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130 lines
2.3 KiB
Coq
130 lines
2.3 KiB
Coq
Set Implicit Arguments.
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Section trc.
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Variable A : Type.
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Variable R : A -> A -> Prop.
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Inductive trc : A -> A -> Prop :=
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| TrcRefl : forall x, trc x x
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| TrcFront : forall x y z,
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R x y
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-> trc y z
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-> trc x z.
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Hint Constructors trc.
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Theorem trc_one : forall x y, R x y
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-> trc x y.
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Proof.
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eauto.
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Qed.
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Hint Resolve trc_one.
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Theorem trc_trans : forall x y, trc x y
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-> forall z, trc y z
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-> trc x z.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trc_trans.
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Inductive trcEnd : A -> A -> Prop :=
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| TrcEndRefl : forall x, trcEnd x x
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| TrcBack : forall x y z,
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trcEnd x y
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-> R y z
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-> trcEnd x z.
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Hint Constructors trcEnd.
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Lemma TrcFront' : forall x y z,
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R x y
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-> trcEnd y z
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-> trcEnd x z.
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Proof.
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induction 2; eauto.
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Qed.
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Hint Resolve TrcFront'.
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Theorem trc_trcEnd : forall x y, trc x y
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-> trcEnd x y.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trc_trcEnd.
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Lemma TrcBack' : forall x y z,
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trc x y
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-> R y z
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-> trc x z.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve TrcBack'.
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Theorem trcEnd_trans : forall x y, trcEnd x y
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-> forall z, trcEnd y z
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-> trcEnd x z.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trcEnd_trans.
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Theorem trcEnd_trc : forall x y, trcEnd x y
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-> trc x y.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trcEnd_trc.
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Inductive trcLiteral : A -> A -> Prop :=
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| TrcLiteralRefl : forall x, trcLiteral x x
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| TrcTrans : forall x y z, trcLiteral x y
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-> trcLiteral y z
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-> trcLiteral x z
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| TrcInclude : forall x y, R x y
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-> trcLiteral x y.
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Hint Constructors trcLiteral.
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Theorem trc_trcLiteral : forall x y, trc x y
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-> trcLiteral x y.
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Proof.
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induction 1; eauto.
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Qed.
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Theorem trcLiteral_trc : forall x y, trcLiteral x y
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-> trc x y.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trc_trcLiteral trcLiteral_trc.
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Theorem trcEnd_trcLiteral : forall x y, trcEnd x y
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-> trcLiteral x y.
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Proof.
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induction 1; eauto.
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Qed.
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Theorem trcLiteral_trcEnd : forall x y, trcLiteral x y
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-> trcEnd x y.
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Proof.
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induction 1; eauto.
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Qed.
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Hint Resolve trcEnd_trcLiteral trcLiteral_trcEnd.
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End trc.
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Notation "R ^*" := (trc R) (at level 0).
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Notation "*^ R" := (trcEnd R) (at level 0).
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Hint Constructors trc.
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