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273 строки
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273 строки
6.2 KiB
Plaintext
!!! Park-CSE-490 !!!
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Valentini - Another Introduction to Martin-Lof Intuitionistic Type Theory
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Judgement - ñóæäåíèå (Park. - A judgement is an object of knowledge, or simply a statement, that may or may not be provable).
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Proposition - ïðåäëîæåíèå (Pierce - óòâåðæäåíèå) (Park - An object of verification whose truth can be checked by inf-rules).
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(A) - IS A PROPOSITION
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(A is true), (A is false), (A maybe true), ... - ARE JUDGEMENTS
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Premise - ïîñûëêà.
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Conclusion - âûâîä.
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Evidence - äîêàçàòåëüñòâî
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judgement - something we may know (evident if we know it)
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In other words: a judgement is evident if we have a proof for it.
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E.x: "A is true", "A is false", "A is true at time t", "A is necessarily true",
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"program M has type \tau"
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Hypothetical Deduction:
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J_1 ... J_n A true B true
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-----------name ex is --------------&I (And intro)
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J A && B true
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J_1 ... J_n |- J (|- - entailment of consequent relation).
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This is the localized-notation.
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Judgements J_1 ... J_n are premises here (assumptions, antecedents),
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J is - conclusion (succedent).
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The collection of antecedends is often marked as a big-greek-gamma (Ã |- J).
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A, B, ... - propositions.
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Example (of hypothetical judgements):
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------ u (under the assumption)
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A true
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.
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.
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.
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B true
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----------- =>I^u (intro under the assumption)
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A => B true
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proposition /
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premice
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----------
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conclusion
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Verification approach: when can I conclude it???
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Pragmatist approach: how can I use it???
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Harmony:
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Local soundness - shows that the elimination rules are "not too strong":
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- we can not gain any new info no matter how we apply the elimination rules.
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in other words - elimination rules do not gain a new knowledge comparing to the inroduction rules
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Local completeness - shows that the elimination rules are "not too weak":
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- there is always a way to apply elimination rules so that we can reconstinute
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a proof of the original proposition from the results by applying introduction rules.
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in other words - elimination rules gain all the knowledge, provided by the introduction rules
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Verification:
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A - the judgement "A has a verification" (denoted by arrow pointed up).
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A ver B ver
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--------------- &I
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(A && B) ver
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Uses:
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!!! here we become pragmatists !!!
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A use - the judgement "A may be used" (denoted by arrow pointed down):
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- either "A true" is a hypothesis
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- or A is deduced from a hypothesis via elimination rules.
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Local soundness provide some evidence that we can't deduce anything incorrect in this manner.
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Conjunction case:
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A ver B ver (A && B) use (A && B) use
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------------ &I ------------ &E_L ------------ &E_R
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(A && B) ver A use B use
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If we can use (A && B) we can use A and we can use B (E_L and E_R).
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Implication case:
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The introduction rule creates a new hypothesis, which we may use in a proof.
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----- u (under the assumption)
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A use
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B ver
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------------- =>^u
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(A => B) ver
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In order to use an implication (A => B) we require a verification of A.
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Just recuring that A may be used would be too weak.
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(A => B) use A ver
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------------------- =>E
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B use
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Disjunction case:
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The verification of disjunction immediately follow from their intro-rules:
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A ver B ver
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------------ ||I_L ------------ ||I_R
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(A || B) ver (A || B) ver
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A disjunction is used in a proof by cases, called here ||E.
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The conclusion of ||E should be a verification (with uses of hypotheses):
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----- u ----- w
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A use B use
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. .
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. .
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(A || B) use C ver C ver
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---------------------------- ||E^(u,w)
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C ver
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Truch case:
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The only verification of truth is the trivial one:
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----- TI
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T ver
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A hypothesis (T use) can not be used because there is no elimination rule for T.
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Falsehood case:
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There is no verification of falsehood because we have no introduction rule.
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We can use falsehood, signifying a contradiction from our current hypothesis,
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to verify any conclusion (C). This is a zero-ary case of disjunction.
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Bot use
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------- Bot E
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C ver
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Atomic propositions case:
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We can not break down the structure of P because there is none, so we can only verify it
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if we already know that P is true (can use P). !!! P need to be atomic !!!
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P use
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----- use-ver
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P ver
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This (use-ver) rule is of a special status - judgemental rule (as opposed to introduction
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and elimination rules).
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***********************************************
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Global soundness (Oregon 10 44:00):
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If an arbitrary proposition A has a verification then we may use A without gaining any information
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A use
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If (A ver) and (C ver) then (C ver).
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***********************************************
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Global soundness (Oregon 11-2 51:00)
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Any proven truth-judgement (A true) can be turned to the verification (A ver).
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Verifications could be thought of as a normal forms (Oregon 11-2 1:03:00).
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Global completeness (Oregon 10 43:00):
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If we may use A then we can construct from this a verification of A:
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A use
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A ver
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Proof normalization:
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Takes an arbitrary proof of (A true) and constructs a verification of A.
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CURRY-HOWARDT
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2010 Pfenning-ProofTheoryFoundation 1 - Implication case (lambda abstraction - 43:00).
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Substitution:
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[N/x]M - N substituted for x in M
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(\x:A.M)N ==>R [N/x]M
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This corresponds to beta-reduction in LC.
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M : (A => B) ==>E \x:A.Mx (x is free/not-bound in M)
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This corresponds to eta-reduction inversed in LC.
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à |- M : A Ã, x : A, Ã' |- N : B
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-------------------------------------
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Ã, Ã' |- [M/x]N : B
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2010 Pfenning-ProofTheoryFoundation 1 - Substitution operation 1:01:00.
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[M/x](x) = M
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[M/x](y) = y, if y != x
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[M/x]<N1,N2> = <[M/x]N1,[M/x]N2>
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[M/x](\y:A.N) = (\y:A. [M/x]N), if y !=x AND y not in FV(M) in order to exclude accidental binding
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Sequent Calculus:
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A Sequent is a particular form of hypothetical judgement.
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(A1 left), ..., (AN left) |- (C right)
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Where (A left) corresponds to (A use), (C right) - (C ver).
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Let us go through all the verification rules and construct appropriate sequent rules.
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1. Atomic use-ver
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P use
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----- use-ver
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P ver
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-------------------- init
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Ã, P left |- P right
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2. Conjunction
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