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| Original file line number | Diff line number | Diff line change |
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| embed | ||
| {{ coq | ||
| Require Export Ascii. | ||
| Require Export String. | ||
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| Hint Resolve ascii_dec : ott_coq_equality. | ||
| }} | ||
| metavar n ::= | ||
| {{ lex numeral }} | ||
| {{ coq nat }} | ||
| {{ coq-equality }} | ||
| {{ com number }} | ||
| metavar x ::= | ||
| {{ lex alphanum }} | ||
| {{ coq string }} | ||
| {{ coq-equality }} | ||
| {{ com variable }} | ||
| grammar | ||
| e :: e_ ::= | ||
| | x :: :: var {{ com variable }} | ||
| | n :: :: num {{ com number }} | ||
| | e + e' :: :: plus {{ com plus }} | ||
| | e * e' :: :: times {{ com times }} | ||
| | let x := e in e' :: :: def (+ bind x in e' +) | ||
| {{ com let }} | ||
| | e [ e' / x ] :: M :: subst | ||
| {{ com substitution }} | ||
| {{ coq (subst_e [[e']] [[x]] [[e]]) }} | ||
| | ( e ) :: S :: parentheses | ||
| {{ coq ([[e]]) }} | ||
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| terminals :: terminals_ ::= | ||
| | -> :: :: red | ||
| {{ tex \rightarrow }} | ||
| | \||/ :: :: down | ||
| {{ tex \Downarrow }} | ||
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| formula :: formula_ ::= | ||
| {{ com formulas }} | ||
| | judgement :: :: judgement | ||
| | n + n' = n'' :: M :: num_plus_eq | ||
| {{ coq ([[n]] + [[n']] = [[n'']]) }} | ||
| | n * n' = n'' :: M :: num_times_eq | ||
| {{ coq ([[n]] * [[n']] = [[n'']]) }} | ||
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| substitutions | ||
| single e x :: subst | ||
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| %freevars | ||
| % e x :: free | ||
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| defns | ||
| operational_semantics :: os_ ::= | ||
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| defn | ||
| e -> e' :: :: red :: red_ | ||
| {{ com reduction step }} by | ||
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| n1 + n2 = n | ||
| ------------ :: plus | ||
| n1 + n2 -> n | ||
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| e1 -> e'1 | ||
| ------------------- :: plus_l | ||
| e1 + e2 -> e'1 + e2 | ||
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| e -> e' | ||
| --------------- :: plus_r | ||
| n + e -> n + e' | ||
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| n1 * n2 = n | ||
| ------------ :: times | ||
| n1 * n2 -> n | ||
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| e1 -> e'1 | ||
| ------------------- :: times_l | ||
| e1 * e2 -> e'1 * e2 | ||
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| e -> e' | ||
| --------------- :: times_r | ||
| n * e -> n * e' | ||
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| e1 -> e'1 | ||
| --------------------------------------- :: let | ||
| let x := e1 in e2 -> let x := e'1 in e2 | ||
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| -------------------------------- :: bind | ||
| let x := n in e2 -> e2 [ n / x ] | ||
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| defn | ||
| e \||/ n :: :: eval :: eval_ | ||
| {{ com evaluates to }} by | ||
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| -------- :: num | ||
| n \||/ n | ||
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| e1 \||/ n1 | ||
| e2 \||/ n2 | ||
| n1 + n2 = n | ||
| -------------- :: plus | ||
| e1 + e2 \||/ n | ||
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| e1 \||/ n1 | ||
| e2 \||/ n2 | ||
| n1 * n2 = n | ||
| -------------- :: times | ||
| e1 * e2 \||/ n | ||
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| e1 \||/ n1 | ||
| e2 [ n1 / x ] \||/ n2 | ||
| ------------------------- :: let | ||
| let x := e1 in e2 \||/ n2 |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,130 @@ | ||
| % all | ||
| metavar termvar, x, y ::= {{ com term variable }} | ||
| {{ isa string}} {{ coq nat}} {{ hol string}} {{ coq-equality }} | ||
| {{ ocaml int}} {{ lex alphanum}} {{ tex \mathit{[[termvar]]} }} | ||
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| grammar | ||
| t :: 't_' ::= {{ com term }} | ||
