$( hol.mm 7-Oct-2014 $)
$(
~~ PUBLIC DOMAIN ~~
This work is waived of all rights, including copyright, according to the CC0
Public Domain Dedication. http://creativecommons.org/publicdomain/zero/1.0/
Mario Carneiro - email: di.gama at gmail.com
$)
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Foundations
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
$( Declare the primitive constant symbols for lambda calculus. $)
$c var $. $( Typecode for variables (syntax) $)
$c type $. $( Typecode for types (syntax) $)
$c term $. $( Typecode for terms (syntax) $)
$c |- $. $( Typecode for theorems (logical) $)
$c : $. $( Typehood indicator $)
$c . $. $( Separator $)
$c |= $. $( Context separator $)
$c bool $. $( Boolean type $)
$c ind $. $( 'Individual' type $)
$c -> $. $( Function type $)
$c ( $. $( Open parenthesis $)
$c ) $. $( Close parenthesis $)
$c , $. $( Context comma $)
$c \ $. $( Lambda expression $)
$c = $. $( Equality term $)
$c T. $. $( Truth term $)
$c [ $. $( Infix operator $)
$c ] $. $( Infix operator $)
$v al $. $( Greek alpha $)
$v be $. $( Greek beta $)
$v ga $. $( Greek gamma $)
$v de $. $( Greek delta $)
$v x y z f g p q $. $( Bound variables $)
$v A B C F R S T $. $( Term variables $)
$( Let variable ` al ` be a type. $)
hal $f type al $.
$( Let variable ` be ` be a type. $)
hbe $f type be $.
$( Let variable ` ga ` be a type. $)
hga $f type ga $.
$( Let variable ` de ` be a type. $)
hde $f type de $.
$( Let variable ` x ` be a var. $)
vx $f var x $.
$( Let variable ` y ` be a var. $)
vy $f var y $.
$( Let variable ` z ` be a var. $)
vz $f var z $.
$( Let variable ` f ` be a var. $)
vf $f var f $.
$( Let variable ` g ` be a var. $)
vg $f var g $.
$( Let variable ` p ` be a var. $)
vp $f var p $.
$( Let variable ` q ` be a var. $)
vq $f var q $.
$( Let variable ` A ` be a term. $)
ta $f term A $.
$( Let variable ` B ` be a term. $)
tb $f term B $.
$( Let variable ` C ` be a term. $)
tc $f term C $.
$( Let variable ` F ` be a term. $)
tf $f term F $.
$( Let variable ` R ` be a term. $)
tr $f term R $.
$( Let variable ` S ` be a term. $)
ts $f term S $.
$( Let variable ` T ` be a term. $)
tt $f term T $.
$( A var is a term. $)
tv $a term x : al $.
$( The type of all functions from type ` al ` to type ` be ` . $)
ht $a type ( al -> be ) $.
$( The type of booleans (true and false). $)
hb $a type bool $.
$( The type of individuals. $)
hi $a type ind $.
$( A combination (function application). $)
kc $a term ( F T ) $.
$( A lambda abstraction. $)
kl $a term \ x : al . T $.
$( The equality term. $)
ke $a term = $.
$( Truth term. $)
kt $a term T. $.
$( Infix operator. $)
kbr $a term [ A F B ] $.
$( Context operator. $)
kct $a term ( A , B ) $.
$c wff $. $( Not used; for mmj2 compatibility $)
$( Internal axiom for mmj2 use. $)
wffMMJ2 $a wff A |= B $.
$( Internal axiom for mmj2 use. $)
wffMMJ2t $a wff A : al $.
${
idi.1 $e |- R |= A $.
$( The identity inference. $)
idi $p |- R |= A $=
( ) C $.
$( [9-Oct-2014] $)
$}
${
idt.1 $e |- A : al $.
$( The identity inference. $)
idt $p |- A : al $=
( ) C $.
$( [9-Oct-2014] $)
$}
${
ax-syl.1 $e |- R |= S $.
ax-syl.2 $e |- S |= T $.
$( Syllogism inference. $)
ax-syl $a |- R |= T $.
$( Syllogism inference. $)
syl $p |- R |= T $=
( ax-syl ) ABCDEF $.
$( [8-Oct-2014] $)
$}
${
ax-jca.1 $e |- R |= S $.
ax-jca.2 $e |- R |= T $.
$( Join common antecedents. $)
ax-jca $a |- R |= ( S , T ) $.
$( Syllogism inference. $)
jca $p |- R |= ( S , T ) $=
( ax-jca ) ABCDEF $.
$( [8-Oct-2014] $)
$}
${
syl2anc.1 $e |- R |= S $.
syl2anc.2 $e |- R |= T $.
syl2anc.3 $e |- ( S , T ) |= A $.
$( Syllogism inference. $)
syl2anc $p |- R |= A $=
( kct jca syl ) BCDHABCDEFIGJ $.
$( [7-Oct-2014] $)
$}
${
ax-simpl.1 $e |- R : bool $.
ax-simpl.2 $e |- S : bool $.
$( Extract an assumption from the context. $)
ax-simpl $a |- ( R , S ) |= R $.
$( Extract an assumption from the context. $)
ax-simpr $a |- ( R , S ) |= S $.
$( Extract an assumption from the context. $)
simpl $p |- ( R , S ) |= R $=
( ax-simpl ) ABCDE $.
$( [8-Oct-2014] $)
$( Extract an assumption from the context. $)
simpr $p |- ( R , S ) |= S $=
( ax-simpr ) ABCDE $.
$( [8-Oct-2014] $)
$}
${
ax-id.1 $e |- R : bool $.
$( The identity inference. $)
ax-id $a |- R |= R $.
$( The identity inference. $)
id $p |- R |= R $=
( ax-id ) ABC $.
$( [7-Oct-2014] $)
$}
${
ax-trud.1 $e |- R : bool $.
$( Deduction form of ~ tru . $)
ax-trud $a |- R |= T. $.
$( Deduction form of ~ tru . $)
trud $p |- R |= T. $=
( ax-trud ) ABC $.
$( [7-Oct-2014] $)
ax-a1i.2 $e |- T. |= A $.
$( Change an empty context into any context. $)
a1i $p |- R |= A $=
( kt ax-trud syl ) BEABCFDG $.
$( [7-Oct-2014] $)
$}
${
ax-cb.1 $e |- R |= A $.
$( A context has type boolean.
This and the next few axioms are not strictly necessary, and are
conservative on any theorem for which every variable has a specified
type, but by adding this axiom we can save some typehood hypotheses in
many theorems. The easy way to see that this axiom is conservative is
to note that every axiom and inference rule that constructs a theorem of
the form ` R |= A ` where ` R ` and ` A ` are type variables, also
ensures that ` R : bool ` and ` A : bool ` . Thus it is impossible to
prove any theorem ` |- R |= A ` unless both ` |- R : bool ` and
` |- A : bool ` had been previously derived, so it is conservative to
deduce ` |- R : bool ` from ` |- R |= A ` . The same remark applies to
the construction of the theorem ` ( A , B ) : bool ` - there is only one
rule that creates a formula of this type, namely ~ wct , and it requires
that ` A : bool ` and ` B : bool ` be previously established, so it is
safe to reverse the process in ~ wctl and ~ wctr . $)
ax-cb1 $a |- R : bool $.
$( A theorem has type boolean. (This axiom is unnecessary;
see ~ ax-cb1 .) $)
ax-cb2 $a |- A : bool $.
$}
${
wctl.1 $e |- ( S , T ) : bool $.
$( Reverse closure for the type of a context. (This axiom is unnecessary;
see ~ ax-cb1 .) $)
wctl $a |- S : bool $.
$( Reverse closure for the type of a context. (This axiom is unnecessary;
see ~ ax-cb1 .) $)
wctr $a |- T : bool $.
$}
${
mpdan.1 $e |- R |= S $.
mpdan.2 $e |- ( R , S ) |= T $.
$( Modus ponens deduction. $)
mpdan $p |- R |= T $=
( ax-cb1 id syl2anc ) CAABABADFGDEH $.
$( [8-Oct-2014] $)
$}
${
syldan.1 $e |- ( R , S ) |= T $.
syldan.2 $e |- ( R , T ) |= A $.
$( Syllogism inference. $)
syldan $p |- ( R , S ) |= A $=
( kct ax-cb1 wctl wctr simpl syl2anc ) ABCGZBDBCBCDMEHZIBCNJKEFL $.
$( [8-Oct-2014] $)
$}
${
simpld.1 $e |- R |= ( S , T ) $.
$( Extract an assumption from the context. $)
simpld $p |- R |= S $=
( kct ax-cb2 wctl wctr simpl syl ) ABCEZBDBCBCKADFZGBCLHIJ $.
$( [8-Oct-2014] $)
$( Extract an assumption from the context. $)
simprd $p |- R |= T $=
( kct ax-cb2 wctl wctr simpr syl ) ABCEZCDBCBCKADFZGBCLHIJ $.
$( [8-Oct-2014] $)
$}
${
trul.1 $e |- ( T. , R ) |= S $.
$( Deduction form of ~ tru . $)
trul $p |- R |= S $=
( kt kct ax-cb1 wctr trud id syl2anc ) BADAADABDAECFGZHAKICJ $.
$( [7-Oct-2014] $)
$}
$( The equality function has type ` al -> al -> bool ` , i.e. it is
polymorphic over all types, but the left and right type must agree. $)
weq $a |- = : ( al -> ( al -> bool ) ) $.
${
ax-refl.1 $e |- A : al $.
$( Reflexivity of equality. $)
ax-refl $a |- T. |= ( ( = A ) A ) $.
$}
$( Truth type. $)
wtru $p |- T. : bool $=
( ke kc kt hb ht weq ax-refl ax-cb1 ) AABABCDDDEEADFGH $.
$( [10-Oct-2014] $)
$( Tautology is provable. $)
tru $p |- T. |= T. $=
( kt wtru id ) ABC $.
$( [7-Oct-2014] $)
${
ax-eqmp.1 $e |- R |= A $.
ax-eqmp.2 $e |- R |= ( ( = A ) B ) $.
$( Modus ponens for equality. $)
ax-eqmp $a |- R |= B $.
$}
${
ax-ded.1 $e |- ( R , S ) |= T $.
ax-ded.2 $e |- ( R , T ) |= S $.
$( Deduction theorem for equality. $)
ax-ded $a |- R |= ( ( = S ) T ) $.
$}
${
wct.1 $e |- S : bool $.
wct.2 $e |- T : bool $.
$( The type of a context. $)
wct $a |- ( S , T ) : bool $.
$}
${
wc.1 $e |- F : ( al -> be ) $.
wc.2 $e |- T : al $.
$( The type of a combination. $)
wc $a |- ( F T ) : be $.
$}
${
ax-ceq.1 $e |- F : ( al -> be ) $.
ax-ceq.2 $e |- T : ( al -> be ) $.
ax-ceq.3 $e |- A : al $.
ax-ceq.4 $e |- B : al $.
$( Equality theorem for combination. $)
ax-ceq $a |- ( ( ( = F ) T ) , ( ( = A ) B ) ) |=
( ( = ( F A ) ) ( T B ) ) $.
$}
${
eqcomx.1 $e |- A : al $.
eqcomx.2 $e |- B : al $.
eqcomx.3 $e |- R |= ( ( = A ) B ) $.
$( Commutativity of equality. $)
eqcomx $p |- R |= ( ( = B ) A ) $=
( ke kc ax-cb1 ax-refl a1i hb ht weq ax-ceq syl2anc wc ax-eqmp ) HBIZBIZH
CIZBIZDUADTCIZDGJZABEKLZHUAIUCIDHTIUBIZUAUGDHHIHIZUDUHDUEAAMNZNHAOZKLGAUI
BCHHUJUJEFPQUFAMBBTUBAUIHBUJERAUIHCUJFREEPQS $.
$( [7-Oct-2014] $)
$}
${
mpbirx.1 $e |- B : bool $.
mpbirx.2 $e |- R |= A $.
mpbirx.3 $e |- R |= ( ( = B ) A ) $.
$( Deduction from equality inference. $)
mpbirx $p |- R |= B $=
( hb ax-cb2 eqcomx ax-eqmp ) ABCEGBACDACEHFIJ $.
$( [7-Oct-2014] $)
$}
${
ancoms.1 $e |- ( R , S ) |= T $.
$( Swap the two elements of a context. $)
ancoms $p |- ( S , R ) |= T $=
( kct ax-cb1 wctr wctl simpr simpl syl2anc ) CBAEABBAABCABEDFZGZABLHZIBAM
NJDK $.
$( [8-Oct-2014] $)
$}
${
adantr.1 $e |- R |= T $.
adantr.2 $e |- S : bool $.
$( Extract an assumption from the context. $)
adantr $p |- ( R , S ) |= T $=
( kct ax-cb1 simpl syl ) ABFACABCADGEHDI $.
$( [8-Oct-2014] $)
$( Extract an assumption from the context. $)
adantl $p |- ( S , R ) |= T $=
( adantr ancoms ) ABCABCDEFG $.
