mirror of
https://github.com/Smaug123/agdaproofs
synced 2025-10-11 14:48:42 +00:00
Make more stuff private (#120)
This commit is contained in:
Patrick Stevens
GitHub
@@ -34,6 +34,7 @@ plus' (ofEmpty i g nonZero) s = prepend i g nonZero s
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plus' (prependLetter i g nonZero x pr) s = prepend i g nonZero (plus' x s)
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_+RP_ : ReducedSequence → ReducedSequence → ReducedSequence
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empty +RP b = b
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nonempty i x +RP b = plus' x b
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@@ -105,28 +106,11 @@ abstract
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prependWD' {i} g1 nz (nonempty j w1) (nonempty k w2) eq | inr i!=j | inr i!=k | inl refl = Equivalence.reflexive (Setoid.eq (S i)) ,, eq
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prependWD' {i} g1 nz (nonempty j w1) (nonempty k w2) eq | inr i!=j | inr i!=k | inr x = exFalso (x refl)
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abstract
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plus'WDlemm : {i : I} (x1 x2 : ReducedSequenceBeginningWith i) (y : ReducedSequence) → (=RP' x1 x2) → plus' x1 y =RP plus' x2 y
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plus'WDlemm (ofEmpty i g nonZero) (ofEmpty .i g₁ nonZero₁) y x1=x2 with decidableIndex i i
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plus'WDlemm (ofEmpty i g nonZero) (ofEmpty .i h nonZero2) y x1=x2 | inl refl = prependWD g h nonZero nonZero2 y x1=x2
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plus'WDlemm (prependLetter i g nonZero {j} x1 x) (prependLetter .i h nonZero2 {j2} x2 pr) y x1=x2 with decidableIndex i i
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... | inl refl with decidableIndex j j2
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plus'WDlemm (prependLetter i g nonZero {j} x1 x) (prependLetter .i h nonZero2 {.j} x2 pr) y x1=x2 | inl refl | inl refl = Equivalence.transitive (Setoid.eq freeProductSetoid) {prepend i g nonZero (plus' x1 y)} {prepend i h nonZero2 (plus' x1 y)} {plus' (prependLetter i h nonZero2 x2 pr) y} (prependWD g h nonZero nonZero2 (plus' x1 y) (_&&_.fst x1=x2)) (prependWD' h nonZero2 (plus' x1 y) (plus' x2 y) (plus'WDlemm x1 x2 y (_&&_.snd x1=x2)))
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... | inr j!=j2 = exFalso (notEqualIfStartDifferent j!=j2 x1 x2 (_&&_.snd x1=x2))
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plus'WDlemm _ _ _ _ | inr bad = exFalso (bad refl)
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abstract
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plus'WD : {i j : I} (x1 : ReducedSequenceBeginningWith i) (x2 : ReducedSequenceBeginningWith j) (y : ReducedSequence) → (=RP' x1 x2) → plus' x1 y =RP plus' x2 y
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plus'WD {i} {j} x1 x2 y x1=x2 with decidableIndex i j
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plus'WD {i} {.i} x1 x2 y x1=x2 | inl refl = plus'WDlemm x1 x2 y x1=x2
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plus'WD {i} {j} x1 x2 y x1=x2 | inr x = exFalso (notEqualIfStartDifferent x x1 x2 x1=x2)
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private
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abstract
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prependLetterWD : {i : I} {h1 h2 : A i} (h1=h2 : Setoid._∼_ (S i) h1 h2) → .(nonZero1 : _) .(nonZero2 : _) {j1 j2 : I} (w1 : ReducedSequenceBeginningWith j1) (w2 : ReducedSequenceBeginningWith j2) → (eq : =RP' w1 w2) → (x1 : i ≡ _ → False) (x2 : i ≡ _ → False) → nonempty i (prependLetter i h1 nonZero1 w1 x1) =RP nonempty i (prependLetter i h2 nonZero2 w2 x2)
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 with decidableIndex i i
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 | inl refl = h1=h2 ,, eq
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 | inr x = exFalso (x refl)
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prependLetterWD : {i : I} {h1 h2 : A i} (h1=h2 : Setoid._∼_ (S i) h1 h2) → .(nonZero1 : _) .(nonZero2 : _) {j1 j2 : I} (w1 : ReducedSequenceBeginningWith j1) (w2 : ReducedSequenceBeginningWith j2) → (eq : =RP' w1 w2) → (x1 : i ≡ _ → False) (x2 : i ≡ _ → False) → nonempty i (prependLetter i h1 nonZero1 w1 x1) =RP nonempty i (prependLetter i h2 nonZero2 w2 x2)
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 with decidableIndex i i
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 | inl refl = h1=h2 ,, eq
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prependLetterWD {i} h1=h2 nonZero1 nonZero2 {j1} {j2} w1 w2 eq x1 x2 | inr x = exFalso (x refl)
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abstract
