Download A user's guide to ALF * Thorsten Altenkirch Veronica Gaspes Bengt
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3FX:C<7>#"+!$'"0<#37>+B0(9¥%N!$0 HLH 0;ZS7T')A´C?AP=A;=C?A;?'I#"0Q0<'6C<7>#"+!$')0#"hU A;7 H P=0<#"'37>;Yn'"+1c')A0#37>+O'"0iIo@ %(PP?-87!A%('37>0;=J0(9L30 H \0(92')!$0;=3'&%;Q'&W%(#" C<7/1P?-/%h+CmXQh\C?$9¬%(F?->'I47>'"M%(;J7>;?7>'37%-61+EFA;=!$|0(9 %(#"YF H A;?'& 1FPP#"+"3+CU|?7G7G7T;=C<7!A%(')+C 0;L"!$#"AA;LX?h*%BgA#)'37!A%-:%(#"#"0(4 t)µv 7>;|9r#"0;?'0(9='" %(#)YF H A;?';=% H 7>;L'"'3hQP_{09:'" !$0;=3'$%(;Q'U ?77¥7>;Q')A;=C?+CL'"0¤7 H P#"0(g #"+%<C%(X?78-7>'3h@?XF'{!A%(;WX=I!)=%(;Y+C|F=b7>;Y|'",!$0 HLH %(;=CW¶·¸¹ º=»» º=¼+½¾5¿\À_Á=Â+ä7T;\'"ÄZÅ>À?¹ H A;QFoU¤Æ0 H 0(9'"*!$0<;=3'&%(;?'& t 1F=!"\%,Ç_È ÉÊ2 v %(#"|C<7/1P?-/%h+CJ0; "!$#"AA;ZF=S7T;YO%(;O$R'"A;=C?+CW9r0;?'+?'"+3|!A%3+ 30<F?-/C[X=|3$-89K´r$RP?-%(;=%('"0#)hU Ë ÌÍ ÎBÏ<:Ð ?7-87TX#$%(#"hO!$0(-8-T+!$'$gA#"h|X=%b7!B'"A0<#37>+{!$0<;=!$A#";?7>;YL'" H 0Q3'!$0 HLH 0;?->hWF=3+CW3A'&'"0YA')A# 47>'"n30 H W$-> H A;?'&%(#"hMP#)0P=A#"'17T+U\`' H %hMX=N7 H P=0<#"'"+Ci%-8->'"0YA')A#27>;?g0q<7>;YÑ¿mÒ¸¼Â0; Ê*ÈÔÓÕÖ×ØÈÚÙQUGÛÓ$Ö?ÕÜ'"A0#"hJP#"0(g7C?+I%(;[7 H P?-> H A;Q'$%('37>0;M0(9'"L3A'&GP#"+3A;?'"+C[7>;n}>Ý+Þ ~ @_'"0FY 47>'"0F'W'"c!$0;=1'&%(;?'&|!$0#"#"+3P_0;=C<7>;Yi'"0M'"J$-87 H 7T;=%'37>0;#"F?->+AUkZ')A0#"hfÕÜ(×5P#)0.g<7C?+ !$0;=3'&%(;?'&G'"0O!"=%(#$%!$'"A#37>A2X?7>;=%(#)hZ#"$-%('37>0;=@_7T'B9r0# H %-8-ThZC?$D=;+¤4 =%('G7>' H +%;=G9K0#B%LX?7>;=%(#"h #"$-%('37>0;M'"0WX=N#"$ß=$RQ7>g@3h HLH A'"#37!(@'"#$%(;=b7>'37>g@oC?+!)7C%(X?->@'"0'$%-p%(;=C\C?$D=;+I')a%<!A!$+"b7>X?-> P=%(#"'I0(9 %W#"$-%('37>0;oU|`;e'"|'"A0#"heÈ àOP#)0P=A#"'17T+¤0(9{P#)0P=0QS7T'17T0<;=%-{+E<F=%-87>'ShM%(#)*P#)0.g<+CU* '"A0#"h\× ÉÜ(Ù(ÕKL!$0<;Q'&%.7T;=C?$D=;?7>'37>0;=2!)=%(#&%!$')A#37>$7T;Y\3 H 7>Y#"0<FP=A@ H 0<;0(7CI%(;=CM!$0 HLH F'&%'37>g Ý+á H 0 ;0(7CAU'"A0#37>+.@?×ÚÈ S5%(;=C+Ü+ÓC?$D=;;=%('"F#&%-?;QF H X=A#$A@-87/1'&5%;=C2%(#"#$%+hA@A')0YA'"A# 47>'"n30 H a9KF;=!$'37>0;=@#"$-%('17T0<;=*%;=CMP#"0<P=A#"'37>+|096')+3U\MW;0.4 -87/1'L'"W'"A0#37>+0(9,'" H 7!$#"0(-87>X#&%(#"hUe`;i'"?7-873'+@')OP#37 H 7>'37>gZ!$0;=3'&%;Q'&|%(#)30(4 ;i%'"Ah3'&%;=C@XF'9K0#|'" C?$D=;+Ck!$0;=3'&%;Q'&0;?