$( [8-Oct-2014] $)
$}
${
ct1.1 $e |- R |= S $.
ct1.2 $e |- T : bool $.
$( Introduce a right conjunct. $)
ct1 $p |- ( R , T ) |= ( S , T ) $=
( kct adantr ax-cb1 simpr jca ) ACFBCACBDEGACBADHEIJ $.
$( Introduce a left conjunct. $)
ct2 $p |- ( T , R ) |= ( T , S ) $=
( kct ax-cb1 simpl adantl jca ) CAFCBCAEBADGHACBDEIJ $.
$}
${
sylan.1 $e |- R |= S $.
sylan.2 $e |- ( S , T ) |= A $.
$( Syllogism inference. $)
sylan $p |- ( R , T ) |= A $=
( kct ax-cb1 wctr adantr simpr syl2anc ) ABDGCDBDCECDACDGFHIZJBDCBEHMKFL
$.
$( [8-Oct-2014] $)
$}
${
an32s.1 $e |- ( ( R , S ) , T ) |= A $.
$( Commutation identity for context. $)
an32s $p |- ( ( R , T ) , S ) |= A $=
( kct ax-cb1 wctl wctr simpl ct1 simpr adantr syl2anc ) ABDFZCFBCFZDOBCBD
BCPDAPDFEGZHZHZPDQIZJBCRIZKOCDBDSTLUAMEN $.
$( [8-Oct-2014] $)
$( Associativity for context. $)
anasss $p |- ( R , ( S , T ) ) |= A $=
( kct ax-cb1 wctl id ancoms sylan an32s ) CDFBAACBDACBFBCFZDBCMMMDAMDFEGH
IJEKLJ $.
$( [8-Oct-2014] $)
$}
${
anassrs.1 $e |- ( R , ( S , T ) ) |= A $.
$( Associativity for context. $)
anassrs $p |- ( ( R , S ) , T ) |= A $=
( kct ax-cb1 wctl wctr simpl adantr simpr ct1 syl2anc ) ABCFZDFBCDFZODBBC
BPABPFEGZHZCDBPQIZHZJCDSIZKOCDBCRTLUAMEN $.
$( [8-Oct-2014] $)
$}
$( The type of a typed variable. $)
wv $a |- x : al : al $.
${
wl.1 $e |- T : be $.
$( The type of a lambda abstraction. $)
wl $a |- \ x : al . T : ( al -> be ) $.
$}
${
ax-beta.1 $e |- A : be $.
$( Axiom of beta-substitution. $)
ax-beta $a |- T. |= ( ( = ( \ x : al . A x : al ) ) A ) $.
ax-distrc.2 $e |- B : al $.
ax-distrc.3 $e |- F : ( be -> ga ) $.
$( Distribution of combination over substitution. $)
ax-distrc $a |- T. |= ( ( = ( \ x : al . ( F A ) B ) )
( ( \ x : al . F B ) ( \ x : al . A B ) ) ) $.
$}
${
$d x R $.
ax-leq.1 $e |- A : be $.
ax-leq.2 $e |- B : be $.
ax-leq.3 $e |- R |= ( ( = A ) B ) $.
$( Equality theorem for abstraction. $)
ax-leq $a |- R |= ( ( = \ x : al . A ) \ x : al . B ) $.
$}
${
$d x y $. $d y B $.
ax-distrl.1 $e |- A : ga $.
ax-distrl.2 $e |- B : al $.
$( Distribution of lambda abstraction over substitution. $)
ax-distrl $a |- T. |=
( ( = ( \ x : al . \ y : be . A B ) ) \ y : be . ( \ x : al . A B ) ) $.
$}
${
wov.1 $e |- F : ( al -> ( be -> ga ) ) $.
wov.2 $e |- A : al $.
wov.3 $e |- B : be $.
$( Type of an infix operator. $)
wov $a |- [ A F B ] : ga $.
$( Infix operator. This is a simple metamath way of cleaning up the syntax
of all these infix operators to make them a bit more readable than the
curried representation. $)
df-ov $a |- T. |= ( ( = [ A F B ] ) ( ( F A ) B ) ) $.
$}
${
dfov1.1 $e |- F : ( al -> ( be -> bool ) ) $.
dfov1.2 $e |- A : al $.
dfov1.3 $e |- B : be $.
${
dfov1.4 $e |- R |= [ A F B ] $.
$( Forward direction of ~ df-ov . $)
dfov1 $p |- R |= ( ( F A ) B ) $=
( kbr kc ke ax-cb1 hb df-ov a1i ax-eqmp ) CDEKZECLDLZFJMSLTLFSFJNABOCDE
GHIPQR $.
$( [8-Oct-2014] $)
$}
dfov2.4 $e |- R |= ( ( F A ) B ) $.
$( Reverse direction of ~ df-ov . $)
dfov2 $p |- R |= [ A F B ] $=
( kc kbr hb wov ke ax-cb1 df-ov a1i mpbirx ) ECKDKZCDELZFABMCDEGHINJOUAKT
KFTFJPABMCDEGHIQRS $.
$( [8-Oct-2014] $)
$}
${
weqi.1 $e |- A : al $.
weqi.2 $e |- B : al $.
$( Type of an equality. $)
weqi $p |- [ A = B ] : bool $=
( hb ke weq wov ) AAFBCGAHDEI $.
$( [8-Oct-2014] $)
$}
${
eqcomi.1 $e |- A : al $.
eqcomi.2 $e |- R |= [ A = B ] $.
$( Deduce equality of types from equality of expressions. (This is
unnecessary but eliminates a lot of hypotheses.) $)
eqtypi $a |- B : al $.
$( Commutativity of equality. $)
eqcomi $p |- R |= [ B = A ] $=
( ke weq eqtypi dfov1 eqcomx dfov2 ) AACBGDAHZABCDEFIZEABCDENAABCGDMENFJK
L $.
$( [7-Oct-2014] $)
$}
${
eqtypri.1 $e |- A : al $.
eqtypri.2 $e |- R |= [ B = A ] $.
$( Deduce equality of types from equality of expressions. (This is
unnecessary but eliminates a lot of hypotheses.) $)
eqtypri $a |- B : al $.
$}
${
mpbi.1 $e |- R |= A $.
mpbi.2 $e |- R |= [ A = B ] $.
$( Deduction from equality inference. $)
mpbi $p |- R |= B $=
( hb ke weq ax-cb2 eqtypi dfov1 ax-eqmp ) ABCDFFABGCFHACDIZFABCMEJEKL $.
$( [7-Oct-2014] $)
$}
${
eqid.1 $e |- R : bool $.
eqid.2 $e |- A : al $.
$( Reflexivity of equality. $)
eqid $p |- R |= [ A = A ] $=
( ke weq kc ax-refl a1i dfov2 ) AABBFCAGEEFBHBHCDABEIJK $.
$( [7-Oct-2014] $)
$}
${
ded.1 $e |- ( R , S ) |= T $.
ded.2 $e |- ( R , T ) |= S $.
$( Deduction theorem for equality. $)
ded $p |- R |= [ S = T ] $=
( hb ke weq kct ax-cb2 ax-ded dfov2 ) FFBCGAFHBACIEJCABIDJABCDEKL $.
$( [7-Oct-2014] $)
$}
${
dedi.1 $e |- S |= T $.
dedi.2 $e |- T |= S $.
$( Deduction theorem for equality. $)
dedi $p |- T. |= [ S = T ] $=
( kt wtru adantl ded ) EABAEBCFGBEADFGH $.
$( [7-Oct-2014] $)
$}
${
eqtru.1 $e |- R |= A $.
$( If a statement is provable, then it is equivalent to truth. $)
eqtru $p |- R |= [ T. = A ] $=
( kt wtru adantr kct ax-cb1 ax-cb2 wct tru a1i ded ) BDABDACEFDBAGBAABCHA
BCIJKLM $.
$( [8-Oct-2014] $)
$}
${
mpbir.1 $e |- R |= A $.
mpbir.2 $e |- R |= [ B = A ] $.
$( Deduction from equality inference. $)
mpbir $p |- R |= B $=
( hb ax-cb2 eqtypri eqcomi mpbi ) ABCDFBACFABCACDGEHEIJ $.
$( [7-Oct-2014] $)
$}
${
ceq12.1 $e |- F : ( al -> be ) $.
ceq12.2 $e |- A : al $.
${
ceq12.3 $e |- R |= [ F = T ] $.
${
ceq12.4 $e |- R |= [ A = B ] $.
$( Equality theorem for combination. $)
ceq12 $p |- R |= [ ( F A ) = ( T B ) ] $=
( kc ke weq wc ht eqtypi dfov1 ax-ceq syl2anc dfov2 ) BBECLZGDLZMFBNA
BECHIOABGDABPZEGFHJQZACDFIKQZOMUBLUCLFMELGLMCLDLUDUDEGMFUDNHUEJRAACDM
FANIUFKRABCDEGHUEIUFSTUA $.
$( [7-Oct-2014] $)
$}
$( Equality theorem for combination. $)
ceq1 $p |- R |= [ ( F A ) = ( T A ) ] $=
( ke kbr ax-cb1 eqid ceq12 ) ABCCDEFGHIACEDFJKEILHMN $.
$( [7-Oct-2014] $)
$}
ceq2.3 $e |- R |= [ A = B ] $.
$( Equality theorem for combination. $)
ceq2 $p |- R |= [ ( F A ) = ( F B ) ] $=
( ht ke kbr ax-cb1 eqid ceq12 ) ABCDEFEGHABJEFCDKLFIMGNIO $.
$( [7-Oct-2014] $)
$}
${
$d x R $.
leq.1 $e |- A : be $.
leq.2 $e |- R |= [ A = B ] $.
$( Equality theorem for lambda abstraction. $)
leq $p |- R |= [ \ x : al . A = \ x : al . B ] $=
( ht kl ke weq wl eqtypi dfov1 ax-leq dfov2 ) ABIZRACDJACEJKFRLABCDGMABCE
BDEFGHNZMABCDEFGSBBDEKFBLGSHOPQ $.
$( [8-Oct-2014] $)
$}
${
beta.1 $e |- A : be $.
$( Axiom of beta-substitution. $)
beta $p |- T. |= [ ( \ x : al . A x : al ) = A ] $=
( kl tv kc ke kt weq wl wv wc ax-beta dfov2 ) BBACDFZACGZHDIJBKABQRABCDEL
ACMNEABCDEOP $.
$( [8-Oct-2014] $)
$}
${
distrc.1 $e |- F : ( be -> ga ) $.
distrc.2 $e |- A : be $.
distrc.3 $e |- B : al $.
$( Distribution of combination over substitution. $)
distrc $p |- T. |= [ ( \ x : al . ( F A ) B ) =
( ( \ x : al . F B ) ( \ x : al . A B ) ) ] $=
( kc kl ke kt weq wc wl ht ax-distrc dfov2 ) CCADGEKZLZFKADGLZFKZADELZFKZ
KMNCOACUBFACDUABCGEHIPQJPBCUDUFABCRZUCFAUGDGHQJPABUEFABDEIQJPPABCDEFGIJHS
T $.
$( [8-Oct-2014] $)
$}
${
$d x y $. $d y B $.
distrl.1 $e |- A : ga $.
distrl.2 $e |- B : al $.
$( Distribution of lambda abstraction over substitution. $)
distrl $p |- T. |=
[ ( \ x : al . \ y : be . A B ) = \ y : be . ( \ x : al . A B ) ] $=
( ht kl kc ke kt weq wl wc ax-distrl dfov2 ) BCJZTADBEFKZKZGLBEADFKZGLZKM
NTOATUBGATDUABCEFHPPIQBCEUDACUCGACDFHPIQPABCDEFGHIRS $.
$( [8-Oct-2014] $)
$}
${
eqtri.1 $e |- A : al $.
eqtri.2 $e |- R |= [ A = B ] $.
eqtri.3 $e |- R |= [ B = C ] $.
$( Transitivity of equality. $)
eqtri $p |- R |= [ A = C ] $=
( ke weq eqtypi kc dfov1 hb ht wc ceq2 mpbi dfov2 ) AABDIEAJZFACDEABCEFGK
ZHKIBLZCLUBDLEAABCIETFUAGMANCDUBEAANOIBTFPUAHQRS $.
$( [7-Oct-2014] $)
$}
${
3eqtr4i.1 $e |- A : al $.
3eqtr4i.2 $e |- R |= [ A = B ] $.
${
3eqtr4i.3 $e |- R |= [ S = A ] $.
3eqtr4i.4 $e |- R |= [ T = B ] $.
$( Transitivity of equality. $)
3eqtr4i $p |- R |= [ S = T ] $=
( eqtypri eqtypi eqcomi eqtri ) AEBFDABEDGIKIABCFDGHAFCDACFDABCDGHLJKJMNN
$.
$( [7-Oct-2014] $)
$}
3eqtr3i.3 $e |- R |= [ A = S ] $.
3eqtr3i.4 $e |- R |= [ B = T ] $.
