Nuprl Lemma : one-rem
∀[m:ℤ]. 1 rem m ~ 1 supposing 1 < m
Proof
Definitions occuring in Statement :
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
squash: ↓T,
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
uiff: uiff(P;Q)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
div_rem_sum,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
nequal_wf,
equal_wf,
squash_wf,
true_wf,
quotient-is-zero,
false_wf,
le_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
iff_weakening_equal,
decidable__equal_int,
add-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
less_than_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
lambdaFormation,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
baseClosed,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
unionElimination,
imageMemberEquality,
productElimination,
pointwiseFunctionality,
rename,
promote_hyp,
baseApply,
closedConclusion,
sqequalAxiom
Latex:
\mforall{}[m:\mBbbZ{}]. 1 rem m \msim{} 1 supposing 1 < m
Date html generated:
2018_05_21-PM-00_25_46
Last ObjectModification:
2017_11_03-PM-01_55_32
Theory : int_2
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