meeting -*- Outline -*-

* properties of functions (4.1)

** equality
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     REASONING ABOUT EQUALITY (4.1)

Equivalence rules:
 _________                  (== reflexive)
 |- t == t

 Phi |- t == t'   Phi' |- t' == t''
 __________________________(== transitive)
 Phi \union Phi' |- t == t''

  Phi |- t == t'
  ______________            (== symmetric)
  Phi |- t' == t

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      Q: what's the type of the t, t', and t'' metavariables?
         can be any type
      Q: How do the assumptions work in the transitive rule?
         use both sets 
      Q: should there be some assumptions used in the reflexivity axiom?
         wouldn't hurt to have Phi |- t == t

      Q: suppose we are given that |- Charity == Love and |- God == Love,
         what is a deduction that proves that Charity == God?

** substitution and congruence
*** substitution
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           SUBSTITUTION

  Phi' |- t1 == t1'
  Phi  |- t[v := t1]
___________________________ (substitution)
  Phi \union Phi' |- t[v := t1'] 
  * if t1 and t1' are free for v in t

  Phi |- t1 == t1'      (substitute equals
______________________________ for equals)
  Phi |- t[v := t1] = t[v := t1']
  * if t1 and t1' are free for v in t

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        Q: What's the meaning of "t1 is free for v in t"?
                 that no free variables of t1 become bound when v t1
                      is substituted for v in t
                 a stronger condition that works is that
                    the free variables in t1 aren't mentioned in the
                    lambda binders of t

        Q: What's the meaning of their use of double quotes in
        derivations where the substitution rule is used?
              it just emphasizes the part of the term that is being changed

*** congruence
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           FOR YOU TO DO
 
Derive the following rules
from the substitution rule:

    Phi |- t' == t''
  ____________________(operand congruence)
  Phi |- t.t' == t.t''

    Phi |- t' == t''
  ___________________(operator congruence)
  Phi |- t'.t == t''.t

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*** abstraction
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        ABSTRACTION

          Phi |- t == t'
  __________________________ (abstraction)
  Phi |- (\ v . t) == (\ v . t')
   * if v is not free in Phi

Example:
  Arith |- 0 + 1 == 1
  __________________________ (abstraction)
  Arith |- (\ v . 0 + 1) == (\v . 1)

Capture (wrong):
  Arith |- z + 1 == 1
  __________________________________
  Arith |- (\ z . z + 1) == (\z . 1)

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        Q: what's wrong with the capture example?

** conversion rules
   standard from the lambda calculus

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            CONVERSION RULES

  _____________________ (alpha conversion)
  |- (\ v . t) == (\ w . t[v := w])
  * w does not occur free in t

  ______________________ (beta conversion)
  |- (\ v . t).t' == t[v := t']
  * t' is free for v in t

  ______________________  (eta conversion)
  |- (\ v . t.v) == t
  * v is not free in t

  Phi |- t.v == t'.v
  __________________      (extensionality)
  Phi |- t == t'
  * v is not free in Phi, t, or t'
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        Q: What does "t' is free for v in t" mean?
           that where v is used in t, the free variables in t' are not bound,
           i.e., no capture will occur when t' is substitued for free
                 occurrences of v in t

        Q: Why all of these side conditions?