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plus'WD' : {i : I} (x : ReducedSequenceBeginningWith i) (y1 y2 : ReducedSequence) → (y1 =RP y2) → plus' x y1 =RP plus' x y2
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@@ -201,13 +185,6 @@ abstract
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plusEmptyRight (prependLetter i g nonZero {j} w1 i!=j) | b | nonempty k w | inr i!=k | inl refl = Equivalence.reflexive (Setoid.eq (S i)) ,, b
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plusEmptyRight (prependLetter i g nonZero {j} w1 i!=j) | b | nonempty k w | inr i!=k | inr x = exFalso (x refl)
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abstract
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plusWD : (m n o p : ReducedSequence) → (m =RP o) → (n =RP p) → (m +RP n) =RP (o +RP p)
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plusWD empty empty empty empty m=o n=p = record {}
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plusWD empty (nonempty i x) empty (nonempty i₁ x₁) m=o n=p = n=p
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plusWD (nonempty i x) empty (nonempty j y) empty m=o record {} = Equivalence.transitive (Setoid.eq freeProductSetoid) {plus' x empty} {nonempty i x} {plus' y empty} (plusEmptyRight x) (Equivalence.transitive (Setoid.eq freeProductSetoid) {nonempty i x} {nonempty j y} {plus' y empty} m=o (Equivalence.symmetric (Setoid.eq freeProductSetoid) {plus' y empty} {nonempty j y} (plusEmptyRight y)))
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plusWD (nonempty i1 x1) (nonempty i2 x2) (nonempty i3 x3) (nonempty i4 x4) m=o n=p = (Equivalence.transitive (Setoid.eq freeProductSetoid)) {plus' x1 (nonempty i2 x2)} {plus' x3 (nonempty i2 x2)} {plus' x3 (nonempty i4 x4)} (plus'WD x1 x3 (nonempty i2 x2) m=o) (plus'WD' x3 (nonempty i2 x2) (nonempty i4 x4) n=p)
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abstract
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prependFrom : {i : I} (g1 g2 : A i) (pr : (Setoid._∼_ (S i) (_+_ i g1 g2) (Group.0G (G i))) → False) (c : ReducedSequence) .(p2 : _) .(p3 : _) → prepend i (_+_ i g1 g2) pr c =RP prepend i g1 p2 (prepend i g2 p3 c)
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prependFrom {i} g1 g2 pr empty _ _ with decidableIndex i i
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@@ -354,33 +331,3 @@ abstract
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prependFrom' {i} g h pr (nonempty j x) _ _ | inr i!=j | inl refl | inl eq1 = =RP'reflex x
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prependFrom' {i} g h pr (nonempty j x) _ _ | inr i!=j | inl refl | inr neq1 = exFalso (neq1 pr)
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prependFrom' {i} g h pr (nonempty j x) _ _ | inr i!=j | inr bad = exFalso (bad refl)
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prependAssocLemma' : {i : I} {g : A i} .(nz : Setoid._∼_ (S i) g (Group.0G (G i)) → False) (b c : ReducedSequence) → (prepend i g nz b +RP c) =RP prepend i g nz (b +RP c)
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prependAssocLemma' {i} {g} nz empty c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ c}
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prependAssocLemma' {i} nz (nonempty k x) c with decidableIndex i k
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl with decidableGroups k ((k + g1) g) (Group.0G (G k))
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend k g1 _ (prepend k g _ c)} {c} (prependFrom' g1 g eq1 c nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl | inr neq1 = (prependFrom g1 g neq1 c nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl with decidableGroups k ((k + g1) g) (Group.0G (G k))
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend k g1 _ (prepend k g _ (plus' x c))} {plus' x c} (prependFrom' g1 g eq1 (plus' x c) nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl | inr neq1 = prependFrom g1 g neq1 (plus' x c) nz nonZero
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prependAssocLemma' {i} {g} nz (nonempty k x) c | inr i!=k = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' x c)}
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abstract
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plusAssocLemma : {i : I} (x : ReducedSequenceBeginningWith i) (b c : ReducedSequence) → (plus' x b +RP c) =RP plus' x (b +RP c)
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plusAssocLemma (ofEmpty i g nonZero) empty c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ c}
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plusAssocLemma (ofEmpty i g nonZero) (nonempty j b) c with decidableIndex i j
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i g₁ nonZero₁)) c | inl refl with decidableIndex i i
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl with decidableGroups i ((i + g) h) (Group.0G (G i))
