Nuprl Lemma : ppcc-test2
∀[T:Type]
∀f:T ⟶ T
∀[Q:T ⟶ ℙ]. ∀[P:T ⟶ T ⟶ ℙ]. ((∀z:T. (Q[z] ⇒ P[z;f[z]])) ⇒ (∀x,y:T. Q[x] ⇒ P[x;y] supposing y = f[x] ∈ T))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
guard: {T},
and: P ∧ Q
Lemmas referenced :
equal_wf,
all_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
axiomEquality,
hypothesis,
thin,
rename,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
setElimination,
productElimination,
setEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
\mforall{}[Q:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}z:T. (Q[z] {}\mRightarrow{} P[z;f[z]])) {}\mRightarrow{} (\mforall{}x,y:T. Q[x] {}\mRightarrow{} P[x;y] supposing y = f[x]))
Date html generated:
2016_10_25-AM-10_43_36
Last ObjectModification:
2016_07_12-AM-06_53_50
Theory : general
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