Nuprl Lemma : triangular-num-le
∀[n,m:ℕ]. t(n) ≤ t(m) supposing n ≤ m
Proof
Definitions occuring in Statement :
triangular-num: t(n),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
all: ∀x:A. B[x],
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
less_than'_wf,
triangular-num_wf,
le_wf,
nat_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
subtype_base_sq,
int_subtype_base,
and_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
triangular-num-add1,
iff_weakening_equal,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
hypothesis,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
isect_memberEquality,
voidElimination,
lambdaFormation,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
dependent_set_memberEquality,
unionElimination,
instantiate,
cumulativity,
setEquality,
hyp_replacement,
applyLambdaEquality,
addEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[n,m:\mBbbN{}]. t(n) \mleq{} t(m) supposing n \mleq{} m
Date html generated:
2019_06_20-PM-02_38_12
Last ObjectModification:
2019_06_12-PM-00_26_39
Theory : num_thy_1
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