mirror of https://github.com/nmvdw/HITs-Examples
Merge remote-tracking branch 'origin/leon' into properties
This commit is contained in:
Niels
commit
2273ecaae0
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@ -3,3 +3,5 @@
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definition.v
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definition.v
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operations.v
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operations.v
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properties.v
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properties.v
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empty_set.v
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ordered.v
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@ -190,3 +190,7 @@ Instance FSet_recursion A : HitRecursion (FSet A) := {
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H_inductor := FSet_ind A; H_recursor := FSet_rec A }.
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H_inductor := FSet_ind A; H_recursor := FSet_rec A }.
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End FSet.
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End FSet.
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Notation "{| x |}" := (L x).
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Infix "∪" := U (at level 8, right associativity).
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Notation "∅" := E.
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@ -0,0 +1,227 @@
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Require Import HoTT.
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Require Import HitTactics.
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Require Import definition.
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Require Import operations.
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Require Import properties.
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Ltac destruct_match := repeat
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match goal with
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| [|- match ?X with | _ => _ end ] => destruct X
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end.
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Ltac destruct_match_1 :=
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repeat match goal with
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| [|- match ?X with | _ => _ end ] => destruct X
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| [|- ?X = ?Y ] => apply path_ishprop
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| [ H: ?x <> E |- Empty ] => destruct H
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| [ H1: ?x = E, H2: ?y = E, H3: ?w ∪ ?q = E |- ?r = E]
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=> rewrite H1, H2 in H3; rewrite nl in H3; rewrite nl in H3
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end.
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Ltac eq_neq_tac :=
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match goal with
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| [ H: ?x <> E, H': ?x = E |- _ ] => destruct H; assumption
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end.
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Section EmptySetProperties.
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Context {A : Type}.
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Context {A_deceq : DecidablePaths A}.
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(*Should go to properties *)
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Lemma union_subset `{Funext} :
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forall x y z: FSet A, x ∪ y ⊆ z = true -> x ⊆ z = true /\ y ⊆ z = true.
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intros x y z Hu.
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split.
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- eapply subset_isIn. intros a Ha.
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eapply subset_isIn in Hu.
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+ instantiate (1 := a) in Hu.
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assumption.
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+ transitivity (a ∈ x || a ∈ y)%Bool .
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apply union_isIn.
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rewrite Ha. cbn; reflexivity.
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- eapply subset_isIn. intros a Ha.
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eapply subset_isIn in Hu.
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+ instantiate (1 := a) in Hu.
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assumption.
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+ rewrite comm. transitivity (a ∈ y || a ∈ x)%Bool .
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apply union_isIn.
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rewrite Ha. cbn. reflexivity.
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Defined.
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Lemma eset_subset_l `{Funext} : forall x: FSet A, x ⊆ ∅ = true -> x = ∅.
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intros x He.
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apply eq_subset.
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split.
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- cbn; reflexivity.
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- assumption.
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Defined.
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Lemma eset_subset_r `{Funext} : forall x: FSet A, x = ∅ -> x ⊆ ∅ = true.
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intros x He.
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apply eq_subset. apply symmetry. assumption.
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Defined.
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Lemma subset_transitive `{Funext}:
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forall x y z, x ⊆ y = true -> y ⊆ z = true -> x ⊆ z = true.
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intros.
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Admitted.
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Lemma eset_union_l `{Funext} : forall x y: FSet A, x ∪ y = ∅ -> x = ∅ .
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Proof.
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intros.
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assert (x ⊆ (x ∪ y) = true).
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apply subset_union_equiv. rewrite assoc. rewrite (union_idem x). reflexivity.
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apply eset_subset_r in X.
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assert (x ⊆ ∅ = true).
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apply (subset_transitive x (U x y)); assumption.
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apply eset_subset_l.
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assumption.
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Defined.
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Lemma eset_union_lr `{Funext} :
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forall x y: FSet A, x ∪ y = ∅ -> ((x = ∅) /\ (y = ∅)).
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Proof.
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intros.
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split.
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apply eset_union_l in X; assumption.
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rewrite comm in X.
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apply eset_union_l in X. assumption.
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Defined.
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Lemma non_empty_union_l `{Funext} :
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forall x y: FSet A, x <> E -> x ∪ y <> E.
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intros x y He.
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intro Hu.
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apply He.
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apply eq_subset in Hu.
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destruct Hu as [_ H1].
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apply union_subset in H1.
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apply eset_subset_l.
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destruct H1.
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assumption.
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Defined.
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Lemma non_empty_union_r `{Funext} :
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forall x y: FSet A, y <> E -> x ∪ y <> E.
