5858 < a id ="CommMonoidTheory.commAssocSwap "> </ a > < a id ="1873 " href ="Cubical.Algebra.CommMonoid.Properties.html#1873 " class ="Function "> commAssocSwap</ a > < a id ="1887 " class ="Symbol "> :</ a > < a id ="1889 " class ="Symbol "> (</ a > < a id ="1890 " href ="Cubical.Algebra.CommMonoid.Properties.html#1890 " class ="Bound "> x</ a > < a id ="1892 " href ="Cubical.Algebra.CommMonoid.Properties.html#1892 " class ="Bound "> y</ a > < a id ="1894 " href ="Cubical.Algebra.CommMonoid.Properties.html#1894 " class ="Bound "> z</ a > < a id ="1896 " href ="Cubical.Algebra.CommMonoid.Properties.html#1896 " class ="Bound "> w</ a > < a id ="1898 " class ="Symbol "> :</ a > < a id ="1900 " href ="Cubical.Algebra.CommMonoid.Properties.html#1452 " class ="Function "> M</ a > < a id ="1901 " class ="Symbol "> )</ a > < a id ="1903 " class ="Symbol "> →</ a > < a id ="1905 " class ="Symbol "> (</ a > < a id ="1906 " href ="Cubical.Algebra.CommMonoid.Properties.html#1890 " class ="Bound "> x</ a > < a id ="1908 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1910 " href ="Cubical.Algebra.CommMonoid.Properties.html#1892 " class ="Bound "> y</ a > < a id ="1911 " class ="Symbol "> )</ a > < a id ="1913 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1915 " class ="Symbol "> (</ a > < a id ="1916 " href ="Cubical.Algebra.CommMonoid.Properties.html#1894 " class ="Bound "> z</ a > < a id ="1918 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1920 " href ="Cubical.Algebra.CommMonoid.Properties.html#1896 " class ="Bound "> w</ a > < a id ="1921 " class ="Symbol "> )</ a > < a id ="1923 " href ="Agda.Builtin.Cubical.Path.html#272 " class ="Function Operator "> ≡</ a > < a id ="1925 " class ="Symbol "> (</ a > < a id ="1926 " href ="Cubical.Algebra.CommMonoid.Properties.html#1890 " class ="Bound "> x</ a > < a id ="1928 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1930 " href ="Cubical.Algebra.CommMonoid.Properties.html#1894 " class ="Bound "> z</ a > < a id ="1931 " class ="Symbol "> )</ a > < a id ="1933 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1935 " class ="Symbol "> (</ a > < a id ="1936 " href ="Cubical.Algebra.CommMonoid.Properties.html#1892 " class ="Bound "> y</ a > < a id ="1938 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1940 " href ="Cubical.Algebra.CommMonoid.Properties.html#1896 " class ="Bound "> w</ a > < a id ="1941 " class ="Symbol "> )</ a >
5959 < a id ="1944 " href ="Cubical.Algebra.CommMonoid.Properties.html#1873 " class ="Function "> commAssocSwap</ a > < a id ="1958 " href ="Cubical.Algebra.CommMonoid.Properties.html#1958 " class ="Bound "> x</ a > < a id ="1960 " href ="Cubical.Algebra.CommMonoid.Properties.html#1960 " class ="Bound "> y</ a > < a id ="1962 " href ="Cubical.Algebra.CommMonoid.Properties.html#1962 " class ="Bound "> z</ a > < a id ="1964 " href ="Cubical.Algebra.CommMonoid.Properties.html#1964 " class ="Bound "> w</ a > < a id ="1966 " class ="Symbol "> =</ a > < a id ="1968 " href ="Cubical.Algebra.Semigroup.Base.html#1031 " class ="Function "> ·Assoc</ a > < a id ="1975 " class ="Symbol "> (</ a > < a id ="1976 " href ="Cubical.Algebra.CommMonoid.Properties.html#1958 " class ="Bound "> x</ a > < a id ="1978 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="1980 " href ="Cubical.Algebra.CommMonoid.Properties.html#1960 " class ="Bound "> y</ a > < a id ="1981 " class ="Symbol "> )</ a > < a id ="1983 " href ="Cubical.Algebra.CommMonoid.Properties.html#1962 " class ="Bound "> z</ a > < a id ="1985 " href ="Cubical.Algebra.CommMonoid.Properties.html#1964 " class ="Bound "> w</ a > < a id ="1987 " href ="Cubical.Foundations.Prelude.html#4007 " class ="Function Operator "> ∙∙</ a > < a id ="1990 " href ="Cubical.Foundations.Prelude.html#1430 " class ="Function "> cong</ a > < a id ="1995 " class ="Symbol "> (</ a > < a id ="1996 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> _·</ a > < a id ="1999 " href ="Cubical.Algebra.CommMonoid.Properties.html#1964 " class ="Bound "> w</ a > < a id ="2000 " class ="Symbol "> )</ a > < a id ="2002 " class ="Symbol "> (</ a > < a id ="2003 " href ="Cubical.Algebra.CommMonoid.Properties.html#1602 " class ="Function "> commAssocr</ a > < a id ="2014 " href ="Cubical.Algebra.CommMonoid.Properties.html#1958 " class ="Bound "> x</ a > < a id ="2016 " href ="Cubical.Algebra.CommMonoid.Properties.html#1960 " class ="Bound "> y</ a > < a id ="2018 " href ="Cubical.Algebra.CommMonoid.Properties.html#1962 " class ="Bound "> z</ a > < a id ="2019 " class ="Symbol "> )</ a >
6060 < a id ="2068 " href ="Cubical.Foundations.Prelude.html#4007 " class ="Function Operator "> ∙∙</ a > < a id ="2071 " href ="Cubical.Foundations.Prelude.html#968 " class ="Function "> sym</ a > < a id ="2075 " class ="Symbol "> (</ a > < a id ="2076 " href ="Cubical.Algebra.Semigroup.Base.html#1031 " class ="Function "> ·Assoc</ a > < a id ="2083 " class ="Symbol "> (</ a > < a id ="2084 " href ="Cubical.Algebra.CommMonoid.Properties.html#1958 " class ="Bound "> x</ a > < a id ="2086 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="2088 " href ="Cubical.Algebra.CommMonoid.Properties.html#1962 " class ="Bound "> z</ a > < a id ="2089 " class ="Symbol "> )</ a > < a id ="2091 " href ="Cubical.Algebra.CommMonoid.Properties.html#1960 " class ="Bound "> y</ a > < a id ="2093 " href ="Cubical.Algebra.CommMonoid.Properties.html#1964 " class ="Bound "> w</ a > < a id ="2094 " class ="Symbol "> )</ a >
61+
62+ < a id ="CommMonoidTheory.hasInverse "> </ a > < a id ="2098 " href ="Cubical.Algebra.CommMonoid.Properties.html#2098 " class ="Function "> hasInverse</ a > < a id ="2109 " class ="Symbol "> :</ a > < a id ="2111 " class ="Symbol "> (</ a > < a id ="2112 " href ="Cubical.Algebra.CommMonoid.Properties.html#2112 " class ="Bound "> x</ a > < a id ="2114 " class ="Symbol "> :</ a > < a id ="2116 " href ="Cubical.Algebra.CommMonoid.Properties.html#1452 " class ="Function "> M</ a > < a id ="2117 " class ="Symbol "> )</ a > < a id ="2119 " class ="Symbol "> →</ a > < a id ="2121 " href ="Agda.Primitive.html#388 " class ="Primitive "> Type</ a > < a id ="2126 " href ="Cubical.Algebra.CommMonoid.Properties.html#1405 " class ="Bound "> ℓ</ a >
63+ < a id ="2129 " href ="Cubical.Algebra.CommMonoid.Properties.html#2098 " class ="Function "> hasInverse</ a > < a id ="2140 " href ="Cubical.Algebra.CommMonoid.Properties.html#2140 " class ="Bound "> x</ a > < a id ="2142 " class ="Symbol "> =</ a > < a id ="2144 " href ="Cubical.Core.Primitives.html#6268 " class ="Function "> Σ[</ a > < a id ="2147 " href ="Cubical.Algebra.CommMonoid.Properties.html#2147 " class ="Bound "> -x</ a > < a id ="2150 " href ="Cubical.Core.Primitives.html#6268 " class ="Function "> ∈</ a > < a id ="2152 " href ="Cubical.Algebra.CommMonoid.Properties.html#1452 " class ="Function "> M</ a > < a id ="2154 " href ="Cubical.Core.Primitives.html#6268 " class ="Function "> ]</ a > < a id ="2156 " href ="Cubical.Algebra.CommMonoid.Properties.html#2147 " class ="Bound "> -x</ a > < a id ="2159 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="2161 " href ="Cubical.Algebra.CommMonoid.Properties.html#2140 " class ="Bound "> x</ a > < a id ="2163 " href ="Agda.Builtin.Cubical.Path.html#272 " class ="Function Operator "> ≡</ a > < a id ="2165 " href ="Cubical.Algebra.CommMonoid.Base.html#991 " class ="Function "> ε</ a >