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl | inl t = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend i g _ (prepend i h _ c)} {c} (prependFrom' g h t c _ _)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl | inr t = prependFrom g h t c _ _
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i g₁ nonZero₁)) c | inl refl | inr bad = exFalso (bad refl)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl with decidableGroups i ((i + g) h) (Group.0G (G i))
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend i g _ (prepend i h _ (plus' b c))} {plus' b c} (prependFrom' g h eq1 (plus' b c) _ _)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl | inr neq1 = prependFrom g h neq1 (plus' b c) nonZero _
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plusAssocLemma (ofEmpty i g nonZero) (nonempty j b) c | inr i!=j = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' b c)}
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plusAssocLemma (prependLetter i g nonZero {j} w i!=j) b c = Equivalence.transitive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' w b) +RP c} {prepend i g _ (plus' w b +RP c)} {prepend i g _ (plus' w (b +RP c))} (prependAssocLemma' nonZero (plus' w b) c) (prependWD' g nonZero (plus' w b +RP c) (plus' w (b +RP c)) (plusAssocLemma w b c))
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plusAssoc : (a b c : ReducedSequence) → ((a +RP b) +RP c) =RP (a +RP (b +RP c))
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plusAssoc empty b c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {b +RP c}
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plusAssoc (nonempty i x) b c = plusAssocLemma x b c
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@@ -16,6 +16,56 @@ open import Groups.FreeProduct.Definition decidableIndex decidableGroups G
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open import Groups.FreeProduct.Setoid decidableIndex decidableGroups G
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open import Groups.FreeProduct.Addition decidableIndex decidableGroups G
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private
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plus'WDlemm : {i : I} (x1 x2 : ReducedSequenceBeginningWith i) (y : ReducedSequence) → (=RP' x1 x2) → plus' x1 y =RP plus' x2 y
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plus'WDlemm (ofEmpty i g nonZero) (ofEmpty .i g₁ nonZero₁) y x1=x2 with decidableIndex i i
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plus'WDlemm (ofEmpty i g nonZero) (ofEmpty .i h nonZero2) y x1=x2 | inl refl = prependWD g h nonZero nonZero2 y x1=x2
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plus'WDlemm (prependLetter i g nonZero {j} x1 x) (prependLetter .i h nonZero2 {j2} x2 pr) y x1=x2 with decidableIndex i i
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... | inl refl with decidableIndex j j2
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plus'WDlemm (prependLetter i g nonZero {j} x1 x) (prependLetter .i h nonZero2 {.j} x2 pr) y x1=x2 | inl refl | inl refl = Equivalence.transitive (Setoid.eq freeProductSetoid) {prepend i g nonZero (plus' x1 y)} {prepend i h nonZero2 (plus' x1 y)} {plus' (prependLetter i h nonZero2 x2 pr) y} (prependWD g h nonZero nonZero2 (plus' x1 y) (_&&_.fst x1=x2)) (prependWD' h nonZero2 (plus' x1 y) (plus' x2 y) (plus'WDlemm x1 x2 y (_&&_.snd x1=x2)))
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... | inr j!=j2 = exFalso (notEqualIfStartDifferent j!=j2 x1 x2 (_&&_.snd x1=x2))
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plus'WDlemm _ _ _ _ | inr bad = exFalso (bad refl)
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plus'WD : {i j : I} (x1 : ReducedSequenceBeginningWith i) (x2 : ReducedSequenceBeginningWith j) (y : ReducedSequence) → (=RP' x1 x2) → plus' x1 y =RP plus' x2 y
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plus'WD {i} {j} x1 x2 y x1=x2 with decidableIndex i j
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plus'WD {i} {.i} x1 x2 y x1=x2 | inl refl = plus'WDlemm x1 x2 y x1=x2
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plus'WD {i} {j} x1 x2 y x1=x2 | inr x = exFalso (notEqualIfStartDifferent x x1 x2 x1=x2)
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plusWD : (m n o p : ReducedSequence) → (m =RP o) → (n =RP p) → (m +RP n) =RP (o +RP p)
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plusWD empty empty empty empty m=o n=p = record {}
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plusWD empty (nonempty i x) empty (nonempty i₁ x₁) m=o n=p = n=p
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plusWD (nonempty i x) empty (nonempty j y) empty m=o record {} = Equivalence.transitive (Setoid.eq freeProductSetoid) {plus' x empty} {nonempty i x} {plus' y empty} (plusEmptyRight x) (Equivalence.transitive (Setoid.eq freeProductSetoid) {nonempty i x} {nonempty j y} {plus' y empty} m=o (Equivalence.symmetric (Setoid.eq freeProductSetoid) {plus' y empty} {nonempty j y} (plusEmptyRight y)))
