Nuprl Lemma : l-ordered-from-upto-lt-nat-true
∀[n,m:ℕ]. (l-ordered(ℕ;x,y.x < y;[n, m)) ⇐⇒ True)
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
from-upto: [n, m),
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
true: True
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
true: True,
prop: ℙ,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
l-ordered: l-ordered(T;x,y.R[x; y];L)
Lemmas referenced :
l_before_wf,
member-less_than,
true_wf,
l-ordered-from-upto-lt-nat,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtype_rel_sets,
less_than_wf,
le_wf,
and_wf,
subtype_rel_list,
from-upto_wf,
nat_wf,
l-ordered_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
natural_numberEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
setElimination,
rename,
hypothesisEquality,
applyEquality,
setEquality,
intEquality,
independent_isectElimination,
sqequalRule,
because_Cache,
lambdaEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n,m:\mBbbN{}]. (l-ordered(\mBbbN{};x,y.x < y;[n, m)) \mLeftarrow{}{}\mRightarrow{} True)
Date html generated:
2016_05_15-PM-04_37_31
Last ObjectModification:
2016_01_16-AM-11_18_45
Theory : general
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