Mercurial > hg > Members > kono > Proof > category
annotate system-f.agda @ 344:45b973f5d89e
ItInt on system F
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 19 Apr 2014 00:29:39 +0900 |
| parents | 716f85bc7259 |
| children | 17acb62419ac |
| rev | line source |
|---|---|
| 315 | 1 open import Level |
| 2 open import Relation.Binary.PropositionalEquality | |
| 3 | |
| 330 | 4 module system-f where |
| 315 | 5 |
| 336 | 6 Bool : {l : Level} (X : Set l) → Set l |
| 7 Bool = λ{l : Level} → λ(X : Set l) → X → X → X | |
| 315 | 8 |
| 336 | 9 T : {l : Level} (X : Set l) → Bool X |
| 10 T X = λ(x y : X) → x | |
| 315 | 11 |
| 336 | 12 F : {l : Level} (X : Set l) → Bool X |
| 13 F X = λ(x y : X) → y | |
| 315 | 14 |
| 336 | 15 D : {l : Level} → {U : Set l} → U → U → Bool U → U |
| 331 | 16 D u v t = t u v |
| 315 | 17 |
| 336 | 18 lemma04 : {l : Level} { U : Set l} {u v : U} → D {_} {U} u v (T U ) ≡ u |
| 315 | 19 lemma04 = refl |
| 20 | |
| 336 | 21 lemma05 : {l : Level} { U : Set l} {u v : U} → D {_} {U} u v (F U ) ≡ v |
| 315 | 22 lemma05 = refl |
| 23 | |
| 336 | 24 _×_ : {l : Level} → Set l → Set l → Set (suc l) |
| 25 _×_ {l} U V = {X : Set l} → (U → V → X) → X | |
| 315 | 26 |
| 336 | 27 <_,_> : {l : Level} {U V : Set l} → U → V → (U × V) |
| 28 <_,_> {l} {U} {V} u v = λ{X} → λ(x : U → V → X) → x u v | |
| 315 | 29 |
| 336 | 30 π1 : {l : Level} {U V : Set l} → (U × V) → U |
| 31 π1 {l} {U} {V} t = t {U} (λ(x : U) → λ(y : V) → x) | |
| 315 | 32 |
| 336 | 33 π2 : {l : Level} {U V : Set l} → (U × V) → V |
| 34 π2 {l} {U} {V} t = t {V} (λ(x : U) → λ(y : V) → y) | |
| 315 | 35 |
| 336 | 36 lemma06 : {l : Level} {U V : Set l } → {u : U } → {v : V} → π1 ( < u , v > ) ≡ u |
| 315 | 37 lemma06 = refl |
| 38 | |
| 336 | 39 lemma07 : {l : Level} {U V : Set l } → {u : U } → {v : V} → π2 ( < u , v > ) ≡ v |
| 315 | 40 lemma07 = refl |
| 41 | |
| 336 | 42 hoge : {l : Level} {U V : Set l} → U → V → (U × V) |
| 315 | 43 hoge u v = < u , v > |
| 44 | |
| 336 | 45 -- lemma08 : {l : Level} {U V : Set l } → {u : U } → (t : U × V) → < π1 t , π2 t > ≡ t |
| 46 -- lemma08 t = refl | |
| 316 | 47 |
| 48 -- Emp definision is still wrong... | |
| 49 | |
| 336 | 50 Emp : {l : Level} {X : Set l} → Set l |
| 51 Emp {l} = λ{X : Set l} → X | |
| 316 | 52 |
| 336 | 53 -- ε : {l : Level} (U : Set l) {l' : Level} {U' : Set l'} → Emp → Emp |
| 327 | 54 -- ε {l} U t = t |
| 316 | 55 |
| 336 | 56 -- lemma09 : {l : Level} {U : Set l} {l' : Level} {U' : Set l} → (t : Emp {l} {U} ) → ε U (ε Emp t) ≡ ε U t |
| 322 | 57 -- lemma09 t = refl |
| 321 | 58 |
| 336 | 59 -- lemma10 : {l : Level} {U V X : Set l} → (t : Emp {_} {U × V}) → U × V |
| 327 | 60 -- lemma10 {l} {U} {V} t = ε (U × V) t |
| 316 | 61 |
| 336 | 62 -- lemma10' : {l : Level} {U V X : Set l} → (t : Emp {_} {U × V}) → Emp |
| 327 | 63 -- lemma10' {l} {U} {V} {X} t = ε (U × V) t |
| 64 | |
| 336 | 65 -- lemma100 : {l : Level} {U V X : Set l} → (t : Emp {_} {U}) → Emp |
| 322 | 66 -- lemma100 {l} {U} {V} t = ε U t |
| 321 | 67 |
| 336 | 68 -- lemma101 : {l k : Level} {U V : Set l} → (t : Emp {_} {U × V}) → π1 (ε (U × V) t) ≡ ε U t |
| 322 | 69 -- lemma101 t = refl |
| 319 | 70 |
| 336 | 71 -- lemma102 : {l k : Level} {U V : Set l} → (t : Emp {_} {U × V}) → π2 (ε (U × V) t) ≡ ε V t |
| 322 | 72 -- lemma102 t = refl |
| 321 | 73 |
| 336 | 74 -- lemma103 : {l : Level} {U V : Set l} → (u : U) → (t : Emp {l} {_} ) → (ε (U → V) t) u ≡ ε V t |
| 322 | 75 -- lemma103 u t = refl |
| 316 | 76 |
| 336 | 77 _+_ : {l : Level} → Set l → Set l → Set (suc l) |
