Step
*
1
4
2
2
1
of Lemma
term-accum_wf_wfterm
1. opr : Type
2. P : Type
3. sort : term(opr) ⟶ ℕ
4. arity : opr ⟶ ((ℕ × ℕ) List)
5. R : P ⟶ wfterm(opr;sort;arity) ⟶ ℙ
6. Q : P
⟶ f:opr
⟶ vs:(varname() List)
⟶ {L:(t:wfterm(opr;sort;arity) × p:P × (R p t)) List|
||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))}
⟶ P
7. varcase : p:P ⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())} ⟶ (R p varterm(v))
8. mktermcase : p:P
⟶ f:opr
⟶ bts:wf-bound-terms(opr;sort;arity;f)
⟶ L:{L:(t:wfterm(opr;sort;arity) × p:P × (R p t)) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||. ((fst(snd(L[i]))) = (Q p f (fst(bts[i])) firstn(i;L)) ∈ P))}
⟶ (R p mkwfterm(f;bts))
9. t : wfterm(opr;sort;arity)
10. p : P
11. ok : ∀f:opr. ∀vs:varname() List. ∀L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List.
Dec(||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))
∧ (∀i:ℕ||L||. (↑wf-term(arity;sort;fst(L[i])))))
12. ∀f:opr. ∀vs:varname() List. ∀L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List.
(↑isl(ok f vs L)
⇐⇒ ||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))
∧ (∀i:ℕ||L||. (↑wf-term(arity;sort;fst(L[i])))))
13. QQ : P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List) ⟶ P
14. (λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi )
= QQ
∈ (P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List) ⟶ P)
15. ∀[varcase:p:P ⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())} ⟶ R p varterm(v) supposing True].
∀[mktermcase:p:P
⟶ f:opr
⟶ bts:(bound-term(opr) List)
⟶ L:{L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||
((fst(snd(L[i])))
= ((λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f (fst(bts[i])) firstn(i;L))
∈ P))}
⟶ R p mkterm(f;bts) supposing ↑wf-term(arity;sort;mkterm(f;bts))]. ∀[t:term(opr)]. ∀[p:P].
(term-accum(t with p)
p,f,vs,tr.(λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f vs tr
varterm(x) with p ⇒ varcase p x
mkterm(f,bts) with p ⇒ trs.mktermcase p f bts trs ∈ R p t supposing ↑wf-term(arity;sort;t))
16. p1 : P
17. f : opr
18. bts : bound-term(opr) List
19. L : (t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List
20. ||L|| = ||bts|| ∈ ℤ
21. ∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr))
22. i : ℕ||L||
23. ||firstn(i;L)|| < ||arity f||
∧ (||fst(bts[i])|| = (fst(arity f[||firstn(i;L)||])) ∈ ℤ)
∧ (∀i1:ℕ||firstn(i;L)||. ((sort (fst(firstn(i;L)[i1]))) = (snd(arity f[i1])) ∈ ℤ))
∧ (∀i1:ℕ||firstn(i;L)||. (↑wf-term(arity;sort;fst(firstn(i;L)[i1]))))
⊢ firstn(i;L) ∈ {L:(t:wfterm(opr;sort;arity) × p:P × (R p t)) List|
||L|| < ||arity f||
∧ (||fst(bts[i])|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))}
BY
{ (MemTypeCD THENL [skip{[tactic]}; Auto; Auto]) }
1
1. opr : Type
2. P : Type
3. sort : term(opr) ⟶ ℕ
4. arity : opr ⟶ ((ℕ × ℕ) List)
5. R : P ⟶ wfterm(opr;sort;arity) ⟶ ℙ
6. Q : P
⟶ f:opr
⟶ vs:(varname() List)
⟶ {L:(t:wfterm(opr;sort;arity) × p:P × (R p t)) List|
||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))}
⟶ P
7. varcase : p:P ⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())} ⟶ (R p varterm(v))
8. mktermcase : p:P
⟶ f:opr
⟶ bts:wf-bound-terms(opr;sort;arity;f)
⟶ L:{L:(t:wfterm(opr;sort;arity) × p:P × (R p t)) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||. ((fst(snd(L[i]))) = (Q p f (fst(bts[i])) firstn(i;L)) ∈ P))}
⟶ (R p mkwfterm(f;bts))
9. t : wfterm(opr;sort;arity)
10. p : P
11. ok : ∀f:opr. ∀vs:varname() List. ∀L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List.
