Coq.Bool.IfProp

# Module Coq.Bool.IfProp

``Require` Bool.

`Inductive` IfProp [A,B:Prop] : bool-> Prop
:= Iftrue : A -> (IfProp A B true)
| Iffalse : B -> (IfProp A B false).

`Hints` Resolve Iftrue Iffalse : bool v62.

`Lemma` Iftrue_inv : (A,B:Prop)(b:bool) (IfProp A B b) -> b=true -> A.
NewDestruct 1; Intros; Auto with bool.
Case diff_true_false; Auto with bool.
`Save`.

`Lemma` Iffalse_inv : (A,B:Prop)(b:bool) (IfProp A B b) -> b=false -> B.
NewDestruct 1; Intros; Auto with bool.
Case diff_true_false; Trivial with bool.
`Save`.

`Lemma` IfProp_true : (A,B:Prop)(IfProp A B true) -> A.
Intros.
Inversion H.
Assumption.
`Save`.

`Lemma` IfProp_false : (A,B:Prop)(IfProp A B false) -> B.
Intros.
Inversion H.
Assumption.
`Save`.

`Lemma` IfProp_or : (A,B:Prop)(b:bool)(IfProp A B b) -> A\/B.
NewDestruct 1; Auto with bool.
`Save`.

`Lemma` IfProp_sum : (A,B:Prop)(b:bool)(IfProp A B b) -> {A}+{B}.
NewDestruct b; Intro H.
Left; Inversion H; Auto with bool.
Right; Inversion H; Auto with bool.
`Save`.

`

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