74 lines
1.1 KiB
Plaintext
74 lines
1.1 KiB
Plaintext
#import "@preview/prooftrees:0.1.0": *
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= 1
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#tree(
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axi[$d in [a =_A b]$],
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uni[$"inv"(d) in ["symm"("symm"(d)) =_[a =_A b] d]$],
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)
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Defined by:
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$
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"inv"(d) equiv "idpeel"(d, (x) id(id(x)))
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$
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Derivation tree:
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#tree(
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axi[],
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uni[$A "set"$],
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)
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= 2
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```
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d∈[a=A b] e∈[b=A c]
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-------------------
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trans(d, e) ∈ [a =A c]
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trans(d,e) ≡ apply(idpeel(d,(x)λy.y),e)
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```
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```
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trans (trans e f) g == trans e (trans f g)
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apply(idpeel(
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apply(idpeel(e, (x)\y.y), f),
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(x)\y.y,
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), g)
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```
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= 3
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#tree(
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axi[$d in [a =_A b]$],
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uni[$"unitl"(a, d) ∈ ["trans"(id(a), d) =_[a =_A b] d]$]
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)
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---
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Simplifying the output type:
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$
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"trans"(id(a), d) &=_[a =_A b] d \
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"apply"("idpeel"(id(a), (x)lambda y.y), d) &=_[a =_A b] d \
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"apply"(((x)lambda y.y)(a), d) &=_[a =_A b] d \
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"apply"((lambda y.y), d) &=_[a =_A b] d \
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d &=_[a =_A b] d \
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$
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This is just a reflexive equality so can be satisfied by $"id"$:
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$ "unitl"(a, d) equiv id(d) $
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= 4
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#tree(
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axi[$d in [a =_A b]$],
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uni[$"unitr"(d, b) in ["trans"(d, id(b)) =_[a =_A b] d]$],
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)
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$
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"unitr"(d, b) equiv "idpeel"(d, (x)id(id(x))) \
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$ |