Nuprl Lemma : iseg_product-positive
∀[i,j:ℕ]. (0 < iseg_product(i;j)) supposing ((i ≤ j) and 0 < i)
Proof
Definitions occuring in Statement :
iseg_product: iseg_product(i;j),
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iseg_product: iseg_product(i;j),
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
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,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
subtype_rel: A ⊆r B
Lemmas referenced :
nat_wf,
less_than_wf,
lelt_wf,
decidable__lt,
iseg_product_wf,
member-less_than,
int_formula_prop_less_lemma,
intformless_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_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,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf,
combinations-positive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
productElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[i,j:\mBbbN{}]. (0 < iseg\_product(i;j)) supposing ((i \mleq{} j) and 0 < i)
Date html generated:
2016_05_15-PM-06_01_37
Last ObjectModification:
2016_01_16-PM-00_39_54
Theory : general
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