### Nuprl Lemma : b-almost-full-intersection

`This is the main technical lemma from Veldman & Bezem's proof`
`of the Intuitionistic Ramsey theorem.`
`We were able to closely follow their proof except that `
`before carrying out some of the reasoning steps we have to`
`"unsquash" some of the hypotheses. `
`The needed "unsquashing" is usually done using lemmas`
`implies-quotient-true or all-quotient-true`
`(making use of the fact that we can prove canonicalizable(StrictInc)).⋅`

`∀R,T:ℕ ⟶ ℕ ⟶ ℙ.  (b-almost-full(n,m.R[n;m]) `` b-almost-full(n,m.T[n;m]) `` ⇃(∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))`

Proof

Definitions occuring in Statement :  b-almost-full: `b-almost-full(n,m.R[n; m])` quotient: `x,y:A//B[x; y]` int_upper: `{i...}` nat: `ℕ` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` true: `True` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` nat: `ℕ` subtype_rel: `A ⊆r B` prop: `ℙ` exists: `∃x:A. B[x]` or: `P ∨ Q` ge: `i ≥ j ` decidable: `Dec(P)` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` and: `P ∧ Q` strict-inc: `StrictInc` guard: `{T}` int_upper: `{i...}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` sq_stable: `SqStable(P)` squash: `↓T` subtract: `n - m` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` int_seg: `{i..j-}` lelt: `i ≤ j < k` strictly-increasing-seq: `strictly-increasing-seq(n;s)` seq-add: `s.x@n` nequal: `a ≠ b ∈ T ` less_than: `a < b` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_uimplies: `rev_uimplies(P;Q)` cand: `A c∧ B` nat_plus: `ℕ+` u-almost-full: `u-almost-full(n.A[n])` compose: `f o g` isr: `isr(x)` outr: `outr(x)` isl: `isl(x)` baf-bar: `baf-bar(n,m.R[n; m];n,m.T[n; m];l;a)` pi1: `fst(t)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin sqequalRule lambdaEquality_alt isectElimination applyEquality hypothesisEquality inhabitedIsType setElimination rename hypothesis setIsType functionIsType universeIsType natural_numberEquality functionExtensionality because_Cache closedConclusion functionEquality productEquality unionEquality dependent_set_memberEquality_alt addEquality unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation productElimination imageMemberEquality baseClosed imageElimination minusEquality instantiate universeEquality intEquality equalityElimination equalityTransitivity equalitySymmetry equalityIstype promote_hyp cumulativity hypothesis_subsumption productIsType applyLambdaEquality pointwiseFunctionality baseApply int_eqReduceTrueSq equalityIsType1 int_eqReduceFalseSq inlFormation_alt unionIsType setEquality multiplyEquality equalityIsType4 inrFormation_alt equalityIsType3 hyp_replacement

Latex:
\mforall{}R,T:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}  {}\mrightarrow{}  \mBbbP{}.
(b-almost-full(n,m.R[n;m])
{}\mRightarrow{}  b-almost-full(n,m.T[n;m])
{}\mRightarrow{}  \00D9(\mexists{}n:\mBbbN{}.  \mexists{}m:\{n  +  1...\}.  (R[n;m]  \mwedge{}  T[n;m])))

Date html generated: 2020_05_19-PM-10_05_50
Last ObjectModification: 2019_10_29-PM-01_46_29

Theory : continuity

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