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intros x y He.
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intro Hu.
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apply He.
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apply eq_subset in Hu.
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destruct Hu as [_ H1].
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apply union_subset in H1.
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apply eset_subset_l.
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destruct H1.
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assumption.
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Defined.
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Theorem contrapositive : forall P Q : Type,
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(P -> Q) -> (not Q -> not P) .
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Proof.
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intros p q H1 H2.
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unfold "~".
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intro H3.
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apply H1 in H3. apply H2 in H3. assumption.
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Defined.
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Lemma non_empty_singleton : forall a: A, L a <> E.
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intros a H.
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enough (false = true).
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contradiction (false_ne_true X).
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transitivity (isIn a E).
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cbn. reflexivity.
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transitivity (a ∈ (L a)).
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apply (ap (fun x => a ∈ x) H^) .
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cbn. destruct (dec (a = a)).
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reflexivity.
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destruct n.
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reflexivity.
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Defined.
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(* Lemma aux `{Funext}: forall x: FSet A, forall p q: x = ∅ -> False, p = q.
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intros. apply path_forall. intro.
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apply path_ishprop.
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Defined.*)
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Lemma fset_eset_dec `{Funext}: forall x: FSet A, x = ∅ \/ x <> ∅.
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hinduction.
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- left; reflexivity.
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- right. apply non_empty_singleton.
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- intros x y [G1 | G2] [G3 | G4].
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+ left. rewrite G1, G3. apply nl.
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+ right. apply non_empty_union_r; assumption.
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+ right. apply non_empty_union_l; assumption.
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+ right. apply non_empty_union_l; assumption.
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- intros. destruct px, py, pz; apply path_sum; destruct_match_1.
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+ rewrite p, p0, p1. rewrite nl. rewrite nl. reflexivity.
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+ assumption.
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+ rewrite p, p0 in p1. rewrite nl in p1. rewrite comm in p1. rewrite nl in p1.
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assumption.
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+ rewrite p in p0. rewrite nl in p0.
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apply (non_empty_union_l y z) in n. eq_neq_tac.
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+ rewrite p, p0 in p1. rewrite nr in p1. rewrite nr in p1. assumption.
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+ rewrite p in p0. rewrite nr in p0.
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apply (non_empty_union_l x z) in n. eq_neq_tac.
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+ rewrite p in p0. rewrite nr in p0.
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apply (non_empty_union_l x y) in n. eq_neq_tac.
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+ apply (non_empty_union_l x y) in n.
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apply (non_empty_union_l (x ∪ y) z) in n. eq_neq_tac.
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- intros. destruct px, py; apply path_sum; destruct_match_1.
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+ rewrite p, p0; apply union_idem.
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+ rewrite p in p0. rewrite nr in p0. assumption.
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+ rewrite p in p0. rewrite nl in p0. assumption.
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+ apply (non_empty_union_r y x) in n. eq_neq_tac.
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- intros x px.
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destruct px. apply path_sum; destruct_match_1.
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+ assumption.
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+ apply path_sum; destruct_match_1. assumption.
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- intros x px.
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destruct px. apply path_sum; destruct_match_1.
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+ assumption.
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+ apply path_sum; destruct_match_1. assumption.
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- intros. cbn. apply path_sum. destruct_match_1.
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+ apply (non_empty_singleton x). apply p.
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Defined.
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Lemma union_non_empty `{Funext}:
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forall X1 X2: FSet A, U X1 X2 <> ∅ -> X1 <> ∅ \/ X2 <> ∅.
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intros X1 X2 G.
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specialize (fset_eset_dec X1).
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intro. destruct X. rewrite p in G. rewrite nl in G.
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right. assumption. left. apply n.
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|
Defined.
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|
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Lemma union_non_empty' `{Funext}:
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forall X1 X2: FSet A, U X1 X2 <> ∅ ->
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(X1 <> ∅ /\ X2 = ∅) \/
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(X1 = ∅ /\ X2 <> ∅) \/
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|
(X1 <> ∅ /\ X2 <> ∅ ).
|
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|
intros X1 X2 G.
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|
specialize (fset_eset_dec X1).
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|
specialize (fset_eset_dec X2).
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|
intros H1 H2.
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|
destruct H1, H2.
|
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|
- rewrite p, p0 in G. destruct G. apply union_idem.
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|
- left; split; assumption.
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|
- right. left. split; assumption.
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||||||
|
- right. right. split; assumption.
|
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|
Defined.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
|
End EmptySetProperties.
