≡ not a ∨ not b
+bool-DeMorgan-∧ {false} {b} = refl
+bool-DeMorgan-∧ {true} {b} = refl
+
+DeMorgan-∧ : (¬′ (ϕ ∧′ ψ)) ~ ((¬′ ϕ) ∨′ (¬′ ψ))
+DeMorgan-∧ {ϕ = ϕ}{ψ = ψ} A =
+ begin
+ A * (¬′ (ϕ ∧′ ψ))
+ ≡⟨⟩
+ not (A * ϕ ∧ A * ψ)
+ ≡⟨ bool-DeMorgan-∧ {A * ϕ} {A * ψ} ⟩
+ not (A * ϕ) ∨ not (A * ψ)
+ ≡⟨⟩
+ A * ((¬′ ϕ) ∨′ (¬′ ψ))
+ ∎
+
+bool-DeMorgan-∨ : ∀ {a b} → not (a ∨ b) ≡ not a ∧ not b
+bool-DeMorgan-∨ {false} {b} = refl
+bool-DeMorgan-∨ {true} {b} = refl
+
+DeMorgan-∨ : (¬′ (ϕ ∨′ ψ)) ~ ((¬′ ϕ) ∧′ (¬′ ψ))
+DeMorgan-∨ {ϕ = ϕ}{ψ = ψ} A =
+ begin
+ A * (¬′ (ϕ ∨′ ψ))
+ ≡⟨⟩
+ not (A * ϕ ∨ A * ψ)
+ ≡⟨ bool-DeMorgan-∨ {A * ϕ} {A * ψ} ⟩
+ not (A * ϕ) ∧ not (A * ψ)
+ ≡⟨⟩
+ A * ((¬′ ϕ) ∧′ (¬′ ψ))
+ ∎
+
+IsBasic : LP σ → Set
+IsBasic ⊥′ = ⊤
+IsBasic (var _ _) = ⊤
+IsBasic (¬′ ϕ) = IsBasic ϕ
+IsBasic (ϕ ∧′ ψ) = IsBasic ϕ × IsBasic ψ
+IsBasic (_ ∨′ _) = ⊥
+IsBasic (_ →′ _) = ⊥
+IsBasic (_ ↔′ _) = ⊥
+
+AreBasicAssumptions : Assumptions σ → Set₁
+AreBasicAssumptions Γ = ∀ ϕ → ϕ ∈ Γ → IsBasic ϕ
+
+{-
+_
+ :
+ → IsConsistent Γ
+ → Dec (IsConsistent (Γ ∪ ⟦ χ ⟧))
+ -- ~ Dec (Γ ⊢ χ)
+-}
+
+{-
+Lemma 3.10.5
+
+We add the precondition that Γ is decidable since we don't have LEM.
