Step
*
2
1
1
1
of Lemma
padic-ring_wf
1. p : {2...}
2. ℤ(p) ∈ RngSig
3. ∀[x,y,z:p-adics(p)]. (x + y + z = x + y + z ∈ p-adics(p))
4. ∀[x:p-adics(p)]. ((x + 0(p) = x ∈ p-adics(p)) ∧ (0(p) + x = x ∈ p-adics(p)))
5. ∀[x:p-adics(p)]. ((x + -(x) = 0(p) ∈ p-adics(p)) ∧ (-(x) + x = 0(p) ∈ p-adics(p)))
6. ∀[x,y,z:p-adics(p)]. (x * y * z = x * y * z ∈ p-adics(p))
7. ∀[x:p-adics(p)]. ((x * 1(p) = x ∈ p-adics(p)) ∧ (1(p) * x = x ∈ p-adics(p)))
8. ∀[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y ∈ p-adics(p)) ∧ (x + y * a = x * a + y * a ∈ p-adics(p)))
9. ∀[x,y:p-adics(p)]. (x * y = y * x ∈ p-adics(p))
10. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. x : padic(p)
13. y : padic(p)
14. z : padic(p)
⊢ bpa-equiv(p;bpa-add(p;x;bpa-add(p;y;z));bpa-add(p;bpa-add(p;x;y);z))
BY
{ (((GenConcl ⌜x = X ∈ basic-padic(p)⌝⋅ THENA Auto) THEN Thin (-1) THEN D -1)
THEN ((GenConcl ⌜y = Y ∈ basic-padic(p)⌝⋅ THENA Auto) THEN Thin (-1) THEN D -1)
THEN ((GenConcl ⌜z = Z ∈ basic-padic(p)⌝⋅ THENA Auto) THEN Thin (-1) THEN D -1)
THEN RepUR ``bpa-equiv bpa-add`` 0
THEN (Assert (Y1 ≤ imax(Y1;Z1)) ∧ (Z1 ≤ imax(Y1;Z1)) BY
Auto)
THEN MoveToConcl (-1)
THEN (GenConcl ⌜imax(Y1;Z1) = M ∈ ℤ⌝⋅ THENA Auto)
THEN (D 0 THENA Auto)
THEN (Assert (X1 ≤ imax(X1;Y1)) ∧ (Y1 ≤ imax(X1;Y1)) BY
Auto)
THEN MoveToConcl (-1)
THEN (GenConcl ⌜imax(X1;Y1) = N ∈ ℤ⌝⋅ THENA Auto)
THEN (D 0 THENA Auto)
THEN (CallByValueReduceOn ⌜M⌝ 0⋅ THENA Auto)
THEN (CallByValueReduceOn ⌜N⌝ 0⋅ THENA Auto)
THEN ((CallByValueReduceOn ⌜p^M - Y1⌝ 0⋅ THENA Auto) THEN (CallByValueReduceOn ⌜p^M - Z1⌝ 0⋅ THENA Auto))
THEN (CallByValueReduceOn ⌜p^N - X1⌝ 0⋅ THENA Auto)
THEN (CallByValueReduceOn ⌜p^N - Y1⌝ 0⋅ THENA Auto)
THEN Reduce 0) }
1
1. p : {2...}
2. ℤ(p) ∈ RngSig
3. ∀[x,y,z:p-adics(p)]. (x + y + z = x + y + z ∈ p-adics(p))
4. ∀[x:p-adics(p)]. ((x + 0(p) = x ∈ p-adics(p)) ∧ (0(p) + x = x ∈ p-adics(p)))
5. ∀[x:p-adics(p)]. ((x + -(x) = 0(p) ∈ p-adics(p)) ∧ (-(x) + x = 0(p) ∈ p-adics(p)))
6. ∀[x,y,z:p-adics(p)]. (x * y * z = x * y * z ∈ p-adics(p))
7. ∀[x:p-adics(p)]. ((x * 1(p) = x ∈ p-adics(p)) ∧ (1(p) * x = x ∈ p-adics(p)))
8. ∀[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y ∈ p-adics(p)) ∧ (x + y * a = x * a + y * a ∈ p-adics(p)))
9. ∀[x,y:p-adics(p)]. (x * y = y * x ∈ p-adics(p))
10. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. ∀a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. x : padic(p)
13. y : padic(p)
14. z : padic(p)
15. X1 : ℕ
16. X2 : p-adics(p)
17. Y1 : ℕ
18. Y2 : p-adics(p)
19. Z1 : ℕ
20. Z2 : p-adics(p)
21. M : ℤ
22. imax(Y1;Z1) = M ∈ ℤ
23. (Y1 ≤ M) ∧ (Z1 ≤ M)
