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symsym.ml
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symsym.ml
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open Testutil.Pmp
let () =
run @@ fun () ->
let uu, _ = synth UU in
let xx = assume "X" uu (* Bottom face *) in
let x000 = assume "x000" xx in
let x001 = assume "x001" xx in
let xx002, _ = synth (id !!"X" !!"x000" !!"x001") in
let x002 = assume "x002" xx002 in
let x010 = assume "x010" xx in
let x011 = assume "x011" xx in
let xx012, _ = synth (id !!"X" !!"x010" !!"x011") in
let x012 = assume "x012" xx012 in
let xx020, _ = synth (id !!"X" !!"x000" !!"x010") in
let xx021, _ = synth (id !!"X" !!"x001" !!"x011") in
let x020 = assume "x020" xx020 in
let x021 = assume "x021" xx021 in
let xx022, _ =
synth
(refl (refl !!"X")
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x020"
$ !!"x021") in
let x022 = assume "x022" xx022 (* Top face *) in
let x100 = assume "x100" xx in
let x101 = assume "x101" xx in
let xx102, _ = synth (id !!"X" !!"x100" !!"x101") in
let x102 = assume "x102" xx102 in
let x110 = assume "x110" xx in
let x111 = assume "x111" xx in
let xx112, _ = synth (id !!"X" !!"x110" !!"x111") in
let x112 = assume "x112" xx112 in
let xx120, _ = synth (id !!"X" !!"x100" !!"x110") in
let xx121, _ = synth (id !!"X" !!"x101" !!"x111") in
let x120 = assume "x120" xx120 in
let x121 = assume "x121" xx121 in
let xx122, _ =
synth
(refl (refl !!"X")
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x120"
$ !!"x121") in
let x122 = assume "x122" xx122 (* Front face *) in
let xx200, _ = synth (id !!"X" !!"x000" !!"x100") in
let x200 = assume "x200" xx200 in
let xx201, _ = synth (id !!"X" !!"x001" !!"x101") in
let x201 = assume "x201" xx201 in
let xx202, _ =
synth
(refl (refl !!"X")
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x200"
$ !!"x201") in
let x202 = assume "x202" xx202 (* Back face *) in
let xx210, _ = synth (id !!"X" !!"x010" !!"x110") in
let x210 = assume "x210" xx210 in
let xx211, _ = synth (id !!"X" !!"x011" !!"x111") in
let x211 = assume "x211" xx211 in
let xx212, _ =
synth
(refl (refl !!"X")
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x210"
$ !!"x211") in
let x212 = assume "x212" xx212 (* Left face *) in
let xx220, _ =
synth
(refl (refl !!"X")
$ !!"x000"
$ !!"x010"
$ !!"x020"
$ !!"x100"
$ !!"x110"
$ !!"x120"
$ !!"x200"
$ !!"x210") in
let x220 = assume "x220" xx220 (* Right face *) in
let xx221, _ =
synth
(refl (refl !!"X")
$ !!"x001"
$ !!"x011"
$ !!"x021"
$ !!"x101"
$ !!"x111"
$ !!"x121"
$ !!"x201"
$ !!"x211") in
let x221 = assume "x221" xx221 (* Center *) in
let xx222, _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x020"
$ !!"x021"
$ !!"x022"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x120"
$ !!"x121"
$ !!"x122"
$ !!"x200"
$ !!"x201"
$ !!"x202"
$ !!"x210"
$ !!"x211"
$ !!"x212"
$ !!"x220"
$ !!"x221") in
let x222 = assume "x222" xx222 (* The two transpositions that can act on a cube *) in
let sym_x222, sym_xx222 = synth (sym !!"x222") in
let apsym_x222, apsym_xx222 =
