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nus/cs3234/labs/week-07_isometries3.v
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(* week-07_isometries3.v *)
(* LPP 2024 - CS3234 2023-2024, Sem2 *)
(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
(* Version of 08 Mar 2024 *)
(* ********** *)
(* A formal study of isometries of the equilateral triangle, *)
(* after Chantal Keller, Damien Pous and Sylvain Chevillard. *)
(* ********** *)
Inductive Rotation : Type :=
R000 : Rotation (* 0 degrees (identity) *)
| R120 : Rotation (* 120 degrees *)
| R240 : Rotation. (* 240 degrees *)
(* ********** *)
(* Performing two rotations in a row, clockwise. *)
(* "RaR" stands for "a Rotation after a Rotation" *)
Definition RaR (r2 r1: Rotation) : Rotation :=
match r1 with
R000 => match r2 with
R000 => R000
| R120 => R120
| R240 => R240
end
| R120 => match r2 with
R000 => R120
| R120 => R240
| R240 => R000
end
| R240 => match r2 with
R000 => R240
| R120 => R000
| R240 => R120
end
end.
(* Some properties: *)
Proposition R000_is_neutral_for_RaR_on_the_left :
forall r : Rotation,
RaR R000 r = r.
Proof.
intros [ | | ].
- unfold RaR.
reflexivity.
- unfold RaR.
reflexivity.
- unfold RaR.
reflexivity.
Qed.
Proposition R000_is_neutral_for_RaR_on_the_right :
forall r : Rotation,
RaR r R000 = r.
Proof.
intros [ | | ]; unfold RaR.
reflexivity.
reflexivity.
reflexivity.
Qed.
Proposition RaR_is_commutative :
forall r1 r2 : Rotation,
RaR r2 r1 = RaR r1 r2.
Proof.
intros [ | | ] [ | | ]; unfold RaR; reflexivity.
Restart.
intros [ | | ] [ | | ]; reflexivity.
Qed.
Proposition RaR_is_associative :
forall r1 r2 r3 : Rotation,
RaR (RaR r3 r2) r1 = RaR r3 (RaR r2 r1).
Proof.
intros [ | | ] [ | | ] [ | | ]; reflexivity.
Abort.
Proposition RaR_is_nilpotent_with_order_3 :
forall r : Rotation,
RaR (RaR r r) (RaR r r) = r.
Proof.
intros [ | | ]; unfold RaR; reflexivity.
Qed.
(* Abort. *)
(* ********** *)
(* The following symmetries are indexed by the invariant vertex: *)
Inductive Reflection : Type :=
S_NN : Reflection (* North, at the top *)
| S_SW : Reflection (* South-West, at the bottom left *)
| S_SE : Reflection. (* South-East, at the bottom right *)
(* These reflections are symmetries here. *)
(* Performing two reflections in a row. *)
(* "SaS" stands for "a Symmetry after a Symmetry" *)
Definition SaS (s2 s1 : Reflection) : Rotation :=
match s1 with
S_NN => match s2 with
S_NN => R000
| S_SW => R120
| S_SE => R240
end
| S_SW => match s2 with
S_NN => R240
| S_SW => R000
| S_SE => R120
end
| S_SE => match s2 with
S_NN => R120
| S_SW => R240
| S_SE => R000
end
end.
(* is SaS commutative? *)
Proposition SaS_is_not_commutative :
exists s1 s2 : Reflection,
SaS s1 s2 <> SaS s2 s1.
Proof.
exists S_NN, S_SW.
unfold SaS.
unfold not.
intro H_absurd.
discriminate H_absurd.
Qed.
(* is SaS associative? *)
(* is SaS nilpotent? *)
(* ********** *)
(* Performing a rotation and then a reflection in a row. *)
(* "SaR" stands for "a Symmetry after a Rotation" *)
Definition SaR (s : Reflection) (r : Rotation) : Reflection :=
match r with
R000 => match s with
S_NN => S_NN
| S_SW => S_SW
| S_SE => S_SE
end
| R120 => match s with
S_NN => S_SE
| S_SW => S_NN
| S_SE => S_SW
end
| R240 => match s with
S_NN => S_SW
| S_SW => S_SE
| S_SE => S_NN
end
end.