24. N : ℤ
25. imax(X1;Y1) = N ∈ ℤ
26. (X1 ≤ N) ∧ (Y1 ≤ N)
⊢ let n,a = eval k = imax(X1;M) in
eval c = p^k - X1 in
eval d = p^k - M in
<k, c(p) * X2 + d(p) * p^M - Y1(p) * Y2 + p^M - Z1(p) * Z2>
in let m,b = eval k = imax(N;Z1) in
eval c = p^k - N in
eval d = p^k - Z1 in
<k, c(p) * p^N - X1(p) * X2 + p^N - Y1(p) * Y2 + d(p) * Z2>
in p^m(p) * a = p^n(p) * b ∈ p-adics(p)
Latex:
Latex:
1. p : \{2...\}
2. \mBbbZ{}(p) \mmember{} RngSig
3. \mforall{}[x,y,z:p-adics(p)]. (x + y + z = x + y + z)
4. \mforall{}[x:p-adics(p)]. ((x + 0(p) = x) \mwedge{} (0(p) + x = x))
5. \mforall{}[x:p-adics(p)]. ((x + -(x) = 0(p)) \mwedge{} (-(x) + x = 0(p)))
6. \mforall{}[x,y,z:p-adics(p)]. (x * y * z = x * y * z)
7. \mforall{}[x:p-adics(p)]. ((x * 1(p) = x) \mwedge{} (1(p) * x = x))
8. \mforall{}[a,x,y:p-adics(p)]. ((a * x + y = a * x + a * y) \mwedge{} (x + y * a = x * a + y * a))
9. \mforall{}[x,y:p-adics(p)]. (x * y = y * x)
10. \mforall{}a,b:basic-padic(p). bpa-equiv(p;bpa-mul(p;a;b);pa-mul(p;a;b))
11. \mforall{}a,b:basic-padic(p). bpa-equiv(p;bpa-add(p;a;b);pa-add(p;a;b))
12. x : padic(p)
13. y : padic(p)
14. z : padic(p)
\mvdash{} bpa-equiv(p;bpa-add(p;x;bpa-add(p;y;z));bpa-add(p;bpa-add(p;x;y);z))
By
Latex:
(((GenConcl \mkleeneopen{}x = X\mkleeneclose{}\mcdot{} THENA Auto) THEN Thin (-1) THEN D -1)
THEN ((GenConcl \mkleeneopen{}y = Y\mkleeneclose{}\mcdot{} THENA Auto) THEN Thin (-1) THEN D -1)
THEN ((GenConcl \mkleeneopen{}z = Z\mkleeneclose{}\mcdot{} THENA Auto) THEN Thin (-1) THEN D -1)
THEN RepUR ``bpa-equiv bpa-add`` 0
THEN (Assert (Y1 \mleq{} imax(Y1;Z1)) \mwedge{} (Z1 \mleq{} imax(Y1;Z1)) BY
Auto)
THEN MoveToConcl (-1)
THEN (GenConcl \mkleeneopen{}imax(Y1;Z1) = M\mkleeneclose{}\mcdot{} THENA Auto)
THEN (D 0 THENA Auto)
THEN (Assert (X1 \mleq{} imax(X1;Y1)) \mwedge{} (Y1 \mleq{} imax(X1;Y1)) BY
Auto)
THEN MoveToConcl (-1)
THEN (GenConcl \mkleeneopen{}imax(X1;Y1) = N\mkleeneclose{}\mcdot{} THENA Auto)
THEN (D 0 THENA Auto)
THEN (CallByValueReduceOn \mkleeneopen{}M\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN (CallByValueReduceOn \mkleeneopen{}N\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN ((CallByValueReduceOn \mkleeneopen{}p\^{}M - Y1\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN (CallByValueReduceOn \mkleeneopen{}p\^{}M - Z1\mkleeneclose{} 0\mcdot{} THENA Auto)
)
THEN (CallByValueReduceOn \mkleeneopen{}p\^{}N - X1\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN (CallByValueReduceOn \mkleeneopen{}p\^{}N - Y1\mkleeneclose{} 0\mcdot{} THENA Auto)
THEN Reduce 0)
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