Nuprl Lemma : exp_preserves_lt
∀[n:ℕ+]. ∀[x,y:ℕ]. x^n < y^n supposing x < y
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
nat: ℕ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
all: ∀x:A. B[x],
nat_plus: ℕ+,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
subtract: n - m,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
int_term_value_mul_lemma,
itermMultiply_wf,
add-subtract-cancel,
minus-zero,
not-equal-2,
mul_preserves_lt,
exp_wf_nat_plus,
le_weakening,
less_than_transitivity2,
mul_bounds_1b,
add-swap,
zero-mul,
int_subtype_base,
subtype_base_sq,
decidable__equal_int,
exp_step,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
less-iff-le,
not-lt-2,
decidable__lt,
nat_plus_subtype_nat,
primrec-wf-nat-plus,
nat_plus_wf,
isect_wf,
uall_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
nat_plus_properties,
nat_wf,
le_wf,
false_wf,
exp_wf2,
member-less_than,
iff_weakening_equal,
exp1,
true_wf,
squash_wf,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
lemma_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
intEquality,
setElimination,
rename,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
isect_memberEquality,
dependent_set_memberEquality,
independent_pairFormation,
lambdaFormation,
because_Cache,
addEquality,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
instantiate,
cumulativity,
multiplyEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbN{}]. x\^{}n < y\^{}n supposing x < y
Date html generated:
2016_05_14-PM-04_26_36
Last ObjectModification:
2016_01_14-PM-11_37_44
Theory : num_thy_1
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