synth
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x020"
$ !!"x021"
$ !!"x022"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x120"
$ !!"x121"
$ !!"x122"
$ !!"x200"
$ !!"x201"
$ !!"x202"
$ !!"x210"
$ !!"x211"
$ !!"x212"
$ !!"x220"
$ !!"x221"
$ !!"x222")
(* They have different types *) in
let () = unequal sym_xx222 apsym_xx222 (* Here are their types. *) in
let sym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x010"
$ !!"x020"
$ !!"x001"
$ !!"x011"
$ !!"x021"
$ !!"x002"
$ !!"x012"
$ sym !!"x022"
$ !!"x100"
$ !!"x110"
$ !!"x120"
$ !!"x101"
$ !!"x111"
$ !!"x121"
$ !!"x102"
$ !!"x112"
$ sym !!"x122"
$ !!"x200"
$ !!"x210"
$ !!"x220"
$ !!"x201"
$ !!"x211"
$ !!"x221"
$ !!"x202"
$ !!"x212") in
let () = equal sym_xx222 sym_xx222' in
let apsym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x200"
$ !!"x201"
$ !!"x202"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x210"
$ !!"x211"
$ !!"x212"
$ !!"x020"
$ !!"x021"
$ !!"x022"
$ !!"x120"
$ !!"x121"
$ !!"x122"
$ sym !!"x220"
$ sym !!"x221") in
let () = equal apsym_xx222 apsym_xx222' (* They both have alternative degeneracy notations. *) in
let sym_x222', sym_xx222' = synth (!!"x222" $^ "213") in
let () = equal sym_xx222 sym_xx222' in
let () = equal sym_x222 sym_x222' in
let apsym_x222', apsym_xx222' = synth (!!"x222" $^ "132") in
let () = equal apsym_xx222 apsym_xx222' in
let () =
equal apsym_x222 apsym_x222'
(* But the two sides of the braid equation are equal. We build them up in stages, checking their types as we go. First the left-hand side sym-apsym-sym. *)
in
let apsym_sym_x222, apsym_sym_xx222 =
synth
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x010"
$ !!"x020"
$ !!"x001"
$ !!"x011"
$ !!"x021"
$ !!"x002"
$ !!"x012"
$ sym !!"x022"
$ !!"x100"
$ !!"x110"
$ !!"x120"
$ !!"x101"
$ !!"x111"
$ !!"x121"
$ !!"x102"
$ !!"x112"
$ sym !!"x122"
$ !!"x200"
$ !!"x210"
$ !!"x220"
$ !!"x201"
$ !!"x211"
$ !!"x221"
$ !!"x202"
$ !!"x212"
$ sym !!"x222") in
let apsym_sym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x010"
$ !!"x020"
$ !!"x100"
$ !!"x110"
$ !!"x120"
$ !!"x200"
$ !!"x210"
$ !!"x220"
$ !!"x001"
$ !!"x011"
$ !!"x021"
$ !!"x101"
$ !!"x111"
$ !!"x121"
$ !!"x201"
$ !!"x211"
$ !!"x221"
$ !!"x002"
$ !!"x012"
$ sym !!"x022"
$ !!"x102"
$ !!"x112"
$ sym !!"x122"
$ sym !!"x202"
$ sym !!"x212") in
let () = equal apsym_sym_xx222 apsym_sym_xx222' in
let sym_apsym_sym_x222, sym_apsym_sym_xx222 =
synth
(sym
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x010"
$ !!"x020"
$ !!"x001"
$ !!"x011"
$ !!"x021"
$ !!"x002"
$ !!"x012"
$ sym !!"x022"
$ !!"x100"
$ !!"x110"
$ !!"x120"
$ !!"x101"
$ !!"x111"
$ !!"x121"
$ !!"x102"
$ !!"x112"
$ sym !!"x122"
$ !!"x200"
$ !!"x210"
$ !!"x220"
$ !!"x201"
$ !!"x211"
$ !!"x221"
$ !!"x202"
$ !!"x212"
$ sym !!"x222")) in
let sym_apsym_sym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x100"
$ !!"x200"
$ !!"x010"
$ !!"x110"
$ !!"x210"
$ !!"x020"
$ !!"x120"
$ sym !!"x220"
$ !!"x001"
$ !!"x101"
$ !!"x201"
$ !!"x011"
$ !!"x111"
$ !!"x211"
$ !!"x021"
$ !!"x121"
$ sym !!"x221"
