Nuprl Lemma : exp-le-iff
∀[n:ℕ+]. ∀[x,y:ℕ]. uiff(x ≤ y;x^n ≤ y^n)
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
exp_preserves_lt,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
nat_properties,
decidable__le,
nat_plus_subtype_nat,
exp_preserves_le,
nat_plus_wf,
nat_wf,
le_wf,
exp_wf2,
less_than'_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesis,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
isect_memberEquality,
independent_isectElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbN{}]. uiff(x \mleq{} y;x\^{}n \mleq{} y\^{}n)
Date html generated:
2016_05_14-PM-04_26_40
Last ObjectModification:
2016_01_14-PM-11_37_05
Theory : num_thy_1
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