Dec(||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))
∧ (∀i:ℕ||L||. (↑wf-term(arity;sort;fst(L[i])))))
12. ∀f:opr. ∀vs:varname() List. ∀L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List.
(↑isl(ok f vs L)
⇐⇒ ||L|| < ||arity f||
∧ (||vs|| = (fst(arity f[||L||])) ∈ ℤ)
∧ (∀i:ℕ||L||. ((sort (fst(L[i]))) = (snd(arity f[i])) ∈ ℤ))
∧ (∀i:ℕ||L||. (↑wf-term(arity;sort;fst(L[i])))))
13. QQ : P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List) ⟶ P
14. (λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi )
= QQ
∈ (P ⟶ opr ⟶ (varname() List) ⟶ ((t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List) ⟶ P)
15. ∀[varcase:p:P ⟶ v:{v:varname()| ¬(v = nullvar() ∈ varname())} ⟶ R p varterm(v) supposing True].
∀[mktermcase:p:P
⟶ f:opr
⟶ bts:(bound-term(opr) List)
⟶ L:{L:(t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List|
(||L|| = ||bts|| ∈ ℤ)
∧ (∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr)))
∧ (∀i:ℕ||L||
((fst(snd(L[i])))
= ((λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f (fst(bts[i])) firstn(i;L))
∈ P))}
⟶ R p mkterm(f;bts) supposing ↑wf-term(arity;sort;mkterm(f;bts))]. ∀[t:term(opr)]. ∀[p:P].
(term-accum(t with p)
p,f,vs,tr.(λp,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f vs tr
varterm(x) with p ⇒ varcase p x
mkterm(f,bts) with p ⇒ trs.mktermcase p f bts trs ∈ R p t supposing ↑wf-term(arity;sort;t))
16. p1 : P
17. f : opr
18. bts : bound-term(opr) List
19. L : (t:term(opr) × p:P × R p t supposing ↑wf-term(arity;sort;t)) List
20. ||L|| = ||bts|| ∈ ℤ
21. ∀i:ℕ||L||. ((fst(L[i])) = (snd(bts[i])) ∈ term(opr))
22. i : ℕ||L||
23. ||firstn(i;L)|| < ||arity f||
∧ (||fst(bts[i])|| = (fst(arity f[||firstn(i;L)||])) ∈ ℤ)
∧ (∀i1:ℕ||firstn(i;L)||. ((sort (fst(firstn(i;L)[i1]))) = (snd(arity f[i1])) ∈ ℤ))
∧ (∀i1:ℕ||firstn(i;L)||. (↑wf-term(arity;sort;fst(firstn(i;L)[i1]))))
⊢ firstn(i;L) ∈ (t:wfterm(opr;sort;arity) × p:P × (R p t)) List
Latex:
Latex:
1. opr : Type
2. P : Type
3. sort : term(opr) {}\mrightarrow{} \mBbbN{}
4. arity : opr {}\mrightarrow{} ((\mBbbN{} \mtimes{} \mBbbN{}) List)
5. R : P {}\mrightarrow{} wfterm(opr;sort;arity) {}\mrightarrow{} \mBbbP{}
6. Q : P
{}\mrightarrow{} f:opr
{}\mrightarrow{} vs:(varname() List)
{}\mrightarrow{} \{L:(t:wfterm(opr;sort;arity) \mtimes{} p:P \mtimes{} (R p t)) List|
||L|| < ||arity f||
\mwedge{} (||vs|| = (fst(arity f[||L||])))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((sort (fst(L[i]))) = (snd(arity f[i]))))\}
{}\mrightarrow{} P
7. varcase : p:P {}\mrightarrow{} v:\{v:varname()| \mneg{}(v = nullvar())\} {}\mrightarrow{} (R p varterm(v))
8. mktermcase : p:P
{}\mrightarrow{} f:opr
{}\mrightarrow{} bts:wf-bound-terms(opr;sort;arity;f)
{}\mrightarrow{} L:\{L:(t:wfterm(opr;sort;arity) \mtimes{} p:P \mtimes{} (R p t)) List|
(||L|| = ||bts||)
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((fst(L[i])) = (snd(bts[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((fst(snd(L[i]))) = (Q p f (fst(bts[i])) firstn(i;L))))\}
{}\mrightarrow{} (R p mkwfterm(f;bts))
9. t : wfterm(opr;sort;arity)
10. p : P
11. ok : \mforall{}f:opr. \mforall{}vs:varname() List.
