Nuprl Lemma : l-ordered-from-upto-lt-nat
∀[n,m:ℕ]. l-ordered(ℕ;x,y.x < y;[n, m))
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]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
exists: ∃x:A. B[x],
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
prop: ℙ,
l-ordered: l-ordered(T;x,y.R[x; y];L),
subtype_rel: A ⊆r B,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
squash: ↓T,
label: ...$L... t,
guard: {T},
sq_type: SQType(T)
Lemmas referenced :
decidable__le,
from-upto-is-nil,
l-ordered-nil-true,
less_than_wf,
nat_wf,
subtract_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
equal_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
member-less_than,
l_before_wf,
from-upto_wf,
from-upto-nil,
l-ordered-nil,
subtype_rel_list,
subtype_rel_sets,
l-ordered_wf,
squash_wf,
true_wf,
list_wf,
from-upto-decomp-last,
decidable__lt,
strong-subtype-equal-lists,
strong-subtype-set3,
strong-subtype-self,
append_wf,
cons_wf,
nil_wf,
iff_weakening_equal,
l-ordered-append,
subtype_base_sq,
int_subtype_base,
l-ordered-single,
member_singleton,
from-upto-member-nat,
l-ordered-cons,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
isectElimination,
because_Cache,
productElimination,
independent_pairFormation,
independent_isectElimination,
sqequalRule,
lambdaEquality,
independent_functionElimination,
natural_numberEquality,
dependent_pairFormation,
dependent_set_memberEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
intWeakElimination,
lambdaFormation,
applyEquality,
setEquality,
productEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
imageMemberEquality,
baseClosed,
instantiate,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[n,m:\mBbbN{}]. l-ordered(\mBbbN{};x,y.x < y;[n, m))
Date html generated:
2018_05_21-PM-07_37_41
Last ObjectModification:
2017_07_26-PM-05_11_58
Theory : general
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