$( Transitivity of equality. $)
3eqtr3i $p |- R |= [ S = T ] $=
( eqcomi eqtypi 3eqtr4i ) ABCDEFGHABEDGIKACFDABCDGHLJKM $.
$( [7-Oct-2014] $)
$}
${
oveq.1 $e |- F : ( al -> ( be -> ga ) ) $.
oveq.2 $e |- A : al $.
oveq.3 $e |- B : be $.
${
oveq123.4 $e |- R |= [ F = S ] $.
oveq123.5 $e |- R |= [ A = C ] $.
oveq123.6 $e |- R |= [ B = T ] $.
$( Equality theorem for binary operation. $)
oveq123 $p |- R |= [ [ A F B ] = [ C S T ] ] $=
( kc kbr wc ke ht ceq12 weq wov ax-cb1 df-ov a1i dfov2 eqtypi 3eqtr4i )
CGDQZEQZIFQZJQZHDEGRZFJIRZBCUKEABCUAZGDKLSZMSZBCEJUKHUMURMAUQDFGHIKLNOU
BPUBCCUOULTHCUCZABCDEGKLMUDUSTUOQULQHGITRHNUEZABCDEGKLMUFUGUHCCUPUNTHUT
ABCFJIAUQUAGIHKNUIZADFHLOUIZBEJHMPUIZUDBCUMJAUQIFVBVCSVDSTUPQUNQHVAABCF
JIVBVCVDUFUGUHUJ $.
$( [7-Oct-2014] $)
$}
${
oveq1.4 $e |- R |= [ A = C ] $.
$( Equality theorem for binary operation. $)
oveq1 $p |- R |= [ [ A F B ] = [ C F B ] ] $=
( ht ke kbr ax-cb1 eqid oveq123 ) ABCDEFGHGEIJKABCMMGHDFNOHLPZIQLBEHSKQ
R $.
$( [7-Oct-2014] $)
oveq12.5 $e |- R |= [ B = T ] $.
$( Equality theorem for binary operation. $)
oveq12 $p |- R |= [ [ A F B ] = [ C F T ] ] $=
( ht ke kbr ax-cb1 eqid oveq123 ) ABCDEFGHGIJKLABCOOGHDFPQHMRJSMNT $.
$( [7-Oct-2014] $)
$}
${
oveq2.4 $e |- R |= [ B = T ] $.
$( Equality theorem for binary operation. $)
oveq2 $p |- R |= [ [ A F B ] = [ A F T ] ] $=
( ke kbr ax-cb1 eqid oveq12 ) ABCDEDFGHIJKADGEHMNGLOJPLQ $.
$( [7-Oct-2014] $)
$}
${
oveq.4 $e |- R |= [ F = S ] $.
$( Equality theorem for binary operation. $)
oveq $p |- R |= [ [ A F B ] = [ A S B ] ] $=
( ke kbr ax-cb1 eqid oveq123 ) ABCDEDFGHEIJKLADGFHMNGLOZJPBEGRKPQ $.
$( [7-Oct-2014] $)
$}
$}
${
ax-hbl1.1 $e |- A : ga $.
ax-hbl1.2 $e |- B : al $.
$( ` x ` is bound in ` \ x A ` . $)
ax-hbl1 $a |- T. |= [ ( \ x : al . \ x : be . A B ) = \ x : be . A ] $.
hbl1.3 $e |- R : bool $.
$( Inference form of ~ ax-hbl1 . $)
hbl1 $p |- R |= [ ( \ x : al . \ x : be . A B ) = \ x : be . A ] $=
( kl kc ke kbr ax-hbl1 a1i ) ADBDEKZKFLQMNGJABCDEFHIOP $.
$( [8-Oct-2014] $)
$}
${
$d x A $.
ax-17.1 $e |- A : be $.
ax-17.2 $e |- B : al $.
$( If ` x ` does not appear in ` A ` , then any substitution to ` A `
yields ` A ` again, i.e. ` \ x A ` is a constant function. $)
ax-17 $a |- T. |= [ ( \ x : al . A B ) = A ] $.
a17i.3 $e |- R : bool $.
$( Inference form of ~ ax-17 . $)
a17i $p |- R |= [ ( \ x : al . A B ) = A ] $=
( kl kc ke kbr ax-17 a1i ) ACDJEKDLMFIABCDEGHNO $.
$( [8-Oct-2014] $)
$}
${
$d x R $.
hbxfr.1 $e |- T : be $.
hbxfr.2 $e |- B : al $.
${
hbxfrf.3 $e |- R |= [ T = A ] $.
hbxfrf.4 $e |- ( S , R ) |= [ ( \ x : al . A B ) = A ] $.
$( Transfer a hypothesis builder to an equivalent expression. $)
hbxfrf $p |- ( S , R ) |= [ ( \ x : al . T B ) = T ] $=
( kl kc kct eqtypi wl ke kbr adantl wc leq ceq1 ax-cb1 wctl 3eqtr4i ) B
ACDMZENZDGFOZACHMZENZHABUGEABCDBHDFIKPQJUALFGUKUHRSABEUJFUGABCHIQJABCHD
FIKUBUCGFUHDRSUILUDUEZTFGHDRSKULTUF $.
$( [8-Oct-2014] $)
$}
hbxfr.3 $e |- R |= [ T = A ] $.
hbxfr.4 $e |- R |= [ ( \ x : al . A B ) = A ] $.
$( Transfer a hypothesis builder to an equivalent expression. $)
hbxfr $p |- R |= [ ( \ x : al . T B ) = T ] $=
( kl kc ke kbr ax-cb1 id adantr hbxfrf syl2anc ) ACGLEMGNOFFFFGDNOFJPZQZU
BABCDEFFGHIJFFACDLEMDNOKUARST $.
$( [8-Oct-2014] $)
$}
${
$d x R $.
hbth.1 $e |- B : al $.
hbth.2 $e |- R |= A $.
$( Hypothesis builder for a theorem. $)
hbth $p |- R |= [ ( \ x : al . A B ) = A ] $=
( hb kt ax-cb2 wtru eqtru eqcomi ax-cb1 a17i hbxfr ) AHBIDECCEGJFHICEKCEG
LMAHBIDEKFCEGNOP $.
$( [8-Oct-2014] $)
$}
${
hbc.1 $e |- F : ( be -> ga ) $.
hbc.2 $e |- A : be $.
hbc.3 $e |- B : al $.
hbc.4 $e |- R |= [ ( \ x : al . F B ) = F ] $.
hbc.5 $e |- R |= [ ( \ x : al . A B ) = A ] $.
$( Hypothesis builder for combination. $)
hbc $p |- R |= [ ( \ x : al . ( F A ) B ) = ( F A ) ] $=
( kc kl wc wl ke kbr ax-cb1 distrc a1i ht eqtypri ceq12 eqtri ) CADGENZOZ
FNZADGOZFNZADEOFNZNZUGHACUHFACDUGBCGEIJPQKPUIUMRSHUKGRSHLTABCDEFGIJKUAUBB
CULEUKHGABCUCZUJFAUNDGIQKPBEULHJMUDLMUEUF $.
$( [8-Oct-2014] $)
$}
${
hbov.1 $e |- F : ( be -> ( ga -> de ) ) $.
hbov.2 $e |- A : be $.
hbov.3 $e |- B : al $.
hbov.4 $e |- C : ga $.
hbov.5 $e |- R |= [ ( \ x : al . F B ) = F ] $.
hbov.6 $e |- R |= [ ( \ x : al . A B ) = A ] $.
hbov.7 $e |- R |= [ ( \ x : al . C B ) = C ] $.
$( Hypothesis builder for binary operation. $)
hbov $p |- R |= [ ( \ x : al . [ A F C ] B ) = [ A F C ] ] $=
( kt kbr kc kl ke ax-cb1 trud wov weq ht wc df-ov dfov2 hbc adantr hbxfrf
wtru mpdan ) JRAEFHISZUAGTUPUBSJAEIUAGTIUBSJOUCUDADEIFTZHTZGRJUPBCDFHIKLN
UEZMDDUPURUBRDUFUSCDUQHBCDUGZIFKLUHZNUHBCDFHIKLNUIUJJRAEURUAGTURUBSACDEHG
UQJVANMABUTEFGIJKLMOPUKQUKUNULUMUO $.
$( [8-Oct-2014] $)
$}
${
$d x y $. $d y B $. $d y R $.
hbl.1 $e |- A : ga $.
hbl.2 $e |- B : al $.
hbl.3 $e |- R |= [ ( \ x : al . A B ) = A ] $.
$( Hypothesis builder for lambda abstraction. $)
hbl $p |- R |= [ ( \ x : al . \ y : be . A B ) = \ y : be . A ] $=
( ht kl kc wl wc ke kbr ax-cb1 distrl a1i leq eqtri ) BCLZADBEFMZMZGNZBEA
DFMZGNZMZUEHAUDUFGAUDDUEBCEFIOOJPUGUJQRHUIFQRHKSABCDEFGIJTUABCEUIFHACUHGA
CDFIOJPKUBUC $.
$( [8-Oct-2014] $)
$}
${
$d x y $. $d y B $. $d y S $.
ax-inst.1 $e |- R |= A $.
ax-inst.2 $e |- T. |= [ ( \ x : al . B y : al ) = B ] $.
ax-inst.3 $e |- T. |= [ ( \ x : al . S y : al ) = S ] $.
ax-inst.4 $e |- [ x : al = C ] |= [ A = B ] $.
ax-inst.5 $e |- [ x : al = C ] |= [ R = S ] $.
$( Instantiate a theorem with a new term. The second and third hypotheses
are the HOL equivalent of set.mm "effectively not free in" predicate
(see set.mm's ax-17, or ~ ax17 ). $)
ax-inst $a |- S |= B $.
$}
${
$d x y R $. $d y B $.
insti.1 $e |- C : al $.
insti.2 $e |- B : bool $.
insti.3 $e |- R |= A $.
insti.4 $e |- T. |= [ ( \ x : al . B y : al ) = B ] $.
insti.5 $e |- [ x : al = C ] |= [ A = B ] $.
$( Instantiate a theorem with a new term. $)
insti $p |- R |= B $=
( hb tv ax-cb1 wv ax-17 ke kbr eqid ax-inst ) ABCDEFGGJKAMBGACNDGJOZACPQL
MGABNFRSZDERSUCLOUBTUA $.
$( [8-Oct-2014] $)
$}
${
$d y A $. $d y B $. $d y C $. $d x y al $.
clf.1 $e |- A : be $.
clf.2 $e |- C : al $.
clf.3 $e |- [ x : al = C ] |= [ A = B ] $.
clf.4 $e |- T. |= [ ( \ x : al . B y : al ) = B ] $.
clf.5 $e |- T. |= [ ( \ x : al . C y : al ) = C ] $.
$( Evaluate a lambda expression. $)
clf $p |- T. |= [ ( \ x : al . A C ) = B ] $=
( kl tv kc ke kbr kt wc hb wl eqtypi weqi ax-cb1 beta a1i wv ht a17i hbl1
weq hbc hbov id ceq2 oveq12 insti ) ACDACEMZACNZOZEPQZURGOZFPQGRIBVBFABUR
GABCEHUAZISZBEFUSGPQZHJUBZUCVARACFMADNZOFPQRKUDZABCEHUEUFABBTCVBVGFPRBUKZ
VDADUGZVFABBTUHUHCPVGRVIVJVHUIAABCGVGURRVCIVJAABCEVGRHVJVHUJLULKUMBBTUTEV
BPVEFVIABURUSVCACUGZSHABUSGURVEVCVKVEAUSGVKIUCUNUOJUPUQ $.
$( [8-Oct-2014] $)
$}
${
$d y A $. $d x y B $. $d x y C $. $d x y al $.
cl.1 $e |- A : be $.
cl.2 $e |- C : al $.
cl.3 $e |- [ x : al = C ] |= [ A = B ] $.
$( Evaluate a lambda expression. $)
cl $p |- T. |= [ ( \ x : al . A C ) = B ] $=
( vy tv ke kbr eqtypi wv ax-17 clf ) ABCJDEFGHIABCEAJKZBDEACKFLMGINAJOZPA
ACFRHSPQ $.
$( [8-Oct-2014] $)
$}
${
$d x B $. $d y C $. $d x y S $. $d y T $. $d x al $. $d y be $.
ovl.1 $e |- A : ga $.
ovl.2 $e |- S : al $.
ovl.3 $e |- T : be $.
ovl.4 $e |- [ x : al = S ] |= [ A = B ] $.
ovl.5 $e |- [ y : be = T ] |= [ B = C ] $.
$( Evaluate a lambda expression in a binary operation. $)
ovl $p |- T. |= [ [ S \ x : al . \ y : be . A T ] = C ] $=
( kl kbr kc kt ke ht wl wov weq wc wtru df-ov a1i dfov2 distrl ceq1 eqtri
tv wv weqi cl ) CIJADBEFPZPZQZBEADFPZIRZPZJRZHSABCIJURABCUAZDUQBCEFKUBUBZ
LMUCZCUSURIRZJRZVCSVFCCUSVHTSCUDVFBCVGJAVDURIVELUEZMUETUSRVHRSUFABCIJURVE
LMUGUHUIBCJVGSVBVIMVGVBTQSUFABCDEFIKLUJUHUKULBCEVAHJACUTIACDFKUBLUEZMCVAG
HBEUMZJTQZVJVAGTQVLBVKJBEUNMUOACDFGIKLNUPUHOULUPUL $.