| 78 _+_ {l} U V = {X : Set l} → ( U → X ) → (V → X) → X | |
| 316 | 79 |
| 336 | 80 ι1 : {l : Level } {U V : Set l} → U → U + V |
| 81 ι1 {l} {U} {V} u = λ{X} → λ(x : U → X) → λ(y : V → X ) → x u | |
| 316 | 82 |
| 336 | 83 ι2 : {l : Level } {U V : Set l} → V → U + V |
| 84 ι2 {l} {U} {V} v = λ{X} → λ(x : U → X) → λ(y : V → X ) → y v | |
| 316 | 85 |
| 336 | 86 δ : {l : Level} { U V R S : Set l } → (R → U) → (S → U) → ( R + S ) → U |
| 87 δ {l} {U} {V} {R} {S} u v t = t {U} (λ(x : R) → u x) ( λ(y : S) → v y) | |
| 316 | 88 |
| 336 | 89 lemma11 : {l : Level} { U V R S : Set _ } → (u : R → U ) (v : S → U ) → (r : R) → δ {l} {U} {V} {R} {S} u v ( ι1 r ) ≡ u r |
| 316 | 90 lemma11 u v r = refl |
| 91 | |
| 336 | 92 lemma12 : {l : Level} { U V R S : Set _ } → (u : R → U ) (v : S → U ) → (s : S) → δ {l} {U} {V} {R} {S} u v ( ι2 s ) ≡ v s |
| 316 | 93 lemma12 u v s = refl |
| 94 | |
| 95 | |
| 336 | 96 _××_ : {l : Level} → Set (suc l) → Set l → Set (suc l) |
| 97 _××_ {l} U V = {X : Set l} → (U → V → X) → X | |
| 322 | 98 |
| 336 | 99 <<_,_>> : {l : Level} {U : Set (suc l) } {V : Set l} → U → V → (U ×× V) |
| 100 <<_,_>> {l} {U} {V} u v = λ{X} → λ(x : U → V → X) → x u v | |
| 316 | 101 |
| 102 | |
| 336 | 103 Int : {l : Level } ( X : Set l ) → Set l |
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104 Int X = X → ( X → X ) → X |
| 322 | 105 |
| 336 | 106 Zero : {l : Level } → { X : Set l } → Int X |
| 107 Zero {l} {X} = λ(x : X ) → λ(y : X → X ) → x | |
| 322 | 108 |
| 336 | 109 S : {l : Level } → { X : Set l } → Int X → Int X |
| 110 S {l} {X} t = λ(x : X) → λ(y : X → X ) → y ( t x y ) | |
| 322 | 111 |
| 336 | 112 n0 : {l : Level} {X : Set l} → Int X |
| 331 | 113 n0 = Zero |
| 326 | 114 |
| 336 | 115 n1 : {l : Level } → { X : Set l } → Int X |
| 116 n1 {_} {X} = λ(x : X ) → λ(y : X → X ) → y x | |
| 322 | 117 |
| 336 | 118 n2 : {l : Level } → { X : Set l } → Int X |
| 119 n2 {_} {X} = λ(x : X ) → λ(y : X → X ) → y (y x) | |
| 322 | 120 |
| 336 | 121 n3 : {l : Level } → { X : Set l } → Int X |
| 122 n3 {_} {X} = λ(x : X ) → λ(y : X → X ) → y (y (y x)) | |
| 323 | 123 |
| 336 | 124 n4 : {l : Level } → { X : Set l } → Int X |
| 125 n4 {_} {X} = λ(x : X ) → λ(y : X → X ) → y (y (y (y x))) | |
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126 |
| 336 | 127 lemma13 : {l : Level } → { X : Set l } → S (S (Zero {_} {X})) ≡ n2 |
| 331 | 128 lemma13 = refl |
| 322 | 129 |
| 336 | 130 It : {l : Level} {U : Set l} → U → ( U → U ) → Int U → U |
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131 It u f t = t u f |
| 316 | 132 |
| 336 | 133 R : {l : Level} { U X : Set l} → U → ( U → Int X → U ) → Int _ → U |
| 331 | 134 R {l} {U} {X} u v t = π1 ( It {suc l} {U × Int X} (< u , Zero >) (λ (x : U × Int X) → < v (π1 x) (π2 x) , S (π2 x) > ) t ) |
| 316 | 135 |
| 344 | 136 ItInt : {l : Level} {X : Set l} → Int X → ( Int X → Int X ) → Int X → Int X |
| 137 ItInt {l} {X} u f t = λ z s → t (u z s) ( λ w → (f (λ z' s' → w )) z s ) | |
| 323 | 138 |
| 344 | 139 -- RInt : {l : Level} { U : Set l} → Int U → ( Int U → Int U → Int U ) → Int U → Int U |
| 140 -- RInt {l} {U} u v t = π1 ( ItInt {suc l} {Int U × Int U} (< u , Zero >) (λ (x : Int U × Int U) → < v (π1 x) (π2 x) , S (π2 x) > ) t ) | |
| 336 | 141 |
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142 -- bad adder which modifies input type |
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143 add' : {l : Level} {X : Set l} → Int (Int X) → Int X → Int X |
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144 add' x y = It y S x |