64+
65+ < a id ="CommMonoidTheory.isPropHasInverse "> </ a > < a id ="2169 " href ="Cubical.Algebra.CommMonoid.Properties.html#2169 " class ="Function "> isPropHasInverse</ a > < a id ="2186 " class ="Symbol "> :</ a > < a id ="2188 " class ="Symbol "> ∀</ a > < a id ="2190 " href ="Cubical.Algebra.CommMonoid.Properties.html#2190 " class ="Bound "> x</ a > < a id ="2192 " class ="Symbol "> →</ a > < a id ="2194 " href ="Cubical.Foundations.Prelude.html#15547 " class ="Function "> isProp</ a > < a id ="2201 " class ="Symbol "> (</ a > < a id ="2202 " href ="Cubical.Algebra.CommMonoid.Properties.html#2098 " class ="Function "> hasInverse</ a > < a id ="2213 " href ="Cubical.Algebra.CommMonoid.Properties.html#2190 " class ="Bound "> x</ a > < a id ="2214 " class ="Symbol "> )</ a >
66+ < a id ="2217 " href ="Cubical.Algebra.CommMonoid.Properties.html#2169 " class ="Function "> isPropHasInverse</ a > < a id ="2234 " href ="Cubical.Algebra.CommMonoid.Properties.html#2234 " class ="Bound "> x</ a > < a id ="2236 " href ="Cubical.Algebra.CommMonoid.Properties.html#2236 " class ="Bound "> yinv</ a > < a id ="2241 " href ="Cubical.Algebra.CommMonoid.Properties.html#2241 " class ="Bound "> zinv</ a >
67+ < a id ="2249 " class ="Symbol "> =</ a > < a id ="2251 " href ="Cubical.Data.Sigma.Properties.html#13825 " class ="Function "> Σ≡Prop</ a > < a id ="2258 " class ="Symbol "> (λ</ a > < a id ="2261 " href ="Cubical.Algebra.CommMonoid.Properties.html#2261 " class ="Bound "> a</ a > < a id ="2263 " class ="Symbol "> →</ a > < a id ="2265 " href ="Cubical.Algebra.Semigroup.Base.html#1010 " class ="Function "> is-set</ a > < a id ="2272 " class ="Symbol "> (</ a > < a id ="2273 " href ="Cubical.Algebra.CommMonoid.Properties.html#2261 " class ="Bound "> a</ a > < a id ="2275 " href ="Cubical.Algebra.CommMonoid.Base.html#1012 " class ="Function Operator "> ·</ a > < a id ="2277 " href ="Cubical.Algebra.CommMonoid.Properties.html#2234 " class ="Bound "> x</ a > < a id ="2278 " class ="Symbol "> )</ a > < a id ="2280 " href ="Cubical.Algebra.CommMonoid.Base.html#991 " class ="Function "> ε</ a > < a id ="2281 " class ="Symbol "> )</ a >
68+ < a id ="2287 " class ="Symbol "> (</ a > < a id ="2288 " href ="Cubical.Data.Sigma.Properties.html#2083 " class ="Function "> PathPΣ</ a > < a id ="2295 " class ="Symbol "> (</ a > < a id ="2296 " href ="Cubical.Algebra.Monoid.Base.html#4323 " class ="Function "> MonoidTheory.isPropHasInverse</ a > < a id ="2326 " class ="Symbol "> (</ a > < a id ="2327 " href ="Cubical.Algebra.CommMonoid.Base.html#2506 " class ="Function "> CommMonoid→Monoid</ a > < a id ="2345 " href ="Cubical.Algebra.CommMonoid.Properties.html#1389 " class ="Bound "> M'</ a > < a id ="2347 " class ="Symbol "> )</ a > < a id ="2349 " href ="Cubical.Algebra.CommMonoid.Properties.html#2234 " class ="Bound "> x</ a >
69+ < a id ="2394 " class ="Symbol "> (</ a > < a id ="2395 " href ="Cubical.Algebra.CommMonoid.Properties.html#2561 " class ="Function "> hasInverseToMonoid</ a > < a id ="2414 " href ="Cubical.Algebra.CommMonoid.Properties.html#2234 " class ="Bound "> x</ a > < a id ="2416 " href ="Cubical.Algebra.CommMonoid.Properties.html#2236 " class ="Bound "> yinv</ a > < a id ="2420 " class ="Symbol "> )</ a >
