Step
*
1
2
1
1
of Lemma
member-gcd-reduce-constraints
1. n : ℕ+
2. u : ℤ
3. [%2] : 1 = n ∈ ℤ
4. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
5. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(gcd-reduce-eq-constraints(sat;v))) ⇒ (cc ∈ sat) ⇒ (cc ∈ outl(gcd-reduce-eq-constraints(sat;v))))
⊢ ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(let s' ⟵ if u=0 then inl [[u] / sat] else (inr ⋅ )
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v)))
⇒ (cc ∈ sat)
⇒ (cc ∈ outl(let s' ⟵ if u=0 then inl [[u] / sat] else (inr ⋅ )
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v))))
BY
{ ((Decide u = 0 ∈ ℤ THENA Auto) THEN Reduce 0) }
1
1. n : ℕ+
2. u : ℤ
3. [%2] : 1 = n ∈ ℤ
4. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
5. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(gcd-reduce-eq-constraints(sat;v))) ⇒ (cc ∈ sat) ⇒ (cc ∈ outl(gcd-reduce-eq-constraints(sat;v))))
6. u = 0 ∈ ℤ
⊢ ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(let s' ⟵ inl [[u] / sat]
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v)))
⇒ (cc ∈ sat)
⇒ (cc ∈ outl(let s' ⟵ inl [[u] / sat]
in accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);s';v))))
2
1. n : ℕ+
2. u : ℤ
3. [%2] : 1 = n ∈ ℤ
4. v : {L:ℤ List| ||L|| = n ∈ ℤ} List
5. ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(gcd-reduce-eq-constraints(sat;v))) ⇒ (cc ∈ sat) ⇒ (cc ∈ outl(gcd-reduce-eq-constraints(sat;v))))
6. ¬(u = 0 ∈ ℤ)
⊢ ∀sat:{L:ℤ List| ||L|| = n ∈ ℤ} List. ∀cc:{L:ℤ List| ||L|| = n ∈ ℤ} .
((↑isl(accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);inr ⋅ ;v)))
⇒ (cc ∈ sat)
⇒ (cc ∈ outl(accumulate_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr ⋅ )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(λx.(x ÷ g);L) in
inl [L' / Ls]
else (inr ⋅ )
else (inl [L / Ls]);inr ⋅ ;v))))
Latex:
Latex:
1. n : \mBbbN{}\msupplus{}
2. u : \mBbbZ{}
3. [\%2] : 1 = n
4. v : \{L:\mBbbZ{} List| ||L|| = n\} List
5. \mforall{}sat:\{L:\mBbbZ{} List| ||L|| = n\} List. \mforall{}cc:\{L:\mBbbZ{} List| ||L|| = n\} .
((\muparrow{}isl(gcd-reduce-eq-constraints(sat;v)))
{}\mRightarrow{} (cc \mmember{} sat)
{}\mRightarrow{} (cc \mmember{} outl(gcd-reduce-eq-constraints(sat;v))))
\mvdash{} \mforall{}sat:\{L:\mBbbZ{} List| ||L|| = n\} List. \mforall{}cc:\{L:\mBbbZ{} List| ||L|| = n\} .
((\muparrow{}isl(let s' \mleftarrow{}{} if u=0 then inl [[u] / sat] else (inr \mcdot{} )
in accumulate\_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr \mcdot{} )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(\mlambda{}x.(x \mdiv{} g);L) in
inl [L' / Ls]
else (inr \mcdot{} )
else (inl [L / Ls]);s';v)))
{}\mRightarrow{} (cc \mmember{} sat)
{}\mRightarrow{} (cc
\mmember{} outl(let s' \mleftarrow{}{} if u=0 then inl [[u] / sat] else (inr \mcdot{} )
in accumulate\_abort(L,Ls.let h,t = L
in if t = Ax then if h=0 then inl [L / Ls] else (inr \mcdot{} )
otherwise eval g = |gcd-list(t)| in
if (1) < (g)
then if h rem g=0
then eval L' = eager-map(...;L) in
inl [L' / Ls]
else (inr \mcdot{} )
else (inl [L / Ls]);s';v))))
By
Latex:
((Decide u = 0 THENA Auto) THEN Reduce 0)
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