| | x :: :: Var {{ com variable}} | ||
| | \ x . t :: :: Lam (+ bind x in t +) {{ com lambda }} | ||
| | t t' :: :: App {{ com app }} | ||
| | ( t ) :: S:: Paren {{ icho [[t]] }} | ||
| | [ t / x ] t' :: M:: Tsub | ||
| {{ icho (tsubst_t [[t]] [[x]] [[t']])}} | ||
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| v :: 'v_' ::= {{ com value }} | ||
| | \ x . t :: :: Lam {{ com lambda }} | ||
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| terminals :: 'terminals_' ::= | ||
| | \ :: :: lambda {{ tex \lambda }} | ||
| | --> :: :: red {{ tex \longrightarrow }} | ||
| | in :: :: in {{ tex \in }} | ||
| | <> :: :: neq {{ tex \neq }} | ||
| | =a :: :: eqa {{ tex \equiv_\alpha }} | ||
| | =b :: :: eqb {{ tex \equiv_\beta }} | ||
| | FV :: :: FV {{ tex \mathrm{FV} }} | ||
| | notin :: :: notin {{ tex \notin }} | ||
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| formula :: formula_ ::= | ||
| {{ com formulas }} | ||
| | judgement :: :: judgement | ||
| | x <> x' :: M :: var_neq | ||
| | x notin FV(t) :: M :: notin_fv | ||
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| subrules | ||
| v <:: t | ||
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| substitutions | ||
| single t x :: tsubst | ||
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| defns | ||
| Jop :: '' ::= | ||
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| defn | ||
| t1 --> t2 :: ::reduce::'' {{ com $[[t1]]$ reduces to $[[t2]]$}} by | ||
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| -------------------------- :: ax_app | ||
| (\x.t12) v2 --> [v2/x]t12 | ||
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| t1 --> t1' | ||
| -------------- :: ctx_app_fun | ||
| t1 t --> t1' t | ||
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| t1 --> t1' | ||
| -------------- :: ctx_app_arg | ||
| v t1 --> v t1' | ||
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| defn | ||
| x in FV ( t ) :: :: fv ::'' {{ com free variable }} by | ||
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| ---------- :: var | ||
| x in FV(x) | ||
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| x in FV(t1) | ||
| -------------- :: app_l | ||
| x in FV(t1 t2) | ||
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| x in FV(t2) | ||
| -------------- :: app_r | ||
| x in FV(t1 t2) | ||
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| x in FV(t) | ||
| x <> y | ||
| ------------- :: lam | ||
| x in FV(\y.t) | ||
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| defn | ||
| t =a t' :: :: aeq :: aeq_ {{ com alpha equivalence }} by | ||
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| ------ :: id | ||
| t =a t | ||
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| t =a t' | ||
| ------- :: sym | ||
| t' =a t | ||
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| t =a t' | ||
| t' =a t'' | ||
| --------- :: trans | ||
| t =a t'' | ||
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| t1 =a t1' | ||
| t2 =a t2' | ||
| ---------------- :: app | ||
| t1 t2 =a t1' t2' | ||
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| t =a t' | ||
| ------------- :: lam | ||
| \x.t =a \x.t' | ||
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| x' notin FV(t) | ||
| ------------------- :: subst | ||
| \x.t =a \x'.[x'/x]t | ||
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| defn | ||
| t =b t' :: :: beq :: beq_ {{ com beta equivalence }} by | ||
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| ------ :: id | ||
| t =b t | ||
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| t =b t' | ||
| ------- :: sym | ||
| t' =b t | ||
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| t =b t' | ||
| t' =b t'' | ||
| --------- :: trans | ||
| t =b t'' | ||
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| t1 =b t1' | ||
| t2 =b t2' | ||
| --------- :: app | ||
| t1 t2 =b t1' t2' | ||
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| t =b t' | ||
| ------------- :: lam | ||
| \x.t =b \x.t' | ||
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| --------------------- :: subst | ||
| (\x.t) t' =b [t'/x] t |
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