70+ < a id ="2465 " class ="Symbol "> (</ a > < a id ="2466 " href ="Cubical.Algebra.CommMonoid.Properties.html#2561 " class ="Function "> hasInverseToMonoid</ a > < a id ="2485 " href ="Cubical.Algebra.CommMonoid.Properties.html#2234 " class ="Bound "> x</ a > < a id ="2487 " href ="Cubical.Algebra.CommMonoid.Properties.html#2241 " class ="Bound "> zinv</ a > < a id ="2491 " class ="Symbol "> ))</ a >
71+ < a id ="2541 " class ="Symbol "> .</ a > < a id ="2542 " href ="Agda.Builtin.Sigma.html#251 " class ="Field "> fst</ a > < a id ="2545 " class ="Symbol "> )</ a >
72+ < a id ="2550 " class ="Keyword "> where</ a >
73+ < a id ="2561 " href ="Cubical.Algebra.CommMonoid.Properties.html#2561 " class ="Function "> hasInverseToMonoid</ a > < a id ="2580 " class ="Symbol "> :</ a > < a id ="2582 " class ="Symbol "> ∀</ a > < a id ="2584 " href ="Cubical.Algebra.CommMonoid.Properties.html#2584 " class ="Bound "> x</ a >
74+ < a id ="2610 " class ="Symbol "> →</ a > < a id ="2612 " href ="Cubical.Algebra.CommMonoid.Properties.html#2098 " class ="Function "> hasInverse</ a > < a id ="2623 " href ="Cubical.Algebra.CommMonoid.Properties.html#2584 " class ="Bound "> x</ a >
75+ < a id ="2649 " class ="Symbol "> →</ a > < a id ="2651 " href ="Cubical.Algebra.Monoid.Base.html#4225 " class ="Function "> MonoidTheory.hasInverse</ a > < a id ="2675 " class ="Symbol "> (</ a > < a id ="2676 " href ="Cubical.Algebra.CommMonoid.Base.html#2506 " class ="Function "> CommMonoid→Monoid</ a > < a id ="2694 " href ="Cubical.Algebra.CommMonoid.Properties.html#1389 " class ="Bound "> M'</ a > < a id ="2696 " class ="Symbol "> )</ a > < a id ="2698 " href ="Cubical.Algebra.CommMonoid.Properties.html#2584 " class ="Bound "> x</ a >
76+ < a id ="2705 " href ="Cubical.Algebra.CommMonoid.Properties.html#2561 " class ="Function "> hasInverseToMonoid</ a > < a id ="2724 " href ="Cubical.Algebra.CommMonoid.Properties.html#2724 " class ="Bound "> x</ a > < a id ="2726 " class ="Symbol "> (</ a > < a id ="2727 " href ="Cubical.Algebra.CommMonoid.Properties.html#2727 " class ="Bound "> y</ a > < a id ="2729 " href ="Agda.Builtin.Sigma.html#235 " class ="InductiveConstructor Operator "> ,</ a > < a id ="2731 " href ="Cubical.Algebra.CommMonoid.Properties.html#2731 " class ="Bound "> yinv</ a > < a id ="2735 " class ="Symbol "> )</ a > < a id ="2737 " class ="Symbol "> =</ a > < a id ="2739 " href ="Cubical.Algebra.CommMonoid.Properties.html#2727 " class ="Bound "> y</ a > < a id ="2741 " href ="Agda.Builtin.Sigma.html#235 " class ="InductiveConstructor Operator "> ,</ a > < a id ="2743 " href ="Cubical.Algebra.CommMonoid.Properties.html#2731 " class ="Bound "> yinv</ a > < a id ="2748 " href ="Agda.Builtin.Sigma.html#235 " class ="InductiveConstructor Operator "> ,</ a > < a id ="2750 " href ="Cubical.Algebra.CommMonoid.Base.html#741 " class ="Function "> ·Comm</ a > < a id ="2756 " href ="Cubical.Algebra.CommMonoid.Properties.html#2724 " class ="Bound "> x</ a > < a id ="2758 " href ="Cubical.Algebra.CommMonoid.Properties.html#2727 " class ="Bound "> y</ a > < a id ="2760 " href ="Cubical.Foundations.Prelude.html#5207 " class ="Function Operator "> ∙</ a > < a id ="2762 " href ="Cubical.Algebra.CommMonoid.Properties.html#2731 " class ="Bound "> yinv</ a >
6177</ pre > </ body > </ html >
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