### Nuprl Lemma : seq-normalize-normalize

`∀[n,m:ℕ]. ∀[s:Top].  (seq-normalize(n;seq-normalize(m;s)) ~ seq-normalize(if (n) < (m)  then n  else m;s))`

Proof

Definitions occuring in Statement :  seq-normalize: `seq-normalize(n;s)` nat: `ℕ` uall: `∀[x:A]. B[x]` top: `Top` less: `if (a) < (b)  then c  else d` sqequal: `s ~ t`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` seq-normalize: `seq-normalize(n;s)` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` cand: `A c∧ B` le: `A ≤ B` has-value: `(a)↓`
Lemmas referenced :  lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int less_than_wf 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 top_wf nat_wf set_subtype_base le_wf int_subtype_base not-lt-2 less-iff-le add_functionality_wrt_le add-associates base_wf add-swap add-commutes le-add-cancel equal-wf-base member_wf less_sqequal has-value_wf_base is-exception_wf bottom-sqle
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule thin rename extract_by_obid sqequalHypSubstitution isectElimination setElimination hypothesisEquality hypothesis lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination because_Cache lessCases sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality natural_numberEquality imageMemberEquality baseClosed imageElimination independent_functionElimination dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity impliesFunctionality applyEquality intEquality lambdaEquality baseApply closedConclusion addEquality productEquality sqequalSqle divergentSqle callbyvalueLess sqleReflexivity lessExceptionCases axiomSqleEquality exceptionSqequal

Latex:
\mforall{}[n,m:\mBbbN{}].  \mforall{}[s:Top].
(seq-normalize(n;seq-normalize(m;s))  \msim{}  seq-normalize(if  (n)  <  (m)    then  n    else  m;s))

Date html generated: 2017_04_14-AM-07_26_42
Last ObjectModification: 2017_02_27-PM-02_56_15

Theory : bar-induction

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