Nuprl Lemma : sqequal-list_ind
∀[F:Base]
∀[G:Base]
∀[H,J:Base].
∀as,b1,b2:Base.
F[rec-case(as) of
[] => b1
h::t =>
r.H[h;t;r]] ~ G[rec-case(as) of
[] => b2
h::t =>
r.J[h;t;r]]
supposing F[b1] ~ G[b2]
supposing (∀x,y,r1,r2:Base. ((F[r1] ≤ G[r2]) ⇒ (F[H[x;y;r1]] ≤ G[J[x;y;r2]])))
∧ (∀x,y,r1,r2:Base. ((G[r1] ≤ F[r2]) ⇒ (G[J[x;y;r1]] ≤ F[H[x;y;r2]])))
supposing strict1(λx.G[x])
supposing strict1(λx.F[x])
Proof
Definitions occuring in Statement :
list_ind: list_ind,
strict1: strict1(F),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
base: Base,
sqle: s ≤ t,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
so_apply: x[s1;s2;s3]
Lemmas referenced :
strict1_wf,
sqle_wf_base,
all_wf,
base_wf,
sqle-list_ind
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalSqle,
sqequalHypSubstitution,
productElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
sqequalRule,
sqleReflexivity,
sqequalAxiom,
sqequalIntensionalEquality,
baseApply,
closedConclusion,
baseClosed,
lambdaEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
productEquality,
functionEquality
Latex:
\mforall{}[F:Base]
\mforall{}[G:Base]
\mforall{}[H,J:Base].
\mforall{}as,b1,b2:Base.
F[rec-case(as) of
[] => b1
h::t =>
r.H[h;t;r]] \msim{} G[rec-case(as) of
[] => b2
h::t =>
r.J[h;t;r]]
supposing F[b1] \msim{} G[b2]
supposing (\mforall{}x,y,r1,r2:Base. ((F[r1] \mleq{} G[r2]) {}\mRightarrow{} (F[H[x;y;r1]] \mleq{} G[J[x;y;r2]])))
\mwedge{} (\mforall{}x,y,r1,r2:Base. ((G[r1] \mleq{} F[r2]) {}\mRightarrow{} (G[J[x;y;r1]] \mleq{} F[H[x;y;r2]])))
supposing strict1(\mlambda{}x.G[x])
supposing strict1(\mlambda{}x.F[x])
Date html generated:
2016_05_14-AM-06_29_05
Last ObjectModification:
2016_01_14-PM-08_25_48
Theory : list_0
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