\mforall{}L:(t:term(opr) \mtimes{} p:P \mtimes{} R p t supposing \muparrow{}wf-term(arity;sort;t)) List.
Dec(||L|| < ||arity f||
\mwedge{} (||vs|| = (fst(arity f[||L||])))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((sort (fst(L[i]))) = (snd(arity f[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. (\muparrow{}wf-term(arity;sort;fst(L[i])))))
12. \mforall{}f:opr. \mforall{}vs:varname() List.
\mforall{}L:(t:term(opr) \mtimes{} p:P \mtimes{} R p t supposing \muparrow{}wf-term(arity;sort;t)) List.
(\muparrow{}isl(ok f vs L)
\mLeftarrow{}{}\mRightarrow{} ||L|| < ||arity f||
\mwedge{} (||vs|| = (fst(arity f[||L||])))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((sort (fst(L[i]))) = (snd(arity f[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. (\muparrow{}wf-term(arity;sort;fst(L[i])))))
13. QQ : P
{}\mrightarrow{} opr
{}\mrightarrow{} (varname() List)
{}\mrightarrow{} ((t:term(opr) \mtimes{} p:P \mtimes{} R p t supposing \muparrow{}wf-term(arity;sort;t)) List)
{}\mrightarrow{} P
14. (\mlambda{}p,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) = QQ
15. \mforall{}[varcase:p:P {}\mrightarrow{} v:\{v:varname()| \mneg{}(v = nullvar())\} {}\mrightarrow{} R p varterm(v) supposing True].
\mforall{}[mktermcase:p:P
{}\mrightarrow{} f:opr
{}\mrightarrow{} bts:(bound-term(opr) List)
{}\mrightarrow{} L:\{L:(t:term(opr) \mtimes{} p:P \mtimes{} R p t supposing \muparrow{}wf-term(arity;sort;t)) List|
(||L|| = ||bts||)
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((fst(L[i])) = (snd(bts[i]))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
((fst(snd(L[i])))
= ((\mlambda{}p,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f (fst(bt\000Cs[i]))
firstn(i;L))))\}
{}\mrightarrow{} R p mkterm(f;bts) supposing \muparrow{}wf-term(arity;sort;mkterm(f;bts))]. \mforall{}[t:term(opr)].
\mforall{}[p:P].
(term-accum(t with p)
p,f,vs,tr.(\mlambda{}p,f,vs,L. if isl(ok f vs L) then Q p f vs L else p fi ) p f vs tr
varterm(x) with p {}\mRightarrow{} varcase p x
mkterm(f,bts) with p {}\mRightarrow{} trs.mktermcase p f bts trs \mmember{} R p t supposing \muparrow{}wf-term(arity;sort;t))
16. p1 : P
17. f : opr
18. bts : bound-term(opr) List
19. L : (t:term(opr) \mtimes{} p:P \mtimes{} R p t supposing \muparrow{}wf-term(arity;sort;t)) List
20. ||L|| = ||bts||
21. \mforall{}i:\mBbbN{}||L||. ((fst(L[i])) = (snd(bts[i])))
22. i : \mBbbN{}||L||
23. ||firstn(i;L)|| < ||arity f||
\mwedge{} (||fst(bts[i])|| = (fst(arity f[||firstn(i;L)||])))
\mwedge{} (\mforall{}i1:\mBbbN{}||firstn(i;L)||. ((sort (fst(firstn(i;L)[i1]))) = (snd(arity f[i1]))))
\mwedge{} (\mforall{}i1:\mBbbN{}||firstn(i;L)||. (\muparrow{}wf-term(arity;sort;fst(firstn(i;L)[i1]))))
\mvdash{} firstn(i;L) \mmember{} \{L:(t:wfterm(opr;sort;arity) \mtimes{} p:P \mtimes{} (R p t)) List|
||L|| < ||arity f||
\mwedge{} (||fst(bts[i])|| = (fst(arity f[||L||])))
\mwedge{} (\mforall{}i:\mBbbN{}||L||. ((sort (fst(L[i]))) = (snd(arity f[i]))))\}
By
Latex:
(MemTypeCD THENL [skip\{[tactic]\}; Auto; Auto])
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