Nuprl Lemma : isr-rep_int_constraint_step
∀[f:IntConstraints ⟶ IntConstraints]. ∀[p:IntConstraints].
(unsat(p)) supposing
((↑isr(rep_int_constraint_step(f;p))) and
((∀p:IntConstraints
(0 < dim(p)
⇒ (dim(f p) < dim(p) ∨ ((dim(f p) = dim(p) ∈ ℤ) ∧ num-eq-constraints(f p) < num-eq-constraints(p)))))
∧ (∀p:IntConstraints. (unsat(f p) ⇒ unsat(p)))))
Proof
Definitions occuring in Statement :
rep_int_constraint_step: rep_int_constraint_step(f;p),
num-eq-constraints: num-eq-constraints(p),
int-problem-dimension: dim(p),
unsat-int-problem: unsat(p),
int-constraint-problem: IntConstraints,
assert: ↑b,
isr: isr(x),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
prop: ℙ,
unsat-int-problem: unsat(p),
not: ¬A,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
nat_plus: ℕ+,
less_than: a < b,
int-constraint-problem: IntConstraints,
rep_int_constraint_step: rep_int_constraint_step(f;p),
callbyvalueall: callbyvalueall,
unit: Unit,
has-value: (a)↓,
has-valueall: has-valueall(a),
bool: 𝔹,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isr: isr(x),
satisfies-int-constraint-problem: xs |= p
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
satisfies-int-constraint-problem_wf,
list_wf,
assert_wf,
isr_wf,
tunion_wf,
nat_wf,
equal-wf-base-T,
unit_wf2,
rep_int_constraint_step_wf,
int-constraint-problem_wf,
int-problem-dimension_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
add_nat_wf,
le_wf,
sq_stable__le,
equal_wf,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
all_wf,
or_wf,
num-eq-constraints_wf,
unsat-int-problem_wf,
subtype_rel-equal,
base_wf,
le_weakening,
le_reflexive,
one-mul,
two-mul,
mul-distributes-right,
minus-zero,
omega-shadow,
list_subtype_base,
int_subtype_base,
equal-wf-T-base,
evalall-reduce,
union-valueall-type,
tunion-valueall-type,
product-valueall-type,
list-valueall-type,
set-valueall-type,
int-valueall-type,
equal-valueall-type,
valueall-type-has-valueall,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_transitivity2,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
lambdaFormation,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
intEquality,
productEquality,
setEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
addEquality,
functionExtensionality,
unionElimination,
independent_pairFormation,
minusEquality,
isect_memberEquality,
voidEquality,
dependent_set_memberEquality,
imageMemberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
functionEquality,
dependent_pairFormation,
sqequalIntensionalEquality,
applyLambdaEquality,
promote_hyp,
addLevel,
levelHypothesis,
callbyvalueReduce,
equalityElimination,
lessCases,
sqequalAxiom,
instantiate,
cumulativity
Latex:
\mforall{}[f:IntConstraints {}\mrightarrow{} IntConstraints]. \mforall{}[p:IntConstraints].
(unsat(p)) supposing
((\muparrow{}isr(rep\_int\_constraint\_step(f;p))) and
((\mforall{}p:IntConstraints
(0 < dim(p)
{}\mRightarrow{} (dim(f p) < dim(p)
\mvee{} ((dim(f p) = dim(p)) \mwedge{} num-eq-constraints(f p) < num-eq-constraints(p)))))
\mwedge{} (\mforall{}p:IntConstraints. (unsat(f p) {}\mRightarrow{} unsat(p)))))
Date html generated:
2017_04_14-AM-09_10_59
Last ObjectModification:
2017_02_27-PM-03_48_31
Theory : omega
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