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

Proof

Definitions occuring in Statement :  consistent-seq: `R-consistent-seq(n)` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` apply: `f a` 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` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` prop: `ℙ` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` seq-add: `s.x@n` sq_stable: `SqStable(P)` squash: `↓T` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` 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` less_than: `a < b` subtract: `n - m` top: `Top` true: `True`
Lemmas referenced :  seq-add_wf int_seg_wf all_wf nat_wf int_seg_subtype_nat false_wf subtype_rel_dep_function subtype_rel_sets and_wf le_wf less_than_wf less_than_transitivity2 le_weakening2 consistent-seq_wf decidable__int_equal sq_stable__le equal_wf subtype_rel_self subtype_rel_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int less_than_transitivity1 le_weakening less_than_irreflexivity eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base iff_transitivity assert_wf bnot_wf not_wf iff_weakening_uiff assert_of_bnot le_antisymmetry_iff less-iff-le condition-implies-le add-associates minus-add minus-one-mul add-swap minus-one-mul-top zero-add add_functionality_wrt_le add-commutes le-add-cancel2 decidable__lt not-lt-2 not-equal-2 le-add-cancel
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalHypSubstitution setElimination thin rename dependent_set_memberEquality introduction extract_by_obid isectElimination cumulativity hypothesisEquality functionExtensionality applyEquality natural_numberEquality hypothesis addEquality because_Cache sqequalRule lambdaEquality independent_isectElimination independent_pairFormation intEquality setEquality productElimination dependent_functionElimination functionEquality universeEquality unionElimination int_eqReduceTrueSq addLevel hyp_replacement equalitySymmetry levelHypothesis independent_functionElimination imageMemberEquality baseClosed imageElimination equalityTransitivity instantiate applyLambdaEquality equalityElimination voidElimination dependent_pairFormation promote_hyp impliesFunctionality int_eqReduceFalseSq isect_memberEquality voidEquality 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{}  (s.t@n  \mmember{}  R-consistent-seq(n  +  1)))

Date html generated: 2017_04_14-AM-07_26_34
Last ObjectModification: 2017_02_27-PM-02_55_55

Theory : bar-induction

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