Step
*
1
3
1
1
1
1
1
2
1
of Lemma
fset-ac-le-distributive-constrained
1. T : Type
2. eq : EqDecider(T)
3. P : fset(T) ⟶ 𝔹
4. ∀x,y:fset(T). (y ⊆ x ⇒ (↑(P x)) ⇒ (↑(P y)))
5. a : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
6. b : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
7. c : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
8. fset-ac-le(eq;glb(P;a;c);glb(P;a;lub(P;b;c)))
9. x : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
10. ∀a1:fset(T). (a1 ∈ glb(P;a;b) ⇒ (¬({y ∈ x | deq-f-subset(eq) y a1} = {} ∈ fset(fset(T)))))
11. ∀a1:fset(T). (a1 ∈ glb(P;a;c) ⇒ (¬({y ∈ x | deq-f-subset(eq) y a1} = {} ∈ fset(fset(T)))))
12. xx : fset(T)
13. as : fset(T)
14. as ∈ a
15. x1 : fset(T)
16. bs : x1 ∈ lub(P;b;c)
17. xx = as ⋃ x1 ∈ fset(T)
18. ↑(P xx)
19. ∀[ys:fset(T)]
↑¬bf-proper-subset-dec(eq;ys;xx)
supposing ys ∈ f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(lub(P;b;c)) s.t. P)
20. {y ∈ x | deq-f-subset(eq) y xx} = {} ∈ fset(fset(T))
21. ∀[ys:fset(T)]. ¬ys ⋃ x1 ⊆≠ xx supposing ys ∈ a ∧ (↑(P ys ⋃ x1))
22. x1 ∈ b
23. ∀[ys:fset(T)]. ¬ys ⊆≠ x1 supposing ys ∈ b ⋃ c
24. ¬xx ∈ glb(P;a;b)
25. z : fset(T)
26. z ∈ glb(P;a;b)
27. z ⊆≠ xx
⊢ False
BY
{ ((FHyp 10 [-2] THENA Auto) THEN D -1) }
1
1. T : Type
2. eq : EqDecider(T)
3. P : fset(T) ⟶ 𝔹
4. ∀x,y:fset(T). (y ⊆ x ⇒ (↑(P x)) ⇒ (↑(P y)))
5. a : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
6. b : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
7. c : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
8. fset-ac-le(eq;glb(P;a;c);glb(P;a;lub(P;b;c)))
9. x : {ac:fset(fset(T))| (↑fset-antichain(eq;ac)) ∧ fset-all(ac;a.P[a])}
10. ∀a1:fset(T). (a1 ∈ glb(P;a;b) ⇒ (¬({y ∈ x | deq-f-subset(eq) y a1} = {} ∈ fset(fset(T)))))
11. ∀a1:fset(T). (a1 ∈ glb(P;a;c) ⇒ (¬({y ∈ x | deq-f-subset(eq) y a1} = {} ∈ fset(fset(T)))))
12. xx : fset(T)
13. as : fset(T)
14. as ∈ a
15. x1 : fset(T)
16. bs : x1 ∈ lub(P;b;c)
17. xx = as ⋃ x1 ∈ fset(T)
18. ↑(P xx)
19. ∀[ys:fset(T)]
↑¬bf-proper-subset-dec(eq;ys;xx)
supposing ys ∈ f-union(deq-fset(eq);deq-fset(eq);a;as.λbs.as ⋃ bs"(lub(P;b;c)) s.t. P)
20. {y ∈ x | deq-f-subset(eq) y xx} = {} ∈ fset(fset(T))
21. ∀[ys:fset(T)]. ¬ys ⋃ x1 ⊆≠ xx supposing ys ∈ a ∧ (↑(P ys ⋃ x1))
22. x1 ∈ b
23. ∀[ys:fset(T)]. ¬ys ⊆≠ x1 supposing ys ∈ b ⋃ c
24. ¬xx ∈ glb(P;a;b)
25. z : fset(T)
26. z ∈ glb(P;a;b)
27. z ⊆≠ xx
⊢ {y ∈ x | deq-f-subset(eq) y z} = {} ∈ fset(fset(T))
Latex:
Latex:
1. T : Type
2. eq : EqDecider(T)
3. P : fset(T) {}\mrightarrow{} \mBbbB{}
4. \mforall{}x,y:fset(T). (y \msubseteq{} x {}\mRightarrow{} (\muparrow{}(P x)) {}\mRightarrow{} (\muparrow{}(P y)))
5. a : \{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\}
6. b : \{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\}
7. c : \{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\}
8. fset-ac-le(eq;glb(P;a;c);glb(P;a;lub(P;b;c)))
9. x : \{ac:fset(fset(T))| (\muparrow{}fset-antichain(eq;ac)) \mwedge{} fset-all(ac;a.P[a])\}
10. \mforall{}a1:fset(T). (a1 \mmember{} glb(P;a;b) {}\mRightarrow{} (\mneg{}(\{y \mmember{} x | deq-f-subset(eq) y a1\} = \{\})))
11. \mforall{}a1:fset(T). (a1 \mmember{} glb(P;a;c) {}\mRightarrow{} (\mneg{}(\{y \mmember{} x | deq-f-subset(eq) y a1\} = \{\})))
12. xx : fset(T)
13. as : fset(T)
14. as \mmember{} a
15. x1 : fset(T)
16. bs : x1 \mmember{} lub(P;b;c)
17. xx = as \mcup{} x1
18. \muparrow{}(P xx)
19. \mforall{}[ys:fset(T)]
\muparrow{}\mneg{}\msubb{}f-proper-subset-dec(eq;ys;xx)
supposing ys \mmember{} f-union(deq-fset(eq);deq-fset(eq);a;as.\mlambda{}bs.as \mcup{} bs"(lub(P;b;c)) s.t. P)
20. \{y \mmember{} x | deq-f-subset(eq) y xx\} = \{\}
21. \mforall{}[ys:fset(T)]. \mneg{}ys \mcup{} x1 \msubseteq{}\mneq{} xx supposing ys \mmember{} a \mwedge{} (\muparrow{}(P ys \mcup{} x1))
22. x1 \mmember{} b
23. \mforall{}[ys:fset(T)]. \mneg{}ys \msubseteq{}\mneq{} x1 supposing ys \mmember{} b \mcup{} c
24. \mneg{}xx \mmember{} glb(P;a;b)
25. z : fset(T)
26. z \mmember{} glb(P;a;b)
27. z \msubseteq{}\mneq{} xx
\mvdash{} False
By
Latex:
((FHyp 10 [-2] THENA Auto) THEN D -1)
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