Nuprl Lemma : fshift_increasing
∀[n:ℕ]. ∀[x:ℤ]. ∀[f:ℕn ⟶ ℤ]. (increasing(fshift(f;x);n + 1)) supposing (x < f 0 and 0 < n and increasing(f;n))
Proof
Definitions occuring in Statement :
fshift: fshift(f;x),
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
fshift: fshift(f;x),
increasing: increasing(f;k),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
int_seg: {i..j-},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
guard: {T},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
false: False,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
label: ...$L... t,
decidable: Dec(P),
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
subtract: n - m,
so_apply: x[s]
Lemmas referenced :
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
add-subtract-cancel,
int_seg_properties,
nat_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
less_than_wf,
decidable__equal_int,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
iff_weakening_equal,
general_arith_equation1,
int_seg_wf,
subtract_wf,
member-less_than,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf,
all_wf,
add-member-int_seg2,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
addEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
promote_hyp,
instantiate,
applyEquality,
imageElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
cumulativity,
functionExtensionality,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[x:\mBbbZ{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(increasing(fshift(f;x);n + 1)) supposing (x < f 0 and 0 < n and increasing(f;n))
Date html generated:
2018_05_21-PM-01_00_11
Last ObjectModification:
2018_05_19-AM-06_38_45
Theory : num_thy_1
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