---

Step * 2 1 of Lemma mFOL-abstract-rename


1. Dom : Type
2. S : FOStruct(Dom)
3. x : ℤ
4. y : ℤ
5. knd : Atom
6. left : mFOL()
7. right : mFOL()
8. ¬(x ∈ mFOL-boundvars(mFOconnect(knd;left;right)))
9. a1 : FOAssignment(mFOL-freevars(mFOL-rename(mFOconnect(knd;left;right);y;x)),Dom)
10. a2 : FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
11. a2
= FOAssigment-rename(a1;mFOconnect(knd;left;right);x;y)
∈ FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
12. ¬(x ∈ mFOL-boundvars(left))
13. ¬(x ∈ mFOL-boundvars(right))
14. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(right;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(right),Dom).
      ((a2 = FOAssigment-rename(a1;right;x;y) ∈ FOAssignment(mFOL-freevars(right),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(right) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(right;y;x)) ∈ ℙ))
15. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(left;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(left),Dom).
      ((a2 = FOAssigment-rename(a1;left;x;y) ∈ FOAssignment(mFOL-freevars(left),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(left) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(left;y;x)) ∈ ℙ))
⊢ Dom,S,a2 |= mFOL-abstract(mFOconnect(knd;left;right))
= Dom,S,a1 |= mFOL-abstract(mFOL-rename(mFOconnect(knd;left;right);y;x))
∈ ℙ
BY
{ (All (Unfold `FOAssigment-rename`) THEN (SplitOnHypITE -5  THENA Auto)) }

1
.....truecase..... 
1. Dom : Type
2. S : FOStruct(Dom)
3. x : ℤ
4. y : ℤ
5. knd : Atom
6. left : mFOL()
7. right : mFOL()
8. ¬(x ∈ mFOL-boundvars(mFOconnect(knd;left;right)))
9. a1 : FOAssignment(mFOL-freevars(mFOL-rename(mFOconnect(knd;left;right);y;x)),Dom)
10. a2 : FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
11. a2 = a1[y := a1 x] ∈ FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
12. ¬(x ∈ mFOL-boundvars(left))
13. ¬(x ∈ mFOL-boundvars(right))
14. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(right;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(right),Dom).
      ((a2 = if y ∈b mFOL-freevars(right) then a1[y := a1 x] else a1 fi  ∈ FOAssignment(mFOL-freevars(right),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(right) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(right;y;x)) ∈ ℙ))
15. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(left;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(left),Dom).
      ((a2 = if y ∈b mFOL-freevars(left) then a1[y := a1 x] else a1 fi  ∈ FOAssignment(mFOL-freevars(left),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(left) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(left;y;x)) ∈ ℙ))
16. (y ∈ mFOL-freevars(mFOconnect(knd;left;right)))
⊢ Dom,S,a2 |= mFOL-abstract(mFOconnect(knd;left;right))
= Dom,S,a1 |= mFOL-abstract(mFOL-rename(mFOconnect(knd;left;right);y;x))
∈ ℙ

2
.....falsecase..... 
1. Dom : Type
2. S : FOStruct(Dom)
3. x : ℤ
4. y : ℤ
5. knd : Atom
6. left : mFOL()
7. right : mFOL()
8. ¬(x ∈ mFOL-boundvars(mFOconnect(knd;left;right)))
9. a1 : FOAssignment(mFOL-freevars(mFOL-rename(mFOconnect(knd;left;right);y;x)),Dom)
10. a2 : FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
11. a2 = a1 ∈ FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
12. ¬(x ∈ mFOL-boundvars(left))
13. ¬(x ∈ mFOL-boundvars(right))
14. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(right;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(right),Dom).
      ((a2 = if y ∈b mFOL-freevars(right) then a1[y := a1 x] else a1 fi  ∈ FOAssignment(mFOL-freevars(right),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(right) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(right;y;x)) ∈ ℙ))
15. ∀a1:FOAssignment(mFOL-freevars(mFOL-rename(left;y;x)),Dom). ∀a2:FOAssignment(mFOL-freevars(left),Dom).
      ((a2 = if y ∈b mFOL-freevars(left) then a1[y := a1 x] else a1 fi  ∈ FOAssignment(mFOL-freevars(left),Dom))
      ⇒ (Dom,S,a2 |= mFOL-abstract(left) = Dom,S,a1 |= mFOL-abstract(mFOL-rename(left;y;x)) ∈ ℙ))
16. ¬(y ∈ mFOL-freevars(mFOconnect(knd;left;right)))
⊢ Dom,S,a2 |= mFOL-abstract(mFOconnect(knd;left;right))
= Dom,S,a1 |= mFOL-abstract(mFOL-rename(mFOconnect(knd;left;right);y;x))
∈ ℙ

---

Latex:

---

Latex:

1.  Dom  :  Type
2.  S  :  FOStruct(Dom)
3.  x  :  \mBbbZ{}
4.  y  :  \mBbbZ{}
5.  knd  :  Atom
6.  left  :  mFOL()
7.  right  :  mFOL()
8.  \mneg{}(x  \mmember{}  mFOL-boundvars(mFOconnect(knd;left;right)))
9.  a1  :  FOAssignment(mFOL-freevars(mFOL-rename(mFOconnect(knd;left;right);y;x)),Dom)
10.  a2  :  FOAssignment(mFOL-freevars(mFOconnect(knd;left;right)),Dom)
11.  a2  =  FOAssigment-rename(a1;mFOconnect(knd;left;right);x;y)
12.  \mneg{}(x  \mmember{}  mFOL-boundvars(left))
13.  \mneg{}(x  \mmember{}  mFOL-boundvars(right))
14.  \mforall{}a1:FOAssignment(mFOL-freevars(mFOL-rename(right;y;x)),Dom).
        \mforall{}a2:FOAssignment(mFOL-freevars(right),Dom).
            ((a2  =  FOAssigment-rename(a1;right;x;y))
            {}\mRightarrow{}  (Dom,S,a2  |=  mFOL-abstract(right)  =  Dom,S,a1  |=  mFOL-abstract(mFOL-rename(right;y;x))))
15.  \mforall{}a1:FOAssignment(mFOL-freevars(mFOL-rename(left;y;x)),Dom).
        \mforall{}a2:FOAssignment(mFOL-freevars(left),Dom).
            ((a2  =  FOAssigment-rename(a1;left;x;y))
            {}\mRightarrow{}  (Dom,S,a2  |=  mFOL-abstract(left)  =  Dom,S,a1  |=  mFOL-abstract(mFOL-rename(left;y;x))))
\mvdash{}  Dom,S,a2  |=  mFOL-abstract(mFOconnect(knd;left;right))
=  Dom,S,a1  |=  mFOL-abstract(mFOL-rename(mFOconnect(knd;left;right);y;x))


By

---

Latex:
(All  (Unfold  `FOAssigment-rename`)  THEN  (SplitOnHypITE  -5    THENA  Auto))

---

Home Index