$ !!"x002"
$ !!"x102"
$ sym !!"x202"
$ !!"x012"
$ !!"x112"
$ sym !!"x212"
$ sym !!"x022"
$ sym !!"x122") in
let () =
equal sym_apsym_sym_xx222 sym_apsym_sym_xx222'
(* Now the right-hand side, apsym-sym-apsym *) in
let sym_apsym_x222, sym_apsym_xx222 =
synth
(sym
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x020"
$ !!"x021"
$ !!"x022"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x120"
$ !!"x121"
$ !!"x122"
$ !!"x200"
$ !!"x201"
$ !!"x202"
$ !!"x210"
$ !!"x211"
$ !!"x212"
$ !!"x220"
$ !!"x221"
$ !!"x222")) in
let sym_apsym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x100"
$ !!"x200"
$ !!"x001"
$ !!"x101"
$ !!"x201"
$ !!"x002"
$ !!"x102"
$ sym !!"x202"
$ !!"x010"
$ !!"x110"
$ !!"x210"
$ !!"x011"
$ !!"x111"
$ !!"x211"
$ !!"x012"
$ !!"x112"
$ sym !!"x212"
$ !!"x020"
$ !!"x120"
$ sym !!"x220"
$ !!"x021"
$ !!"x121"
$ sym !!"x221"
$ !!"x022"
$ !!"x122") in
let () = equal sym_apsym_xx222 sym_apsym_xx222' in
let apsym_sym_apsym_x222, apsym_sym_apsym_xx222 =
synth
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x100"
$ !!"x200"
$ !!"x001"
$ !!"x101"
$ !!"x201"
$ !!"x002"
$ !!"x102"
$ sym !!"x202"
$ !!"x010"
$ !!"x110"
$ !!"x210"
$ !!"x011"
$ !!"x111"
$ !!"x211"
$ !!"x012"
$ !!"x112"
$ sym !!"x212"
$ !!"x020"
$ !!"x120"
$ sym !!"x220"
$ !!"x021"
$ !!"x121"
$ sym !!"x221"
$ !!"x022"
$ !!"x122"
$ sym
(refl
("y00"
@-> "y01"
@-> "y02"
@-> "y10"
@-> "y11"
@-> "y12"
@-> "y20"
@-> "y21"
@-> "y22"
@-> sym !!"y22"
<:> ("y00", !!"X")
@=> ("y01", !!"X")
@=> ("y02", id !!"X" !!"y00" !!"y01")
@=> ("y10", !!"X")
@=> ("y11", !!"X")
@=> ("y12", id !!"X" !!"y10" !!"y11")
@=> ("y20", id !!"X" !!"y00" !!"y10")
@=> ("y21", id !!"X" !!"y01" !!"y11")
@=> ( "y22",
refl (refl !!"X")
$ !!"y00"
$ !!"y01"
$ !!"y02"
$ !!"y10"
$ !!"y11"
$ !!"y12"
$ !!"y20"
$ !!"y21" )
@=> (refl (refl !!"X")
$ !!"y00"
$ !!"y10"
$ !!"y20"
$ !!"y01"
$ !!"y11"
$ !!"y21"
$ !!"y02"
$ !!"y12"))
$ !!"x000"
$ !!"x001"
$ !!"x002"
$ !!"x010"
$ !!"x011"
$ !!"x012"
$ !!"x020"
$ !!"x021"
$ !!"x022"
$ !!"x100"
$ !!"x101"
$ !!"x102"
$ !!"x110"
$ !!"x111"
$ !!"x112"
$ !!"x120"
$ !!"x121"
$ !!"x122"
$ !!"x200"
$ !!"x201"
$ !!"x202"
$ !!"x210"
$ !!"x211"
$ !!"x212"
$ !!"x220"
$ !!"x221"
$ !!"x222")) in
let apsym_sym_apsym_xx222', _ =
synth
(refl (refl (refl !!"X"))
$ !!"x000"
$ !!"x100"
$ !!"x200"
$ !!"x010"
$ !!"x110"
$ !!"x210"
$ !!"x020"
$ !!"x120"
$ sym !!"x220"
$ !!"x001"
$ !!"x101"
$ !!"x201"
$ !!"x011"
$ !!"x111"
$ !!"x211"
$ !!"x021"
$ !!"x121"
$ sym !!"x221"
$ !!"x002"
$ !!"x102"
$ sym !!"x202"
$ !!"x012"
$ !!"x112"
$ sym !!"x212"
$ sym !!"x022"
$ sym !!"x122") in
let () =
equal apsym_sym_apsym_xx222 apsym_sym_apsym_xx222'
(* Now we check that both sides have the same types and are equal *) in
let () = equal sym_apsym_sym_xx222 apsym_sym_apsym_xx222 in
let () =
equal sym_apsym_sym_x222 apsym_sym_apsym_x222
(* And they have an alternative, much simpler, degeneracy notation *) in
let braid', braidty' = synth (!!"x222" $^ "321") in
let () = equal sym_apsym_sym_xx222 braidty' in
let () = equal sym_apsym_sym_x222 braid' in
()