(* ********** *)
(* Performing a reflection and then a rotation in a row. *)
(* "RaS" stands for "a Rotation after a Symmetry" *)
Definition RaS (r : Rotation) (s : Reflection) : Reflection :=
match s with
S_NN => match r with
R000 => S_NN
| R120 => S_SW
| R240 => S_SE
end
| S_SW => match r with
R000 => S_SW
| R120 => S_SE
| R240 => S_NN
end
| S_SE => match r with
R000 => S_SE
| R120 => S_NN
| R240 => S_SW
end
end.
(* ********** *)
Inductive Isomorphism : Type :=
| IR : Rotation -> Isomorphism
| IS : Reflection -> Isomorphism.
(* Identity: *)
Definition Id : Isomorphism := IR R000.
(* Composition: *)
Definition C (i2 i1 : Isomorphism) : Isomorphism :=
match i1 with
IR r1 => match i2 with
IR r2 => IR (RaR r2 r1)
| IS s2 => IS (SaR s2 r1)
end
| IS s1 => match i2 with
IR r2 => IS (RaS r2 s1)
| IS s2 => IR (SaS s2 s1)
end
end.
Proposition Id_is_neutral_on_the_left_of_C :
forall i : Isomorphism,
C Id i = i.
Proof.
intros [[ | | ] | [ | | ]]; reflexivity.
Qed.
Proposition Id_is_neutral_on_the_right_of_C :
forall i : Isomorphism,
C i Id = i.
Proof.
intros [[ | | ] | [ | | ]]; reflexivity.
Qed.
Proposition C_is_associative :
forall i1 i2 i3 : Isomorphism,
C i3 (C i2 i1) = C (C i3 i2) i1.
Proof.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]]
[[ | | ] | [ | | ]] ; reflexivity.
Abort.
Lemma composing_an_isomorphism_is_injective_on_the_right :
forall i x y : Isomorphism,
C i x = C i y -> x = y.
Proof.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]]
[[ | | ] | [ | | ]]; unfold C, RaR, SaR, RaS, SaS.
- intro H.
reflexivity.
- intro H_absurd.
discriminate H_absurd.
Restart.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]]
[[ | | ] | [ | | ]];
intro H; (reflexivity || discriminate H).
Qed.
Lemma composing_an_isomorphism_is_injective_on_the_left :
forall i x y : Isomorphism,
C x i = C y i -> x = y.
Proof.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]]
[[ | | ] | [ | | ]];
intro H; (reflexivity || discriminate H).
Qed.
Lemma composing_an_isomorphism_is_surjective_on_the_right :
forall i2 i3 : Isomorphism,
exists i1 : Isomorphism,
C i2 i1 = i3.
Proof.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]].
- exists Id.
reflexivity.
- exists (IR R120).
reflexivity.
- exists (IR R240).
reflexivity.
Restart.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]];
(
(exists (IR R000); reflexivity )
||
(exists (IR R120); reflexivity )
||
(exists (IR R240); reflexivity )
||
(exists (IS S_NN); reflexivity )
||
(exists (IS S_SW); reflexivity )
||
(exists (IS S_SE); reflexivity )
).
Show Proof.
Qed.
Lemma composing_an_isomorphism_is_surjective_on_the_left :
forall i1 i3 : Isomorphism,
exists i2 : Isomorphism,
C i2 i1 = i3.
Proof.
intros [[ | | ] | [ | | ]]
[[ | | ] | [ | | ]];
(
(exists (IR R000); reflexivity )
||
(exists (IR R120); reflexivity )
||
(exists (IR R240); reflexivity )
||
(exists (IS S_NN); reflexivity )
||
(exists (IS S_SW); reflexivity )
||
(exists (IS S_SE); reflexivity )
).
Qed.
Proposition C_over_rotations_is_nilpotent_with_order_3 :
forall r : Rotation,
C (C (IR r) (IR r)) (IR r) = Id.
Proof.
Abort.
Proposition C_over_symmetries_is_nilpotent_with_order_2 :
forall s : Reflection,
C (IS s) (IS s) = Id.
