Here are collected some results about the type sumbool (see INIT/Specif.v) sumbool A B , which is written {A}+{B} , is the informative disjunction "A or B", where A and B are logical propositions. Its extraction is isomorphic to the type of booleans.
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A boolean is either true or false , and this is decidable
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Lemma
sumbool_of_bool : (b:bool) {b=true}+{b=false}.
Proof
.
Induction b; Auto.
Save
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Hints
Resolve sumbool_of_bool : bool.
Lemma
bool_eq_rec : (b:bool)(P:bool->Set)
((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Save
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Lemma
bool_eq_ind : (b:bool)(P:bool->Prop)
((b=true)->(P true))->((b=false)->(P false))->(P b).
Induction b; Auto.
Save
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Logic connectives on type sumbool
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Section
connectives.
Variables
A,B,C,D : Prop.
Hypothesis
H1 : {A}+{B}.
Hypothesis
H2 : {C}+{D}.
Lemma
sumbool_and : {A/\C}+{B\/D}.
Proof
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Case H1; Case H2; Auto.
Save
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Lemma
sumbool_or : {A\/C}+{B/\D}.
Proof
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Case H1; Case H2; Auto.
Save
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Lemma
sumbool_not : {B}+{A}.
Proof
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Case H1; Auto.
Save
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End
connectives.
Hints
Resolve sumbool_and sumbool_or sumbool_not : core.