### Nuprl Lemma : p-shift_wf

`∀[p:ℕ+]. ∀[a:p-adics(p)]. ∀[k:ℕ+].  p-shift(p;a;k) ∈ p-adics(p) supposing (a k) = 0 ∈ ℤ`

Proof

Definitions occuring in Statement :  p-shift: `p-shift(p;a;k)` p-adics: `p-adics(p)` nat_plus: `ℕ+` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` member: `t ∈ T` apply: `f a` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` prop: `ℙ` p-adics: `p-adics(p)` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` nat_plus: `ℕ+` p-shift: `p-shift(p;a;k)` all: `∀x:A. B[x]` int_upper: `{i...}` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` and: `P ∧ Q` sq_type: `SQType(T)` guard: `{T}` eqmod: `a ≡ b mod m` divides: `b | a` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` int_nzero: `ℤ-o` nequal: `a ≠ b ∈ T ` less_than: `a < b` squash: `↓T` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination thin intEquality applyEquality setElimination rename hypothesisEquality lambdaEquality natural_numberEquality baseClosed isect_memberEquality because_Cache functionExtensionality dependent_functionElimination dependent_set_memberEquality addEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation instantiate cumulativity productElimination universeEquality pointwiseFunctionality promote_hyp baseApply closedConclusion applyLambdaEquality setEquality lambdaFormation imageElimination multiplyEquality minusEquality divideEquality imageMemberEquality

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