$( [8-Oct-2014] $)
$}
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Add propositional calculus definitions
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
$c F. $. $( Contradiction term $)
$c /\ $. $( Conjunction term $)
$c ~ $. $( Negation term $)
$c ==> $. $( Implication term $)
$c ! $. $( For all term $)
$c ? $. $( There exists term $)
$c \/ $. $( Disjunction term $)
$c ?! $. $( There exists unique term $)
$( Contradiction term. $)
tfal $a term F. $.
$( Conjunction term. $)
tan $a term /\ $.
$( Negation term. $)
tne $a term ~ $.
$( Implication term. $)
tim $a term ==> $.
$( For all term. $)
tal $a term ! $.
$( There exists term. $)
tex $a term ? $.
$( Disjunction term. $)
tor $a term \/ $.
$( There exists unique term. $)
teu $a term ?! $.
${
$d f p q x y $.
$( Define the for all operator. $)
df-al $a |- T. |=
[ ! = \ p : ( al -> bool ) . [ p : ( al -> bool ) = \ x : al . T. ] ] $.
$( Define the constant false. $)
df-fal $a |- T. |= [ F. = ( ! \ p : bool . p : bool ) ] $.
$( Define the 'and' operator. $)
df-an $a |- T. |=
[ /\ = \ p : bool . \ q : bool . [ \ f : ( bool -> ( bool -> bool ) ) .
[ p : bool f : ( bool -> ( bool -> bool ) ) q : bool ] =
\ f : ( bool -> ( bool -> bool ) ) .
[ T. f : ( bool -> ( bool -> bool ) ) T. ] ] ] $.
$( Define the implication operator. $)
df-im $a |- T. |= [ ==> =
\ p : bool . \ q : bool . [ [ p : bool /\ q : bool ] = p : bool ] ] $.
$( Define the negation operator. $)
df-not $a |- T. |= [ ~ = \ p : bool . [ p : bool ==> F. ] ] $.
$( Define the existence operator. $)
df-ex $a |- T. |= [ ? = \ p : ( al -> bool ) . ( ! \ q : bool . [ ( ! \ x : al .
[ ( p : ( al -> bool ) x : al ) ==> q : bool ] ) ==> q : bool ] ) ] $.
$( Define the 'or' operator. $)
df-or $a |- T. |= [ \/ = \ p : bool . \ q : bool . ( ! \ x : bool .
[ [ p : bool ==> x : bool ] ==>
[ [ q : bool ==> x : bool ] ==> x : bool ] ] ) ] $.
$( Define the 'exists unique' operator. $)
df-eu $a |- T. |= [ ?! = \ p : ( al -> bool ) . ( ? \ y : al . ( ! \ x : al .
[ ( p : ( al -> bool ) x : al ) = [ x : al = y : al ] ] ) ) ] $.
$}
${
$d f p q x y $.
$( For all type. $)
wal $p |- ! : ( ( al -> bool ) -> bool ) $=
( vp vx hb ht tv kt kl ke kbr tal wv wtru wl weqi df-al eqtypri ) ADEZDER
BRBFZACGHZIJZHKGRDBUARSTRBLADCGMNONACBPQ $.
$( [8-Oct-2014] $)
$( Contradiction type. $)
wfal $p |- F. : bool $=
( vp hb tal tv kl kc tfal kt ht wal wv wl wc df-fal eqtypri ) BCBABADZEZF
GHBBIBCQBJBBAPBAKLMANO $.
$( [8-Oct-2014] $)
$( Conjunction type. $)
wan $p |- /\ : ( bool -> ( bool -> bool ) ) $=
( vp vq vf hb ht tv kbr kl kt ke tan wv wov wl wtru weqi df-an eqtypri )
DDDEZEZDADBTCDAFZDBFZTCFZGZHZTCIIUCGZHZJGZHZHKIDSAUIDDBUHTDEUEUGTDCUDDDDU
AUBUCTCLZDALDBLMNTDCUFDDDIIUCUJOOMNPNNCABQR $.
$( [8-Oct-2014] $)
$( Implication type. $)
wim $p |- ==> : ( bool -> ( bool -> bool ) ) $=
( vp vq hb ht tv tan kbr ke kl tim kt wan wv wov weqi wl df-im eqtypri )
CCCDZDCACBCAEZCBEZFGZTHGZIZIJKCSAUDCCBUCCUBTCCCTUAFLCAMZCBMNUEOPPABQR $.
$( [8-Oct-2014] $)
$( Negation type. $)
wnot $p |- ~ : ( bool -> bool ) $=
( vp hb ht tv tfal tim kbr kl tne kt wim wv wfal wov wl df-not eqtypri )
BBCBABADZEFGZHIJBBASBBBREFKBALMNOAPQ $.
$( [8-Oct-2014] $)
$( There exists type. $)
wex $p |- ? : ( ( al -> bool ) -> bool ) $=
( vp vq vx hb ht tal tv kc tim kbr kl tex kt wal wim wv wc wov wl eqtypri
df-ex ) AEFZEFUCBGECGADUCBHZADHZIZECHZJKZLZIZUGJKZLZIZLMNUCEBUMEEFEGULEOE
ECUKEEEUJUGJPUCEGUIAOAEDUHEEEUFUGJPAEUDUEUCBQADQRECQZSTRUNSTRTADBCUBUA $.
$( [8-Oct-2014] $)
$( Disjunction type. $)
wor $p |- \/ : ( bool -> ( bool -> bool ) ) $=
( vp vq vx hb ht tal tv tim kbr kl kc tor kt wal wim wv wov wl wc df-or
eqtypri ) DDDEZEDADBFDCDAGZDCGZHIZDBGZUDHIZUDHIZHIZJZKZJZJLMDUBAULDDBUKUB
DFUJDNDDCUIDDDUEUHHODDDUCUDHODAPDCPZQDDDUGUDHODDDUFUDHODBPUMQUMQQRSRRCABT
UA $.
$( [8-Oct-2014] $)
$( There exists unique type. $)
weu $p |- ?! : ( ( al -> bool ) -> bool ) $=
( vp vy vx hb ht tex tal tv kc ke kbr kl teu kt wex wv wc weqi wl eqtypri
wal df-eu ) AEFZEFUDBGACHADUDBIZADIZJZUFACIZKLZKLZMZJZMZJZMNOUDEBUNUDEGUM
APAECULUDEHUKAUBAEDUJEUGUIAEUEUFUDBQADQZRAUFUHUOACQSSTRTRTADCBUCUA $.
$( [8-Oct-2014] $)
$}
${
$d p q x y al $. $d p q y F $.
alval.1 $e |- F : ( al -> bool ) $.
$( Value of the for all predicate. $)
alval $p |- T. |= [ ( ! F ) = [ F = \ x : al . T. ] ] $=
( vp hb tal kc ht tv kt kl ke kbr wal wc df-al ceq1 wv weqi wtru wl oveq1
weq id cl eqtri ) FGCHAFIZEUHEJZABKLZMNZLZCHCUJMNZKUHFGCAOZDPUHFCGKULUNDA
BEQRUHFEUKUMCUHUIUJUHESZAFBKUAUBZTDUHUHFUIUJCMUICMNZUHUDUOUPUQUHUICUODTUE
UCUFUG $.
$( [8-Oct-2014] $)
$d x F $.
$( Value of the 'there exists' predicate. $)
exval $p |- T. |= [ ( ? F ) = ( ! \ q : bool . [ ( ! \ x : al .
[ ( F x : al ) ==> q : bool ] ) ==> q : bool ] ) ] $=
( vp hb tex kc ht tal tv tim kbr kl kt wc ceq1 wim wv wex df-ex wal wl ke
wov weqi id oveq1 leq ceq2 cl eqtri ) GHDIAGJZFKGCKABUNFLZABLZIZGCLZMNZOZ
IZURMNZOZIZOZDIKGCKABDUPIZURMNZOZIZURMNZOZIZPUNGHDAUAZEQUNGDHPVEVMEABFCUB
RUNGFVDVLDGGJZGKVCGUCZGGCVBGGGVAURMSUNGKUTAUCZAGBUSGGGUQURMSAGUOUPUNFTZAB
TZQZGCTZUFZUDZQZVTUFZUDZQEVNGVCVKKUODUENZVOWEGGCVBVJWFWDGGGVAURVIMWFSWCVT
UNGUTVHKWFVPWBAGBUSVGWFWAGGGUQURVFMWFSVSVTAGUPUOWFDVQVRWFUNUODVQEUGUHRUIU
JUKUIUJUKULUM $.
$( [8-Oct-2014] $)
$( Value of the 'exists unique' predicate. $)
euval $p |- T. |= [ ( ?! F ) = ( ? \ y : al . ( ! \ x : al .
[ ( F x : al ) = [ x : al = y : al ] ] ) ) ] $=
( vp hb teu kc tex tal tv ke kbr kl kt wc ceq1 wv weqi ht weu wex wal weq
df-eu wl id oveq1 leq ceq2 cl eqtri ) GHDIAGUAZFJACKABUNFLZABLZIZUPACLZMN
ZMNZOZIZOZIZOZDIJACKABDUPIZUSMNZOZIZOZIZPUNGHDAUBZEQUNGDHPVEVLEABCFUFRUNG
FVDVKDUNGJVCAUCZAGCVBUNGKVAAUDZAGBUTGUQUSAGUOUPUNFSZABSZQZAUPURVPACSTZTZU
GZQZUGZQEUNGVCVJJUODMNZVMWBAGCVBVIWCWAUNGVAVHKWCVNVTAGBUTVGWCVSGGGUQUSVFM
WCGUEVQVRAGUPUOWCDVOVPWCUNUODVOETUHRUIUJUKUJUKULUM $.
$( [8-Oct-2014] $)
$}
${
$d f p q x A $. $d f q x B $.
imval.1 $e |- A : bool $.
$( Value of negation. $)
notval $p |- T. |= [ ( ~ A ) = [ A ==> F. ] ] $=
( vp hb tne kc tv tfal tim kbr kl kt wnot wc df-not ceq1 wim wv wfal wov
ke weqi id oveq1 cl eqtri ) DEAFDCDCGZHIJZKZAFAHIJZLDDEAMBNDDAELUIMBCOPDD
CUHUJADDDUGHIQDCRZSTBDDDUGHAIUGAUAJZQUKSULDUGAUKBUBUCUDUEUF $.
$( [8-Oct-2014] $)
imval.2 $e |- B : bool $.
$( Value of the implication. $)
imval $p |- T. |= [ [ A ==> B ] = [ [ A /\ B ] = A ] ] $=
( vp vq hb tim kbr tv tan ke kl kt wim wov wan wv weqi id df-im weq oveq1
oveq oveq12 oveq2 ovl eqtri ) GABHIABGEGFGEJZGFJZKIZUILIZMMZIABKIZALIZNGG
GABHOCDPGGGABHNUMOCDEFUAUDGGGEFULAUJKIZALIUOABGUKUIGGGUIUJKQGERZGFRZPZUQS
CDGGGUKUIUPLUIALIZAGUBZUSUQGGGUIUJAKUTQUQURUTGUIAUQCSTZUCVBUEGGGUPAUNLUJB
LIZVAGGGAUJKQCURPCGGGAUJKVCBQCURVCGUJBURDSTUFUCUGUH $.
$( [9-Oct-2014] $)
$( Value of the disjunction. $)
orval $p |- T. |= [ [ A \/ B ] = ( ! \ x : bool .
[ [ A ==> x : bool ] ==> [ [ B ==> x : bool ] ==> x : bool ] ] ) ] $=
( vp vq hb tor kbr tal tv tim kl kc kt wov wim wv oveq1 wor df-or oveq ht
wal wl wc ke weqi id leq ceq2 oveq2 ovl eqtri ) HBCIJBCHFHGKHAHFLZHALZMJZ
HGLZUQMJZUQMJZMJZNZOZNNZJKHABUQMJZCUQMJZUQMJZMJZNZOZPHHHBCIUADEQHHHBCIPVE
UADEAFGUBUCHHHFGVDKHAVFVAMJZNZOVKBCHHUDZHKVCHUEZHHAVBHHHURVAMRHHHUPUQMRHF
SZHASZQZHHHUTUQMRHHHUSUQMRHGSZVQQZVQQZQZUFZUGDEVNHVCVMKUPBUHJZVOWCHHAVBVL
WDWBHHHURVAVFMWDRVRWAHHHUPUQBMWDRVPVQWDHUPBVPDUIUJTTUKULVNHVMVJKUSCUHJZVO
HHAVLHHHVFVAMRHHHBUQMRDVQQZWAQZUFHHAVLVIWEWGHHHVFVAMWEVHRWFWAHHHUTUQVGMWE
RVTVQHHHUSUQCMWERVSVQWEHUSCVSEUIUJTTUMUKULUNUO $.
