Nuprl Lemma : assert-ctt-is-fibrant
∀[t:CttTerm]
∀X:?CubicalContext. [[X;t]] ⊆r Provisional''''(cttType(context-set(X))) supposing context-ok(X)
supposing ↑ctt-is-fibrant(t)
Proof
Definitions occuring in Statement :
ctt-is-fibrant: ctt-is-fibrant(t),
ctt_meaning: [[ctxt;t]],
ctt-term: CttTerm,
context-set: context-set(ctxt),
context-ok: context-ok(ctxt),
cubical-context: ?CubicalContext,
ctt-type-meaning: cttType(X),
assert: ↑b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
provisional-type: Provisional(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
ctt_meaning: [[ctxt;t]],
prop: ℙ,
implies: P ⇒ Q,
ctt-meaning-type: ctt-meaning-type{i:l}(X;t),
ctt-term: CttTerm,
wfterm: wfterm(opr;sort;arity),
ctt-is-fibrant: ctt-is-fibrant(t),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
assert: ↑b,
bfalse: ff,
false: False,
band: p ∧b q,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
eq_atom: x =a y,
bool: 𝔹,
unit: Unit,
it: ⋅,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
uimplies_subtype,
provisional-type_wf,
ctt-meaning-type_wf,
context-set_wf,
context-ok_wf,
provisional-subtype,
ctt-type-meaning_wf,
isvarterm_wf,
ctt-op_wf,
ctt_meaning_wf,
cubical-context_wf,
istype-assert,
ctt-is-fibrant_wf,
ctt-term_wf,
assert_wf,
bnot_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bool_cases_sqequal,
eqff_to_assert,
assert-bnot,
eq_atom_wf,
ctt-op-sort_wf,
term-opr_wf,
bfalse_wf,
assert_elim,
btrue_neq_bfalse,
not_wf,
equal-wf-base,
set_subtype_base,
l_member_wf,
cons_wf,
nil_wf,
istype-atom,
atom_subtype_base,
assert_of_bnot,
istype-void,
subtype_rel-equal,
ifthenelse_wf,
ctt-term-meaning_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_eq_atom
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
cut,
hypothesisEquality,
applyEquality,
thin,
instantiate,
introduction,
extract_by_obid,
isectElimination,
independent_isectElimination,
hypothesis,
because_Cache,
cumulativity,
universeIsType,
sqequalRule,
inhabitedIsType,
setElimination,
rename,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
productElimination,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
tokenEquality,
productEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
applyLambdaEquality,
closedConclusion,
atomEquality,
baseClosed,
isect_memberEquality_alt,
functionIsType,
sqequalBase,
universeEquality,
equalityElimination
Latex:
\mforall{}[t:CttTerm]
\mforall{}X:?CubicalContext. [[X;t]] \msubseteq{}r Provisional''''(cttType(context-set(X))) supposing context-ok(X)
supposing \muparrow{}ctt-is-fibrant(t)
Date html generated:
2020_05_21-AM-10_36_16
Last ObjectModification:
2020_05_18-AM-00_17_54
Theory : cubical!type!theory
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