Nuprl Lemma : sqle_n_subtype_rel
∀[a,b:Base]. ∀[n:ℕ]. a ≤n b supposing a ≤n + 1 b
Proof
Definitions occuring in Statement :
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
add: n + m,
natural_number: $n,
base: Base,
sqle_n: s ≤n t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
sqle_n_wf,
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
le_wf,
nat_wf,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
Error :inhabitedIsType,
sqlenSubtypeRel
Latex:
\mforall{}[a,b:Base]. \mforall{}[n:\mBbbN{}]. a \mleq{}n b supposing a \mleq{}n + 1 b
Date html generated:
2019_06_20-AM-11_33_48
Last ObjectModification:
2018_09_26-AM-11_31_53
Theory : int_1
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