Nuprl Lemma : exp_functionality_wrt_le_1
∀[b:ℕ+]. ∀[x,y:ℕ]. b^x ≤ b^y supposing x ≤ y
Proof
Definitions occuring in Statement :
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uiff: uiff(P;Q),
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
guard: {T},
sq_type: SQType(T),
and: P ∧ Q,
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
nat_plus: ℕ+,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_plus_wf,
istype-nat,
le_witness_for_triv,
false_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
exp_wf4,
mul_preserves_le,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
exp_wf_nat_plus,
iff_weakening_equal,
subtype_rel_self,
exp_add,
exp_wf2,
true_wf,
squash_wf,
istype-le,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformle_wf,
intformand_wf,
decidable__le,
int_formula_prop_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
subtract_wf,
decidable__equal_int,
nat_plus_properties,
nat_properties,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
nat_wf,
subtype_base_sq
Rules used in proof :
isectIsTypeImplies,
equalityIstype,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
multiplyEquality,
applyLambdaEquality,
productElimination,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
applyEquality,
lambdaFormation_alt,
inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
unionElimination,
addEquality,
dependent_functionElimination,
rename,
setElimination,
hypothesisEquality,
natural_numberEquality,
lambdaEquality_alt,
intEquality,
sqequalRule,
independent_isectElimination,
hypothesis,
cumulativity,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[b:\mBbbN{}\msupplus{}]. \mforall{}[x,y:\mBbbN{}]. b\^{}x \mleq{} b\^{}y supposing x \mleq{} y
Date html generated:
2019_10_15-AM-10_25_18
Last ObjectModification:
2019_10_01-PM-02_17_59
Theory : num_thy_1
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