Proof.
intros [ | | ]; unfold C, SaS, Id; reflexivity.
Qed.
(* Proposition C_is_nilpotent_with_order_??? : *)
(* forall i : Isomorphism, *)
(* ... *)
(* Proof. *)
(* Abort. *)
(* ********** *)
(* Let us now introduce the vertices: *)
Inductive Vertex : Type := (* enumerated clockwise *)
NN : Vertex
| SW : Vertex
| SE : Vertex.
(* And let us define the effect of applying an isomorphism
to a vertex: *)
Definition A (i : Isomorphism) (v : Vertex) : Vertex :=
match i with
IR r => match r with
R000 => match v with
NN => NN
| SW => SW
| SE => SE
end
| R120 => match v with
NN => SW
| SW => SE
| SE => NN
end
| R240 => match v with
NN => SE
| SE => SW
| SW => NN
end
end
| IS s => match s with
S_NN => match v with
NN => NN
| SW => SE
| SE => SW
end
| S_SE => match v with
NN => SW
| SW => NN
| SE => SE
end
| S_SW => match v with
NN => SE
| SW => SW
| SE => NN
end
end
end.
(*
Proposition A_is_equivalent_to_C :
forall (i1 i2 : Isomorphism) (v : Vertex),
A i2 (A i1 v) = A (C i2 i1) v.
Proof.
Abort.
Proposition applying_an_isomorphism_is_injective :
forall (i : Isomorphism) (v1 v2 : Vertex),
(A i v1 = A i v2) -> v1 = v2.
Proof.
Abort.
Proposition applying_an_isomorphism_is_surjective :
forall (i : Isomorphism) (v2 : Vertex),
exists v1 : Vertex,
A i v1 = v2.
Proof.
Abort.
*)
(* ********** *)
(* Intensional equality:
two isomorphisms are equal
iff
they are are constructed alike.
*)
Definition intensional_equality (i1 i2: Isomorphism) : Prop :=
i1 = i2.
(* Extensional equality:
two isomorphisms are equal
iff
their graphs are the same.
*)
Definition extensional_equality (i1 i2: Isomorphism) : Prop :=
forall v : Vertex,
A i1 v = A i2 v.
(* The two notions of equalities coincide: *)
Proposition the_two_equalities_are_the_same :
forall i1 i2 : Isomorphism,
intensional_equality i1 i2 <-> extensional_equality i1 i2.
Proof.
Abort.
(* ********** *)
(*
Lemma isomorphism_equality_in_context_on_the_left :
forall x y i : Isomorphism,
x = y -> C x i = C y i.
Proof.
Abort.
Proposition take_five :
forall i : Isomorphism,
extensional_equality (C (C (C (C i i) i) i) i) Id
->
i = Id.
Proof.
Abort.
Proposition characteristic_property_of_Id :
forall i : Isomorphism,
i = Id <-> forall v : Vertex, A i v = v.
Proof.
Abort.
Proposition one_for_the_road :
forall i : Isomorphism,
(forall v : Vertex, A i v = v)
->
C i i = Id.
Proof.
Abort.
Proposition notable_property_of_Id :
exists i : Isomorphism,
exists v : Vertex,
not (A i v = v -> i = Id).
Proof.
Abort.
Proposition notable_property_of_Id' :
not (forall (i : Isomorphism) (v : Vertex),
A i v = v -> i = Id).
Proof.
Abort.
Proposition notable_property_of_symmetries :
forall (i : Isomorphism)
(v : Vertex),
A i v = v ->
((i = IR R0)
\/
(exists s : Reflection,
i = IS s)).
Proof.
Abort.
Proposition notable_property_of_symmetries' :
forall i : Isomorphism,
(exists v : Vertex,
A i v = v) ->
((i = IR R0)
\/
(exists s : Reflection,
i = IS s)).
Proof.
Abort.
Proposition one_more_for_the_road :
forall (i : Isomorphism) (v : Vertex),
A i v = v -> C i i = Id.
Proof.
Abort.
*)
(* ********** *)
(* end of week-07_isometries3.v *)
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