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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