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|
|
@ -19,7 +19,6 @@ hrecursion.
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- intros a'. compute. destruct (A_deceq a a'); reflexivity.
|
- intros a'. compute. destruct (A_deceq a a'); reflexivity.
|
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Defined.
|
Defined.
|
||||||
|
|
||||||
Infix "∈" := isIn (at level 9, right associativity).
|
|
||||||
|
|
||||||
Definition comprehension :
|
Definition comprehension :
|
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(A -> Bool) -> FSet A -> FSet A.
|
(A -> Bool) -> FSet A -> FSet A.
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|
|
@ -54,15 +53,16 @@ Proof.
|
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intros X Y.
|
intros X Y.
|
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hrecursion X.
|
hrecursion X.
|
||||||
- exact true.
|
- exact true.
|
||||||
- exact (fun a => (a ∈ Y)).
|
- exact (fun a => (isIn a Y)).
|
||||||
- exact andb.
|
- exact andb.
|
||||||
- intros. compute. destruct x; reflexivity.
|
- intros. compute. destruct x; reflexivity.
|
||||||
- intros x y; compute; destruct x, y; reflexivity.
|
- intros x y; compute; destruct x, y; reflexivity.
|
||||||
- intros x; compute; destruct x; reflexivity.
|
- intros x; compute; destruct x; reflexivity.
|
||||||
- intros x; compute; destruct x; reflexivity.
|
- intros x; compute; destruct x; reflexivity.
|
||||||
- intros x; cbn; destruct (x ∈ Y); reflexivity.
|
- intros x; cbn; destruct (isIn x Y); reflexivity.
|
||||||
Defined.
|
Defined.
|
||||||
|
|
||||||
Notation "⊆" := subset.
|
|
||||||
|
|
||||||
End operations.
|
End operations.
|
||||||
|
|
||||||
|
Infix "∈" := isIn (at level 9, right associativity).
|
||||||
|
Infix "⊆" := subset (at level 10, right associativity).
|
||||||
|
|
@ -0,0 +1,574 @@
|
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|
Require Import HoTT.
|
||||||
|
Require Import HitTactics.
|
||||||
|
Require Import definition.
|
||||||
|
Require Import operations.
|
||||||
|
Require Import properties.
|
||||||
|
Require Import empty_set.
|
||||||
|
Class Antisymmetric {A} (R : relation A) :=
|
||||||
|
antisymmetry : forall x y, R x y -> R y x -> x = y.
|
||||||
|
|
||||||
|
|
||||||
|
Class Total {A} (R : relation A) :=
|
||||||
|
total : forall x y, x = y \/ R x y \/ R y x.
|
||||||
|
|
||||||
|
Class TotalOrder {A} (R : relation A) :=
|
||||||
|
{ TotalOrder_Reflexive : Reflexive R | 2 ;
|
||||||
|
TotalOrder_Antisymmetric : Antisymmetric R | 2;
|
||||||
|
TotalOrder_Transitive : Transitive R | 2;
|
||||||
|
TotalOrder_Total : Total R | 2; }.
|
||||||
|
|
||||||
|
Context {A : Type0}.
|
||||||
|
Context {A_deceq : DecidablePaths A}.
|
||||||
|
Context {R: relation A}.
|
||||||
|
Context {A_ordered : TotalOrder R}.
|
||||||
|
|
||||||
|
|
||||||
|
Ltac eq_neq_tac :=
|
||||||
|
match goal with
|
||||||
|
| [ H: ?x <> E, H': ?x = E |- _ ] => destruct H; assumption
|
||||||
|
end.
|
||||||
|
|
||||||
|
|
||||||
|
Ltac destruct_match_1 :=
|
||||||
|
repeat match goal with
|
||||||
|
| [|- match ?X with | _ => _ end ] => destruct X
|
||||||
|
| [|- ?X = ?Y ] => apply path_ishprop
|
||||||
|
| [ H: ?x <> E |- Empty ] => destruct H
|
||||||
|
| [ H1: ?x = E, H2: ?y = E, H3: ?w ∪ ?q = E |- ?r = E]
|
||||||
|
=> rewrite H1, H2 in H3; rewrite nl in H3; rewrite nl in H3
|
||||||
|
end.
|
||||||
|
|
||||||
|
|
||||||
|
Lemma transport_dom_eq (D1 D2 C: Type) (P: D1 = D2) (f: D1 -> C) :
|
||||||
|
transport (fun T: Type => T -> C) P f = fun y => f (transport (fun X => X) P^ y).
|
||||||
|
Proof.
|
||||||
|
induction P.
