Nuprl Lemma : divides_subtract
∀x,y,z:ℤ. ((x | y) ⇒ (x | z) ⇒ (x | (y - z)))
Proof
Definitions occuring in Statement :
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
divides: b | a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
exists_wf,
equal_wf,
int_formula_prop_wf,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermVar_wf,
itermSubtract_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
subtract_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
because_Cache,
unionElimination,
natural_numberEquality,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
multiplyEquality
Latex:
\mforall{}x,y,z:\mBbbZ{}. ((x | y) {}\mRightarrow{} (x | z) {}\mRightarrow{} (x | (y - z)))
Date html generated:
2016_05_14-PM-04_16_28
Last ObjectModification:
2016_01_14-PM-11_42_28
Theory : num_thy_1
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