include "basics/logic.ma".
(* void *)
-inductive void : Type[0] ≝.
+\ 5img class="anchor" src="icons/tick.png" id="void"\ 6inductive void : Type[0] ≝.
(* unit *)
-inductive unit : Type[0] ≝ it: unit.
+\ 5img class="anchor" src="icons/tick.png" id="unit"\ 6inductive unit : Type[0] ≝ it: unit.
-(* Prod *)
-inductive Prod (A,B:Type[0]) : Type[0] ≝
-pair : A → B → Prod A B.
+(* sum *)
+\ 5img class="anchor" src="icons/tick.png" id="Sum"\ 6inductive Sum (A,B:Type[0]) : Type[0] ≝
+ inl : A → Sum A B
+| inr : B → Sum A B.
-interpretation "Pair construction" 'pair x y = (pair ? ? x y).
+interpretation "Disjoint union" 'plus A B = (Sum A B).
-interpretation "Product" 'product x y = (Prod x y).
+(* option *)
+\ 5img class="anchor" src="icons/tick.png" id="option"\ 6inductive option (A:Type[0]) : Type[0] ≝
+ None : option A
+ | Some : A → option A.
+
+\ 5img class="anchor" src="icons/tick.png" id="option_map"\ 6definition option_map : ∀A,B:Type[0]. (A → B) → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 B ≝
+λA,B,f,o. match o with [ None ⇒ \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 B | Some a ⇒ \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 B (f a) ].
+
+\ 5img class="anchor" src="icons/tick.png" id="option_map_none"\ 6lemma option_map_none : ∀A,B,f,x.
+ \ 5a href="cic:/matita/basics/types/option_map.def(1)"\ 6option_map\ 5/a\ 6 A B f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 B → x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 A.
+#A #B #f * [ // | #a #E whd in E:(??%?); destruct ]
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="option_map_some"\ 6lemma option_map_some : ∀A,B,f,x,v.
+ \ 5a href="cic:/matita/basics/types/option_map.def(1)"\ 6option_map\ 5/a\ 6 A B f x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 B v → \ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6y. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 ? y \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 f y \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 v.
+#A #B #f *
+[ #v normalize #E destruct
+| #y #v normalize #E %{y} destruct % //
+] qed.
-definition fst ≝ λA,B.λp:A \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 B.
- match p with [pair a b ⇒ a].
+\ 5img class="anchor" src="icons/tick.png" id="option_map_def"\ 6definition option_map_def : ∀A,B:Type[0]. (A → B) → B → \ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A → B ≝
+λA,B,f,d,o. match o with [ None ⇒ d | Some a ⇒ f a ].
-definition snd ≝ λA,B.λp:A \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 B.
- match p with [pair a b ⇒ b].
+\ 5img class="anchor" src="icons/tick.png" id="refute_none_by_refl"\ 6lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. ∀x:\ 5a href="cic:/matita/basics/types/option.ind(1,0,1)"\ 6option\ 5/a\ 6 A. ∀H:x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,1,1)"\ 6None\ 5/a\ 6 ? → \ 5a href="cic:/matita/basics/logic/False.ind(1,0,0)"\ 6False\ 5/a\ 6.
+ (∀v. x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a href="cic:/matita/basics/types/option.con(0,2,1)"\ 6Some\ 5/a\ 6 ? v → Q (P v)) →
+ Q (match x return λy.x \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 y → ? with [ Some v ⇒ λ_. P v | None ⇒ λE. match H E in False with [ ] ] (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ? x)).
+#A #B #P #Q *
+[ #H cases (H (\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 ??))
+| #a #H #p normalize @p @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6
+] qed.
+
+(* sigma *)
+\ 5img class="anchor" src="icons/tick.png" id="Sig"\ 6record Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝ {
+ pi1: A
+ ; pi2: f pi1
+ }.
+
+interpretation "Sigma" 'sigma x = (Sig ? x).
+
+notation "hvbox(« term 19 a, break term 19 b»)"
+with precedence 90 for @{ 'dp $a $b }.
+
+interpretation "mk_Sig" 'dp x y = (mk_Sig ?? x y).
+
+(* Prod *)
+
+\ 5img class="anchor" src="icons/tick.png" id="Prod"\ 6record Prod (A,B:Type[0]) : Type[0] ≝ {
+ fst: A
+ ; snd: B
+ }.
