Nuprl Lemma : csm-case-type
∀[phi,A,B,s:Top]. (((if phi then A else B))s ~ (if (phi)s then (A)s else (B)s))
Proof
Definitions occuring in Statement :
case-type: (if phi then A else B),
csm-ap-term: (t)s,
csm-ap-type: (AF)s,
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
case-type: (if phi then A else B),
case-cube: case-cube(phi;A;B;I;rho),
csm-ap-type: (AF)s,
csm-ap-term: (t)s,
cubical-term-at: u(a),
csm-ap: (s)x,
cubical-type-at: A(a),
pi1: fst(t),
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
top: Top,
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cubical-type-ap-morph: (u a f),
pi2: snd(t)
Lemmas referenced :
top_wf,
lifting-strict-spread,
has-value_wf_base,
base_wf,
is-exception_wf,
strict4-spread
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
baseClosed,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueApply,
baseApply,
closedConclusion,
applyExceptionCases,
inrFormation,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}[phi,A,B,s:Top]. (((if phi then A else B))s \msim{} (if (phi)s then (A)s else (B)s))
Date html generated:
2017_01_10-AM-08_51_28
Last ObjectModification:
2016_12_27-PM-01_48_21
Theory : cubical!type!theory
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