+-}
+hintikka-has-model
+ : ∀ {σ}{Γ : Assumptions σ}
+ → Decidable Γ
+ → AreBasicAssumptions Γ
+ → IsHintikka Γ
+ → ∃[ A ] IsModel A Γ
+hintikka-has-model {σ = σ}{Γ = Γ} dec-Γ is-basic is-hintikka =
+ B , a
+ where
+ B : Structure σ
+ B p prf with dec-Γ (var p prf)
+ ... | yes _ = true
+ ... | no _ = false
+
+ a : ∀ ϕ → ϕ ∈ Γ → B * ϕ ≡ true
+ b : ∀ ϕ → ¬′ ϕ ∈ Γ → B * ϕ ≡ false
+
+ b→a : ∀ ϕ → B * ϕ ≡ false → B * (¬′ ϕ) ≡ true
+ b→a ϕ prf =
+ begin
+ B * (¬′ ϕ)
+ ≡⟨⟩
+ not (B * ϕ)
+ ≡⟨ cong not prf ⟩
+ not false
+ ≡⟨⟩
+ true
+ ∎
+
+ a ⊥′ prf = ⊥-elim (IsHintikka.bot is-hintikka prf)
+ a (var p prf′) prf with dec-Γ (var p prf′)
+ ... | yes _ = refl
+ ... | no ¬prf = ⊥-elim (¬prf prf)
+ a (¬′ ϕ) prf = b→a ϕ (b ϕ prf)
+ a (ϕ ∧′ ψ) prf =
+ let
+ ϕ-prf , ψ-prf = IsHintikka.and is-hintikka prf
+ ind-hyp₁ = a ϕ ϕ-prf
+ ind-hyp₂ = a ψ ψ-prf
+ in
+ begin
+ B * (ϕ ∧′ ψ)
+ ≡⟨⟩
+ B * ϕ ∧ B * ψ
+ ≡⟨ cong₂ _∧_ ind-hyp₁ ind-hyp₂ ⟩
+ true
+ ∎
+ a ϕ@(_ ∨′ _) prf = ⊥-elim (is-basic ϕ prf)
+ a ϕ@(_ →′ _) prf = ⊥-elim (is-basic ϕ prf)
+ a ϕ@(_ ↔′ _) prf = ⊥-elim (is-basic ϕ prf)
+
+ b ⊥′ prf = refl
+ b (var p prf′) prf₁ with dec-Γ (var p prf′)
+ ... | yes prf₂ =
+ ⊥-elim (IsHintikka.vars is-hintikka p prf′ (prf₂ , prf₁))
+ ... | no _ = refl
+ b (¬′ ϕ) prf =
+ let
+ ind-hyp : (B * ϕ) ≡ true
+ ind-hyp = a ϕ (IsHintikka.neg is-hintikka prf)
+ in
+ begin
+ B * (¬′ ϕ)
+ ≡⟨⟩
+ not (B * ϕ)
+ ≡⟨ cong not ind-hyp ⟩
+ not true
+ ≡⟨⟩
+ false
+ ∎
+ b (ϕ ∧′ ψ) prf with IsHintikka.or is-hintikka prf
+ ... | inj₁ prf₁ =
+ let
+ ind-hyp : B * ϕ ≡ false
+ ind-hyp = b ϕ prf₁
+ in
+ begin
+ B * (ϕ ∧′ ψ)
+ ≡⟨⟩
+ B * ϕ ∧ B * ψ
+ ≡⟨ cong (_∧ B * ψ) ind-hyp ⟩
+ false ∧ B * ψ
+ ≡⟨⟩
+ false
+ ∎
+ ... | inj₂ prf₂ =
+ let
+ ind-hyp : B * ψ ≡ false
+ ind-hyp = b ψ prf₂
+ in
+ begin
+ B * (ϕ ∧′ ψ)
+ ≡⟨⟩
+ B * ϕ ∧ B * ψ
+ ≡⟨ cong (B * ϕ ∧_) ind-hyp ⟩
+ B * ϕ ∧ false
+ ≡⟨ ∧-comm (B * ϕ) false ⟩
+ false ∧ B * ϕ
+ ≡⟨⟩
+ false
+ ∎
+ b ϕ@(_ ∨′ _) prf = ⊥-elim (is-basic (¬′ ϕ) prf)
+ b ϕ@(_ →′ _) prf = ⊥-elim (is-basic (¬′ ϕ) prf)
+ b ϕ@(_ ↔′ _) prf = ⊥-elim (is-basic (¬′ ϕ) prf)
+
+module _ (lem : ExcludedMiddle 0ℓ) where
+
+ {-
+ Lemma 3.10.6
+ -}
+ consistent→hintikka
+ : ∀ {Γ : Assumptions σ}
+ → IsConsistent Γ
+ → AreBasicAssumptions Γ
+ → Σ[ Δ ∈ Assumptions σ ] IsHintikka Δ × Γ ⊆ Δ
+ consistent→hintikka consistent is-basic = {!!}
+
+ consistent-have-models→adequacy
