Nuprl Lemma : test-add-member-elim
∀x,y:Base. ∀d:ℤ. ((x + y ~ d) ⇒ (0 < x ∨ (x ≤ 0)))
Proof
Definitions occuring in Statement :
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
add: n + m,
natural_number: $n,
int: ℤ,
base: Base,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
has-value: (a)↓,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_or_lemma,
int_formula_prop_not_lemma,
intformle_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformor_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
decidable__lt,
le_wf,
less_than_wf,
decidable__or,
int-value-type,
value-type-has-value,
base_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalIntensionalEquality,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
thin,
lemma_by_obid,
hypothesis,
sqequalHypSubstitution,
intEquality,
isectElimination,
independent_isectElimination,
callbyvalueAdd,
productElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_functionElimination,
dependent_functionElimination,
because_Cache,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}x,y:Base. \mforall{}d:\mBbbZ{}. ((x + y \msim{} d) {}\mRightarrow{} (0 < x \mvee{} (x \mleq{} 0)))
Date html generated:
2016_05_15-PM-07_49_52
Last ObjectModification:
2016_01_16-AM-09_33_05
Theory : general
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