Library LinAlg.support.cast_between_subsets
Set Implicit Arguments.
Unset Strict Implicit.
Require Export Map_embed.
Require Export algebra_omissions.
Unset Strict Implicit.
Require Export Map_embed.
Require Export algebra_omissions.
- Various casting functions between equal but non-convertible types
- This one maps elements of a subset to the same subset under a different name
Definition map_between_equal_subsets :
∀ (A : Setoid) (X Y : part_set A), X =' Y in _ → X → Y.
intros A X Y H x.
inversion_clear x.
apply (Build_subtype (E:=A) (P:=Y) (subtype_elt:=subtype_elt)).
simpl in H.
red in H.
generalize (H subtype_elt).
intro H'.
inversion_clear H'.
apply H0; auto with algebra.
Defined.
Lemma subtype_elt_eats_map_between_equal_subsets :
∀ (A : Setoid) (X Y : part_set A) (H : X =' Y in _) (x : X),
subtype_elt (map_between_equal_subsets H x) =' subtype_elt x in _.
intros.
elim x.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_map_between_equal_subsets: algebra.
Lemma map_between_equal_subsets_inj :
∀ (A : Setoid) (X Y : part_set A) (H H' : X =' Y in _) (x x' : X),
map_between_equal_subsets H x =' map_between_equal_subsets H' x' in _ →
x =' x' in _.
simpl in |- ×.
unfold subtype_image_equal in |- *; simpl in |- ×.
intros.
apply Trans with (subtype_elt (map_between_equal_subsets H x));
auto with algebra.
apply Trans with (subtype_elt (map_between_equal_subsets H' x'));
auto with algebra.
Qed.
∀ (A : Setoid) (X Y : part_set A), X =' Y in _ → X → Y.
intros A X Y H x.
inversion_clear x.
apply (Build_subtype (E:=A) (P:=Y) (subtype_elt:=subtype_elt)).
simpl in H.
red in H.
generalize (H subtype_elt).
intro H'.
inversion_clear H'.
apply H0; auto with algebra.
Defined.
Lemma subtype_elt_eats_map_between_equal_subsets :
∀ (A : Setoid) (X Y : part_set A) (H : X =' Y in _) (x : X),
subtype_elt (map_between_equal_subsets H x) =' subtype_elt x in _.
intros.
elim x.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_map_between_equal_subsets: algebra.
Lemma map_between_equal_subsets_inj :
∀ (A : Setoid) (X Y : part_set A) (H H' : X =' Y in _) (x x' : X),
map_between_equal_subsets H x =' map_between_equal_subsets H' x' in _ →
x =' x' in _.
simpl in |- ×.
unfold subtype_image_equal in |- *; simpl in |- ×.
intros.
apply Trans with (subtype_elt (map_between_equal_subsets H x));
auto with algebra.
apply Trans with (subtype_elt (map_between_equal_subsets H' x'));
auto with algebra.
Qed.
- This one turns into whenever
Definition Map_to_equal_subsets :
∀ (A B : Setoid) (X Y : part_set A), X =' Y in _ → MAP B X → MAP B Y.
intros.
apply (Build_Map (Ap:=fun b : B ⇒ map_between_equal_subsets H (X0 b))).
red in |- *; simpl in |- *; red in |- ×.
intros.
unfold map_between_equal_subsets in |- ×.
generalize (Map_compatible_prf X0 H0).
case (X0 x); case (X0 y).
simpl in |- ×.
auto.
Defined.
Lemma subtype_elt_eats_Map_to_equal_subsets :
∀ (A B : Setoid) (X Y : part_set A) (H : X =' Y in _)
(b : B) (M : Map B X),
subtype_elt (Map_to_equal_subsets H M b) =' subtype_elt (M b) in _.
intros.
simpl in |- ×.
case (M b).
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_Map_to_equal_subsets: algebra.
Lemma Map_embed_eats_Map_to_equal_subsets :
∀ (A B : Setoid) (X Y : part_set A) (H : X =' Y in _) (M : Map B X),
Map_embed (Map_to_equal_subsets H M) =' Map_embed M in _.
intros.
simpl in |- ×.
red in |- ×.
intros b.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_Map_to_equal_subsets: algebra.
Lemma Map_to_equal_subsets_inj :
∀ (A B : Setoid) (X Y : part_set A) (H H' : X =' Y in _)
(f g : Map B X),
Map_to_equal_subsets H f =' Map_to_equal_subsets H' g in _ →
f =' g in MAP _ _.
unfold Map_to_equal_subsets in |- *; simpl in |- *; unfold Map_eq in |- *;
simpl in |- *; unfold subtype_image_equal in |- *;
simpl in |- ×.
intros.
apply subtype_elt_comp.
apply map_between_equal_subsets_inj with Y H H'; auto with algebra.
simpl in |- *; red in |- *; simpl in |- ×.
auto.
