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1 module agda-term where
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2
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3 open import Data.Nat.Base
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4 open import Relation.Binary.PropositionalEquality
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5
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6 +zero : {y : @$\mathbb{N}$@} @$\rightarrow$@ y + zero @$\equiv$@ y
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7 +zero {zero} = refl
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8 +zero {suc y} = cong (@$\lambda$@ yy @$\rightarrow$@ suc yy) (+zero {y})
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9
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10 +-suc : {x y : @$\mathbb{N}$@} @$\rightarrow$@ x + suc y @$\equiv$@ suc (x + y)
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11 +-suc {zero} {y} = refl
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12 +-suc {suc x} {y} = cong suc (+-suc {x} {y})
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13
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14 +-comm : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
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15 +-comm zero y rewrite (+zero {y}) = refl
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16 +-comm (suc x) y = let open @$\equiv$@-Reasoning in
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17 begin
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18 suc (x + y) @$\equiv$@@$\langle$@@$\rangle$@
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19 suc (x + y) @$\equiv$@@$\langle$@ cong suc (+-comm x y) @$\rangle$@
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20 suc (y + x) @$\equiv$@@$\langle$@ sym (+-suc {y} {x}) @$\rangle$@
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21 y + suc x @$\blacksquare$@
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22
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23 +-come : (x y : @$\mathbb{N}$@) @$\rightarrow$@ x + y @$\equiv$@ y + x
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24 +-come zero y rewrite (+zero {y}) = refl
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25 +-come (suc x) y
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26 rewrite (cong suc (+-come x y)) | sym (+-suc {y} {x}) = refl
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27
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