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145 |
| 336 | 146 add : {l : Level} {X : Set l} → Int X → Int X → Int X |
| 344 | 147 add x y = λ z s → x (y z s) s |
| 336 | 148 |
| 339 | 149 -- bad adder which modifies input type |
| 150 mul' : {l : Level } {X : Set l} → Int X → Int (Int X) → Int X | |
| 151 mul' {l} {X} x y = It Zero (add x) y | |
| 324 | 152 |
| 339 | 153 mul : {l : Level } {X : Set l} → Int X → Int X → Int X |
| 344 | 154 mul {l} {X} x y = λ z s → x z ( λ w → y w s ) |
| 155 | |
| 156 mul'' : {l : Level } {X : Set l} → Int X → Int X → Int X | |
| 157 mul'' {l} {X} x y = ItInt Zero (add x) y | |
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158 |
| 336 | 159 fact : {l : Level} {X : Set l} → Int _ → Int X |
| 339 | 160 fact {l} {X} n = R (S Zero) (λ z → λ w → mul z (S w)) n |
| 326 | 161 |
| 336 | 162 lemma13' : {l : Level} {X : Set l} → fact {l} {X} n4 ≡ mul n4 ( mul n2 n3) |
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163 lemma13' = refl |
| 324 | 164 |
| 344 | 165 |
| 336 | 166 -- lemma14 : {l : Level} {X : Set l} → (x y : Int X) → mul x y ≡ mul y x |
| 326 | 167 -- lemma14 x y = It {!!} {!!} {!!} |
| 323 | 168 |
| 336 | 169 lemma15 : {l : Level} {X : Set l} (x y : Int X) → mul {l} {X} n2 n3 ≡ mul {l} {X} n3 n2 |
| 324 | 170 lemma15 x y = refl |
| 323 | 171 |
| 344 | 172 lemma15' : {l : Level} {X : Set l} (x y : Int X) → mul'' {l} {X} n2 n3 ≡ mul'' {l} {X} n3 n2 |
| 173 lemma15' x y = refl | |
| 174 | |
| 336 | 175 lemma16 : {l : Level} {X U : Set l} → (u : U ) → (v : U → Int X → U ) → R u v Zero ≡ u |
| 324 | 176 lemma16 u v = refl |
| 177 | |
| 336 | 178 -- lemma17 : {l : Level} {X U : Set l} → (u : U ) → (v : U → Int → U ) → (t : Int ) → R u v (S t) ≡ v ( R u v t ) t |
| 324 | 179 -- lemma17 u v t = refl |
| 180 | |
| 336 | 181 -- postulate lemma17 : {l : Level} {X U : Set l} → (u : U ) → (v : U → Int X → U ) → (t : Int X ) → R u v (S t) ≡ v ( R u v t ) t |
| 316 | 182 |
| 336 | 183 List : {l : Level} (U X : Set l) → Set l |
| 184 List {l} = λ( U X : Set l) → X → ( U → X → X ) → X | |
| 323 | 185 |
| 336 | 186 Nil : {l : Level} {U : Set l} {X : Set l} → List U X |
| 187 Nil {l} {U} {X} = λ(x : X) → λ(y : U → X → X) → x | |
| 325 | 188 |
| 336 | 189 Cons : {l : Level} {U : Set l} {X : Set l} → U → List U X → List U X |
| 190 Cons {l} {U} {X} u t = λ(x : X) → λ(y : U → X → X) → y u (t x y ) | |
| 325 | 191 |
| 336 | 192 l0 : {l : Level} {X X' : Set l} → List (Int X) (X') |
| 325 | 193 l0 = Nil |
| 194 | |
| 336 | 195 l1 : {l : Level} {X X' : Set l} → List (Int X) (X') |
| 325 | 196 l1 = Cons n1 Nil |
| 197 | |
| 336 | 198 l2 : {l : Level} {X X' : Set l} → List (Int X) (X') |
| 325 | 199 l2 = Cons n2 l1 |
| 323 | 200 |
| 336 | 201 l3 : {l : Level} {X X' : Set l} → List (Int X) (X') |
| 330 | 202 l3 = Cons n3 l2 |
| 203 | |
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204 ListIt : {l : Level} {U W : Set l} → (X : Set l) → W → ( U → W → W ) → List U W → W |
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205 ListIt _ w f t = t w f |
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206 |
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207 -- Car : {l : Level} {U : Set l} {X : Set l} → List U _ → U → U |
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208 -- Car x z = ListIt z ( λ u w → u ) x |
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209 |
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210 -- Cdr : {l : Level} {U : Set l} {X : Set l} → List U _ → List U X |
