Nuprl Lemma : TI-weak-TI
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[Q:T ⟶ ℙ]. (TI(T;x,y.R[x;y];t.Q[t]) ⇒ weak-TI(T;x,y.R[x;y];t.Q[t]))
Proof
Definitions occuring in Statement :
weak-TI: weak-TI(T;x,y.R[x; y];t.Q[t]),
TI: TI(T;x,y.R[x; y];t.Q[t]),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
TI: TI(T;x,y.R[x; y];t.Q[t]),
weak-TI: weak-TI(T;x,y.R[x; y];t.Q[t]),
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
guard: {T}
Lemmas referenced :
all_wf,
TI_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
independent_functionElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
setEquality,
because_Cache,
sqequalRule,
lambdaEquality,
setElimination,
rename,
functionEquality,
cumulativity,
universeEquality,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. (TI(T;x,y.R[x;y];t.Q[t]) {}\mRightarrow{} weak-TI(T;x,y.R[x;y];t.Q[t]))
Date html generated:
2016_05_13-PM-03_50_02
Last ObjectModification:
2015_12_26-AM-10_17_33
Theory : bar-induction
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