43 lines
1,010 B
Coq
43 lines
1,010 B
Coq
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From Coq Require Import Arith.Arith.
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From Coq Require Import Bool.Bool.
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From Coq Require Import Logic.FunctionalExtensionality.
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From Coq Require Import Lists.List.
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From Coq Require Import Strings.String.
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Import ListNotations.
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Set Default Goal Selector "!".
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Definition relation (X: Type) := X -> X -> Prop.
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Definition partial_function {X: Type} (R: relation X) :=
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forall x y1 y2 : X, R x y1 -> R x y2 -> y1 = y2.
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Inductive next_nat : nat -> nat -> Prop :=
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| nn n : next_nat n (S n).
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Check next_nat : relation nat.
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Theorem next_nat_partial_function: partial_function next_nat.
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Proof.
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unfold partial_function.
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intros x y1 y2 H1 H2.
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inversion H1.
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inversion H2.
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reflexivity.
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Qed.
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Theorem le_not_a_partial_function :
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~ (partial_function le).
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Proof.
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unfold not.
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unfold partial_function.
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intros Hc.
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assert (0 = 1) as Nonsense. {
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apply Hc with (x := 0).
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- apply le_n.
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- apply le_S. apply le_n.
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}
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discriminate Nonsense.
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Qed.
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Inductive total_relation : nat -> nat -> Prop :=
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