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--# -path=.:prelude
concrete ProofEng of Proof = FormulaSymb ** open Prelude, Precedence in {
flags startcat=Text ; unlexer=text ; lexer = text ;
lincat
Text, Proof = {s : Str} ;
[Formula] = {s : Str ; isnil : Bool} ;
lin
Start prems concl proof = {
s = ifNotNil (\p -> "Suppose" ++ p ++ ".") prems ++
["We will show that"] ++ useTop concl ++ "." ++
proof.s
} ;
Hypo = {
s = new ++ ["But this holds trivially ."]
} ;
Implic prems concl proof = {
s = new ++ ["It is enough to"] ++
ifNotNil (\p -> "assume" ++ p ++ "and") prems ++
"show" ++ concl.s ++
"." ++
proof.s ;
} ;
RedAbs form proof =
let neg = useTop (prefixR 4 "~" form) in {
s = new ++ ["To that end , we will assume"] ++
neg ++
["and show"] ++ "_|_" ++
"." ++
new ++ ["So let us assume that"] ++
neg ++
"." ++
proof.s
} ;
ExFalso form = {
s = new ++ ["Hence we get"] ++
useTop form ++ ["as desired"] ++
"."
} ;
ConjSplit a b c proof = {
s = new ++ ["So by splitting we get"] ++
useTop a ++ "," ++ useTop b ++
toProve c ++
"." ++
proof.s
} ;
ModPon prems concl proof = {
s = new ++ ["Then , by Modus ponens , we get"] ++ prems.s ++
toProve concl ++
"." ++
proof.s
} ;
Forget prems concl proof = {
s = new ++ ["In particular we get"] ++ prems.s ++
toProve concl ++
"." ++
proof.s
} ;
--
DeMorgan1 = \form, concl, proof -> {
s = new ++ ["Then , by the first de Morgan's law , we get"] ++ useTop form ++
toProve concl ++
"." ++
proof.s
} ;
DeMorgan2 = \form, concl, proof -> {
s = new ++ ["Then , by the second de Morgan's law , we get"] ++ useTop form ++
toProve concl ++
"." ++
proof.s
} ;
ImplicNeg forms concl proof = {
s = new ++ ["By implication negation , we get"] ++ forms.s ++
toProve concl ++
"." ++
proof.s
} ;
NegRewrite form forms proof =
let
neg = useTop (prefixR 4 "~" form) ;
impl = useTop (infixR 1 "->" form (constant "_|_")) ;
in {
s = new ++ ["Thus , by rewriting"] ++
neg ++
ifNotNil (\p -> ["we may assume"] ++ p ++ "and") forms ++
"show" ++ impl ++
"." ++
proof.s
} ;
--
BaseFormula = {
s = [] ;
isnil = True
} ;
ConsFormula f fs = {
s = useTop f ++ ifNotNil (\p -> "," ++ p) fs ;
isnil = False
} ;
oper
ifNotNil : (Str -> Str) -> {s : Str ; isnil : Bool} -> Str = \cons,prems ->
case prems.isnil of {
True => prems.s ;
False => cons prems.s
} ;
new = variants {"&-" ; []} ; -- unlexing and parsing, respectively
toProve : PrecExp -> Str = \c ->
variants {
[] ; -- for generation
["to prove"] ++ useTop c -- for parsing
} ;
}
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