$( [9-Oct-2014] $)
$( Value of the conjunction. $)
anval $p |- T. |= [ [ A /\ B ] = [ \ f : ( bool -> ( bool -> bool ) ) .
[ A f : ( bool -> ( bool -> bool ) ) B ] =
\ f : ( bool -> ( bool -> bool ) ) .
[ T. f : ( bool -> ( bool -> bool ) ) T. ] ] ] $=
( vp vq hb tan kbr ht tv kl kt ke wov wv wl weqi oveq1 wan df-an oveq weq
wtru id leq oveq2 ovl eqtri ) HBCIJBCHFHGHHHKKZAHFLZHGLZUKALZJZMZUKANNUNJ
ZMZOJZMMZJUKABCUNJZMZUROJZNHHHBCIUADEPHHHBCINUTUADEAFGUBUCHHHFGUSUKABUMUN
JZMZUROJVCBCUKHKZUPURUKHAUOHHHULUMUNUKAQZHFQZHGQZPZRZUKHAUQHHHNNUNVGUEUEP
RZSDEVFVFHUPURVEOULBOJZVFUDZVKVLUKHAUOVDVMVJHHHULUMBUNVMVGVHVIVMHULBVHDSU
FTUGTVFVFHVEURVBOUMCOJZVNUKHAVDHHHBUMUNVGDVIPZRVLUKHAVDVAVOVPHHHBUMUNVOCV
GDVIVOHUMCVIESUFUHUGTUIUJ $.
$( [9-Oct-2014] $)
$}
${
$d p F $. $d p al $.
ax4g.1 $e |- F : ( al -> bool ) $.
ax4g.2 $e |- A : al $.
$( If ` F ` is true for all ` x : al ` , then it is true for ` A ` . $)
ax4g $p |- ( ! F ) |= ( F A ) $=
( vp kt kc tal hb ht wal wc trud kl ke kbr ax-cb1 id alval mpbi ceq1 hbth
a1i eqtri mpbir ) GCBHZICHZUHAJKJICALDMNZJUGAFGOZBHGUHAJCBDEMAJBCUHUJDEUH
CUJPQZUHUHGUHUIRZSUHUKPQUHULAFCDTUDUAUBAFGBUHEUIUCUEUF $.
$( [9-Oct-2014] $)
$}
${
ax4.1 $e |- A : bool $.
$( If ` A ` is true for all ` x : al ` , then it is true for ` A ` . $)
ax4 $p |- ( ! \ x : al . A ) |= A $=
( kl tv kc tal hb wl wv ax4g ke kbr ax-cb1 beta a1i mpbi ) ABCEZABFZGZCHS
GZATSAIBCDJABKLZUACMNUBUAUBUCOAIBCDPQR $.
$( [9-Oct-2014] $)
$}
${
$d x R $. $d x al $.
alrimiv.1 $e |- R |= A $.
$( If one can prove ` R |= A ` where ` R ` does not contain ` x ` , then
` A ` is true for all ` x ` . $)
alrimiv $p |- R |= ( ! \ x : al . A ) $=
( kl kt ke kbr tal kc hb ax-cb2 wtru eqtru eqcomi leq ax-cb1 wl alval a1i
mpbir ) ABCFZABGFHIZJUCKZDALBCGDCDEMZLGCDNCDEOPQUEUDHIDCDERABUCALBCUFSTUA
UB $.
$( [9-Oct-2014] $)
$}
${
$d x B $. $d x C $. $d x al $.
cla4v.1 $e |- A : bool $.
cla4v.2 $e |- B : al $.
cla4v.3 $e |- [ x : al = B ] |= [ A = C ] $.
$( If ` A ( x ) ` is true for all ` x : al ` , then it is true for
` C = A ( B ) ` . $)
cla4v $p |- ( ! \ x : al . A ) |= C $=
( kl kc tal hb wl ax4g ke kbr ax-cb1 cl a1i mpbi ) ABCIZDJZEKUAJZADUAALBC
FMGNZUBEOPUCUBUCUDQALBCEDFGHRST $.
$( [9-Oct-2014] $)
$}
${
$d p A $.
pm2.21.1 $e |- A : bool $.
$( A falsehood implies anything. $)
pm2.21 $p |- F. |= A $=
( vp tfal tal hb tv kl kc wfal id ke kbr df-fal a1i mpbi weqi cla4v syl
wv ) DEFCFCGZHIZADUBDDJKDUBLMDJCNOPFCUAAAFCTZBUAALMFUAAUCBQKRS $.
$( [9-Oct-2014] $)
$}
${
$d f x y A $. $d f x y B $.
dfan2.1 $e |- A : bool $.
dfan2.2 $e |- B : bool $.
$( An alternative defintion of the "and" term in terms of the context
conjunction. $)
dfan2 $p |- T. |= [ [ A /\ B ] = ( A , B ) ] $=
( vf vx vy kbr kt hb kl kc wl ke id a1i weqi oveq eqid wtru tan kct ht tv
wan wov trud wv wc anval mpbi ceq1 ovl eqtri cl 3eqtr3i mpbir simpl eqtru
jca simpr oveq12 eqcomi leq ax-cb1 dedi ) ABUAHZABUBZVGABIAVGVGJJJABUAUEC
DUFZUGZJJJJUCZUCZEABVLEUDZHZKZJFJGJFUDZKZKZLZVLEIIVMHZKZVRLZVGAIVLJVOVRVL
JEVNJJJABVMVLEUHZCDUFZMZJVKFVQJJGVPJFUHZMMZUIVLJVRVOVGWAWEWGVGVOWANHZVGVG
VIOVGWHNHZVGVIEABCDUJZPUKZULVSANHVGVIVLJEVNAVRWDWGJVNABVRHZAVMVRNHZWDJJJA
BVMWMVRWCCDWMVLVMVRWCWGQZOZRWLANHWMWNJJJFGVPAAABWFCDVPANHZJVPAWFCQZOJAJGU
DZBNHZJWRBJGUHZDQZCSUMPUNUOPWBINHVGVIVLJEVTIVRJJJIIVMWCTTUFZWGJVTIIVRHZIW
MXBJJJIIVMWMVRWCTTWORXCINHWMWNJJJFGVPIIIIWFTTVPINHZJVPIWFTQZOJIWRINHZJWRI
WTTQZTSUMPUNUOPUPUQIBVGVJJVOJFJGWRKZKZLZWAXILZVGBIVLJVOXIWEJVKFXHJJGWRWTM
MZUIZVLJXIVOVGWAWEXLWKULXJBNHVGVIJXJABXIHZBIXMVLJEVNXNXIWDXLJJJABVMVMXINH
ZXIWCCDXOVLVMXIWCXLQZOZRUOXNBNHITJJJFGWRWRBABWTCDJWRWPWQWTSWSXAOUMPUNPXKI
NHVGVIVLJEVTIXIXBXLJVTIIXIHZIXOXBJJJIIVMXOXIWCTTXQRXRINHXOXPJJJFGWRWRIIIW
TTTJWRXDXEWTSXFXGOUMPUNUOPUPUQUTWHVGVHVLJEVNVTVHWDJVTVNVHXBJJJIIAVMVHBWCT
TAVHABCDURZUSBVHABCDVAUSVBVCVDWIVHAVHXSVEWJPUQVF $.
$( [9-Oct-2014] $)
$}
${
hbct.1 $e |- A : bool $.
hbct.2 $e |- B : al $.
hbct.3 $e |- C : bool $.
hbct.4 $e |- R |= [ ( \ x : al . A B ) = A ] $.
hbct.5 $e |- R |= [ ( \ x : al . C B ) = C ] $.
$( Hypothesis builder for context conjunction. $)
hbct $p |- R |= [ ( \ x : al . ( A , C ) B ) = ( A , C ) ] $=
( kt kl kc ke kbr hb tan wan ht kct ax-cb1 trud wct wov dfan2 eqcomi a17i
hbov wtru adantr hbxfrf mpdan ) FLABCEUAZMDNUNOPFABCMDNCOPFJUBZUCAQBCERPZ
DLFUNCEGIUDHQUPUNLQQQCERSGIUECEGIUFUGFLABUPMDNUPOPAQQQBCDERFSGHIAQQQTTBRD
FSHUOUHJKUIUJUKULUM $.
$( [8-Oct-2014] $)
$}
${
mp.1 $e |- T : bool $.
mp.2 $e |- R |= S $.
mp.3 $e |- R |= [ S ==> T ] $.
$( Modus ponens. $)
mpd $p |- R |= T $=
( tan kbr kct tim ke ax-cb1 ax-cb2 imval a1i mpbi mpbir dfan2 simprd ) AB
CBCGHZBCIZABTAEBCJHZTBKHZAFUBUCKHABAELZBCBAEMZDNOPQTUAKHAUDBCUEDROPS $.
$( [9-Oct-2014] $)
$}
${
imp.1 $e |- S : bool $.
imp.2 $e |- T : bool $.
imp.3 $e |- R |= [ S ==> T ] $.
$( Importation deduction. $)
imp $p |- ( R , S ) |= T $=
( kct tim kbr ax-cb1 simpr adantr mpd ) ABGBCEABBCHIZAFJDKABNFDLM $.
$( [9-Oct-2014] $)
$}
${
ex.1 $e |- ( R , S ) |= T $.
$( Exportation deduction. $)
ex $p |- R |= [ S ==> T ] $=
( tan kbr ke tim kct wan ax-cb1 wctr ax-cb2 wov wctl dfan2 a1i wct simpr
hb simpld jca ded eqtri imval mpbir ) BCEFZBGFZBCHFZATUGBCIZBATTTBCEJABCA
BIZDKZLZCUKDMZNUGUJGFAABULOZBCUMUNPQAUJBAUJIBCAUJUOBCUMUNRSUAUKBCABUOUMSD
UBUCUDZUIUHGFAUHAUPKBCUMUNUEQUF $.
$( [9-Oct-2014] $)
$}
${
notval2.1 $e |- A : bool $.
$( Another way two write ` ~ A ` , the negation of ` A ` . $)
notval2 $p |- T. |= [ ( ~ A ) = [ A = F. ] ] $=
( hb tne kc tfal tim kbr ke kt wnot wc notval kct wim wov simpr simpl mpd
wfal pm2.21 adantl ded ax-cb2 mpbi ex dedi eqtri ) CDAEAFGHZAFIHZJCCDAKBL
ABMUIUJUIAFUIANAFTUIACCCAFGOBTPZBQUIAUKBRSFUIAABUAUKUBUCZUJAFAFUJANUJAUJU
IULUDZBQUJAUMBRUEUFUGUH $.
$( [9-Oct-2014] $)
$( One side of ~ notnot . $)
notnot1 $p |- A |= ( ~ ( ~ A ) ) $=
( tne kc tfal tim kbr kct wfal hb wnot simpl simpr ax-cb1 notval a1i mpbi
wc ke mpd ex mpbir ) CADZEFGZCUCDZAAUCEAUCHZAEIAUCBJJCAKBRZLZUCAEFGZUFAUCB
UGMUCUISGUFAUFUHNABOPQTUAUEUDSGABUCUGOPUB $.
$( [10-Oct-2014] $)
$}
${
con2d.1 $e |- T : bool $.
con2d.2 $e |- ( R , S ) |= ( ~ T ) $.
$( A contraposition deduction. $)
con2d $p |- ( R , T ) |= ( ~ S ) $=
( tfal tim kbr tne kc kct wfal ke ax-cb1 notval a1i mpbi imp an32s ex wct
wctl wctr mpbir ) BFGHZIBJZACKZUGBFFABCABKZCFDLICJZCFGHZUHEUIUJMHUHUIUHEN
ZCDOPQRSTUFUEMHUGACABUKUBDUABABUKUCOPUD $.
$( [10-Oct-2014] $)
$}
${
con3d.1 $e |- ( R , S ) |= T $.
$( A contraposition deduction. $)
con3d $p |- ( R , ( ~ T ) ) |= ( ~ S ) $=
( tne kc hb wnot kct ax-cb2 wc notnot1 syl con2d ) ABECFZGGECHCABIZDJZKPC
EOFDCQLMN $.
$( [10-Oct-2014] $)
$}
${
$d x A $. $d x B $. $d x T $.
ecase.1 $e |- A : bool $.
ecase.2 $e |- B : bool $.
ecase.3 $e |- T : bool $.
ecase.4 $e |- R |= [ A \/ B ] $.
ecase.5 $e |- ( R , A ) |= T $.
ecase.6 $e |- ( R , B ) |= T $.
