Open partizanos opened 6 years ago
only the cut therem is different i think it is the essence of the deduction that permits as to prove something from the Theorems, based on the "enchanced" cut on induction theorems But I am not sure.. I speak for the ADT slide 40
There is a remark in slide 39 mentioning the following but it is not so clear to me. Remark : Th(Spec) ⊆ ThInd (Spec) and the induction rule define the inductive theories.
The main diff?rence is the induction rule that says that proving P(x) for any t/x then the Formula \forall x, (P(x) is valid. It means that we can synthetize universally quantified variables from proof on specific values.
The rest of the axioms are the same obviously because they reflect properties of equality or functions.
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Le 28 janv. 2018 ? 16:56, Dimitrios Proios notifications@github.com<mailto:notifications@github.com> a ?crit :
i dont get differnece in inductive and equational proofs: they both have the same theorems inductive reflexivity: Reflexivity : ?t ? T?,s ,t = t ? ThInd (Spec) equational reflexivity: Reflexivity : ?t ? T?,s ,t = t ? Th(Spec) i dont get the difference: ThInd (Spec) and Th(Spec)
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In fact the cut is just here for the conditional axioms.
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Le 28 janv. 2018 ? 17:04, Dimitrios Proios notifications@github.com<mailto:notifications@github.com> a ?crit :
only the cut therem is different i think it is the essence of the deduction that permits as to prove something from the Theorems, based on the "enchanced" cut on induction theorems But I am not sure.. I speak for the ADT slide 40
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This inclusion assert that there are more theorems that we can deduce with the inductive theories, those that are defined with universal quantification that cannot be Obtained with equational theories. This is the case for instance with commutativity In natural specificationn or with not(not(x))=x . It is not strictly greater because in some Very specific cases inductive and equationbal theories can coincide.
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Le 28 janv. 2018 ? 17:16, Dimitrios Proios notifications@github.com<mailto:notifications@github.com> a ?crit :
There is a remark in slide 39 mentioning the following but it is not so clear to me. Remark : Th(Spec) ? ThInd (Spec) and the induction rule define the inductive theories.
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Thank you a lot for your answers.
Mini-questions: 1.(P=stands for proof? , t/x stands for combinations of terms and variables?)
2.how can i read this:
4.P(x)is valid means P(x)= True?
On 29 Jan 2018, at 03:47, Dimitrios Proios notifications@github.com<mailto:notifications@github.com> wrote:
Thank you a lot for your answers.
Mini-questions: 1.(P=stands for proof? , t/x stands for combinations of terms and variables?)
P stands for a predicate, it is parameterised suc as P(x)=x>0 or x<=0
2.how can i read this:
you are right, x is a variable so t is a value that can replace it, P(x)[x/3], in our case it means P(3)=3>0 or 3<=0
4.P(x)is valid means P(x)= True?
sort of, but a predicate is not a function, it is a relation.
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i dont get differnece in inductive and equational proofs: they both have the same theorems inductive reflexivity: Reflexivity : ∀t ∈ TΣ,s ,t = t ∈ ThInd (Spec) equational reflexivity: Reflexivity : ∀t ∈ TΣ,s ,t = t ∈ Th(Spec) i dont get the difference: ThInd (Spec) and Th(Spec)