+
+interpretation "Pair construction" 'pair x y = (mk_Prod ? ? x y).
+
+interpretation "Product" 'product x y = (Prod x y).
interpretation "pair pi1" 'pi1 = (fst ? ?).
interpretation "pair pi2" 'pi2 = (snd ? ?).
interpretation "pair pi1" 'pi1b x y = (fst ? ? x y).
interpretation "pair pi2" 'pi2b x y = (snd ? ? x y).
-theorem eq_pair_fst_snd: ∀A,B.∀p:A \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 B.
- p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p, \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p 〉.
+notation "π1" with precedence 10 for @{ (proj1 ??) }.
+notation "π2" with precedence 10 for @{ (proj2 ??) }.
+
+(* Yeah, I probably ought to do something more general... *)
+notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c\rangle)"
+with precedence 90 for @{ 'triple $a $b $c}.
+interpretation "Triple construction" 'triple x y z = (mk_Prod ? ? (mk_Prod ? ? x y) z).
+
+notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c, break term 19 d\rangle)"
+with precedence 90 for @{ 'quadruple $a $b $c $d}.
+interpretation "Quadruple construction" 'quadruple w x y z = (mk_Prod ? ? (mk_Prod ? ? w x) (mk_Prod ? ? y z)).
+
+
+\ 5img class="anchor" src="icons/tick.png" id="eq_pair_fst_snd"\ 6theorem eq_pair_fst_snd: ∀A,B.∀p:A \ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6 B.
+ p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6 \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 p, \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 p \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
#A #B #p (cases p) // qed.
-(* sum *)
-inductive Sum (A,B:Type[0]) : Type[0] ≝
- inl : A → Sum A B
-| inr : B → Sum A B.
+\ 5img class="anchor" src="icons/tick.png" id="fst_eq"\ 6lemma fst_eq : ∀A,B.∀a:A.∀b:B. \ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 a.
+// qed.
-interpretation "Disjoint union" 'plus A B = (Sum A B).
+\ 5img class="anchor" src="icons/tick.png" id="snd_eq"\ 6lemma snd_eq : ∀A,B.∀a:A.∀b:B. \ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 b.
+// qed.
-(* option *)
-inductive option (A:Type[0]) : Type[0] ≝
- None : option A
- | Some : A → option A.
+notation > "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t with [ mk_Prod ${ident x} ${ident y} ⇒ $s ] }.
-(* sigma *)
-inductive Sig (A:Type[0]) (f:A→Type[0]) : Type[0] ≝
- dp: ∀a:A.(f a)→Sig A f.
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp≝ break t \nbsp 'in' \nbsp) break s)"
+ with precedence 10
+for @{ match $t with [ mk_Prod (${ident x}:$X) (${ident y}:$Y) ⇒ $s ] }.
-interpretation "Sigma" 'sigma x = (Sig ? x).
+(* Also extracts an equality proof (useful when not using Russell). *)
+notation > "hvbox('let' 〈ident x,ident y〉 'as' ident E ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t return λx.x = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
+ λ${ident E}.$s ] (refl ? $t) }.
+
+(* Prop sigma *)
+\ 5img class="anchor" src="icons/tick.png" id="PSig"\ 6record PSig (A:Type[0]) (P:A→Prop) : Type[0] ≝
+ {elem:>A; eproof: P elem}.
+
+interpretation "subset type" 'sigma x = (PSig ? x).
+
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y〉 \nbsp 'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+ with precedence 10
+for @{ match $t return λ${ident k}:$X.$eq $T $k $t → ? with [ mk_Prod (${ident x}:$U) (${ident y}:$W) ⇒
+ λ${ident E}:$e.$s ] ($refl $T $t) }.
+
+notation > "hvbox('let' 〈ident x,ident y,ident z〉 'as' ident E ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t return λx.x = $t → ? with [ mk_Prod ${fresh xy} ${ident z} ⇒
+ match ${fresh xy} return λx. ? = $t → ? with [ mk_Prod ${ident x} ${ident y} ⇒
+ λ${ident E}.$s ] ] (refl ? $t) }.