$( Elimination by cases. $)
ecase $p |- R |= T $=
( vx tim kbr ex hb wim wov ke oveq2 oveq12 tal tv kl tor ax-cb1 orval a1i
kc mpbi wv weqi id cla4v syl mpd ) CBDLMZDGCBDJNCADLMZUPDLMZOOOUPDLPOOOBD
LPFGQGQCADINCUAOKAOKUBZLMZBUSLMZUSLMZLMZUCUHZUQURLMZABUDMZVDCHVFVDRMCVFCH
UEKABEFUFUGUIOKVCDVEOOOUTVBLPOOOAUSLPEOKUJZQZOOOVAUSLPOOOBUSLPFVGQZVGQZQG
OOOUTVBUQLUSDRMZURPVHVJOOOAUSLVKDPEVGVKOUSDVGGUKULZSOOOVAUSUPLVKDPVIVGOOO
BUSLVKDPFVGVLSVLTTUMUNUOUO $.
$( [9-Oct-2014] $)
$}
${
$d x A $. $d x B $.
olc.1 $e |- A : bool $.
olc.2 $e |- B : bool $.
$( Or introduction. $)
olc $p |- B |= [ A \/ B ] $=
( vx hb tv tim kbr kl kt ke tor wim wv wov wtru kct simpl ex simpr adantr
mpd eqtru eqcomi leq tal kc wor orval wl alval eqtri a1i mpbir ) FEAFEGZH
IZBUPHIZUPHIZHIZJZFEKJLIZABMIZBFFEUTKBFFFUQUSHNFFFAUPHNCFEOZPZFFFURUPHNFF
FBUPHNDVDPZVDPPZFKUTBQUTBBUQUSBUQUSBURUPBURRBUPVDBURDVFSBURDVFUAUCTVEUBTU
DUEUFVCVBLIBDFVCUGVAUHVBKFFFABMUICDPEABCDUJFEVAFFEUTVGUKULUMUNUO $.
$( [9-Oct-2014] $)
$( Or introduction. $)
orc $p |- A |= [ A \/ B ] $=
( vx hb tv tim kbr kl kt ke tor wim wv wov wtru kct simpl ex simpr adantr
mpd eqtru eqcomi leq tal kc wor orval wl alval eqtri a1i mpbir ) FEAFEGZHI
ZBUPHIZUPHIZHIZJZFEKJLIZABMIZAFFEUTKAFFFUQUSHNFFFAUPHNCFEOZPZFFFURUPHNFFF
BUPHNDVDPZVDPPZFKUTAQUTAAUQUSAUQRZURUPVHURUPVHAUPVDAUQCVESAUQCVEUAUCVFUBT
TUDUEUFVCVBLIACFVCUGVAUHVBKFFFABMUICDPEABCDUJFEVAFFEUTVGUKULUMUNUO $.
$( [9-Oct-2014] $)
$}
${
$d p x F $. $d x R $. $d p x T $. $d p x al $.
exlimdv2.1 $e |- F : ( al -> bool ) $.
exlimdv2.2 $e |- ( R , ( F x : al ) ) |= T $.
$( Existential elimination. $)
exlimdv2 $p |- ( R , ( ? F ) ) |= T $=
( vp tex kc kct tal tv tim kbr kl hb wc ke wim ax-cb2 ex ht adantr ax-cb1
alrimiv wex wctl simpr wct exval a1i mpbi wal wv wov wl weqi id oveq2 leq
ceq2 oveq12 cla4v syl mpd ) DICJZKZLABCABMZJZENOZPZJZEEDVJKZGUAZDVGVMABVKD
DVJEGUBUFAQUCZQICAUGFRZUDVHLQHLABVJQHMZNOZPZJZVRNOZPJZVMENOZVGWCVHDVGDVJE
VNGUEUHZVQUIVGWCSOVHDVGWEVQUJABHCFUKULUMQHWBEWDQQQWAVRNTVPQLVTAUNZAQBVSQQ
QVJVRNTAQCVIFABUORZQHUOZUPZUQZRZWHUPVOQQQWAVRVMNVRESOZETWKWHVPQVTVLLWLWFW
JAQBVSVKWLWIQQQVJVRNWLETWGWHWLQVREWHVOURUSZUTVAVBWMVCVDVEVF $.
$( [9-Oct-2014] $)
$}
${
$d y z A $. $d x y z R $. $d x y z T $. $d x y z al $.
exlimdv.1 $e |- ( R , A ) |= T $.
$( Existential elimination. $)
exlimdv $p |- ( R , ( ? \ x : al . A ) ) |= T $=
( vy vz hb tv kc wv wc tim kbr wim kt ht ax-17 ke kl kct ax-cb1 wl ax-cb2
wctr wov ex ax-hbl1 hbc hbov weqi beta eqcomi a1i id ceq2 eqtri oveq1 imp
insti exlimdv2 ) AGABCUAZDEAIBCDCEDCUBZFUCUFZUDZDVCAGJZKZEAIVCVGVFAGLZMZE
VDFUEZABHCENOVHENOVGDVIIIIVHENPVJVKUGDCEFUHAIIIBVHAHJZENQPVJAHLZVKAIIIRRB
NVLPVMSAAIBVGVLVCQVFVIVMAAIBCVLVEVMUIAABVGVLVIVMSUJAIBEVLVKVMSUKIIICEVHNA
BJZVGTOZPVEVKICVCVNKZVHVOVECVPTOVOAVNVGABLZVIULZIVPCQAIVCVNVFVQMAIBCVEUMU
NUOAIVNVGVCVOVFVQVOVRUPUQURUSVAUTVB $.
$( [9-Oct-2014] $)
$}
${
$d p x A $. $d p x F $. $d p x al $.
ax4e.1 $e |- F : ( al -> bool ) $.
ax4e.2 $e |- A : al $.
$( Existential introduction. $)
ax4e $p |- ( F A ) |= ( ? F ) $=
( vp vx tal hb tv kc tim kbr kl tex kct wv wc wim ke ht wal wl simpl weqi
wov id ceq2 oveq1 cla4v adantl mpd ex alrimiv exval a1i mpbir ) HIFHAGCAG
JZKZIFJZLMZNZKZUTLMZNKZOCKZCBKZIFVDVGVGVCUTVGVCPVGUTIFQZVGVCAICBDERZAIUAI
HVBAUBAIGVAIIIUSUTLSAICURDAGQZRZVHUFZUCRUDVCVGVGUTLMZAGVABVMVLEIIIUSUTVGL
URBTMZSVKVHAIURBCVNDVJVNAURBVJEUEUGUHUIUJVIUKULUMUNVFVETMVGVIAGFCDUOUPUQ
$.
$( [9-Oct-2014] $)
$}
${
$d x B $. $d x C $. $d x al $.
cla4ev.1 $e |- A : bool $.
cla4ev.2 $e |- B : al $.
cla4ev.3 $e |- [ x : al = B ] |= [ A = C ] $.
$( Existential introduction. $)
cla4ev $p |- C |= ( ? \ x : al . A ) $=
( kl kc tex hb tv ke kbr eqtypi id cl a1i mpbir wl ax4e syl ) EABCIZDJZKU
DJEUEEELCEABMDNOFHPZQUEENOEUFALBCEDFGHRSTADUDALBCFUAGUBUC $.
$( [9-Oct-2014] $)
$}
${
19.8a.1 $e |- A : bool $.
$( Existential introduction. $)
19.8a $p |- A |= ( ? \ x : al . A ) $=
( kl tv kc tex ax-id ke kbr hb beta a1i mpbir wl wv ax4e syl ) CABCEZABFZ
GZHTGCUBCCDIUBCJKCDALBCDMNOAUATALBCDPABQRS $.
$( [9-Oct-2014] $)
$}
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Type definition mechanism
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
$c typedef $.
$( Internal axiom for mmj2 use. $)
wffMMJ2d $a wff typedef al ( A , B ) C $.
${
$d x A $. $d x R $. $d x F $.
ax-tdef.1 $e |- B : al $.
ax-tdef.2 $e |- F : ( al -> bool ) $.
ax-tdef.3 $e |- T. |= ( F B ) $.
ax-tdef.4 $e |- typedef be ( A , R ) F $.
$( Type of the abstraction function. $)
wabs $a |- A : ( al -> be ) $.
$( Type of the representation function. $)
wrep $a |- R : ( be -> al ) $.
$( The type definition axiom. The last hypothesis corresponds to the actual
definition one wants to make; here we are defining a new type ` be ` and
the definition will provide us with pair of bijections ` A , R ` mapping
the new type ` be ` to the subset of the old type ` al ` such that
` F x ` is true. In order for this to be a valid (conservative)
extension, we must ensure that the new type is non-empty, and for that
purpose we need a witness ` B ` that ` F ` is not always false. $)
ax-tdef $a |- T. |= ( ( ! \ x : be . [ ( A ( R x : be ) ) = x : be ] ) ,
( ! \ x : al . [ ( F x : al ) = [ ( R ( A x : al ) ) = x : al ] ] ) ) $.
$}
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Extensionality
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
${
$d f x $.
$( The eta-axiom: a function is determined by its values. $)
ax-eta $a |- T. |= ( ! \ f : ( al -> be ) .
[ \ x : al . ( f : ( al -> be ) x : al ) = f : ( al -> be ) ] ) $.
$}
${
$d f x F $. $d f x al $. $d f x be $.
eta.1 $e |- F : ( al -> be ) $.
$( The eta-axiom: a function is determined by its values. $)
eta $p |- T. |= [ \ x : al . ( F x : al ) = F ] $=
( vf kt tal ht tv kc kl ke kbr ax-eta hb weq wv wc wl weqi id ceq1 oveq12
wov leq cla4v syl ) GHABIZFACUIFJZACJZKZLZUJMNZLKACDUKKZLZDMNZABCFOUIFUND
UQUIUIPUMUJMUIQZABCULABUJUKUIFRZACRZSZTZUSUEEUIUIPUMUJUPMUJDMNZDURVBUSABC
ULUOVCVAABUKUJVCDUSUTVCUIUJDUSEUAUBZUCUFVDUDUGUH $.
$( [8-Oct-2014] $)
$}
${
$d p z A $. $d p z B $. $d p x y z al $. $d p be $.
cbvf.1 $e |- A : be $.
cbvf.2 $e |- T. |= [ ( \ y : al . A z : al ) = A ] $.
cbvf.3 $e |- T. |= [ ( \ x : al . B z : al ) = B ] $.
cbvf.4 $e |- [ x : al = y : al ] |= [ A = B ] $.
$( Change bound variables in a lambda abstraction. $)
cbvf $p |- T. |= [ \ x : al . A = \ y : al . B ] $=
( vp ht kl tv kc kt wc ke kbr wl eqtypi weqi ax-17 clf weq hbl hbc ax-cb1
wv hb hbl1 hbov ceq2 beta eqcomi a1i eqtri oveq12 insti leq eta 3eqtr3i
id ) ABMALACFNZALOZPZNALADGNZVFPZNQVEVHABLVGABVEVFABCFHUAZALUJZRZUAABLVGV
IQVLADEVEADOZPZGSTVGVISTVFQVKBVGVIVLABVHVFABDGBFGACOVMSTHKUBZUAZVKRZUCABC
EFGVMHADUJZKJAACVMAEOZVRAEUJZUDUEABBUKDVGVSVISQBUFZVLVTVQABBUKMMDSVSWAVTU
DAABDVFVSVEQVJVKVTAABDCFVSQHVTIUGAADVFVSVKVTUDZUHAABDVFVSVHQVPVKVTAABDGVS
QVOVTADFNVSPFSTQIUIULWBUHUMBBUKVNGVGSVMVFSTZVIWAABVEVMVJVRRVOABVMVFVEWCVJ
VRWCAVMVFVRVKUCVDZUNZBGVHVMPZVIWCVOGWFSTWCVNVGSTWCWEUIBWFGQABVHVMVPVRRABD
GVOUOUPUQABVMVFVHWCVPVRWDUNURUSUTVAABLVEVJVBABLVHVPVBVC $.
$( [8-Oct-2014] $)
$}
${
$d y z A $. $d x z B $. $d x y z al $. $d y be $.
cbv.1 $e |- A : be $.
cbv.2 $e |- [ x : al = y : al ] |= [ A = B ] $.
$( Change bound variables in a lambda abstraction. $)
cbv $p |- T. |= [ \ x : al . A = \ y : al . B ] $=
( vz tv wv ax-17 ke kbr eqtypi cbvf ) ABCDIEFGABDEAIJZGAIKZLABCFQBEFACJAD
JMNGHORLHP $.
$( [8-Oct-2014] $)
$}
${
$d y z A $. $d y z B $. $d y z R $. $d x y z al $. $d z be $.
leqf.1 $e |- A : be $.
leqf.2 $e |- R |= [ A = B ] $.
leqf.3 $e |- T. |= [ ( \ x : al . R y : al ) = R ] $.
