98 lines
3.2 KiB
Text
98 lines
3.2 KiB
Text
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/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: hit.circle
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Authors: Floris van Doorn
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Declaration of the circle
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-/
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import .sphere
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open eq suspension bool sphere_index
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definition circle [reducible] := suspension bool --redefine this as sphere 1
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namespace circle
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definition base1 : circle := !north
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definition base2 : circle := !south
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definition seg1 : base1 = base2 := merid tt
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definition seg2 : base2 = base1 := (merid ff)⁻¹
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definition base : circle := base1
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definition loop : base = base := seg1 ⬝ seg2
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definition rec2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
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(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) (x : circle) : P x :=
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begin
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fapply (suspension.rec_on x),
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{ exact Pb1},
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{ exact Pb2},
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{ intro b, cases b,
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apply tr_eq_of_eq_inv_tr, exact Ps2⁻¹,
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exact Ps1},
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end
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definition rec2_on [reducible] {P : circle → Type} (x : circle) (Pb1 : P base1) (Pb2 : P base2)
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(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1) : P x :=
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circle.rec2 Pb1 Pb2 Ps1 Ps2 x
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definition rec2_seg1 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
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(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
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: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
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sorry
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definition rec2_seg2 {P : circle → Type} (Pb1 : P base1) (Pb2 : P base2)
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(Ps1 : seg1 ▹ Pb1 = Pb2) (Ps2 : seg2 ▹ Pb2 = Pb1)
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: apD (rec2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
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sorry
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definition elim2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) (x : circle) : P :=
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rec2 Pb1 Pb2 (!tr_constant ⬝ Ps1) (!tr_constant ⬝ Ps2) x
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definition elim2_on [reducible] {P : Type} (x : circle) (Pb1 Pb2 : P)
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(Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1) : P :=
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elim2 Pb1 Pb2 Ps1 Ps2 x
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definition elim2_seg1 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
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: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg1 = sorry ⬝ Ps1 ⬝ sorry :=
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sorry
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definition elim2_seg2 {P : Type} (Pb1 Pb2 : P) (Ps1 : Pb1 = Pb2) (Ps2 : Pb2 = Pb1)
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: ap (elim2 Pb1 Pb2 Ps1 Ps2) seg2 = sorry ⬝ Ps2 ⬝ sorry :=
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sorry
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protected definition rec {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase)
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(x : circle) : P x :=
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begin
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fapply (rec2_on x),
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{ exact Pbase},
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{ exact (transport P seg1 Pbase)},
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{ apply idp},
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{ apply eq_tr_of_inv_tr_eq, rewrite -tr_con, apply Ploop},
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end
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protected definition rec_on [reducible] {P : circle → Type} (x : circle) (Pbase : P base)
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(Ploop : loop ▹ Pbase = Pbase) : P x :=
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circle.rec Pbase Ploop x
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protected definition elim {P : Type} (Pbase : P) (Ploop : Pbase = Pbase)
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(x : circle) : P :=
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circle.rec Pbase (tr_constant loop Pbase ⬝ Ploop) x
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protected definition elim_on [reducible] {P : Type} (x : circle) (Pbase : P)
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(Ploop : Pbase = Pbase) : P :=
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elim Pbase Ploop x
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definition rec_loop {P : circle → Type} (Pbase : P base) (Ploop : loop ▹ Pbase = Pbase) :
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ap (circle.rec Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
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sorry
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definition elim_loop {P : Type} (Pbase : P) (Ploop : Pbase = Pbase) :
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ap (circle.elim Pbase Ploop) loop = sorry ⬝ Ploop ⬝ sorry :=
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sorry
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end circle
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