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Cleanup the Pauli group example

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Dan Frumin
2017-09-24 17:55:19 +02:00
parent d5e08c43bf
commit 206bf9edb6
1 changed files with 9 additions and 17 deletions
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26
FiniteSets/misc/dec_fset.v
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@@ -1,5 +1,5 @@
Require Import HoTT HitTactics. Require Import HoTT HitTactics.
Require Export FSets lattice_examples. Require Export FSets lattice_examples k_finite.
Section quantifiers. Section quantifiers.
Context {A : Type} `{Univalence}. Context {A : Type} `{Univalence}.
@@ -132,9 +132,8 @@ Section simple_example.
Defined. Defined.
End simple_example. End simple_example.
Require Import k_finite.
Section pauli. Section pauli.
Context `{Univalence}.
Inductive Pauli : Type := Inductive Pauli : Type :=
| XP : Pauli | XP : Pauli
@@ -193,32 +192,25 @@ Section pauli.
apply _. apply _.
Defined. Defined.
Context `{Univalence}.
Definition Pauli_list : FSet Pauli := {|XP|} ∪ {|YP|} ∪ {|ZP|} ∪ {|IP|}. Definition Pauli_list : FSet Pauli := {|XP|} ∪ {|YP|} ∪ {|ZP|} ∪ {|IP|}.
Ltac solve_in_list :=
apply tr; try apply idpath;
first [ apply inl ; solve_in_list | apply inr ; solve_in_list ].
Theorem Pauli_finite : Kf Pauli. Theorem Pauli_finite : Kf Pauli.
Proof. Proof.
unfold Kf, Kf_sub, Kf_sub_intern. apply Kf_unfold.
exists Pauli_list. exists Pauli_list.
apply path_forall.
unfold map. unfold map.
intros [ | | | ] ; rewrite ?union_isIn ; apply path_iff_hprop ; try constructor ; intros []. intros [ | | | ]; rewrite !union_isIn; solve_in_list.
- apply (tr(inl(tr idpath))).
- apply (tr(inr(tr(inl(tr idpath))))).
- apply (tr(inr(tr(inr(tr(inl(tr idpath))))))).
- refine (tr(inr(tr(inr(tr(inr(tr idpath))))))).
Defined. Defined.
Theorem Pauli_all (P : Pauli -> hProp) : all P Pauli_list -> forall (x : Pauli), P x. Theorem Pauli_all (P : Pauli -> hProp) : all P Pauli_list -> forall (x : Pauli), P x.
Proof. Proof.
intros HP x. intros HP x.
refine (all_elim P Pauli_list x HP _). refine (all_elim P Pauli_list x HP _).
destruct x ; rewrite ?union_isIn ; try constructor. destruct x ; rewrite ?union_isIn; solve_in_list.
- apply (tr(inl(tr idpath))).
- apply (tr(inr(tr(inl(tr idpath))))).
- apply (tr(inr(tr(inr(tr(inl(tr idpath))))))).
- refine (tr(inr(tr(inr(tr(inr(tr idpath))))))).
Defined. Defined.
Definition comm x y : hProp := BuildhProp(Pauli_mult x y = Pauli_mult y x). Definition comm x y : hProp := BuildhProp(Pauli_mult x y = Pauli_mult y x).