|
||||||
|
hott_simpl.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma transport_dom_eq_gen (Ty: Type) (D1 D2: Ty) (C: Type) (P: D1 = D2)
|
||||||
|
(Q : Ty -> Type) (f: Q D1 -> C) :
|
||||||
|
transport (fun X: Ty => Q X -> C) P f = fun y => f (transport Q P^ y).
|
||||||
|
Proof.
|
||||||
|
induction P.
|
||||||
|
hott_simpl.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
Lemma min {HFun: Funext} (x: FSet A): x <> ∅ -> A.
|
||||||
|
Proof.
|
||||||
|
hrecursion x.
|
||||||
|
- intro H. destruct H. reflexivity.
|
||||||
|
- intros. exact a.
|
||||||
|
- intros x y rx ry H.
|
||||||
|
apply union_non_empty' in H.
|
||||||
|
destruct H.
|
||||||
|
+ destruct p. specialize (rx fst). exact rx.
|
||||||
|
+ destruct s.
|
||||||
|
* destruct p. specialize (ry snd). exact ry.
|
||||||
|
* destruct p. specialize (rx fst). specialize (ry snd).
|
||||||
|
destruct (TotalOrder_Total rx ry) as [Heq | [ Hx | Hy ]].
|
||||||
|
** exact rx.
|
||||||
|
** exact rx.
|
||||||
|
** exact ry.
|
||||||
|
- intros. rewrite transport_dom_eq_gen.
|
||||||
|
apply path_forall. intro y0.
|
||||||
|
destruct ( union_non_empty' x y ∪ z
|
||||||
|
(transport (fun X : FSet A => X <> ∅) (assoc x y z)^ y0))
|
||||||
|
as [[ G1 G2] | [[ G3 G4] | [G5 G6]]].
|
||||||
|
+ pose (G2' := G2). apply eset_union_lr in G2'; destruct G2'. cbn.
|
||||||
|
destruct (union_non_empty' x ∪ y z y0) as
|
||||||
|
[[H'x H'y] | [ [H'a H'b] | [H'c H'd]]]; try eq_neq_tac.
|
||||||
|
destruct (union_non_empty' x y H'x).
|
||||||
|
** destruct p. assert (G1 = fst0). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
** destruct s; destruct p; eq_neq_tac.
|
||||||
|
+ destruct (union_non_empty' y z G4) as
|
||||||
|
[[H'x H'y] | [ [H'a H'b] | [H'c H'd]]]; try eq_neq_tac.
|
||||||
|
destruct (union_non_empty' x ∪ y z y0).
|
||||||
|
** destruct p. cbn. destruct (union_non_empty' x y fst).
|
||||||
|
*** destruct p; eq_neq_tac.
|
||||||
|
*** destruct s. destruct p.
|
||||||
|
**** assert (H'x = snd0). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
**** destruct p. eq_neq_tac.
|
||||||
|
** destruct s; destruct p; try eq_neq_tac.
|
||||||
|
** destruct (union_non_empty' x ∪ y z y0).
|
||||||
|
*** destruct p. eq_neq_tac.
|
||||||
|
*** destruct s. destruct p.
|
||||||
|
**** assert (H'b = snd). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
**** destruct p. assert (x ∪ y = E).
|
||||||
|
rewrite H'a, G3. apply union_idem. eq_neq_tac.
|
||||||
|
** cbn. destruct (TotalOrder_Total (py H'c) (pz H'd)).
|
||||||
|
*** destruct (union_non_empty' x ∪ y z y0).
|
||||||
|
**** destruct p0. eq_neq_tac.
|
||||||
|
**** destruct s.
|
||||||
|
***** destruct p0. rewrite G3, nl in fst. eq_neq_tac.
|
||||||
|
***** destruct p0. destruct (union_non_empty' x y fst).
|
||||||
|
****** destruct p0. eq_neq_tac.
|
||||||
|
****** destruct s.
|
||||||
|
******* destruct p0.
|
||||||
|
destruct (TotalOrder_Total (py snd0) (pz snd)).
|
||||||
|
f_ap. apply path_forall. intro.
|
||||||
|
apply path_ishprop.
|
||||||
|
destruct s. f_ap. apply path_forall. intro.
