Nuprl Lemma : div_rec_case
∀[a:ℕ]. ∀[n:ℕ+]. (a ÷ n) = (((a - n) ÷ n) + 1) ∈ ℤ supposing a ≥ n
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
divide: n ÷ m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
ge: i ≥ j ,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
div_nrel: Div(a;n;q),
lelt: i ≤ j < k
Lemmas referenced :
nat_wf,
nat_plus_wf,
ge_wf,
div_elim,
subtract_wf,
nat_plus_properties,
nat_properties,
decidable__le,
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,
equal-wf-base-T,
int_subtype_base,
equal-wf-T-base,
div_unique,
itermAdd_wf,
int_term_value_add_lemma,
add_cancel_in_le,
full-omega-unsat,
itermMultiply_wf,
itermMinus_wf,
istype-int,
istype-void,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
add_cancel_in_lt,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
Error :universeIsType,
hypothesis,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
dependent_set_memberEquality,
productElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
applyLambdaEquality,
hyp_replacement,
multiplyEquality,
minusEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (a \mdiv{} n) = (((a - n) \mdiv{} n) + 1) supposing a \mgeq{} n
Date html generated:
2019_06_20-PM-01_14_48
Last ObjectModification:
2019_01_15-PM-02_51_35
Theory : int_2
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