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211 -- Cdr w = λ x → λ y → w x (λ x y → y) |
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212 |
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213 -- lemma181 :{l : Level} {U : Set l} {X : Set l} → Car Zero l2 ≡ n2 |
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214 -- lemma181 = refl |
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215 |
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216 -- lemma182 :{l : Level} {U : Set l} {X : Set l} → Cdr l2 ≡ l1 |
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217 -- lemma182 = refl |
| 323 | 218 |
| 336 | 219 Nullp : {l : Level} {U : Set (suc l)} { X : Set (suc l)} → List U (Bool X) → Bool _ |
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220 Nullp {_} {_} {X} list = ListIt X (T X) (λ u w → (F X)) list |
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221 |
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222 -- bad append |
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223 Append' : {l : Level} {U X : Set l} → List U (List U X) → List U X → List U X |
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224 Append' {_} {_} {X} x y = ListIt X y Cons x |
| 325 | 225 |
| 336 | 226 Append : {l : Level} {U : Set l} {X : Set l} → List U X → List U X → List U X |
| 227 Append x y = λ s t → x (y s t) t | |
| 325 | 228 |
| 336 | 229 lemma18 :{l : Level} {U : Set l} {X : Set l} → Append {_} {Int U} {X} l1 l2 ≡ Cons n1 (Cons n2 (Cons n1 Nil)) |
| 328 | 230 lemma18 = refl |
| 326 | 231 |
| 336 | 232 Reverse : {l : Level} {U : Set l} {X : Set l} → List U (List U X) → List U X |
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233 Reverse {l} {U} {X} x = ListIt X Nil ( λ u w → Append w (Cons u Nil) ) x |
| 330 | 234 |
| 336 | 235 lemma19 :{l : Level} {U : Set l} {X : Set l} → Reverse {_} {Int U} {X} l3 ≡ Cons n1 (Cons n2 (Cons n3 Nil)) |
| 330 | 236 lemma19 = refl |
| 237 | |
| 336 | 238 Tree : {l : Level} → Set l → Set l → Set l |
| 239 Tree {l} = λ( U X : Set l) → X → ( (U → X) → X ) → X | |
| 325 | 240 |
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241 NilTree : {l : Level} {U : Set l} {X : Set l} → Tree U X |
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242 NilTree {l} {U} {X} = λ(x : X) → λ(y : (U → X) → X) → x |
| 325 | 243 |
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244 Collect : {l : Level} {U : Set l} {X : Set l} → (U → Tree U X ) → Tree U X |
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245 Collect {l} {U} {X} f = λ(x : X) → λ(y : (U → X) → X) → y (λ(z : U) → f z x y ) |
| 325 | 246 |
| 336 | 247 TreeIt : {l : Level} {U W X : Set l} → W → ( (U → W) → W ) → Tree U W → W |
| 331 | 248 TreeIt w h t = t w h |
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249 |
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250 t0 : {l : Level} {X X' : Set l} → Tree (Bool X) X' |
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251 t0 {l} {X} {X'} = NilTree |
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252 |
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336
diff
changeset
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253 t1 : {l : Level} {X X' : Set l} → Tree (Bool X) X' |
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203593ff1e60
add Tree example ( not yet worked )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
336
diff
changeset
|
254 t1 {l} {X} {X'} = NilTree -- Collect (λ b → D b NilTree (λ c → Collect NilTree NilTree )) |