Nuprl Lemma : fadd_increasing
∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ]. (increasing(fadd(f;g);n)) supposing (nondecreasing(g;n) and increasing(f;n))
Proof
Definitions occuring in Statement :
fadd: fadd(f;g),
nondecreasing: nondecreasing(f;k),
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 :
fadd: fadd(f;g),
increasing: increasing(f;k),
nondecreasing: nondecreasing(f;k),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
guard: {T},
nat: ℕ,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
subtract: n - m,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_wf,
less_than_wf,
le_wf,
all_wf,
add-member-int_seg2,
member-less_than,
int_seg_wf,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
itermAdd_wf,
intformle_wf,
decidable__le,
lelt_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
subtract_wf,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
lemma_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
productElimination,
addEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(increasing(fadd(f;g);n)) supposing (nondecreasing(g;n) and increasing(f;n))
Date html generated:
2016_05_14-PM-09_30_07
Last ObjectModification:
2016_01_14-PM-11_33_33
Theory : num_thy_1
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