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite p. f_ap. apply path_forall. intro.
|
||||||
|
apply path_ishprop.
|
||||||
|
******* destruct p0. eq_neq_tac.
|
||||||
|
*** destruct (union_non_empty' x ∪ y z y0).
|
||||||
|
**** destruct p. eq_neq_tac.
|
||||||
|
**** destruct s0. destruct p. rewrite comm in fst.
|
||||||
|
apply eset_union_l in fst. eq_neq_tac.
|
||||||
|
destruct p.
|
||||||
|
destruct (union_non_empty' x y fst).
|
||||||
|
***** destruct p; eq_neq_tac.
|
||||||
|
***** destruct s0. destruct p.
|
||||||
|
destruct (TotalOrder_Total (py snd0) (pz snd));
|
||||||
|
destruct s; try (f_ap; apply path_forall; intro;
|
||||||
|
apply path_ishprop).
|
||||||
|
rewrite p. f_ap; apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
destruct s0. f_ap; apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (snd0 = H'c). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (snd = H'd). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite <- X0 in r. rewrite X in r0.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
destruct s0.
|
||||||
|
assert (snd0 = H'c). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (snd = H'd). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite <- X in r. rewrite X0 in r0.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
f_ap; apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
destruct p; eq_neq_tac.
|
||||||
|
+ cbn. destruct (union_non_empty' y z G6).
|
||||||
|
** destruct p. destruct ( union_non_empty' x ∪ y z y0).
|
||||||
|
*** destruct p. destruct (union_non_empty' x y fst0).
|
||||||
|
**** destruct p; eq_neq_tac.
|
||||||
|
**** destruct s; destruct p. eq_neq_tac.
|
||||||
|
assert (fst1 = G5). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (fst = snd1). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite X, X0.
|
||||||
|
destruct (TotalOrder_Total (px G5) (py snd1)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
*** destruct s; destruct p; eq_neq_tac.
|
||||||
|
** destruct (union_non_empty' x ∪ y z y0).
|
||||||
|
*** destruct p. destruct s; destruct p; eq_neq_tac.
|
||||||
|
*** destruct s. destruct p. destruct s0. destruct p.
|
||||||
|
apply eset_union_l in fst0. eq_neq_tac.
|
||||||
|
**** destruct p.
|
||||||
|
assert (snd = snd0). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
|
||||||
|
destruct (union_non_empty' x y fst0).
|
||||||
|
destruct p.
|
||||||
|
assert (fst1 = G5). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (fst = snd1). apply set_path2.
|
||||||
|
***** rewrite X0. rewrite <- X. reflexivity.
|
||||||
|
***** destruct s; destruct p; eq_neq_tac.
|
||||||
|
**** destruct s0. destruct p0. destruct p.
|
||||||
|
***** apply eset_union_l in fst. eq_neq_tac.
|
||||||
|
***** destruct p, p0.
|
||||||
|
assert (snd0 = snd). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite X.
|
||||||
|
destruct (union_non_empty' x y fst0).
|
||||||
|
destruct p; eq_neq_tac.
|
||||||
|
destruct s. destruct p; eq_neq_tac.
|
||||||
|
destruct p.
|
||||||
|
assert (fst = snd1). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
assert (fst1 = G5). apply path_forall; intro;
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite <- X0. rewrite X1.
|
||||||
|
destruct (TotalOrder_Total (py fst) (pz snd)).
|
||||||
|
****** rewrite <- p.
|
||||||
|
destruct (TotalOrder_Total (px G5) (py fst)).
|
||||||
|
rewrite <- p0.
|
||||||
|
destruct (TotalOrder_Total (px G5) (px G5)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
destruct s. destruct (TotalOrder_Total (px G5) (py fst)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
destruct (TotalOrder_Total (py fst) (py fst)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s;
|
||||||
|
reflexivity.
|
||||||
|
****** destruct s.
|
||||||
|
destruct (TotalOrder_Total (px G5) (py fst)).
|
||||||
|
destruct (TotalOrder_Total (px G5) (pz snd)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity. rewrite <- p in r.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
destruct s.
|
||||||
|
destruct ( TotalOrder_Total (px G5) (pz snd)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
apply (TotalOrder_Transitive (px G5)) in r.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
assumption.
|
||||||
|
destruct (TotalOrder_Total (py fst) (pz snd)). reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
*******
|
||||||
|
destruct ( TotalOrder_Total (px G5) (py fst)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s. destruct (TotalOrder_Total (px G5) (pz snd)).
|
||||||
|
reflexivity. destruct s; reflexivity.
|
||||||
|
destruct ( TotalOrder_Total (px G5) (pz snd)).
|
||||||
|
rewrite <- p.
|
||||||
|
destruct (TotalOrder_Total (py fst) (px G5)).
|
||||||
|
apply symmetry; assumption.
|
||||||
|
destruct s. rewrite <- p in r.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
reflexivity. destruct s.
