diff --git a/src/Framework/Proof/Transitive.lagda.md b/src/Framework/Proof/Transitive.lagda.md index 64be00d4..f3c86121 100644 --- a/src/Framework/Proof/Transitive.lagda.md +++ b/src/Framework/Proof/Transitive.lagda.md @@ -81,53 +81,53 @@ expressiveness-by-completeness-and-soundness comp sound eˢ with sound eˢ ... | eᶜ , v≅⟦eᶜ⟧ = eᶜ , ≅-trans (≅-sym v≅⟦eˢ⟧) v≅⟦eᶜ⟧ ``` -If a language `L₊` is complete and another language `L₋` is incomplete then `L₋` less expressive than `L₊`. +If a language `L` is complete and another language `M` is incomplete then `M` less expressive than `L`. **Proof sketch:** -Assuming `L₋` is as expressive as `L₊`, and knowing that `L₊` is complete, we can conclude that also `L₋` is complete (via `completeness-by-exoressiveness` above). -Yet, we already know that L₋ is incomplete. +Assuming `M` is as expressive as `L`, and knowing that `L` is complete, we can conclude that also `M` is complete (via `completeness-by-exoressiveness` above). +Yet, we already know that M is incomplete. This yields a contradiction. ```agda -less-expressive-from-completeness : ∀ {L₊ L₋ : VariabilityLanguage V} - → Complete L₊ - → Incomplete L₋ +less-expressive-from-completeness : ∀ {L M : VariabilityLanguage V} + → Complete L + → Incomplete M ------------------------------ - → L₋ ⋡ L₊ -less-expressive-from-completeness L₊-comp L₋-incomp L₋-as-expressive-as-L₊ = - L₋-incomp (completeness-by-expressiveness L₊-comp L₋-as-expressive-as-L₊) + → M ⋡ L +less-expressive-from-completeness L-comp M-incomp M-as-expressive-as-L = + M-incomp (completeness-by-expressiveness L-comp M-as-expressive-as-L) ``` ```agda -less-expressive-from-soundness : ∀ {L₊ L₋ : VariabilityLanguage V} - → Sound L₊ - → Unsound L₋ +less-expressive-from-soundness : ∀ {L M : VariabilityLanguage V} + → Sound L + → Unsound M ------------------------------ - → L₊ ⋡ L₋ -less-expressive-from-soundness L₊-sound L₋-unsound L₋≽L₊ = - L₋-unsound (soundness-by-expressiveness L₊-sound L₋≽L₊) + → L ⋡ M +less-expressive-from-soundness L-sound M-unsound M≽L = + M-unsound (soundness-by-expressiveness L-sound M≽L) ``` -Combined with `expressiveness-by-completeness` we can even further conclude that L₊ is more expressive than L₋: +Combined with `expressiveness-by-completeness` we can even further conclude that L is more expressive than M: ```agda -more-expressive-by-completeness : ∀ {L₊ L₋ : VariabilityLanguage V} - → Complete L₊ - → Sound L₋ - → Incomplete L₋ +more-expressive-by-completeness : ∀ {L M : VariabilityLanguage V} + → Complete L + → Sound M + → Incomplete M ------------- - → L₊ ≻ L₋ -more-expressive-by-completeness L₊-comp L₋-sound L₋-incomp = - expressiveness-by-completeness-and-soundness L₊-comp L₋-sound - , less-expressive-from-completeness L₊-comp L₋-incomp - -more-expressive-by-soundness : ∀ {L₊ L₋ : VariabilityLanguage V} - → Sound L₊ - → Complete L₋ - → Unsound L₋ + → L ≻ M +more-expressive-by-completeness L-comp M-sound M-incomp = + expressiveness-by-completeness-and-soundness L-comp M-sound + , less-expressive-from-completeness L-comp M-incomp + +more-expressive-by-soundness : ∀ {L M : VariabilityLanguage V} + → Sound L + → Complete M + → Unsound M ----------- - → L₋ ≻ L₊ -more-expressive-by-soundness L₊-sound L₋-comp L₋-unsound = - expressiveness-by-completeness-and-soundness L₋-comp L₊-sound - , less-expressive-from-soundness L₊-sound L₋-unsound + → M ≻ L +more-expressive-by-soundness L-sound M-comp M-unsound = + expressiveness-by-completeness-and-soundness M-comp L-sound + , less-expressive-from-soundness L-sound M-unsound ``` ```agda