->hf'"Z;=% H c%(;=C 'Sh?P?7>;YM7W30(4 ; t '"ZC?$D=;?7>'37>0;=O'" H 3$->g+L=%g X=AA;Z?7CC?A;o@_0; H %h|7>;=3P=+!$'B'" H XQhN7 H P=0<#"'37>;YL'"B'"A0#37>+ v U`;O4 #17T'17T;YN'"?7-87TX#$%(#"hW4{ =%+g<B9K0(-8->0(4+CO'"B9r0(-8->0.467T;YO!$0;?gA;?'37>0;=A ÝU;=% H + 0(93A'& 3 '$%(#"'47>'"Z%L!A%(P?7>'&%.-o->A'"'"A#+@ U;=% H + 0(9$-T H A ;Q'&G0(93A'&3'&%#"'47>'"c%L H %-8-o->A'"'"A#+@ ÞU C<7oA#"A;?' 40#&C7>;c%a;=% H 2%(#"2!$0 HLH A ;=!$+CO467T')Z%L!A%(P?7>'&%-o->A'"')A#+@ U9r0#23A'&B467T')c0<;!$0;=3'"#)F=!$'"0#B'"L"% H |;=% H a7/¤F=3+CJX=0')\9K0<#,'"W3A'2%(;=CJ'"!$0;´ 3')#"F=!$'"0#+U core Empty ∈ Set univ ∈ (↓A ∈ Set; Empty) A Not ∈ (Set) Set not ∈ (↓A ∈ Set; (A) Empty) Not(A) Unit ∈ Set unit ∈ Unit Bool ∈ Set tt ∈ Bool ff ∈ Bool Σ ∈ (A ∈ Set; B ∈ (A) Set) Set pair ∈ (↓A ∈ Set; ↓B ∈ (A) Set; a ∈ A; b ∈ B(a)) Σ(A, B) Prod ∈ (Set; Set) Set prod ∈ (↓A, ↓B ∈ Set; a ∈ A; b ∈ B) Prod(A, B) Plus ∈ (A, B ∈ Set) Set inl ∈ (↓A, ↓B ∈ Set; a ∈ A) Plus(A, B) inr ∈ (↓A, ↓B ∈ Set; b ∈ B) Plus(A, B) Lift ∈ (Set) Set bot ∈ (A ∈ Set) Lift(A) in ∈ (↓A ∈ Set; A) Lift(A) Π ∈ (A ∈ Set; B ∈ (A) Set) Set λ ∈ (↓A ∈ Set; B ∈ (A) Set; f ∈ (a ∈ A) B(a)) Π(A, B) Fun ∈ (A, B ∈ Set) Set fun ∈ (↓A, ↓B ∈ Set; f ∈ (A) B) Fun(A, B) Id ∈ (↓A ∈ Set; a, b ∈ A) Set id ∈ (↓A ∈ Set; x ∈ A) Id(x, x) I ∈ (↓A ∈ Set; A) A o ∈ (↓A, ↓B, ↓C ∈ Set; (B) C; (A) B; A) C Ý rel Refl ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set refl ∈ (↓A ∈ Set; R ∈ (A; A) Set; (a ∈ A) R(a, a)) Refl(R) Sym ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set sym ∈ (↓A ∈ Set; ↓R ∈ (A; A) Set; (a, a’ ∈ A; R(a, a’)) R(a’, a)) Sym(R) Trans ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set trans ∈ (↓A ∈ Set; ↓R ∈ (A; A) Set; (a, a’, a’’ ∈ A; R(a, a’); R(a’, a’’)) R(a, a’’)) Trans(R) Dec ∈ (↓A ∈ Set; P ∈ (A) Set) Set dec ∈ (↓A ∈ Set; ↓P ∈ (A) Set; (a ∈ A) Plus(P(a), Not(P(a)))) Dec(P) DecR ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set decR ∈ (↓A ∈ Set; ↓R ∈ (A; A) Set; (a, a’ ∈ A) Plus(R(a, a’), Not(R(a, a’)))) DecR(R) Tot ∈ (↓A ∈ Set; R ∈ (A; A) Set) Set tot ∈ (↓A ∈ Set; ↓R ∈ (A; A) Set; (a, a’ ∈ A) Plus(R(a, a’), R(a’, a))) Tot(R) Acc ∈ (↓A ∈ Set; (A; A) Set; A) Set acc ∈ (↓A ∈ Set; R ∈ (A; A) Set; a ∈ A; (b ∈ A; R(b, a)) Acc(R, b)) Acc(R, a) id substId ∈ (↓A ∈ Set; ↓a, ↓b ∈ A; Id(a, b); C ∈ (A) Set; C(a)) C(b) respId ∈ (↓A, ↓B ∈ Set; ↓a, ↓a’ ∈ A; f ∈ (A) B; Id(a, a’)) Id(f(a), f(a’)) reflId ∈ (A ∈ Set) Refl(Id()) symId1 ∈ (A ∈ Set; a, b ∈ A; Id(a, b)) Id(b, a) symId ∈ (A ∈ Set) Sym(Id()) transId1 ∈ (A ∈ Set; a, b, c ∈ A; Id(a, b); Id(b, c)) Id(a, c) transId ∈ (A ∈ Set) Trans(Id()) Ý algebra Assoc ∈ (A ∈ Set; op ∈ (A; A) A) Set assoc ∈ (A ∈ Set; op ∈ (A; A) A; (x, y, z ∈ A) Id(op(op(x, y), z), op(x, op(y, z)))) Assoc(A, op) Comm ∈ (A ∈ Set; op ∈ (A; A) A) Set comm ∈ (A ∈ Set; op ∈ (A; A) A; (x, y ∈ A) Id(op(x, y), op(y, x))) Comm(A, op) WithUnitL ∈ (A ∈ Set; op ∈ (A; A) A) Set withUnitL ∈ (A ∈ Set; op ∈ (A; A) A; e ∈ A; (x ∈ A) Id(op(e, x), x)) WithUnitL(A, op) Semigroup ∈ (A ∈ Set; op ∈ (A; A) A) Set semigroup ∈ (A ∈ Set; op ∈ (A; A) A; Assoc(A, op)) Semigroup(A, op) WithUnitR ∈ (A ∈ Set; op ∈ (A; A) A) Set withUnitR ∈ (A ∈ Set; op ∈ (A; A) A; e ∈ A; (x ∈ A) Id(op(x, e), x)) WithUnitR(A, op) Monoid ∈ (A ∈ Set; op ∈ (A; A) A) Set monoid ∈ (A ∈ Set; op ∈ (A; A) A; Assoc(A, op); WithUnitL(A, op); WithUnitR(A, op)) Monoid(A, op) CommMonoid ∈ (A ∈ Set; op ∈ (A; A) A) Set commMonoid ∈ (A ∈ Set; op ∈ (A; A) A; Assoc(A, op); Comm(A, op); WithUnitL(A, op); WithUnitR(A, op)) CommMonoid(A, op) Ý nat Nat ∈ Set 0 ∈ Nat s ∈ (Nat) Nat Lt ∈ (Nat; Nat) Set lt0 ∈ (i ∈ Nat) Lt(0, s(i)) lt1 ∈ (↓i, ↓j ∈ Nat; Lt(i, j)) Lt(s(i), s(j)) add ∈ (Nat; Nat) Nat injS ∈ (↓i, ↓j ∈ Nat; Id(s(i), s(j))) Id(i, j) noConfNat ∈ (↓i ∈ Nat; Id(0, s(i))) Empty eqNat1 ∈ (i, j ∈ Nat) Plus(Id(i, j), Not(Id(i, j))) eqNat ∈ DecR(Id()) Le ∈ (Nat; Nat) Set ltLem1 ∈ (i ∈ Nat; Lt(i, 0)) Empty ltLem2 ∈ (↓i, ↓j ∈ Nat; Lt(i, s(j))) Plus(Id(i, j), Lt(i, j)) accLtaux ∈ (i ∈ Nat; (j ∈ Nat; Lt(j, i)) Acc(Lt, j); k ∈ Nat; Lt(k, s(i))) Acc(Lt, k) accLt ∈ (i ∈ Nat) Acc(Lt, i) ltLem3 ∈ (m, n ∈ Nat; Lt(s(m), s(n))) Lt(m, n) decRLt1 ∈ (m, n ∈ Nat) Plus(Lt(m, n), Not(Lt(m, n))) decRLt ∈ DecR(Lt) trichLt ∈ (m, n ∈ Nat) Plus(Lt(m, n), Plus(Lt(n, m), Id(n, m))) totalLe ∈ (m, n ∈ Nat) Plus(Le(m, n), Le(n, m)) leSuccLem ∈ (m, n ∈ Nat; Le(m, n)) Lt(m, s(n)) ltSuccLem ∈ (n ∈ Nat) Lt(n, s(n)) assocAdd1 ∈ (x, y, z ∈ Nat) Id(add(add(x, y), z), add(x, add(y, z))) commAddLem1 ∈ (x ∈ Nat) Id(x, add(x, 0)) commAddLem2 ∈ (x, y ∈ Nat) Id(s(add(x, y)), add(x, s(y))) commAdd1 ∈ (x, y ∈ Nat) Id(add(x, y), add(y, x)) withUnitLAdd1 ∈ (x ∈ Nat) Id(add(0, x), x) withUnitRAdd1 ∈ (x ∈ Nat) Id(add(x, 0), x) assocAdd ∈ Assoc(Nat, add) commAdd ∈ Comm(Nat, add) withUnitLAdd ∈ WithUnitL(Nat, add) withUnitRAdd ∈ WithUnitR(Nat, add) addCommMonoid ∈ CommMonoid(Nat, add) Ý lists List ∈ (A ∈ Set) Set nil ∈ (↓A ∈ Set) List(A) cons ∈ (↓A ∈ Set; a ∈ A; l ∈ List(A)) List(A) map ∈ (↓A, ↓B ∈ Set; f ∈ (A) B; l ∈ List(A)) List(B) append ∈ (↓A ∈ Set; List(A); List(A)) List(A) ListAll ∈ (↓A ∈ Set; P ∈ (A) Set; List(A)) Set la0 ∈ (↓A ∈ Set; P ∈ (A) Set) ListAll(P, nil()) la1 ∈ (↓A ∈ Set; ↓P ∈ (A) Set; ↓a ∈ A; ↓l ∈ List(A); P(a); ListAll(P, l)) ListAll(P, cons(a, l)) length ∈ (↓A ∈ Set; List(A)) Nat listIdLem1 ∈ (↓A ∈ Set; ↓a ∈ A; ↓x ∈ List(A); Id(nil(), cons(a, x))) Empty listIdLem2 ∈ (↓A ∈ Set; ↓a, ↓b ∈ A; ↓x, ↓y ∈ List(A); Id(cons(a, x), cons(b, y))) Id(a, b) listIdLem3 ∈ (↓A ∈ Set; ↓a, ↓b ∈ A; ↓x, ↓y ∈ List(A); Id(cons(a, x), cons(b, y))) Id(x, y) decIdList1 ∈ (↓A ∈ Set; d ∈ (a, b ∈ A) Plus(Id(a, b), Not(Id(a, b))); x, y ∈ List(A)) Plus(Id(x, y), Not(Id(x, y))) decIdList ∈ (A ∈ Set; d ∈ DecR(Id())) DecR(Id()) assocAppend1 ∈ (A ∈ Set; x, y, z ∈ List(A)) Id(append(append(x, y), z), append(x, append(y, z))) withUnitLAppend1 ∈ (A ∈ Set; x ∈ List(A)) Id(append(nil(), x), x) withUnitRAppend1 ∈ (A ∈ Set; x ∈ List(A)) Id(append(x, nil()), x) assocAppend ∈ (A ∈ Set) Assoc(List(A), append()) withUnitLAppend ∈ (A ∈ Set) WithUnitL(List(A), append()) withUnitRAppend ∈ (A ∈ Set) WithUnitR(List(A), append()) monoidAppend ∈ (A ∈ Set) Monoid(List(A), append()) appendLength ∈ (A ∈ Set; x, y ∈ List(A)) Id(length(append(x, y)), add(length(x), length(y))) vec Fin ∈ (Nat) Set 0Fin ∈ (n ∈ Nat) Fin(s(n)) sFin ∈ (↓n ∈ Nat; Fin(n)) Fin(s(n)) finToNat ∈ (↓n ∈ Nat; Fin(n)) Nat emb ∈ (↓n ∈ Nat; Fin(n)) Fin(s(n)) addFin ∈ (↓m, ↓n ∈ Nat; 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