|
||||||
|
assert ((py fst) = (pz snd)). apply TotalOrder_Antisymmetric.
|
||||||
|
apply (TotalOrder_Transitive (py fst) (px G5)); assumption.
|
||||||
|
assumption. rewrite X2. assert (px G5 = pz snd).
|
||||||
|
apply TotalOrder_Antisymmetric. assumption.
|
||||||
|
apply (TotalOrder_Transitive (pz snd) (py fst)); assumption.
|
||||||
|
rewrite X3.
|
||||||
|
destruct ( TotalOrder_Total (pz snd) (pz snd)).
|
||||||
|
reflexivity. destruct s; reflexivity.
|
||||||
|
destruct (TotalOrder_Total (py fst) (pz snd)).
|
||||||
|
apply TotalOrder_Antisymmetric. assumption. rewrite p.
|
||||||
|
apply (TotalOrder_Reflexive). destruct s.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption. reflexivity.
|
||||||
|
- intros. rewrite transport_dom_eq_gen.
|
||||||
|
apply path_forall. intro y0. cbn.
|
||||||
|
destruct
|
||||||
|
(union_non_empty' x y
|
||||||
|
(transport (fun X : FSet A => X <> ∅) (comm x y)^ y0)) as
|
||||||
|
[[Hx Hy] | [ [Ha Hb] | [Hc Hd]]];
|
||||||
|
destruct (union_non_empty' y x y0) as
|
||||||
|
[[H'x H'y] | [ [H'a H'b] | [H'c H'd]]];
|
||||||
|
try (eq_neq_tac).
|
||||||
|
assert (Hx = H'b). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
assert (Hb = H'x). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
assert (Hd = H'c). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X.
|
||||||
|
assert (H'd = Hc). apply path_forall. intro.
|
||||||
|
apply path_ishprop.
|
||||||
|
rewrite X0. rewrite <- X.
|
||||||
|
destruct
|
||||||
|
(TotalOrder_Total (px Hc) (py Hd)) as [G1 | [G2 | G3]];
|
||||||
|
destruct
|
||||||
|
(TotalOrder_Total (py Hd) (px Hc)) as [T1 | [T2 | T3]];
|
||||||
|
try (assumption);
|
||||||
|
try (reflexivity);
|
||||||
|
try (apply symmetry; assumption);
|
||||||
|
try (apply TotalOrder_Antisymmetric; assumption).
|
||||||
|
|
||||||
|
- intros. rewrite transport_dom_eq_gen.
|
||||||
|
apply path_forall. intro y.
|
||||||
|
destruct (union_non_empty' ∅ x (transport (fun X : FSet A => X <> ∅) (nl x)^ y)).
|
||||||
|
destruct p. eq_neq_tac.
|
||||||
|
destruct s.
|
||||||
|
destruct p.
|
||||||
|
assert (y = snd).
|
||||||
|
apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
destruct p. destruct fst.
|
||||||
|
- intros. rewrite transport_dom_eq_gen.
|
||||||
|
apply path_forall. intro y.
|
||||||
|
destruct (union_non_empty' x ∅ (transport (fun X : FSet A => X <> ∅) (nr x)^ y)).
|
||||||
|
destruct p. assert (y = fst). apply path_forall. intro.
|
||||||
|
apply path_ishprop. rewrite X. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
destruct p.
|
||||||
|
eq_neq_tac.
|
||||||
|
destruct p.
|
||||||
|
destruct snd.
|
||||||
|
- intros. rewrite transport_dom_eq_gen.
|
||||||
|
apply path_forall. intro y.
|
||||||
|
destruct ( union_non_empty' {|x|} {|x|} (transport (fun X : FSet A => X <> ∅) (idem x)^ y)).
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity.
|
||||||
|
destruct p.
|
||||||
|
cbn. destruct (TotalOrder_Total x x). reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
|
||||||
|
Definition minfset {HFun: Funext} :
|
||||||
|
FSet A -> { Y: (FSet A) & (Y = E) + { a: A & Y = L a } }.
|
||||||
|
intro X.
|
||||||
|
hinduction X.
|
||||||
|
- exists E. left. reflexivity.
|
||||||
|
- intro a. exists (L a). right. exists a. reflexivity.
|
||||||
|
- intros IH1 IH2.
|
||||||
|
destruct IH1 as [R1 HR1].
|
||||||
|
destruct IH2 as [R2 HR2].
|
||||||
|
destruct HR1.
|
||||||
|
destruct HR2.
|
||||||
|
exists E; left. reflexivity.
