├── .agda-lib ├── .dir-locals.el ├── .github └── workflows │ └── ci.yml ├── .gitignore ├── .gitmodules ├── .stignore ├── .travis.yml.disabled ├── 2D ├── Circle.agda ├── Everything.agda ├── Frac.agda ├── Group.agda ├── Iter.agda ├── Obsolete │ └── opsem.agda ├── Order.agda ├── Order │ └── Old.agda ├── Order2Lemmas.agda ├── Power.agda ├── Prim.agda ├── ProgMorphisms.agda ├── Sing.agda ├── SingIter.agda ├── ToDo.txt ├── Types.agda ├── Val.agda ├── foldSwap.agda ├── g.dot └── opsem3.agda ├── 2dtypes.bib ├── 2dtypesZot.bib ├── A.hs ├── AM.hs ├── AM2.hs ├── Action.agda ├── AutGroup.agda ├── B.agda ├── B2.agda ├── B5.agda ├── Comonad.agda ├── DPi.agda ├── Equiv.agda ├── EquivEquiv.agda ├── FW.agda ├── FW └── PiBoolCat.agda ├── GroupAction.agda ├── Makefile ├── Makefile.fractypes ├── Makefile.upi ├── Musings.agda ├── Perm.agda ├── PermMonad ├── Comonad.agda ├── PermMonad.agda └── PermPi.agda ├── Pi.agda ├── Pi ├── .gitignore ├── Common │ ├── Arithmetic.agda │ ├── Extra.agda │ ├── FinHelpers.agda │ ├── InequalityEquiv.agda │ ├── LList.agda │ ├── ListFinLListEquiv.agda │ ├── ListN.agda │ ├── Misc.agda │ └── README.md ├── Coxeter │ ├── Coxeter.agda │ ├── GeneratedGroup.agda │ ├── GeneratedGroupGeneralised.agda │ ├── GeneratedGroupIso.agda │ ├── GeneratedGroupIsoGeneralised.agda │ ├── Group.agda │ ├── InvTransform.agda │ ├── Lehmer2CoxeterEquiv.agda │ ├── Lehmer2SnEquiv.agda │ ├── LehmerCoxeterEquiv.agda │ ├── LehmerImmersion.agda │ ├── LehmerSnEquiv.agda │ ├── NonParametrized │ │ ├── Coxeter.agda │ │ ├── CoxeterMCoxeterEquiv.agda │ │ ├── CritPairsLong.agda │ │ ├── Diamond.agda │ │ ├── ExchangeLemmas+.agda │ │ ├── ExchangeLemmas.agda │ │ ├── ImpossibleLists.agda │ │ ├── LehmerCanonical.agda │ │ ├── LehmerReduction.agda │ │ ├── MCoxeter.agda │ │ ├── MCoxeterS.agda │ │ ├── README.md │ │ ├── ReductionRel+.agda │ │ └── ReductionRel.agda │ ├── Norm.agda │ ├── NormEquiv.agda │ ├── Parametrized │ │ ├── CoxeterMCoxeterEquiv.agda │ │ ├── MCoxeter.agda │ │ ├── README.md │ │ └── ReductionRel.agda │ ├── README.md │ └── Sn.agda ├── Equiv │ ├── Equiv0.agda │ ├── Equiv0Hat.agda │ ├── Equiv0Norm.agda │ ├── Equiv1.agda │ ├── Equiv1Hat.agda │ ├── Equiv1Norm.agda │ ├── Equiv1NormHelpers.agda │ ├── Equiv2.agda │ ├── Equiv2Hat.agda │ ├── Equiv2HatHelpers.agda │ ├── Equiv2Norm.agda │ ├── README.md │ └── Translation2.agda ├── Everything.agda ├── Examples │ ├── Adder.agda │ ├── Base.agda │ ├── Copy.agda │ ├── Examples.agda │ ├── ExamplesL.lagda │ ├── ExpMod.agda │ ├── Incr.agda │ ├── Interp.agda │ ├── Parity.agda │ ├── README.md │ ├── Reset.agda │ └── Toffoli.agda ├── Experiments │ ├── CoxeterHIT.agda │ ├── Equiv │ │ ├── Equiv0.agda │ │ ├── Equiv1.agda │ │ ├── Equiv1Norm.agda │ │ ├── Equiv2.agda │ │ ├── Level0.agda │ │ └── Util.agda │ ├── PiLevel0.agda │ ├── README.md │ ├── ShiftHIT.agda │ └── Translation.agda ├── FSMG │ ├── Conjectures.agda │ ├── FSMG.agda │ ├── FSMG │ │ └── Properties.agda │ ├── M.agda │ ├── M │ │ └── Properties.agda │ ├── Paths.agda │ ├── README.md │ ├── SMG.agda │ └── UFin.agda ├── Lehmer │ ├── FinExcept.agda │ ├── LICENSE_cubical │ ├── Lehmer.agda │ ├── Lehmer2.agda │ ├── Lehmer2FinEquiv.agda │ ├── LehmerFinEquiv.agda │ └── README.md ├── Makefile ├── README.md ├── Syntax │ ├── Pi+ │ │ ├── Indexed.agda │ │ └── NonIndexed.agda │ ├── Pi.agda │ ├── Pi^.agda │ ├── Pi^Helpers.agda │ └── README.md ├── UFin │ ├── BAut.agda │ ├── Base.agda │ ├── Misc.agda │ ├── Monoidal.agda │ ├── Paths.agda │ ├── README.md │ ├── UFin.agda │ ├── UFinLehmer2Equiv.agda │ ├── UFinLehmerEquiv.agda │ └── Univ.agda └── paper │ ├── .gitignore │ ├── 2dtypesZot.bib │ ├── ACM-Reference-Format.bst │ ├── ExamplesL.tex │ ├── Makefile │ ├── T.lagda │ ├── abstract.md │ ├── acmart.cls │ ├── agda.sty │ ├── applications.tex │ ├── arxiv.tex │ ├── chen.bib │ ├── discussion.tex │ ├── equivalence.tex │ ├── examples.tex │ ├── examples.tex-e │ ├── extra.tex │ ├── finite.tex │ ├── hott.sty │ ├── informalOLD.tex │ ├── inkscape │ ├── let2pi.svg │ ├── metalang.svg │ ├── partiality.svg │ ├── pi.svg │ ├── qis1.pdf │ ├── qis1.svg │ ├── qis2.pdf │ ├── qis2.svg │ └── trace_plus.svg │ ├── introduction.tex │ ├── latex │ ├── Pi │ │ ├── Examples │ │ │ └── ExamplesL.tex │ │ └── Indexed │ │ │ └── ExamplesL.tex │ └── agda.sty │ ├── macros.sty │ ├── main.tex │ ├── notes.org │ ├── outline.png │ ├── popl22-rev-strip.tex │ ├── popl22-rev.tex │ ├── qasm │ ├── qtree.sty │ ├── quiver.sty │ ├── reversible1.tex │ ├── reversibleOLD.tex │ ├── reversibleOr.png │ ├── reversibleOr.qasm │ ├── reversibleOr2.png │ ├── reversibleOr2.qasm │ ├── survey.bib │ ├── syntax.sty │ ├── tikzit.sty │ ├── tikzit.tikzstyles │ ├── ufin.tex │ └── univalentOLD.tex ├── Pi1.agda ├── Pi2 ├── BadSyntax.agda ├── Semantics.agda ├── SemanticsBasic.agda ├── Syntax.agda ├── TwoUniverse.agda └── UniFibExamples.agda ├── PiFin+ ├── Level0.agda ├── Level1.agda ├── ListPerm.agda ├── Semantics.agda ├── Semantics0.agda ├── Semantics1.agda ├── Semantics2.agda ├── Semantics3.agda ├── Semantics4.agda ├── Syntax.agda └── cchm │ ├── equiv.ctt │ ├── prelude.ctt │ ├── semantics.ctt │ ├── syntax.ctt │ └── tests.ctt ├── PiFin ├── CPerm.agda ├── Dec.agda ├── EMSpace.agda ├── EmbeddingsInUniverse.agda ├── Fin.agda ├── Level0.agda ├── Level1.agda ├── Lift.agda ├── List.agda ├── NatEx.agda ├── Perm.agda ├── Syntax.agda ├── Transposition.agda └── paper │ ├── .gitignore │ ├── macros.sty │ └── main.tex ├── PiS1 ├── Semantics.agda ├── Syntax.agda ├── bool.ctt ├── circle.ctt ├── discor.ctt ├── equiv.ctt ├── hedberg.ctt ├── int.ctt ├── nat.ctt ├── prelude.ctt ├── retract.ctt ├── semantics.ctt └── univalence.ctt ├── R.agda ├── R2.agda ├── R3.agda ├── README.md ├── RF.agda ├── RF2.agda ├── Rational+.agda ├── Rational.agda ├── TruncationExamples.agda ├── TypeEquiv.agda ├── Types'.agda ├── Types.agda ├── Uni-fib.agda ├── Univ ├── Cats.agda ├── IdSystem.agda ├── Interval.agda ├── Level0.agda ├── Level1.agda ├── Pi.agda ├── Syntax.agda └── Universe.agda ├── acmart.cls ├── agda.sty ├── appendix.lagda ├── 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