### Nuprl Lemma : seq-append_wf_consistent

`∀T:Type. ∀R:n:ℕ ⟶ (ℕn ⟶ T) ⟶ T ⟶ ℙ. ∀n:ℕ. ∀s:R-consistent-seq(n). ∀t:T.`
`  ((R n s t) `` (seq-append(n;1;s;λi.t) ∈ R-consistent-seq(n + 1)))`

Proof

Definitions occuring in Statement :  consistent-seq: `R-consistent-seq(n)` seq-append: `seq-append(n;m;s1;s2)` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` consistent-seq: `R-consistent-seq(n)` uall: `∀[x:A]. B[x]` nat: `ℕ` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` seq-append: `seq-append(n;m;s1;s2)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` less_than: `a < b` top: `Top` true: `True` squash: `↓T` bfalse: `ff` exists: `∃x:A. B[x]` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_stable: `SqStable(P)` subtract: `n - m`
Lemmas referenced :  seq-append_wf false_wf le_wf int_seg_wf all_wf nat_wf int_seg_subtype_nat subtype_rel_dep_function subtype_rel_sets and_wf less_than_wf less_than_transitivity2 le_weakening2 consistent-seq_wf decidable__int_equal lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_transitivity1 le_weakening less_than_irreflexivity eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot sq_stable__le subtype_rel_self subtype_rel_wf not-lt-2 less-iff-le le_antisymmetry_iff add_functionality_wrt_le add-associates add-swap add-commutes le-add-cancel condition-implies-le minus-add minus-one-mul minus-one-mul-top zero-add le-add-cancel2 not-equal-2 decidable__lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution setElimination thin rename dependent_set_memberEquality introduction extract_by_obid isectElimination cumulativity hypothesisEquality natural_numberEquality sqequalRule independent_pairFormation hypothesis functionExtensionality applyEquality lambdaEquality because_Cache addEquality independent_isectElimination intEquality setEquality productElimination dependent_functionElimination functionEquality universeEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry lessCases isect_memberFormation sqequalAxiom isect_memberEquality voidElimination voidEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_pairFormation promote_hyp instantiate impliesFunctionality addLevel hyp_replacement levelHypothesis applyLambdaEquality minusEquality

Latex:
\mforall{}T:Type.  \mforall{}R:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}.  \mforall{}n:\mBbbN{}.  \mforall{}s:R-consistent-seq(n).  \mforall{}t:T.
((R  n  s  t)  {}\mRightarrow{}  (seq-append(n;1;s;\mlambda{}i.t)  \mmember{}  R-consistent-seq(n  +  1)))

Date html generated: 2017_04_14-AM-07_26_31
Last ObjectModification: 2017_02_27-PM-02_56_00

Theory : bar-induction

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