Proof of Nuprl Lemma : iterated-classrel-progress

Step * 1 1 1 of Lemma iterated-classrel-progress

1. [Info] : Type
2. [A] : Type
3. [S] : Type
4. init : Id ─→ bag(S)@i
5. f : A ─→ S ─→ S@i
6. X : EClass(A)@i'
7. es : EO+(Info)@i'
8. R : S ─→ S ─→ ℙ@i'
9. P : A ─→ S ─→ ℙ@i'
10. e1 : E@i
11. v1 : S@i
12. single-valued-classrel(es;X;A)@i
13. single-valued-bag(init loc(e1);S)@i
14. ∀a:A. ∀s:S.  Dec(P[a;s])@i
15. Trans(S;x,y.R[x;y])@i
16. iterated-classrel(es;S;A;f;init;X;e1;v1)@i
17. e2 : E@i
18. ∀e':E
      ((e' < e2)
      ⇒ (∀a:A. ∀e:E. ∀s:S.
            ((e1 <loc e)
            ⇒ e ≤loc e'
            ⇒ a ∈ X(e)
            ⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
            ⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ S)))))
      ⇒ (e1 <loc e')
      ⇒ (∀v2:S
            (iterated-classrel(es;S;A;f;init;X;e';v2)
            ⇒ (((∃e:E
                   ∃s:S
                    ((e1 <loc e)
                    ∧ e ≤loc e'
                    ∧ iterated-classrel(es;S;A;f;init;X;pred(e);s)
                    ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))
               ⇒ R[v1;v2])
               ∧ ((∀e:E. ∀s:S.
                     ((e1 <loc e)
                     ⇒ e ≤loc e'
                     ⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
                     ⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
                 ⇒ (v1 = v2 ∈ S))))))
19. ∀a:A. ∀e:E. ∀s:S.
      ((e1 <loc e)
      ⇒ e ≤loc e2
      ⇒ a ∈ X(e)
      ⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
      ⇒ ((P[a;s] ⇒ R[s;f a s]) ∧ ((¬P[a;s]) ⇒ (s = (f a s) ∈ S))))@i
20. (e1 <loc e2)@i
21. v2 : S@i
22. z : S@i
23. iterated-classrel(es;S;A;f;init;X;pred(e2);z)@i

24. (∃a:A. (a ∈ X(e2) ∧ (v2 = (f a z) ∈ S))) ∨ ((∀a:A. (¬a ∈ X(e2))) ∧ (v2 = z ∈ S))@i

25. ¬↑first(e2)
26. (e1 <loc pred(e2))
27. (∃e:E
∃s:S

       ((e1 <loc e) ∧ e ≤loc pred(e2)  ∧ iterated-classrel(es;S;A;f;init;X;pred(e);s) ∧ (∃a:A. (a ∈ X(e) ∧ P[a;s]))))

⇒ R[v1;z]
28. (∀e:E. ∀s:S.
       ((e1 <loc e)
       ⇒ e ≤loc pred(e2)
       ⇒ iterated-classrel(es;S;A;f;init;X;pred(e);s)
       ⇒ (∀a:A. ((¬a ∈ X(e)) ∨ (¬P[a;s])))))
⇒ (v1 = z ∈ S)
29. e : E@i
30. s : S@i
31. (e1 <loc e)@i
32. (e <loc e2)@i
33. iterated-classrel(es;S;A;f;init;X;pred(e);s)@i
34. a : A@i
35. a ∈ X(e)@i
36. P[a;s]@i
⊢ R[v1;v2]
BY
{ ((D (-10) THENA (InstConcl [⌈e⌉;⌈s⌉]⋅ THEN Auto))
   THEN D (-13)
   THEN Auto
   THEN ExRepD
   THEN (InstHyp [⌈a1⌉;⌈e2⌉;⌈z⌉] (-20)⋅ THENA Auto)
   THEN RepD
   THEN (InstHyp [⌈a1⌉;⌈z⌉] (-27)⋅ THENA Auto)
   THEN D (-1)
   THEN Auto
   THEN D (-2)
   THEN Auto) }

