Proof of Nuprl Lemma : iterated_classrel

Step * 2 1 3 of Lemma iterated_classrel_mem

1. Info : Type
2. A : Type
3. S : Type
4. init : Id ─→ bag(S)
5. f : A ─→ S ─→ S
6. X : EClass(A)
7. es : EO+(Info)@i'
8. R : A ─→ S ─→ S ─→ ℙ@i'
9. e1 : E@i
10. e2 : E@i
11. ∀e':E
      ((e' < e2)
      ⇒ (∀v1,v2:S. ∀a:A.
            (single-valued-classrel(es;X;A)
            ⇒ single-valued-bag(init loc(e1);S)
            ⇒ (∀s1,s2:S. ∀a:A.  SqStable(R[a;s1;s2]))
            ⇒ (∀a1,a2:A. ∀s:S. ∀e,e2:E.
                  (e1 ≤loc e
                  ⇒ (e <loc e2)
                  ⇒ e2 ≤loc e'
                  ⇒ a1 ∈ X(e)
                  ⇒ iterated_classrel(es;S;A;f;init;X;e;s)
                  ⇒ a2 ∈ X(e2)
                  ⇒ R[a1;s;f a2 s]))
            ⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e1,e2:E.
                  ((e1 <loc e2)
                  ⇒ a1 ∈ X(e1)
                  ⇒ iterated_classrel(es;S;A;f;init;X;e1;s1)
                  ⇒ a2 ∈ X(e2)
                  ⇒ iterated_classrel(es;S;A;f;init;X;pred(e2);s2)
                  ⇒ R[a1;s1;s2]
                  ⇒ R[a1;s1;f a2 s2]))
            ⇒ (e1 <loc e')
            ⇒ a ∈ X(e1)
            ⇒ iterated_classrel(es;S;A;f;init;X;e1;v1)
            ⇒ iterated_classrel(es;S;A;f;init;X;e';v2)
            ⇒ (((∃e:E. ((e1 <loc e) ∧ e ≤loc e'  ∧ (∃a:A. a ∈ X(e)))) ⇒ R[a;v1;v2])
               ∧ ((∀e:E. ((e1 <loc e) ⇒ e ≤loc e'  ⇒ (∀a:A. (¬a ∈ X(e))))) ⇒ (v1 = v2 ∈ S))))))
12. v1 : S@i
13. v2 : S@i
14. a : A@i
15. single-valued-classrel(es;X;A)@i
16. single-valued-bag(init loc(e1);S)@i
17. ∀s1,s2:S. ∀a:A.  SqStable(R[a;s1;s2])@i
18. ∀a1,a2:A. ∀s:S. ∀e,e':E.
      (e1 ≤loc e
      ⇒ (e <loc e')
      ⇒ e' ≤loc e2
      ⇒ a1 ∈ X(e)
      ⇒ iterated_classrel(es;S;A;f;init;X;e;s)
      ⇒ a2 ∈ X(e')
      ⇒ R[a1;s;f a2 s])@i
19. ∀a1,a2:A. ∀s1,s2:S. ∀e1,e2:E.
      ((e1 <loc e2)
      ⇒ a1 ∈ X(e1)
      ⇒ iterated_classrel(es;S;A;f;init;X;e1;s1)
      ⇒ a2 ∈ X(e2)
      ⇒ iterated_classrel(es;S;A;f;init;X;pred(e2);s2)
      ⇒ R[a1;s1;s2]
      ⇒ R[a1;s1;f a2 s2])@i
20. (e1 <loc e2)@i
21. a ∈ X(e1)@i
22. iterated_classrel(es;S;A;f;init;X;e1;v1)@i
23. z : S@i
24. iterated_classrel(es;S;A;f;init;X;e1;z)
25. ∀a:A. (¬a ∈ X(e2))@i
26. v2 = z ∈ S@i
27. ¬↑first(e2)
28. e1 = pred(e2) ∈ E
29. v1 = z ∈ S
30. e : E@i
31. (e1 <loc e)@i
32. e ≤loc e2 @i
33. a1 : A@i
34. a1 ∈ X(e)@i
⊢ R[a;v1;v2]
BY
{ ((InstHyp [⌈a1⌉] (-10)⋅ THENA Auto)
   THEN (Assert ⌈False⌉⋅ THEN Auto)
   THEN UseEsLoclTri ⌈es⌉⌈e⌉⌈e2⌉⋅
   THEN D (-1)
   THEN Try (Complete ((D (-1) THEN Auto)))
   THEN InstLemma `es-pred_property` [⌈es⌉;⌈e2⌉]⋅
   THEN Auto
   THEN (InstHyp [⌈e⌉] (-1)⋅ THENA Auto)
   THEN D (-1)
   THEN Try (Complete (Auto))) }