|
||||||
|
destruct s as [a Ha]. exists (L a). right.
|
||||||
|
exists a. reflexivity.
|
||||||
|
destruct HR2.
|
||||||
|
destruct s as [a Ha].
|
||||||
|
exists (L a). right. exists a. reflexivity.
|
||||||
|
destruct s as [a1 Ha1].
|
||||||
|
destruct s0 as [a2 Ha2].
|
||||||
|
assert (a1 = a2 \/ R a1 a2 \/ R a2 a1).
|
||||||
|
apply TotalOrder_Total.
|
||||||
|
destruct X.
|
||||||
|
exists (L a1). right. exists a1. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
exists (L a1). right. exists a1. reflexivity.
|
||||||
|
exists (L a2). right. exists a2. reflexivity.
|
||||||
|
- cbn. intros R1 R2 R3.
|
||||||
|
destruct R1 as [Res1 HR1].
|
||||||
|
destruct HR1 as [HR1E | HR1S].
|
||||||
|
destruct R2 as [Res2 HR2].
|
||||||
|
destruct HR2 as [HR2E | HR2S].
|
||||||
|
destruct R3 as [Res3 HR3].
|
||||||
|
destruct HR3 as [HR3E | HR3S].
|
||||||
|
+ cbn. reflexivity.
|
||||||
|
+ cbn. reflexivity.
|
||||||
|
+ cbn. destruct R3 as [Res3 HR3].
|
||||||
|
destruct HR3 as [HR3E | HR3S].
|
||||||
|
* cbn. reflexivity.
|
||||||
|
* destruct HR2S as [a2 Ha2].
|
||||||
|
destruct HR3S as [a3 Ha3].
|
||||||
|
destruct (TotalOrder_Total a2 a3).
|
||||||
|
** cbn. reflexivity.
|
||||||
|
** destruct s. cbn. reflexivity.
|
||||||
|
cbn. reflexivity.
|
||||||
|
+ destruct HR1S as [a1 Ha1].
|
||||||
|
destruct R2 as [Res2 HR2].
|
||||||
|
destruct HR2 as [HR2E | HR2S].
|
||||||
|
destruct R3 as [Res3 HR3].
|
||||||
|
destruct HR3 as [HR3E | HR3S].
|
||||||
|
* cbn. reflexivity.
|
||||||
|
* destruct HR3S as [a3 Ha3].
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
* destruct HR2S as [a2 Ha2].
|
||||||
|
destruct R3 as [Res3 HR3].
|
||||||
|
destruct HR3 as [HR3E | HR3S].
|
||||||
|
cbn. destruct (TotalOrder_Total a1 a2).
|
||||||
|
cbn. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
cbn. reflexivity.
|
||||||
|
cbn. reflexivity.
|
||||||
|
destruct HR3S as [a3 Ha3].
|
||||||
|
destruct (TotalOrder_Total a2 a3).
|
||||||
|
** rewrite p.
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
rewrite p0.
|
||||||
|
destruct ( TotalOrder_Total a3 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
destruct s. cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
cbn. destruct (TotalOrder_Total a3 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
** destruct s.
|
||||||
|
*** cbn. destruct (TotalOrder_Total a1 a2).
|
||||||
|
cbn. destruct (TotalOrder_Total a1 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
rewrite <- p in r.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
destruct s. cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
assert (R a1 a3).
|
||||||
|
apply (TotalOrder_Transitive a1 a2); assumption.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X0. reflexivity.
|
||||||
|
cbn. destruct (TotalOrder_Total a2 a3).
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity.
|
||||||
|
assert (a2 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
*** cbn. destruct (TotalOrder_Total a1 a3).
|
||||||
|
rewrite p. destruct (TotalOrder_Total a3 a2).
|
||||||
|
cbn. destruct (TotalOrder_Total a3 a3).
|
||||||
|
reflexivity. destruct s; reflexivity.
|
||||||
|
destruct s. cbn.
|
||||||
|
destruct (TotalOrder_Total a3 a3).
|
||||||
|
reflexivity. destruct s; reflexivity.
|
||||||
|
cbn. destruct (TotalOrder_Total a2 a3).
|
||||||
|
rewrite p0.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a2 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a2).
|
||||||
|
cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
reflexivity.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric. assumption.
|
||||||
|
rewrite <- p in r. assumption.
|
||||||
|
destruct s. reflexivity. rewrite X. reflexivity.
|
||||||
|
destruct s. cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3). reflexivity.
|
||||||
|
destruct s. reflexivity.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
cbn. destruct (TotalOrder_Total a2 a3).
