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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