Nuprl Lemma : l_all_from-upto
∀n,m:ℤ. ∀P:ℤ ⟶ ℙ. ((∀x∈[n, m).P[x]) ⇐⇒ ∀x:ℤ. ((n ≤ x) ⇒ x < m ⇒ P[x]))
Proof
Definitions occuring in Statement :
from-upto: [n, m),
l_all: (∀x∈L.P[x]),
less_than: a < b,
prop: ℙ,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
uimplies: b supposing a,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top
Lemmas referenced :
all_wf,
set_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
squash_wf,
member-less_than,
sq_stable__less_than,
sq_stable__le,
sq_stable__and,
l_all_wf,
member-from-upto,
l_member_wf,
from-upto_wf,
less_than_wf,
le_wf,
and_wf,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
intEquality,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
productElimination,
independent_functionElimination,
dependent_set_memberEquality,
isect_memberEquality,
introduction,
because_Cache,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}P:\mBbbZ{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}x\mmember{}[n, m).P[x]) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:\mBbbZ{}. ((n \mleq{} x) {}\mRightarrow{} x < m {}\mRightarrow{} P[x]))
Date html generated:
2016_05_14-PM-02_02_36
Last ObjectModification:
2016_01_15-AM-08_07_38
Theory : list_1
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