Proof of Nuprl Lemma : iterated_classrel

Step * 2 1 1 1 1 of Lemma iterated_classrel_progress

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 : S ─→ S ─→ ℙ@i'
9. P : A ─→ S ─→ ℙ@i'
10. e1 : E@i
11. e2 : E@i
12. ∀e':E
      ((e' < e2)
      ⇒ (∀v1,v2:S.
            (single-valued-classrel(es;X;A)
            ⇒ (∀a:A. ∀s:S.  Dec(P[a;s]))
            ⇒ (∀x,y:S.  SqStable(R[x;y]))
            ⇒ Trans(S;x,y.R[x;y])
            ⇒ (∀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)))))
            ⇒ single-valued-bag(init loc(e1);S)
            ⇒ (e1 <loc e')
            ⇒ iterated_classrel(es;S;A;f;init;X;e1;v1)
            ⇒ 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))))))
13. v1 : S@i
14. v2 : S@i
15. single-valued-classrel(es;X;A)@i
16. ∀a:A. ∀s:S.  Dec(P[a;s])@i
17. ∀x,y:S.  SqStable(R[x;y])@i
18. Trans(S;x,y.R[x;y])@i
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. single-valued-bag(init loc(e1);S)@i
21. (e1 <loc e2)@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@i
26. a ∈ X(e2)@i
27. v2 = (f a z) ∈ S@i
28. ¬↑first(e2)
29. e1 = pred(e2) ∈ E
30. v1 = z ∈ S
31. e : E@i
32. s : S@i
33. (e1 <loc e)@i
34. e ≤loc e2 @i
35. iterated_classrel(es;S;A;f;init;X;pred(e);s)@i
36. a1 : A@i
37. a1 ∈ X(e)@i
38. P[a1;s]@i
39. P[a;z] ⇒ R[z;f a z]
40. (¬P[a;z]) ⇒ (z = (f a z) ∈ S)
41. e = e2 ∈ E
⊢ R[v1;v2]
BY
{ ((HypSubst' (-1) (-5) THENA Auto)
   THEN (RWO "-1" (-7) THENA Auto)
   THEN UseSingleVal (-16) (-5)
   THEN (RevHypSubst' (-14) (-8) THENA Auto)
   THEN (FLemma `iterated_classrel-single-val` [-19;-8] THENA Auto)
   THEN D (-5)
   THEN 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 : S {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i' 9. P : A {}\mrightarrow{} S {}\mrightarrow{} \mBbbP{}@i' 10. e1 : E@i 11. e2 : E@i 12. \mforall{}e':E ((e' < e2) {}\mRightarrow{} (\mforall{}v1,v2:S. (single-valued-classrel(es;X;A) {}\mRightarrow{} (\mforall{}a:A. \mforall{}s:S. Dec(P[a;s])) {}\mRightarrow{} (\mforall{}x,y:S. SqStable(R[x;y])) {}\mRightarrow{} Trans(S;x,y.R[x;y]) {}\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{} single-valued-bag(init loc(e1);S) {}\mRightarrow{} (e1 <loc e') {}\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 \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)))))) 13. v1 : S@i 14. v2 : S@i 15. single-valued-classrel(es;X;A)@i 16. \mforall{}a:A. \mforall{}s:S. Dec(P[a;s])@i 17. \mforall{}x,y:S. SqStable(R[x;y])@i 18. Trans(S;x,y.R[x;y])@i 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. single-valued-bag(init loc(e1);S)@i 21. (e1 <loc e2)@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@i 26. a \mmember{} X(e2)@i 27. v2 = (f a z)@i 28. \mneg{}\muparrow{}first(e2) 29. e1 = pred(e2) 30. v1 = z 31. e : E@i 32. s : S@i 33. (e1 <loc e)@i 34. e \mleq{}loc e2 @i 35. iterated\_classrel(es;S;A;f;init;X;pred(e);s)@i 36. a1 : A@i 37. a1 \mmember{} X(e)@i 38. P[a1;s]@i 39. P[a;z] {}\mRightarrow{} R[z;f a z] 40. (\mneg{}P[a;z]) {}\mRightarrow{} (z = (f a z)) 41. e = e2 \mvdash{} R[v1;v2] By ((HypSubst' (-1) (-5) THENA Auto) THEN (RWO "-1" (-7) THENA Auto) THEN UseSingleVal (-16) (-5) THEN (RevHypSubst' (-14) (-8) THENA Auto) THEN (FLemma `iterated\_classrel-single-val` [-19;-8] THENA Auto) THEN D (-5) THEN Auto) Home Index