### Nuprl Lemma : sqn+1type_dep_product

`∀[T:Type]. ∀[S:T ⟶ Type]. ∀[n:ℕ].  (sqntype(n + 1;t:T × S[t])) supposing ((∀t:T. sqntype(n;S[t])) and sqntype(n;T))`

Proof

Definitions occuring in Statement :  sqntype: `sqntype(n;T)` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` product: `x:A × B[x]` add: `n + m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` uimplies: `b supposing a` sqntype: `sqntype(n;T)` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_apply: `x[s]` prop: `ℙ` pi1: `fst(t)` pi2: `snd(t)` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` sq_stable: `SqStable(P)` squash: `↓T` uiff: `uiff(P;Q)` and: `P ∧ Q` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` le: `A ≤ B` not: `¬A` less_than': `less_than'(a;b)` true: `True` false: `False` guard: `{T}`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  sqequalHypSubstitution sqequalRule Error :lambdaFormation_alt,  introduction Error :axiomSqequalN,  Error :equalityIsType4,  Error :productIsType,  Error :universeIsType,  hypothesisEquality applyEquality Error :inhabitedIsType,  cut extract_by_obid hypothesis Error :functionIsType,  isectElimination thin universeEquality equalityTransitivity equalitySymmetry productElimination Error :equalityIsType1,  dependent_functionElimination independent_functionElimination applyLambdaEquality setElimination rename natural_numberEquality because_Cache addEquality unionElimination sqequal_n rule imageMemberEquality baseClosed imageElimination independent_isectElimination Error :lambdaEquality_alt,  Error :isect_memberEquality_alt,  voidElimination minusEquality sqequalZero baseApply closedConclusion

Latex:
\mforall{}[T:Type].  \mforall{}[S:T  {}\mrightarrow{}  Type].  \mforall{}[n:\mBbbN{}].
(sqntype(n  +  1;t:T  \mtimes{}  S[t]))  supposing  ((\mforall{}t:T.  sqntype(n;S[t]))  and  sqntype(n;T))

Date html generated: 2019_06_20-AM-11_34_00
Last ObjectModification: 2018_09_29-PM-10_42_41

Theory : int_1

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