+
+notation < "hvbox('let' \nbsp hvbox(〈ident x,ident y,ident z〉 \nbsp'as'\nbsp ident E\nbsp ≝ break t \nbsp 'in' \nbsp) break s)"
+ with precedence 10
+for @{ match $t return λ${ident x}.$eq $T $x $t → $U with [ mk_Prod (${fresh xy}:$V) (${ident z}:$Z) ⇒
+ match ${fresh xy} return λ${ident y}. $eq $R $r $t → ? with [ mk_Prod (${ident x}:$L) (${ident y}:$I) ⇒
+ λ${ident E}:$J.$s ] ] ($refl $A $t) }.
+
+notation > "hvbox('let' 〈ident w,ident x,ident y,ident z〉 ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t with [ mk_Prod ${fresh wx} ${fresh yz} ⇒ match ${fresh wx} with [ mk_Prod ${ident w} ${ident x} ⇒ match ${fresh yz} with [ mk_Prod ${ident y} ${ident z} ⇒ $s ] ] ] }.
+
+notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)"
+ with precedence 10
+for @{ match $t with [ mk_Prod ${fresh xy} ${ident z} ⇒ match ${fresh xy} with [ mk_Prod ${ident x} ${ident y} ⇒ $s ] ] }.
+
+(* This appears to upset automation (previously provable results require greater
+ depth or just don't work), so use example rather than lemma to prevent it
+ being indexed. *)
+\ 5img class="anchor" src="icons/tick.png" id="contract_pair"\ 6example contract_pair : ∀A,B.∀e:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. (let 〈a,b〉 ≝ e in \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 e.
+#A #B * // qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="extract_pair"\ 6lemma extract_pair : ∀A,B,C,D. ∀u:A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B. ∀Q:A → B → C\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6D. ∀x,y.
+((let 〈a,b〉 ≝ u in Q a b) \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6x,y\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6) →
+\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6∃\ 5/a\ 6a,b\ 5a title="exists" href="cic:/fakeuri.def(1)"\ 6.\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6a,b\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 u \ 5a title="logical and" href="cic:/fakeuri.def(1)"\ 6∧\ 5/a\ 6 Q a b \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6x,y\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6.
+#A #B #C #D * #a #b #Q #x #y normalize #E1 %{a} %{b} % try @\ 5a href="cic:/matita/basics/logic/eq.con(0,1,2)"\ 6refl\ 5/a\ 6 @E1 qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="breakup_pair"\ 6lemma breakup_pair : ∀A,B,C:Type[0].∀x. ∀R:C → Prop. ∀P:A → B → C.
+ R (P (\ 5a title="pair pi1" href="cic:/fakeuri.def(1)"\ 6\fst\ 5/a\ 6 x) (\ 5a title="pair pi2" href="cic:/fakeuri.def(1)"\ 6\snd\ 5/a\ 6 x)) → R (let 〈a,b〉 ≝ x in P a b).
+#A #B #C *; normalize /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="pair_elim"\ 6lemma pair_elim:
+ ∀A,B,C: Type[0].
+ ∀T: A → B → C.
+ ∀p.
+ ∀P: A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B → C → Prop.
+ (∀lft, rgt. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → P \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (T lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt).
+ #A #B #C #T * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+qed.
+
+\ 5img class="anchor" src="icons/tick.png" id="pair_elim2"\ 6lemma pair_elim2:
+ ∀A,B,C,C': Type[0].
+ ∀T: A → B → C.
+ ∀T': A → B → C'.
+ ∀p.
+ ∀P: A\ 5a title="Product" href="cic:/fakeuri.def(1)"\ 6×\ 5/a\ 6B → C → C' → Prop.
+ (∀lft, rgt. p \ 5a title="leibnitz's equality" href="cic:/fakeuri.def(1)"\ 6=\ 5/a\ 6 \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 → P \ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〈\ 5/a\ 6lft,rgt\ 5a title="Pair construction" href="cic:/fakeuri.def(1)"\ 6〉\ 5/a\ 6 (T lft rgt) (T' lft rgt)) →
+ P p (let 〈lft, rgt〉 ≝ p in T lft rgt) (let 〈lft, rgt〉 ≝ p in T' lft rgt).
+ #A #B #C #C' #T #T' * /\ 5span class="autotactic"\ 62\ 5span class="autotrace"\ 6 trace \ 5/span\ 6\ 5/span\ 6/
+qed.
\ No newline at end of file
projects / helm.git / blobdiff