Nuprl Lemma : int_formula-subtype-base
int_formula() ⊆r Base
Proof
Definitions occuring in Statement :
int_formula: int_formula(),
subtype_rel: A ⊆r B,
base: Base
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
intformless: (left "<" right),
int_formula_size: int_formula_size(p),
pi1: fst(t),
pi2: snd(t),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
intformle: left "≤" right,
intformeq: left "=" right,
intformand: left "∧" right,
sq_stable: SqStable(P),
squash: ↓T,
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
intformor: left "or" right,
intformimplies: left "=>" right,
intformnot: "¬"form,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
le_wf,
int_formula_size_wf,
int_formula_wf,
int_formula-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
int_term_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
add_nat_wf,
nat_wf,
sq_stable__le,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-one-mul-top,
minus-add,
minus-minus,
add-swap,
add-commutes,
set_subtype_base,
int_subtype_base,
add-is-int-iff,
not-le-2,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
cut,
thin,
isect_memberFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
promote_hyp,
productElimination,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
imageMemberEquality,
imageElimination,
independent_pairFormation,
voidEquality,
intEquality,
minusEquality,
sqequalIntensionalEquality,
multiplyEquality
Latex:
int\_formula() \msubseteq{}r Base
Date html generated:
2017_09_29-PM-05_55_51
Last ObjectModification:
2017_05_31-PM-02_46_26
Theory : omega
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