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plusWD (nonempty i1 x1) (nonempty i2 x2) (nonempty i3 x3) (nonempty i4 x4) m=o n=p = (Equivalence.transitive (Setoid.eq freeProductSetoid)) {plus' x1 (nonempty i2 x2)} {plus' x3 (nonempty i2 x2)} {plus' x3 (nonempty i4 x4)} (plus'WD x1 x3 (nonempty i2 x2) m=o) (plus'WD' x3 (nonempty i2 x2) (nonempty i4 x4) n=p)
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prependAssocLemma' : {i : I} {g : A i} .(nz : Setoid._∼_ (S i) g (Group.0G (G i)) → False) (b c : ReducedSequence) → (prepend i g nz b +RP c) =RP prepend i g nz (b +RP c)
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prependAssocLemma' {i} {g} nz empty c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ c}
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prependAssocLemma' {i} nz (nonempty k x) c with decidableIndex i k
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl with decidableGroups k ((k + g1) g) (Group.0G (G k))
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend k g1 _ (prepend k g _ c)} {c} (prependFrom' g1 g eq1 c nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (ofEmpty .k g nonZero)) c | inl refl | inr neq1 = (prependFrom g1 g neq1 c nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl with decidableGroups k ((k + g1) g) (Group.0G (G k))
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend k g1 _ (prepend k g _ (plus' x c))} {plus' x c} (prependFrom' g1 g eq1 (plus' x c) nz nonZero)
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prependAssocLemma' {.k} {g1} nz (nonempty k (prependLetter .k g nonZero x x₁)) c | inl refl | inr neq1 = prependFrom g1 g neq1 (plus' x c) nz nonZero
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prependAssocLemma' {i} {g} nz (nonempty k x) c | inr i!=k = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' x c)}
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plusAssocLemma : {i : I} (x : ReducedSequenceBeginningWith i) (b c : ReducedSequence) → (plus' x b +RP c) =RP plus' x (b +RP c)
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plusAssocLemma (ofEmpty i g nonZero) empty c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ c}
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plusAssocLemma (ofEmpty i g nonZero) (nonempty j b) c with decidableIndex i j
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i g₁ nonZero₁)) c | inl refl with decidableIndex i i
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl with decidableGroups i ((i + g) h) (Group.0G (G i))
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl | inl t = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend i g _ (prepend i h _ c)} {c} (prependFrom' g h t c _ _)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i h nonZero₁)) c | inl refl | inl refl | inr t = prependFrom g h t c _ _
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (ofEmpty .i g₁ nonZero₁)) c | inl refl | inr bad = exFalso (bad refl)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl with decidableGroups i ((i + g) h) (Group.0G (G i))
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl | inl eq1 = Equivalence.symmetric (Setoid.eq freeProductSetoid) {prepend i g _ (prepend i h _ (plus' b c))} {plus' b c} (prependFrom' g h eq1 (plus' b c) _ _)
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plusAssocLemma (ofEmpty i g nonZero) (nonempty .i (prependLetter .i h nonZero₁ b x)) c | inl refl | inr neq1 = prependFrom g h neq1 (plus' b c) nonZero _
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plusAssocLemma (ofEmpty i g nonZero) (nonempty j b) c | inr i!=j = Equivalence.reflexive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' b c)}
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plusAssocLemma (prependLetter i g nonZero {j} w i!=j) b c = Equivalence.transitive (Setoid.eq freeProductSetoid) {prepend i g _ (plus' w b) +RP c} {prepend i g _ (plus' w b +RP c)} {prepend i g _ (plus' w (b +RP c))} (prependAssocLemma' nonZero (plus' w b) c) (prependWD' g nonZero (plus' w b +RP c) (plus' w (b +RP c)) (plusAssocLemma w b c))
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plusAssoc : (a b c : ReducedSequence) → ((a +RP b) +RP c) =RP (a +RP (b +RP c))
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plusAssoc empty b c = Equivalence.reflexive (Setoid.eq freeProductSetoid) {b +RP c}
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plusAssoc (nonempty i x) b c = plusAssocLemma x b c
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private
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inv' : {i : I} (x : ReducedSequenceBeginningWith i) → ReducedSequence
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inv' (ofEmpty i g nonZero) = nonempty i (ofEmpty i (Group.inverse (G i) g) (λ pr → nonZero (Equivalence.transitive (Setoid.eq (S i)) (swapInv (G i) pr) (invIdent (G i)))))
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