Nuprl Lemma : lifting-strict-less
∀[F:Base]. ∀[p,q,r:Top].
∀[a,b,c,d:Top]. (F[if (a) < (b) then c else d;p;q;r] ~ if (a) < (b) then F[c;p;q;r] else F[d;p;q;r])
supposing strict4(λx,y,z,w. F[x;y;z;w])
Proof
Definitions occuring in Statement :
strict4: strict4(F),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2;s3;s4],
less: if (a) < (b) then c else d,
lambda: λx.A[x],
base: Base,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
so_apply: x[s1;s2;s3;s4]
Lemmas referenced :
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-void,
has-value_wf_base,
is-exception_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
strict4_wf,
top_wf,
base_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalSqle,
sqleRule,
thin,
divergentSqle,
sqequalHypSubstitution,
sqequalRule,
productElimination,
hypothesis,
dependent_functionElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_functionElimination,
callbyvalueLess,
extract_by_obid,
isectElimination,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :universeIsType,
independent_pairFormation,
voidElimination,
natural_numberEquality,
imageMemberEquality,
imageElimination,
sqleReflexivity,
Error :dependent_pairFormation_alt,
Error :equalityIsType2,
promote_hyp,
instantiate,
cumulativity,
Error :equalityIsType1,
axiomSqleEquality,
lessExceptionCases,
exceptionSqequal,
exceptionLess
Latex:
\mforall{}[F:Base]. \mforall{}[p,q,r:Top].
\mforall{}[a,b,c,d:Top].
(F[if (a) < (b) then c else d;p;q;r] \msim{} if (a) < (b) then F[c;p;q;r] else F[d;p;q;r])
supposing strict4(\mlambda{}x,y,z,w. F[x;y;z;w])
Date html generated:
2019_06_20-AM-11_27_10
Last ObjectModification:
2018_09_28-PM-03_23_48
Theory : computation
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