Latex:

1. Info : Type 2. A : Type 3. S : Type 4. init : Id {}\mrightarrow{} bag(S) 5. f : A {}\mrightarrow{} S {}\mrightarrow{} S 6. X : EClass(A) 7. es : EO+(Info)@i' 8. R : A {}\mrightarrow{} S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i' 9. e1 : E@i 10. e2 : E@i 11. \mforall{}e':E ((e' < e2) {}\mRightarrow{} (\mforall{}v1,v2:S. \mforall{}a:A. (single-valued-classrel(es;X;A) {}\mRightarrow{} single-valued-bag(init loc(e1);S) {}\mRightarrow{} (\mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2])) {}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e2:E. (e1 \mleq{}loc e {}\mRightarrow{} (e <loc e2) {}\mRightarrow{} e2 \mleq{}loc e' {}\mRightarrow{} a1 \mmember{} X(e) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e;s) {}\mRightarrow{} a2 \mmember{} X(e2) {}\mRightarrow{} R[a1;s;f a2 s])) {}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E. ((e1 <loc e2) {}\mRightarrow{} a1 \mmember{} X(e1) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;s1) {}\mRightarrow{} a2 \mmember{} X(e2) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e2);s2) {}\mRightarrow{} R[a1;s1;s2] {}\mRightarrow{} R[a1;s1;f a2 s2])) {}\mRightarrow{} (e1 <loc e') {}\mRightarrow{} a \mmember{} X(e1) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;v1) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e';v2) {}\mRightarrow{} (((\mexists{}e:E. ((e1 <loc e) \mwedge{} e \mleq{}loc e' \mwedge{} (\mexists{}a:A. a \mmember{} X(e)))) {}\mRightarrow{} R[a;v1;v2]) \mwedge{} ((\mforall{}e:E. ((e1 <loc e) {}\mRightarrow{} e \mleq{}loc e' {}\mRightarrow{} (\mforall{}a:A. (\mneg{}a \mmember{} X(e))))) {}\mRightarrow{} (v1 = v2)))))) 12. v1 : S@i 13. v2 : S@i 14. a : A@i 15. single-valued-classrel(es;X;A)@i 16. single-valued-bag(init loc(e1);S)@i 17. \mforall{}s1,s2:S. \mforall{}a:A. SqStable(R[a;s1;s2])@i 18. \mforall{}a1,a2:A. \mforall{}s:S. \mforall{}e,e':E. (e1 \mleq{}loc e {}\mRightarrow{} (e <loc e') {}\mRightarrow{} e' \mleq{}loc e2 {}\mRightarrow{} a1 \mmember{} X(e) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e;s) {}\mRightarrow{} a2 \mmember{} X(e') {}\mRightarrow{} R[a1;s;f a2 s])@i 19. \mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e1,e2:E. ((e1 <loc e2) {}\mRightarrow{} a1 \mmember{} X(e1) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;e1;s1) {}\mRightarrow{} a2 \mmember{} X(e2) {}\mRightarrow{} iterated\_classrel(es;S;A;f;init;X;pred(e2);s2) {}\mRightarrow{} R[a1;s1;s2] {}\mRightarrow{} R[a1;s1;f a2 s2])@i 20. (e1 <loc e2)@i 21. a \mmember{} X(e1)@i 22. iterated\_classrel(es;S;A;f;init;X;e1;v1)@i 23. z : S@i 24. iterated\_classrel(es;S;A;f;init;X;e1;z) 25. \mforall{}a:A. (\mneg{}a \mmember{} X(e2))@i 26. v2 = z@i 27. \mneg{}\muparrow{}first(e2) 28. e1 = pred(e2) 29. v1 = z 30. e : E@i 31. (e1 <loc e)@i 32. e \mleq{}loc e2 @i 33. a1 : A@i 34. a1 \mmember{} X(e)@i \mvdash{} R[a;v1;v2] By ((InstHyp [\mkleeneopen{}a1\mkleeneclose{}] (-10)\mcdot{} THENA Auto) THEN (Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto) THEN UseEsLoclTri \mkleeneopen{}es\mkleeneclose{}\mkleeneopen{}e\mkleeneclose{}\mkleeneopen{}e2\mkleeneclose{}\mcdot{} THEN D (-1) THEN Try (Complete ((D (-1) THEN Auto))) THEN InstLemma `es-pred\_property` [\mkleeneopen{}es\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}]\mcdot{} THEN Auto THEN (InstHyp [\mkleeneopen{}e\mkleeneclose{}] (-1)\mcdot{} THENA Auto) THEN D (-1) THEN Try (Complete (Auto))) Home Index