|
||||||
|
rewrite p in r1.
|
||||||
|
assert (a2 = a1).
|
||||||
|
transitivity a3.
|
||||||
|
assumption.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a1 = a2).
|
||||||
|
apply TotalOrder_Antisymmetric.
|
||||||
|
apply (TotalOrder_Transitive a1 a3); assumption.
|
||||||
|
assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric.
|
||||||
|
assumption.
|
||||||
|
apply (TotalOrder_Transitive a3 a2); assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
destruct ( TotalOrder_Total a1 a2).
|
||||||
|
cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3).
|
||||||
|
rewrite p0.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity. reflexivity.
|
||||||
|
destruct s.
|
||||||
|
cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3 ).
|
||||||
|
rewrite p.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a1 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity. reflexivity.
|
||||||
|
cbn.
|
||||||
|
destruct (TotalOrder_Total a1 a3 ).
|
||||||
|
assert (a2 = a3).
|
||||||
|
rewrite p in r1.
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. destruct (TotalOrder_Total a3 a3). reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
destruct s.
|
||||||
|
destruct (TotalOrder_Total a2 a3).
|
||||||
|
rewrite p.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a2 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity.
|
||||||
|
reflexivity.
|
||||||
|
cbn. destruct (TotalOrder_Total a2 a3).
|
||||||
|
rewrite p.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a2 = a3).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X. reflexivity. reflexivity.
|
||||||
|
- cbn. intros R1 R2.
|
||||||
|
destruct R1 as [La1 HR1].
|
||||||
|
destruct HR1 as [HR1E | HR1S].
|
||||||
|
destruct R2 as [La2 HR2].
|
||||||
|
destruct HR2 as [HR2E | HR2S].
|
||||||
|
reflexivity.
|
||||||
|
reflexivity.
|
||||||
|
destruct R2 as [La2 HR2].
|
||||||
|
destruct HR2 as [HR2E | HR2S].
|
||||||
|
reflexivity.
|
||||||
|
destruct HR1S as [a1 Ha1].
|
||||||
|
destruct HR2S as [a2 Ha2].
|
||||||
|
destruct (TotalOrder_Total a1 a2).
|
||||||
|
rewrite p.
|
||||||
|
destruct (TotalOrder_Total a2 a2).
|
||||||
|
reflexivity.
|
||||||
|
destruct s; reflexivity.
|
||||||
|
destruct s.
|
||||||
|
destruct (TotalOrder_Total a2 a1).
|
||||||
|
rewrite p.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
assert (a1 = a2).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X.
|
||||||
|
reflexivity.
|
||||||
|
reflexivity.
|
||||||
|
destruct (TotalOrder_Total a2 a1).
|
||||||
|
rewrite p.
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity.
|
||||||
|
assert (a1 = a2).
|
||||||
|
apply TotalOrder_Antisymmetric; assumption.
|
||||||
|
rewrite X.
|
||||||
|
reflexivity.
|
||||||
|
- cbn. intro R. destruct R as [La HR].
|
||||||
|
destruct HR. rewrite <- p. reflexivity.
|
||||||
|
destruct s as [a1 H].
|
||||||
|
apply (path_sigma' _ H^).
|
||||||
|
rewrite transport_sum.
|
||||||
|
f_ap.
|
||||||
|
rewrite transport_sigma.
|
||||||
|
simpl.
|
||||||
|
simple refine (path_sigma' _ _ _ ).
|
||||||
|
apply transport_const.
|
||||||
|
apply set_path2.
|
||||||
|
|
||||||
|
- intros R. cbn.
|
||||||
|
destruct R as [ R HR].
|
||||||
|
destruct HR as [HE | Ha ].
|
||||||
|
rewrite <- HE. reflexivity.
|
||||||
|
destruct Ha as [a Ha].
|
||||||
|
apply (path_sigma' _ Ha^).
|
||||||
|
rewrite transport_sum.
|
||||||
|
f_ap.
|
||||||
|
rewrite transport_sigma.
|
||||||
|
simpl.
|
||||||
|
simple refine (path_sigma' _ _ _ ).
|
||||||
|
apply transport_const.
|
||||||
|
apply set_path2.
|
||||||
|
- cbn. intro.
|
||||||
|
destruct (TotalOrder_Total x x).
|
||||||
|
reflexivity.
|
||||||
|
destruct s.
|
||||||
|
reflexivity.
|
||||||
|
reflexivity.
|
||||||
|
Defined.
|
||||||
|
|
||||||
|
|
||||||
|
|
||||||
Loading…
Reference in New Issue