Nuprl Lemma : sqequal-list_accum

`∀[F:Base]`
`  ∀[G:Base]`
`    ∀[H,J:Base].`
`      ∀as,b1,b2:Base.`
`        F[accumulate (with value v and list item a):`
`           H[v;a]`
`          over list:`
`            as`
`          with starting value:`
`           b1)] ~ G[accumulate (with value v and list item a):`
`                     J[v;a]`
`                    over list:`
`                      as`
`                    with starting value:`
`                     b2)] `
`        supposing F[b1] ~ G[b2] `
`      supposing (∀a,r1,r2:Base.  ((F[r1] ≤ G[r2]) `` (F[H[r1;a]] ≤ G[J[r2;a]])))`
`      ∧ (∀a,r1,r2:Base.  ((G[r1] ≤ F[r2]) `` (G[J[r1;a]] ≤ F[H[r2;a]]))) `
`    supposing strict1(λx.G[x]) `
`  supposing strict1(λx.F[x])`

Proof

Definitions occuring in Statement :  list_accum: list_accum strict1: `strict1(F)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` 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]`
Lemmas referenced :  strict1_wf sqle_wf_base all_wf base_wf sqle-list_accum
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[accumulate  (with  value  v  and  list  item  a):
H[v;a]
over  list:
as
with  starting  value:
b1)]  \msim{}  G[accumulate  (with  value  v  and  list  item  a):
J[v;a]
over  list:
as
with  starting  value:
b2)]
supposing  F[b1]  \msim{}  G[b2]
supposing  (\mforall{}a,r1,r2:Base.    ((F[r1]  \mleq{}  G[r2])  {}\mRightarrow{}  (F[H[r1;a]]  \mleq{}  G[J[r2;a]])))
\mwedge{}  (\mforall{}a,r1,r2:Base.    ((G[r1]  \mleq{}  F[r2])  {}\mRightarrow{}  (G[J[r1;a]]  \mleq{}  F[H[r2;a]])))
supposing  strict1(\mlambda{}x.G[x])
supposing  strict1(\mlambda{}x.F[x])

Date html generated: 2016_05_14-AM-06_27_53
Last ObjectModification: 2016_01_14-PM-08_26_18

Theory : list_0

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