$( Equality theorem for lambda abstraction, using bound variable instead of
distinct variables. $)
leqf $p |- R |= [ \ x : al . A = \ x : al . B ] $=
( vz ht kl tv kc wc ke kbr a1i hb wl wv ax-cb1 beta eqtypi 3eqtr4i kt weq
ax-17 ax-hbl1 hbc hbov weqi id ceq2 oveq12 eqid ax-inst leq eta 3eqtr3i )
ABLAKACEMZAKNZOZMZAKACFMZVCOZMZGVBVFABKVDABVBVCABCEHUAZAKUBZPZUAABKVDVGGV
KACDVBACNZOZVFVLOZQRVDVGQRVCGGBEFGVMVNHIVMEQRGEFQRGIUCZABCEHUDSVNFQRGVOAB
CFBEFGHIUEZUDSUFABBTCVDADNZVGQUGBUHZVKADUBZABVFVCABCFVPUAZVJPABBTLLCQVQVR
VSUIAABCVCVQVBUGVIVJVSAABCEVQHVSUJAACVCVQVJVSUIZUKAABCVCVQVFUGVTVJVSAABCF
VQVPVSUJWAUKULJBBTVMVNVDQVLVCQRZVGVRABVBVLVIACUBZPABVFVLVTWCPABVLVCVBWBVI
WCWBAVLVCWCVJUMZUNZUOABVLVCVFWBVTWCWEUOUPTGWBWDVOUQURUSVEVBQRGVOABKVBVIUT
SVHVFQRGVOABKVFVTUTSVA $.
$( [8-Oct-2014] $)
$}
${
$d y A $. $d y R $. $d x y al $.
alrimi.1 $e |- R |= A $.
alrimi.2 $e |- T. |= [ ( \ x : al . R y : al ) = R ] $.
$( If one can prove ` R |= A ` where ` R ` does not contain ` x ` , then
` A ` is true for all ` x ` . $)
alrimi $p |- R |= ( ! \ x : al . A ) $=
( kl kt ke kbr tal kc hb ax-cb2 wtru eqtru eqcomi leqf ax-cb1 alval mpbir
wl a1i ) ABDHZABIHJKZLUEMZEANBCDIEDEFOZNIDEPDEFQRGSUGUFJKEDEFTABUEANBDUHU
CUAUDUB $.
$( [9-Oct-2014] $)
$}
${
$d y z A $. $d y z R $. $d y z T $. $d x y z al $.
exlimd.1 $e |- ( R , A ) |= T $.
exlimd.2 $e |- T. |= [ ( \ x : al . R y : al ) = R ] $.
exlimd.3 $e |- T. |= [ ( \ x : al . T y : al ) = T ] $.
$( Existential elimination. $)
exlimd $p |- ( R , ( ? \ x : al . A ) ) |= T $=
( vz kl kc tal tv tim kbr hb wim wov kt tex ax-cb2 ax-cb1 wctr wl wv wctl
kct wc id ex wtru adantl ax-17 ax-hbl1 hbc hbov weqi beta eqcomi a1i ceq2
ht ke eqtri oveq1 oveq2 insti mpd alrimiv wex adantr simpr exval mpbi wal
leq oveq12 cla4v syl ) EUAABDKZLZUHZMAJWAAJNZLZFOPZKZLZFFEDUHZGUBZEWBWHAJ
WFEEEWFQQQWEFORAQWAWDAQBDEDFWIGUCZUDZUEZAJUFZUIZWJSZEEDWKUGZUJEWFOPZEWQAB
CEDFOPZOPWRWDTWNQQQEWFORWQWPSTEWSETWSEDFGUKULUMUKAQQQBEACNZWFOTRWQACUFZWP
AQQQVCVCBOWTRXAUNZHAQQQBWEWTFOTRWOXAWJXBAAQBWDWTWATWMWNXAAAQBDWTWLXAUOAAB
WDWTWNXAUNUPIUQUQQQQEWSOABNZWDVDPZWFRWQQQQDFORWLWJSQQQDFWEOXDRWLWJQDWAXCL
ZWEXDWLDXEVDPXDAXCWDABUFZWNURZQXEDTAQWAXCWMXFUIAQBDWLUSUTVAAQXCWDWAXDWMXF
XDXGUJVBVEVFVGVHVAVIVJAQVCZQUAWAAVKWMUIZVLZWCMQCMAJWEQCNZOPZKZLZXKOPZKLZW
HFOPZWBXPWCEWBWQXIVMWBXPVDPWCWHWCXJUCAJCWAWMVNVAVOQCXOFXQQQQXNXKORXHQMXMA
VPZAQJXLQQQWEXKORWOQCUFZSZUEZUIZXSSWJQQQXNXKWHOXKFVDPZFRYBXSXHQXMWGMYCXRY
AAQJXLWFYCXTQQQWEXKOYCFRWOXSYCQXKFXSWJURUJZVGVQVBYDVRVSVTVI $.
$( [9-Oct-2014] $)
$}
${
$d y A $. $d y B $. $d x y R $. $d x y al $.
alimdv.1 $e |- ( R , A ) |= B $.
$( Deduction from Theorem 19.20 of [Margaris] p. 90. $)
alimdv $p |- ( R , ( ! \ x : al . A ) ) |= ( ! \ x : al . B ) $=
( vy tal kl kc kct ax-cb1 wctr ax4 wctl adantl kt hb ht ax-17 tv wv wl wc
syldan wal ax-hbl1 hbc hbct alrimi ) ABGDEHABCIZJZKDEULCULECABCECDECKFLZM
ZNECUMOZPFUEABEAGUAZULQUOAGUBZARSZRHUKAUFZARBCUNUCZUDARBEUPUOUQTAURRBUKUP
HQUSUTUQAURRSBHUPUSUQTAARBCUPUNUQUGUHUIUJ $.
$( [9-Oct-2014] $)
$( Deduction from Theorem 19.22 of [Margaris] p. 90. $)
eximdv $p |- ( R , ( ? \ x : al . A ) ) |= ( ? \ x : al . B ) $=
( vy tex kl kc kct ax-cb2 19.8a syl hb tv ax-cb1 wctl ax-17 ht wv ax-hbl1
kt wex wl hbc exlimd ) ABGCEHABDIZJZECKZDUIFABDDUJFLZMNAOBEAGPZECDUJFQRAG
UAZSAAOTZOBUHULHUCAUDZAOBDUKUEUMAUNOTBHULUOUMSAAOBDULUKUMUBUFUG $.
$( [9-Oct-2014] $)
$}
${
$d y A $. $d x y al $.
alnex1.1 $e |- A : bool $.
$( Theorem 19.7 of [Margaris] p. 89. $)
alnex $p |- T. |= [ ( ! \ x : al . ( ~ A ) ) = ( ~ ( ? \ x : al . A ) ) ] $=
( vy tal tne kc kl tex tfal tim kbr wfal hb wnot ht kt ax-17 hbc ax4 mpbi
wc ke ax-cb1 notval a1i imp tv wal wl wv ax-hbl1 exlimd ex wex mpbir wtru
19.8a adantl con3d trul alrimi dedi ) FABGCHZIZHZGJABCIZHZHZVIKLMZVJVGVGV
IKABECVGKVGCKDNVECKLMZVGABVEOOGCPDUCZUAZVEVLUDMVGVEVGVNUEZCDUFUGUBUHAAOQZ
OBVFAEUIZFRAUJZAOBVEVMUKAEULZAVPOQZBFVQVRVSSAAOBVEVQVMVSUMTAOBKVQNVSSUNUO
VJVKUDMVGVOVIVPOJVHAUPZAOBCDUKZUCZUFUGUQABEVEVJVJVERCVICRVIABCDUSURUTVAVB
AOOBVIVQGRPWCVSAOOQBGVQPVSSAVPOBVHVQJRWAWBVSAVTBJVQWAVSSAAOBCVQDVSUMTTVCV
D $.
$( [10-Oct-2014] $)
$( Forward direction of ~ exnal . $)
exnal1 $p |- ( ? \ x : al . ( ~ A ) ) |= ( ~ ( ! \ x : al . A ) ) $=
( tex tne kc kl tal kt hb ht wex wnot wc wl kct notnot1 wtru adantl id ke
alimdv wal kbr alnex a1i mpbi syl con2d trul ) EABFCGZHZGZFIABCHGZGJUOUNA
KLZKEUMAMAKBULKKFCNDOZPOJUOQIABFULGZHZGZFUNGZABCURJCJURCDRSTUCUTVAUTUTUPK
IUSAUDAKBURKKFULNUQOPOZUAUTVAUBUEUTVBABULUQUFUGUHUIUJUK $.
$( [10-Oct-2014] $)
isfree.2 $e |- T. |= [ ( \ x : al . A y : al ) = A ] $.
$( Derive the metamath "is free" predicate in terms of the HOL "is free"
predicate. $)
isfree $p |- T. |= [ A ==> ( ! \ x : al . A ) ] $=
( kt tal kl kc id alrimi tv ke kbr ax-cb1 adantl ex ) GDHABDIZJZDGTABCDDD
EKFLSACMJDNOGFPQR $.
$( [9-Oct-2014] $)
$}
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Axioms of infinity and choice
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
$c 1-1 $. $( One-to-one function $)
$c onto $. $( Onto function $)
$c @ $. $( Indefinite descriptor $)
$( One-to-one function. $)
tf11 $a term 1-1 $.
$( Onto function. $)
tfo $a term onto $.
$( Indefinite descriptor. $)
tat $a term @ $.
$( The type of the indefinite descriptor. $)
wat $a |- @ : ( ( al -> bool ) -> al ) $.
${
$d f p x y $.
$( Define a one-to-one function. $)
df-f11 $a |- T. |= [ 1-1 = \ f : ( al -> be ) . ( ! \ x : al . ( ! \ y : al .
[ [ ( f : ( al -> be ) x : al ) = ( f : ( al -> be ) y : al ) ] ==>
[ x : al = y : al ] ] ) ) ] $.
$( Define an onto function. $)
df-fo $a |- T. |= [ onto = \ f : ( al -> be ) . ( ! \ y : be . ( ? \ x : al .
[ y : be = ( f : ( al -> be ) x : al ) ] ) ) ] $.
$( Defining property of the indefinite descriptor: it selects an element
from any type. This is equivalent to global choice in ZF. $)
ax-ac $a |- T. |= ( ! \ p : ( al -> bool ) .
( ! \ x : al . [ ( p : ( al -> bool ) x : al ) ==>
( p : ( al -> bool ) ( @ p : ( al -> bool ) ) ) ] ) ) $.
$}
${
$d x A $. $d p x F $. $d p x al $.
ac.1 $e |- F : ( al -> bool ) $.
ac.2 $e |- A : al $.
$( Defining property of the indefinite descriptor: it selects an element
from any type. This is equivalent to global choice in ZF. $)
ac $p |- ( F A ) |= ( F ( @ F ) ) $=
( vx vp kc tat hb wc id tim kbr kt tal tv kl wim ceq2 ht wat ax-cb1 ax-ac
wal wv wov wl weqi ceq1 ceq12 oveq12 leq cla4v syl eqtypi oveq1 a1i mpd
ke ) CBHZVACICHZHZAJCVBDAJUAZAICAUBZDKKZVAAJCBDEKLZVAVCMNZVAVAVAVGUCOPAFC
AFQZHZVCMNZRZHZVHOPVDGPAFVDGQZVIHZVNIVNHZHZMNZRZHZRHVMAFGUDVDGVTCVMVDJPVS
AUEZAJFVRJJJVOVQMSAJVNVIVDGUFZAFUFZKZAJVNVPWBVDAIVNVEWBKZKZUGZUHZKDVDJVSV
LPVNCUTNZWAWHAJFVRVKWIWGJJJVOVQVJMWIVCSWDWFAJVIVNWICWBWCWIVDVNCWBDUILZUJA
JVPVBVNWICWBWEWJVDAVNCIWIVEWBWJTUKULZUMTUNUOAFVKBVHJVRVKWIWGWKUPEJJJVJVCV
AMVIBUTNZSAJCVIDWCKVFAJVIBCWLDWCWLAVIBWCEUILTUQUNUOURUS $.
$( [10-Oct-2014] $)
$}
${
$d x F $. $d x al $.
dfex2.1 $e |- F : ( al -> bool ) $.
$( Alternative definition of the "there exists" quantifier. $)
dfex2 $p |- T. |= [ ( ? F ) = ( F ( @ F ) ) ] $=
( vx kt tex kc tat tv wv ac wtru adantl exlimdv2 hb ht wat wc ax4e ded )
EFBGZBHBGZGZADBEUCCBADIZGEUCAUDBCADJKLMNUCEUAAUBBCAOPAHBAQCRSLMT $.
$( [10-Oct-2014] $)
$}
${
$d x y A $. $d x al $.
exmid.1 $e |- A : bool $.
