Proof of Nuprl Lemma : iterated-classrel-progress

Step * 2 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;e1;z)
24. a : A@i
25. a ∈ X(e2)@i
26. v2 = (f a z) ∈ S@i
27. ¬↑first(e2)
28. e1 = pred(e2) ∈ E
29. v1 = z ∈ S
30. ∃e:E

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

⊢ R[v1;v2]
BY
{ ((InstHyp [⌈a⌉;⌈e2⌉;⌈z⌉] (-12)⋅ THENA Auto)
   THEN D (-1)
   THEN ExRepD
   THEN D (-7)
   THEN Try (Complete (((Assert ⌈False⌉⋅ THEN Auto)

                        THEN (InstLemma `es-pred_property` [⌈es⌉;⌈e2⌉]⋅ THENA Auto)

                        THEN RepD
                        THEN (InstHyp [⌈e⌉] (-1)⋅ THENA Auto)
                        THEN D (-1)
                        THEN Auto)))
   THEN (HypSubst' (-7) (-4) THENA Auto)
   THEN UseSingleVal (-4) (-15)
   THEN (RWO "-8" (-7) THENA Auto)
   THEN (HypSubst' (-13) (-18) THENA Auto)
   THEN FLemma `iterated-classrel-single-val` [-18;-7]
   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;e1;z) 24. a : A@i 25. a \mmember{} X(e2)@i 26. v2 = (f a z)@i 27. \mneg{}\muparrow{}first(e2) 28. e1 = pred(e2) 29. v1 = z 30. \mexists{}e:E \mexists{}s:S ((e1 <loc e) \mwedge{} e \mleq{}loc 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])))@i \mvdash{} R[v1;v2] By ((InstHyp [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{}] (-12)\mcdot{} THENA Auto) THEN D (-1) THEN ExRepD THEN D (-7) THEN Try (Complete (((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto) THEN (InstLemma `es-pred\_property` [\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}]\mcdot{} THENA Auto) THEN RepD THEN (InstHyp [\mkleeneopen{}e\mkleeneclose{}] (-1)\mcdot{} THENA Auto) THEN D (-1) THEN Auto))) THEN (HypSubst' (-7) (-4) THENA Auto) THEN UseSingleVal (-4) (-15) THEN (RWO "-8" (-7) THENA Auto) THEN (HypSubst' (-13) (-18) THENA Auto) THEN FLemma `iterated-classrel-single-val` [-18;-7] THEN Auto) Home Index