Latex:

1. [Info] : Type 2. [A] : Type 3. [S] : Type 4. init : Id {}\mrightarrow{} bag(S)@i 5. f : A {}\mrightarrow{} S {}\mrightarrow{} S@i 6. X : EClass(A)@i' 7. es : EO+(Info)@i' 8. R : S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i' 9. P : A {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i' 10. e1 : E@i 11. v1 : S@i 12. single-valued-classrel(es;X;A)@i 13. single-valued-bag(init loc(e1);S)@i 14. \mforall{}a:A. \mforall{}s:S. Dec(P[a;s])@i 15. Trans(S;x,y.R[x;y])@i 16. iterated-classrel(es;S;A;f;init;X;e1;v1)@i 17. e2 : E@i 18. \mforall{}e':E ((e' < e2) {}\mRightarrow{} (\mforall{}a:A. \mforall{}e:E. \mforall{}s:S. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e' {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s)))))) {}\mRightarrow{} (e1 <loc e') {}\mRightarrow{} (\mforall{}v2:S (iterated-classrel(es;S;A;f;init;X;e';v2) {}\mRightarrow{} (((\mexists{}e:E \mexists{}s:S ((e1 <loc e) \mwedge{} e \mleq{}loc e' \mwedge{} iterated-classrel(es;S;A;f;init;X;pred(e);s) \mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s])))) {}\mRightarrow{} R[v1;v2]) \mwedge{} ((\mforall{}e:E. \mforall{}s:S. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e' {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s]))))) {}\mRightarrow{} (v1 = v2)))))) 19. \mforall{}a:A. \mforall{}e:E. \mforall{}s:S. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} ((P[a;s] {}\mRightarrow{} R[s;f a s]) \mwedge{} ((\mneg{}P[a;s]) {}\mRightarrow{} (s = (f a s)))))@i 20. (e1 <loc e2)@i 21. v2 : S@i 22. z : S@i 23. iterated-classrel(es;S;A;f;init;X;pred(e2);z)@i 24. (\mexists{}a:A. (a \mmember{} X(e2) \mwedge{} (v2 = (f a z)))) \mvee{} ((\mforall{}a:A. (\mneg{}a \mmember{} X(e2))) \mwedge{} (v2 = z))@i 25. \mneg{}\muparrow{}first(e2) 26. (e1 <loc pred(e2)) 27. (\mexists{}e:E \mexists{}s:S ((e1 <loc e) \mwedge{} e \mleq{}loc pred(e2) \mwedge{} iterated-classrel(es;S;A;f;init;X;pred(e);s) \mwedge{} (\mexists{}a:A. (a \mmember{} X(e) \mwedge{} P[a;s])))) {}\mRightarrow{} R[v1;z] 28. (\mforall{}e:E. \mforall{}s:S. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc pred(e2) {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e);s) {}\mRightarrow{} (\mforall{}a:A. ((\mneg{}a \mmember{} X(e)) \mvee{} (\mneg{}P[a;s]))))) {}\mRightarrow{} (v1 = z) 29. e : E@i 30. s : S@i 31. (e1 <loc e)@i 32. (e <loc e2)@i 33. iterated-classrel(es;S;A;f;init;X;pred(e);s)@i 34. a : A@i 35. a \mmember{} X(e)@i 36. P[a;s]@i \mvdash{} R[v1;v2] By ((D (-10) THENA (InstConcl [\mkleeneopen{}e\mkleeneclose{};\mkleeneopen{}s\mkleeneclose{}]\mcdot{} THEN Auto)) THEN D (-13) THEN Auto THEN ExRepD THEN (InstHyp [\mkleeneopen{}a1\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}] (-20)\mcdot{} THENA Auto) THEN RepD THEN (InstHyp [\mkleeneopen{}a1\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}] (-27)\mcdot{} THENA Auto) THEN D (-1) THEN Auto THEN D (-2) THEN Auto) Home Index