Proof of Nuprl Lemma : iterated-classrel-mem

Step * 2 2 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)
            ⇒ (∀a:A. ∀s:S. ∀e:E.
                  (e1 ≤loc e
                  ⇒ e ≤loc e'
                  ⇒ a ∈ X(e)
                  ⇒ prior-iterated-classrel(es;A;S;s;X;f;init;e)
                  ⇒ R[a;s;f a s]))
            ⇒ (∀a1,a2:A. ∀s1,s2:S. ∀e,e2:E.
                  (e1 ≤loc e
                  ⇒ (e <loc e2)
                  ⇒ e2 ≤loc e'
                  ⇒ a1 ∈ X(e)
                  ⇒ prior-iterated-classrel(es;A;S;s1;X;f;init;e)
                  ⇒ 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)
            ⇒ prior-iterated-classrel(es;A;S;v1;X;f;init;e1)
            ⇒ iterated-classrel(es;S;A;f;init;X;e';v2)
            ⇒ R[a;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. ∀a:A. ∀s:S. ∀e:E.
      (e1 ≤loc e  ⇒ e ≤loc e2  ⇒ a ∈ X(e) ⇒ prior-iterated-classrel(es;A;S;s;X;f;init;e) ⇒ R[a;s;f a s])@i
18. ∀a1,a2:A. ∀s1,s2:S. ∀e,e':E.
      (e1 ≤loc e
      ⇒ (e <loc e')
      ⇒ e' ≤loc e2
      ⇒ a1 ∈ X(e)
      ⇒ prior-iterated-classrel(es;A;S;s1;X;f;init;e)
      ⇒ a2 ∈ X(e')
      ⇒ iterated-classrel(es;S;A;f;init;X;pred(e');s2)
      ⇒ R[a1;s1;s2]
      ⇒ R[a1;s1;f a2 s2])@i
19. e1 = e2 ∈ E@i
20. a ∈ X(e1)@i
21. prior-iterated-classrel(es;A;S;v1;X;f;init;e1)@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)
⊢ R[a;v1;v2]
BY
{ (D (-2)
   THEN ExRepD
   THEN Try (Complete ((Assert ⌈False⌉⋅ THEN Auto)))
   THEN D (-7)
   THEN Auto
   THEN (RWO "-10<" (-5) THEN Auto)
   THEN (FLemma `iterated-classrel-single-val` [-7;-5] THENA Auto)
   THEN UseSingleVal (-10) (-4)
   THEN InstHyp [⌈a⌉;⌈z⌉;⌈e2⌉] (-14)⋅
   THEN Auto
   THEN Unfold `prior-iterated-classrel` 0
   THEN OrRight
   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 : 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{}a:A. \mforall{}s:S. \mforall{}e:E. (e1 \mleq{}loc e {}\mRightarrow{} e \mleq{}loc e' {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} prior-iterated-classrel(es;A;S;s;X;f;init;e) {}\mRightarrow{} R[a;s;f a s])) {}\mRightarrow{} (\mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e,e2:E. (e1 \mleq{}loc e {}\mRightarrow{} (e <loc e2) {}\mRightarrow{} e2 \mleq{}loc e' {}\mRightarrow{} a1 \mmember{} X(e) {}\mRightarrow{} prior-iterated-classrel(es;A;S;s1;X;f;init;e) {}\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 \mleq{}loc e' {}\mRightarrow{} a \mmember{} X(e1) {}\mRightarrow{} prior-iterated-classrel(es;A;S;v1;X;f;init;e1) {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;e';v2) {}\mRightarrow{} R[a;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{}a:A. \mforall{}s:S. \mforall{}e:E. (e1 \mleq{}loc e {}\mRightarrow{} e \mleq{}loc e2 {}\mRightarrow{} a \mmember{} X(e) {}\mRightarrow{} prior-iterated-classrel(es;A;S;s;X;f;init;e) {}\mRightarrow{} R[a;s;f a s])@i 18. \mforall{}a1,a2:A. \mforall{}s1,s2:S. \mforall{}e,e':E. (e1 \mleq{}loc e {}\mRightarrow{} (e <loc e') {}\mRightarrow{} e' \mleq{}loc e2 {}\mRightarrow{} a1 \mmember{} X(e) {}\mRightarrow{} prior-iterated-classrel(es;A;S;s1;X;f;init;e) {}\mRightarrow{} a2 \mmember{} X(e') {}\mRightarrow{} iterated-classrel(es;S;A;f;init;X;pred(e');s2) {}\mRightarrow{} R[a1;s1;s2] {}\mRightarrow{} R[a1;s1;f a2 s2])@i 19. e1 = e2@i 20. a \mmember{} X(e1)@i 21. prior-iterated-classrel(es;A;S;v1;X;f;init;e1)@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) \mvdash{} R[a;v1;v2] By (D (-2) THEN ExRepD THEN Try (Complete ((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto))) THEN D (-7) THEN Auto THEN (RWO "-10<" (-5) THEN Auto) THEN (FLemma `iterated-classrel-single-val` [-7;-5] THENA Auto) THEN UseSingleVal (-10) (-4) THEN InstHyp [\mkleeneopen{}a\mkleeneclose{};\mkleeneopen{}z\mkleeneclose{};\mkleeneopen{}e2\mkleeneclose{}] (-14)\mcdot{} THEN Auto THEN Unfold `prior-iterated-classrel` 0 THEN OrRight THEN Auto) Home Index