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July 12, 2015 01:10
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BCoPL theorem 2.5
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| -- theorem 2.5 | |
| associativity-plus : ∀ {n₁ n₂ n₃ n₄ n₅} → n₁ plus n₂ is n₄ → n₄ plus n₃ is n₅ → | |
| ∃ λ n₆ → n₂ plus n₃ is n₆ → n₁ plus n₆ is n₅ | |
| associativity-plus {Z} {Z} {Z} {Z} {Z} P-Zero P-Zero = Z , (λ x → x) | |
| associativity-plus {Z} {Z} {Z} {Z} {S n₅} P-Zero () | |
| associativity-plus {Z} {Z} {Z} {S n₄} {Z} () l₂ | |
| associativity-plus {Z} {Z} {Z} {S n₄} {S n₅} () l₂ | |
| associativity-plus {Z} {Z} {S n₃} {Z} {Z} P-Zero () | |
| associativity-plus {Z} {Z} {S n₃} {Z} {S .n₃} P-Zero P-Zero = S n₃ , (λ x → x) | |
| associativity-plus {Z} {Z} {S n₃} {S n₄} {Z} () l₂ | |
| associativity-plus {Z} {Z} {S n₃} {S n₄} {S n₅} () l₂ | |
| associativity-plus {Z} {S n₂} {Z} {Z} {Z} () l₂ | |
| associativity-plus {Z} {S n₂} {Z} {Z} {S n₅} () l₂ | |
| associativity-plus {Z} {S n₂} {Z} {S .n₂} {Z} P-Zero () | |
| associativity-plus {Z} {S n₂} {Z} {S .n₂} {S n₅} P-Zero (P-Succ l₂) = S n₅ , (λ x → P-Zero) | |
| associativity-plus {Z} {S n₂} {S n₃} {Z} {Z} () l₂ | |
| associativity-plus {Z} {S n₂} {S n₃} {Z} {S n₅} () l₂ | |
| associativity-plus {Z} {S n₂} {S n₃} {S .n₂} {Z} P-Zero () | |
| associativity-plus {Z} {S n₂} {S n₃} {S .n₂} {S n₅} P-Zero (P-Succ l₂) = S n₅ , (λ x → P-Zero) | |
| associativity-plus {S n₁} {Z} {Z} {Z} {Z} () l₂ | |
| associativity-plus {S n₁} {Z} {Z} {Z} {S n₅} () l₂ | |
| associativity-plus {S n₁} {Z} {Z} {S n₄} {Z} (P-Succ l₁) () | |
| associativity-plus {S n₁} {Z} {Z} {S n₄} {S n₅} (P-Succ l₁) (P-Succ l₂) = S n₅ , (λ ()) | |
| associativity-plus {S n₁} {Z} {S n₃} {Z} {Z} () l₂ | |
| associativity-plus {S n₁} {Z} {S n₃} {Z} {S n₅} () l₂ | |
| associativity-plus {S n₁} {Z} {S n₃} {S n₄} {Z} (P-Succ l₁) () | |
| associativity-plus {S n₁} {Z} {S n₃} {S n₄} {S n₅} (P-Succ l₁) (P-Succ l₂) = Z , (λ ()) | |
| associativity-plus {S n₁} {S n₂} {Z} {Z} {Z} () l₂ | |
| associativity-plus {S n₁} {S n₂} {Z} {Z} {S n₅} () l₂ | |
| associativity-plus {S n₁} {S n₂} {Z} {S n₄} {Z} (P-Succ l₁) () | |
| associativity-plus {S n₁} {S n₂} {Z} {S n₄} {S n₅} (P-Succ l₁) (P-Succ l₂) = Z , (λ ()) | |
| associativity-plus {S n₁} {S n₂} {S n₃} {Z} {Z} () l₂ | |
| associativity-plus {S n₁} {S n₂} {S n₃} {Z} {S n₅} () l₂ | |
| associativity-plus {S n₁} {S n₂} {S n₃} {S n₄} {Z} (P-Succ l₁) () | |
| associativity-plus {S n₁} {S n₂} {S n₃} {S n₄} {S n₅} (P-Succ l₁) (P-Succ l₂) = Z , (λ ()) |
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