Nuprl Lemma : compose_increasing
∀[k,m:ℕ]. ∀[f:ℕk ⟶ ℕm]. ∀[g:ℕm ⟶ ℤ]. (increasing(g o f;k)) supposing (increasing(g;m) and increasing(f;k))
Proof
Definitions occuring in Statement :
compose: f o g,
increasing: increasing(f;k),
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
increasing: increasing(f;k),
all: ∀x:A. B[x],
compose: f o g,
guard: {T},
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
increasing_implies,
int_seg_wf,
subtract_wf,
member-less_than,
compose_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
nat_wf,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel2,
lelt_wf,
add-member-int_seg2,
decidable__le,
not-le-2,
zero-add,
add-zero,
increasing_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalRule,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
setElimination,
rename,
lambdaEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
intEquality,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
unionElimination,
voidElimination,
independent_functionElimination,
addEquality,
minusEquality,
isect_memberEquality,
voidEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[k,m:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbN{}m]. \mforall{}[g:\mBbbN{}m {}\mrightarrow{} \mBbbZ{}].
(increasing(g o f;k)) supposing (increasing(g;m) and increasing(f;k))
Date html generated:
2016_05_13-PM-04_05_25
Last ObjectModification:
2015_12_26-AM-11_05_22
Theory : fun_1
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