Qed.
∀ (A B : Setoid) (X Y : part_set A), X =' Y in _ → MAP B X → MAP B Y.
intros.
apply (Build_Map (Ap:=fun b : B ⇒ map_between_equal_subsets H (X0 b))).
red in |- *; simpl in |- *; red in |- ×.
intros.
unfold map_between_equal_subsets in |- ×.
generalize (Map_compatible_prf X0 H0).
case (X0 x); case (X0 y).
simpl in |- ×.
auto.
Defined.
Lemma subtype_elt_eats_Map_to_equal_subsets :
∀ (A B : Setoid) (X Y : part_set A) (H : X =' Y in _)
(b : B) (M : Map B X),
subtype_elt (Map_to_equal_subsets H M b) =' subtype_elt (M b) in _.
intros.
simpl in |- ×.
case (M b).
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_Map_to_equal_subsets: algebra.
Lemma Map_embed_eats_Map_to_equal_subsets :
∀ (A B : Setoid) (X Y : part_set A) (H : X =' Y in _) (M : Map B X),
Map_embed (Map_to_equal_subsets H M) =' Map_embed M in _.
intros.
simpl in |- ×.
red in |- ×.
intros b.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_Map_to_equal_subsets: algebra.
Lemma Map_to_equal_subsets_inj :
∀ (A B : Setoid) (X Y : part_set A) (H H' : X =' Y in _)
(f g : Map B X),
Map_to_equal_subsets H f =' Map_to_equal_subsets H' g in _ →
f =' g in MAP _ _.
unfold Map_to_equal_subsets in |- *; simpl in |- *; unfold Map_eq in |- *;
simpl in |- *; unfold subtype_image_equal in |- *;
simpl in |- ×.
intros.
apply subtype_elt_comp.
apply map_between_equal_subsets_inj with Y H H'; auto with algebra.
simpl in |- *; red in |- *; simpl in |- ×.
auto.
Qed.
- if for then can be seen as . This is done by cast_map_to_subset.
Definition cast_to_subset_fun :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A),
(∀ i : B, in_part (v i) W) → (B → W:Type).
intros A B v.
elim v.
intros vseq vprf; intros.
generalize X; clear X.
simpl in |- ×.
exact (fun i : B ⇒ Build_subtype (H i)).
Defined.
Lemma cast_doesn't_change :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W) (i : B),
subtype_elt (cast_to_subset_fun H i) =' v i in _.
intros A B v.
elim v.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve cast_doesn't_change: algebra.
Definition cast_map_to_subset :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A),
(∀ i : B, in_part (v i) W) → MAP B W.
intros.
cut (fun_compatible (cast_to_subset_fun H)).
intro.
exact (Build_Map H0).
red in |- ×.
simpl in |- ×.
red in |- ×.
intros.
apply Trans with (v x); auto with algebra.
apply Trans with (v y); auto with algebra.
Defined.
Lemma cast_map_to_subset_doesn't_change :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W) (i : B),
subtype_elt (cast_map_to_subset H i) =' v i in _.
intros.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve cast_map_to_subset_doesn't_change: algebra.
Lemma Map_embed_cast_map_to_subset_inv :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W),
Map_embed (cast_map_to_subset H) =' v in _.
intros.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve Map_embed_cast_map_to_subset_inv: algebra.
Lemma Map_embed_eats_cast_map_to_subset :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C),
Map_embed (cast_map_to_subset H) =' Map_embed v in _.
intros.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_cast_map_to_subset: algebra.
Lemma seq_castable :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A),
(∀ i : B, in_part (v i) W) → ∃ w : MAP B W, Map_embed w =' v in _.
intros.
∃ (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve seq_castable: algebra.
Lemma subset_seq_castable :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C),
∃ w : MAP D C, Map_embed w =' Map_embed v in _.
intros.
∃ (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve subset_seq_castable: algebra.
Lemma cast_seq_nice :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W) (P : Predicate (MAP B A)),
Pred_fun P v → Pred_fun P (Map_embed (cast_map_to_subset H)).
destruct P.
intros.
red in Pred_compatible_prf.
simpl in |- ×.
simpl in H0.
apply Pred_compatible_prf with v; auto with algebra.
Qed.
Hint Resolve cast_seq_nice: algebra.
Lemma cast_subset_seq_nice :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C) (P : Predicate (MAP D A)),
Pred_fun P (Map_embed v) → Pred_fun P (Map_embed (cast_map_to_subset H)).
intros.
auto with algebra.
Qed.
Hint Resolve cast_subset_seq_nice: algebra.
Lemma cast_respects_predicates_per_elt :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(P : Predicate A) (H : ∀ i : D, in_part (Map_embed v i) C)
(i : D),
Pred_fun P (Map_embed v i) →
Pred_fun P (Map_embed (cast_map_to_subset H) i).
intros.
generalize H0; clear H0; elim P.
intros Pf pc H0.
simpl in |- ×.
simpl in H0.
auto.
