Step
*
1
1
of Lemma
es-empty-fset-at
1. es : EO
2. i : Id
3. ∀s:fset(E). ((∀e:E. (e ∈ s ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ s@i))) ∧ no_repeats(E;s@i) ∧ sorted-by(λe,e'. e ≤loc e' ;s@i)\000C)
4. (∀e:E. (e ∈ {} ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ {}@i))) ∧ no_repeats(E;{}@i) ∧ sorted-by(λe,e'. e ≤loc e' ;{}@i)
5. ∀e:E. (¬(e ∈ {}@i))
⊢ {}@i ~ []
BY
{ ((MoveToConcl (-1) THEN GenConclAtAddr [2;1]) THEN (D 0 THENA Auto) THEN D -3) }
1
1. es : EO
2. i : Id
3. ∀s:fset(E). ((∀e:E. (e ∈ s ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ s@i))) ∧ no_repeats(E;s@i) ∧ sorted-by(λe,e'. e ≤loc e' ;s@i)\000C)
4. (∀e:E. (e ∈ {} ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ {}@i))) ∧ no_repeats(E;{}@i) ∧ sorted-by(λe,e'. e ≤loc e' ;{}@i)
5. {}@i = [] ∈ (E List)@i
6. ∀e:E. (¬(e ∈ []))@i
⊢ [] ~ []
2
1. es : EO
2. i : Id
3. ∀s:fset(E). ((∀e:E. (e ∈ s ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ s@i))) ∧ no_repeats(E;s@i) ∧ sorted-by(λe,e'. e ≤loc e' ;s@i)\000C)
4. (∀e:E. (e ∈ {} ∧ (loc(e) = i ∈ Id)
⇐⇒ (e ∈ {}@i))) ∧ no_repeats(E;{}@i) ∧ sorted-by(λe,e'. e ≤loc e' ;{}@i)
5. u : E
6. v : E List
7. {}@i = [u / v] ∈ (E List)@i
8. ∀e:E. (¬(e ∈ [u / v]))@i
⊢ [u / v] ~ []
Latex:
1. es : EO
2. i : Id
3. \mforall{}s:fset(E)
((\mforall{}e:E. (e \mmember{} s \mwedge{} (loc(e) = i) \mLeftarrow{}{}\mRightarrow{} (e \mmember{} s@i)))
\mwedge{} no\_repeats(E;s@i)
\mwedge{} sorted-by(\mlambda{}e,e'. e \mleq{}loc e' ;s@i))
4. (\mforall{}e:E. (e \mmember{} \{\} \mwedge{} (loc(e) = i) \mLeftarrow{}{}\mRightarrow{} (e \mmember{} \{\}@i)))
\mwedge{} no\_repeats(E;\{\}@i)
\mwedge{} sorted-by(\mlambda{}e,e'. e \mleq{}loc e' ;\{\}@i)
5. \mforall{}e:E. (\mneg{}(e \mmember{} \{\}@i))
\mvdash{} \{\}@i \msim{} []
By
((MoveToConcl (-1) THEN GenConclAtAddr [2;1]) THEN (D 0 THENA Auto) THEN D -3)
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