+ : (∀ (Γ : Assumptions σ) → IsConsistent Γ → ∃[ A ] IsModel A Γ)
+ → Adequacy
+ consistent-have-models→adequacy x ψ get-model = {!!}
+
+ adequate : Adequacy
+ adequate = consistent→model→adequacy
+ λ Γ cons → hintikka-has-model
+ {!!}
+ {!!}
+ (proj₁ (proj₂ (consistent→hintikka cons {!!})))
M Examples.agda => Examples.agda +3 -0
@@ 19,6 19,9 @@ open import Relation.Unary
open import Relation.Nullary using (¬_; yes; no)
open import PropositionalLogic funExt
+open import Signature
+open import Structure funExt
+open import Derivation funExt
module example-3-5-4 where
σ : Signature
M PropositionalLogic.agda => PropositionalLogic.agda +8 -4
@@ 86,8 86,8 @@ f * (χ₁ ↔′ χ₂) = (f * χ₁) ⇔b (f * χ₂)
{-
Definition 3.5.7 (tautology)
-}
-⊨ : (ϕ : LP σ) → Set
-⊨ ϕ = ∀ A → A * ϕ ≡ true
+IsTautology : (ϕ : LP σ) → Set
+IsTautology ϕ = ∀ A → A * ϕ ≡ true
{-
Lemma 3.6.1
@@ 100,7 100,7 @@ module lemma-3-6-1 (ϕ ψ : LP σ) where
II = ∀ A → A * ϕ ≡ A * ψ
III : Set
- III = ⊨ (ϕ ↔′ ψ)
+ III = IsTautology (ϕ ↔′ ψ)
Lemma : Set
Lemma = I ⇔ II × II ⇔ III
@@ 533,6 533,10 @@ posts-theorem
posts-theorem n g with posts-theorem-on-tables (fun→table g)
... | ψ , prf = ψ , get-table-fun→table n ψ g prf
+{-
+Definitions (3.8.6)
+-}
+
data IsLiteral {σ} : LP σ → Set₁ where
isLit-var : ∀ {p prf} → IsLiteral (var p prf)
isLit-neg-var : ∀ {p prf} → IsLiteral (¬′ var p prf)
@@ 541,7 545,7 @@ data IsBasicConjunction : LP σ → Set₁ where
isConj-lit : IsLiteral ψ → IsBasicConjunction ψ
isConj-conj : IsLiteral ψ → IsBasicConjunction ϕ → IsBasicConjunction (ψ ∧′ ϕ)
-data IsBasicDisjunction {σ} : LP σ → Set₁ where
+data IsBasicDisjunction : LP σ → Set₁ where
isDisj-lit : IsLiteral ψ → IsBasicDisjunction ψ
isDisj-disj : IsLiteral ψ → IsBasicDisjunction ϕ → IsBasicDisjunction (ψ ∨′ ϕ)
M Signature.agda => Signature.agda +3 -4
@@ 5,7 5,9 @@ module Signature where
open import Level
open import Data.Nat
open import Data.Sum
-open import Relation.Unary hiding (_∈_)
+open import Relation.Unary
+open import Relation.Unary using (_∈_; _∉_) public
+open import Relation.Nullary
Signature : Set₁
Signature = Pred ℕ 0ℓ
@@ 13,9 15,6 @@ Signature = Pred ℕ 0ℓ
finite-signature : ℕ → Signature
finite-signature n = λ k → k < n
-_∈_ : {a ℓ : Level} {A : Set a} (a : A) → Pred A ℓ → Set ℓ
-p ∈ σ = σ p
-
sets-lemma : ∀ {τ σ : Signature} (p : ℕ) → τ ⊆ σ → p ∈ (σ ∪ τ) → p ∈ σ
sets-lemma p τ⊆σ p∈σ∪τ with p∈σ∪τ
... | inj₁ p∈σ = p∈σ