Qed.
Lemma cast_respects_all_elt_predicates :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(P : Predicate A) (H : ∀ i : D, in_part (Map_embed v i) C),
(∀ i : D, Pred_fun P (Map_embed v i)) →
∀ j : D, Pred_fun P (Map_embed (cast_map_to_subset H) j).
intros.
generalize H0; clear H0; elim P.
intros Pf pc H0.
simpl in |- ×.
simpl in H0.
auto.
Qed.
Hint Resolve cast_respects_predicates_per_elt
cast_respects_all_elt_predicates: algebra.
∀ (A B : Setoid) (v : MAP B A) (W : part_set A),
(∀ i : B, in_part (v i) W) → MAP B W.
intros.
cut (fun_compatible (cast_to_subset_fun H)).
intro.
exact (Build_Map H0).
red in |- ×.
simpl in |- ×.
red in |- ×.
intros.
apply Trans with (v x); auto with algebra.
apply Trans with (v y); auto with algebra.
Defined.
Lemma cast_map_to_subset_doesn't_change :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W) (i : B),
subtype_elt (cast_map_to_subset H i) =' v i in _.
intros.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve cast_map_to_subset_doesn't_change: algebra.
Lemma Map_embed_cast_map_to_subset_inv :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W),
Map_embed (cast_map_to_subset H) =' v in _.
intros.
simpl in |- ×.
red in |- ×.
simpl in |- ×.
auto with algebra.
Qed.
Hint Resolve Map_embed_cast_map_to_subset_inv: algebra.
Lemma Map_embed_eats_cast_map_to_subset :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C),
Map_embed (cast_map_to_subset H) =' Map_embed v in _.
intros.
auto with algebra.
Qed.
Hint Resolve Map_embed_eats_cast_map_to_subset: algebra.
Lemma seq_castable :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A),
(∀ i : B, in_part (v i) W) → ∃ w : MAP B W, Map_embed w =' v in _.
intros.
∃ (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve seq_castable: algebra.
Lemma subset_seq_castable :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C),
∃ w : MAP D C, Map_embed w =' Map_embed v in _.
intros.
∃ (cast_map_to_subset H).
auto with algebra.
Qed.
Hint Resolve subset_seq_castable: algebra.
Lemma cast_seq_nice :
∀ (A B : Setoid) (v : MAP B A) (W : part_set A)
(H : ∀ i : B, in_part (v i) W) (P : Predicate (MAP B A)),
Pred_fun P v → Pred_fun P (Map_embed (cast_map_to_subset H)).
destruct P.
intros.
red in Pred_compatible_prf.
simpl in |- ×.
simpl in H0.
apply Pred_compatible_prf with v; auto with algebra.
Qed.
Hint Resolve cast_seq_nice: algebra.
Lemma cast_subset_seq_nice :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : ∀ i : D, in_part (Map_embed v i) C) (P : Predicate (MAP D A)),
Pred_fun P (Map_embed v) → Pred_fun P (Map_embed (cast_map_to_subset H)).
intros.
auto with algebra.
Qed.
Hint Resolve cast_subset_seq_nice: algebra.
Lemma cast_respects_predicates_per_elt :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(P : Predicate A) (H : ∀ i : D, in_part (Map_embed v i) C)
(i : D),
Pred_fun P (Map_embed v i) →
Pred_fun P (Map_embed (cast_map_to_subset H) i).
intros.
generalize H0; clear H0; elim P.
intros Pf pc H0.
simpl in |- ×.
simpl in H0.
auto.
Qed.
Lemma cast_respects_all_elt_predicates :
∀ (A D : Setoid) (B C : part_set A) (v : MAP D B)
(P : Predicate A) (H : ∀ i : D, in_part (Map_embed v i) C),
(∀ i : D, Pred_fun P (Map_embed v i)) →
∀ j : D, Pred_fun P (Map_embed (cast_map_to_subset H) j).
intros.
generalize H0; clear H0; elim P.
intros Pf pc H0.
simpl in |- ×.
simpl in H0.
auto.
Qed.
Hint Resolve cast_respects_predicates_per_elt
cast_respects_all_elt_predicates: algebra.
- Similarly, if are subsets of , then is also .
Definition Map_include :
∀ (A D : Setoid) (B C : part_set A),
included B C → MAP D B → MAP D C.
intros.
apply (cast_map_to_subset (v:=Map_embed X)).
red in H.
intro.
apply H; auto with algebra.
apply Map_embed_prop; auto with algebra.
Defined.
Definition Map_include_map :
∀ (A D : Setoid) (B C : part_set A),
included B C → MAP (MAP D B) (MAP D C).
intros.
simpl in |- ×.
apply Build_Map with (Map_include (D:=D) H).
red in |- ×.
intuition.
Defined.