$( Diaconescu's theorem, which derives the law of the excluded middle from
the axiom of choice. $)
exmid $p |- T. |= [ A \/ ( ~ A ) ] $=
( vx vy tat hb tor kbr kc kt tne wor wc wnot wtru ke weqi id tfal wfal tv
kl ht wat wv wov wl orc oveq1 cl mpbir ac syl ax-17 ax-hbl1 hbc hbov mpbi
clf ceq2 tim simpr notval eqtru eqcomi a1i eqtri kct wct simpl simpld olc
ex leq adantl simprd ax-cb1 mpd ecase ) EFCFCUAZAGHZUBZIZAJAKAIZGHZFFUCZF
EWBFUDZFFCWAFFFVTAGLFCUEZBUFZUGZMZBFFFAWDGLBFFKANBMZUFZWBWCIZWCAGHZJJWBJI
ZWNJAGHZWPJJAOBUHZFFCWAWQJWIOFFFVTAJGVTJPHZLWHBWSFVTJWHOQRUIUJUKFJWBWJOUL
UMFFCDWAWOWCWIWKFFFVTAWCGVTWCPHZLWHBWTFVTWCWHWKQRUIFFFFCWCFDUAZAGJLWKFDUE
ZBFFWFUCCGXALXBUNZFWFFCWBXAEJWGWJXBFWFFUCCEXAWGXBUNZFFFCWAXAWIXBUOUPZFFCA
XABXBUNZUQXEUSURWCJWEKEFCKVTIZAGHZUBZIZIZAWCWEFFKXJNWFFEXIWGFFCXHFFFXGAGL
FFKVTNWHMZBUFZUGZMZMZBWMXKAGHZWCWKXIXJIZXQJJXISIZXRWQXSJWRFFCXHWQSXMTFFFX
GAJGVTSPHZLXLBFXGKSIZJXTXLFFVTSKXTNWHXTFVTSWHTQZRUTYAJPHXTYBFJYAJOYAJSSVA
HYAJJSSJSOTVBVMSTVCUKVDVEVFVGUIUJUKFSXIXNTULUMFFCDXHXQXJXMXOFFFXGAXKGVTXJ
PHZLXLBFFVTXJKYCNWHYCFVTXJWHXOQRUTUIFFFFCXKXAAGJLXPXBBXCFFFCXJXAKJNXOXBFW
FCKXANXBUNFWFFCXIXAEJWGXNXBXDFFFCXHXAXMXBUOUPZUPXFUQYDUSURVFWCXKVHZWDWEAS
VAHZWDYEYEASYEAVHZXJSTWCXJYGYGWCXKYEAWCXKWKXPVIZBVJZVKAYEWCXJPHWFFWBXIEAW
GWJFFCWAXHAWIFWAJXHAWIFJWAAOWAAVTAWHBVLVDVEXHAXGAXLBVLVDVGVNUTYHVOURXKXJS
VAHZYGYGWCXKYIVPXKYJPHYGYEYGYIVQXJXOVCVFURVRVMWDYFPHYEYHABVCVFUKAWDBWLVLU
MAWCWEAWDBWLUHZWKVOVSOVOAJWEYKOVOVS $.
$( [10-Oct-2014] $)
$( Rule of double negation. $)
notnot $p |- T. |= [ A = ( ~ ( ~ A ) ) ] $=
( tne kc notnot1 hb wnot tor kbr ax-cb2 exmid a1i simpr kct tfal wfal tim
wc id ke notval mpbi imp pm2.21 syl ecase dedi ) ACCADZDZABEZAUHUIABFFCAG
BRZBAUHHIUIUIAUJJZABKLUIAULBMUIUHNOAUIUHOUKPUIUHOQIZUIUIULSUIUMTIUIULUHUK
UALUBUCABUDUEUFUG $.
$( [10-Oct-2014] $)
$( Theorem 19.14 of [Margaris] p. 90. $)
exnal $p |- T. |= [ ( ? \ x : al . ( ~ A ) ) = ( ~ ( ! \ x : al . A ) ) ] $=
( hb tne tex kc kl tal kt wnot ht wex wc wl wal alnex ceq2 notnot 3eqtr4i
eqcomi leq ) EFFGABFCHZIZHZHZHFJABFUDHZIZHZHKUFFJABCIZHZHEEFUGLEEFUFLAEMZ
EGUEANAEBUDEEFCLDOZPOZOZOEEUGUJFKLUPEUJUGKUMEJUIAQZAEBUHEEFUDLUNOPOABUDUN
RUBSUFUOTEEULUJFKLUMEJUKUQAEBCDPZOUMEUKUIJKUQURAEBCUHKDCDTUCSSUA $.
$( [10-Oct-2014] $)
$}
$( The axiom of infinity: the set of "individuals" is not Dedekind-finite.
Using the axiom of choice, we can show that this is equivalent
to an embedding of the natural numbers in ` ind ` . $)
ax-inf $a |- T. |= ( ? \ f : ( ind -> ind ) .
[ ( 1-1 f : ( ind -> ind ) ) /\ ( ~ ( onto f : ( ind -> ind ) ) ) ] ) $.
$(
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
Rederive the Metamath axioms
#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#*#
$)
${
ax1.1 $e |- R : bool $.
ax1.2 $e |- S : bool $.
$( Axiom _Simp_. Axiom A1 of [Margaris] p. 49. $)
ax1 $p |- T. |= [ R ==> [ S ==> R ] ] $=
( kt tim kbr kct wtru simpr adantr ex ) EABAFGEAHZBAMBAEAICJDKLL $.
$( [9-Oct-2014] $)
ax2.3 $e |- T : bool $.
$( Axiom _Frege_. Axiom A2 of [Margaris] p. 49. $)
ax2 $p |- T. |=
[ [ R ==> [ S ==> T ] ] ==> [ [ R ==> S ] ==> [ R ==> T ] ] ] $=
( kt tim kbr kct hb wim wov wct simpr simpl simprd mpd simpld ex adantl
wtru ) GABCHIZHIZABHIZACHIZHIZUDGUGUDUEUFUDUEJZACUHAJZBCFUIABEUHAUDUEKKKA
UCHLDKKKBCHLEFMZMKKKABHLDEMNZDOZUIUDUEUHAUKDPZQRUIAUCUJULUIUDUEUMSRRTTUBU
AT $.
$( [9-Oct-2014] $)
$}
${
ax3.1 $e |- R : bool $.
ax3.2 $e |- S : bool $.
$( Axiom _Transp_. Axiom A3 of [Margaris] p. 49. $)
ax3 $p |- T. |= [ [ ( ~ R ) ==> ( ~ S ) ] ==> [ S ==> R ] ] $=
( kt tne kc tim kbr kct hb wnot wc tor wim a1i ax-cb1 tfal imp ex wov wct
exmid simpr wfal id ke notval mpbi an32s pm2.21 syl ecase wtru adantl ) E
FAGZFBGZHIZBAHIZUREUSURBAAUPURBJZACKKFALCMZCAUPNIZUTURBKKKUPUQHOVAKKFBLDM
ZUAZDUBACUCPZUTAVBUTVEQCUDUTUPJRARURUPBURUPJZBRDUEUQBRHIZVFURUPUQVAVCURVD
UFSZUQVGUGIVFUQVFVHQBDUHPUISUJACUKULUMTUNUOT $.
$( [10-Oct-2014] $)
$}
${
axmp.1 $e |- S : bool $.
axmp.2 $e |- T. |= R $.
axmp.3 $e |- T. |= [ R ==> S ] $.
$( Rule of Modus Ponens. The postulated inference rule of propositional
calculus. See e.g. Rule 1 of [Hamilton] p. 73. $)
axmp $p |- T. |= S $=
( kt mpd ) FABCDEG $.
$( [10-Oct-2014] $)
$}
${
$d y R $. $d y S $. $d x y al $.
ax5.1 $e |- R : bool $.
ax5.2 $e |- S : bool $.
$( Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105. $)
ax5 $p |- T. |= [ ( ! \ x : al . [ R ==> S ] ) ==>
[ ( ! \ x : al . R ) ==> ( ! \ x : al . S ) ] ] $=
( vy kt tal tim kbr kl kc ax4 hb ht wl adantl ax-hbl1 hbc kct wal wim wov
wc ax-cb1 adantr mpd tv wv ax-17 hbct alrimi ex wtru ) HIABCDJKZLZMZIABCL
ZMZIABDLMZJKZURHVBURUTVAABGDURUTUAZVCCDFUTURCABCENZAOPZOIUQAUBZAOBUPOOOCD
JUCEFUDZQZUEZRURUTUPABUPVGNCUTVDUFZUGUHABURAGUIZUTHVIAGUJZVJAVEOBUQVKIHVF
VHVLAVEOPBIVKVFVLUKZAAOBUPVKVGVLSTAVEOBUSVKIHVFAOBCEQVLVMAAOBCVKEVLSTULUM
UNUORUN $.
$( [10-Oct-2014] $)
$}
${
$d y R $. $d x y al $.
ax6.1 $e |- R : bool $.
$( Axiom of Quantified Negation. Axiom C5-2 of [Monk2] p. 113. $)
ax6 $p |- T. |= [ ( ~ ( ! \ x : al . R ) ) ==>
( ! \ x : al . ( ~ ( ! \ x : al . R ) ) ) ] $=
( vy tne tal kl kc hb wnot ht wal wl wc tv kt wv ax-17 hbc ax-hbl1 isfree
) ABEFGABCHZIZIJJFUDKAJLZJGUCAMZAJBCDNZOZOAJJBUDAEPZFQKUHAERZAJJLBFUIKUJS
AUEJBUCUIGQUFUGUJAUEJLBGUIUFUJSAAJBCUIDUJUATTUB $.
$( [10-Oct-2014] $)
$}
${
$d z R $. $d x y z al $.
ax7.1 $e |- R : bool $.
$( Axiom of Quantifier Commutation. Axiom scheme C6' in [Megill] p. 448
(p. 16 of the preprint). $)
ax7 $p |- T. |= [ ( ! \ x : al . ( ! \ y : al . R ) ) ==>
( ! \ y : al . ( ! \ x : al . R ) ) ] $=
( vz kt tal kl kc hb ht wal wl wc ax4 ax-17 ax-hbl1 hbc alrimi syl tv hbl
wv wtru adantl ex ) GHABHACDIZJZIZJZHACHABDIJZIJZUKGUMACFULUKABFDUKUKUIDA
BUIAKLZKHUHAMZAKCDENZOZPACDEPUAAUNKBUJAFUBZHGUOAKBUIUQNZAFUDZAUNKLZBHURUO
UTQAAKBUIURUQUTRSTAUNKCUJURHGUOUSUTAVACHURUOUTQZAAKCBUIURGUQUTAUNKCUHURHG
UOUPUTVBAAKCDUREUTRSUCSTUEUFUG $.
$( [10-Oct-2014] $)
$}
${
$d x al $.
axgen.1 $e |- T. |= R $.
$( Rule of Generalization. See e.g. Rule 2 of [Hamilton] p. 74. $)
axgen $p |- T. |= ( ! \ x : al . R ) $=
( kt alrimiv ) ABCEDF $.
$( [10-Oct-2014] $)
$}
${
ax8.1 $e |- A : al $.
ax8.2 $e |- B : al $.
ax8.3 $e |- C : al $.
$( Axiom of Equality. Axiom scheme C8' in [Megill] p. 448
(p. 16 of the preprint). Also appears as Axiom C7 of
[Monk2] p. 105. $)
ax8 $p |- T. |= [ [ A = B ] ==> [ [ A = C ] ==> [ B = C ] ] ] $=
( kt ke kbr tim kct weqi simpl eqcomi simpr eqtri ex wtru adantl ) HBCIJZ
BDIJZCDIJZKJZUAHUDUAUBUCACBDUAUBLZFABCUEEUAUBABCEFMZABDEGMZNOUAUBUFUGPQRS
TR $.
$( [10-Oct-2014] $)
$}
${
$d y A $. $d x y al $.
ax9.1 $e |- A : al $.
$( Axiom of Equality. Axiom scheme C8' in [Megill] p. 448
(p. 16 of the preprint). Also appears as Axiom C7 of
[Monk2] p. 105. $)
ax9 $p |- T. |= ( ~ ( ! \ x : al . ( ~ [ x : al = A ] ) ) ) $=
( vy tne tex tv kl kc tal kt wv hb ht wl ax-17 wtru wc wnot ke weqi 19.8a
kbr wex ax-hbl1 hbc eqid eqtru eqcomi ax-inst notnot1 syl wal alnex mpbir
id ceq2 ) FFGABABHZCUAUDZIZJZJZJZFKABFUTJZIZJZJLLVBVDABEVBVBCUTLABUTAUSCA
BMDUBZUCAANOZNBVAAEHZGLAUEZANBUTVHPZAEMZAVINOBGVJVKVMQAANBUTVJVHVMUFUGANB
LVJRVMQNVBUTVHVINGVAVKVLSZUHNLUTUTRUTUTUTVHUQUIUJUKVBVNULUMNNVGVCFLTVINKV
FAUNANBVENNFUTTVHSPSABUTVHUOURUP $.
$( [10-Oct-2014] $)
$}
${
$d x y z al $.
$( Axiom of Quantifier Substitution. Appears as Lemma L12 in
[Megill] p. 445 (p. 12 of the preprint). $)
ax10 $p |- T. |= [ ( ! \ x : al . [ x : al = y : al ] ) ==>
( ! \ y : al . [ y : al = x : al ] ) ] $=
( vz kt tal tv ke kbr kl kc wv weqi hb weq wov id oveq1 cla4v wl eqtri ht
ax4 eqcomi alrimiv wal wc cbv ceq2 a1i mpbir wtru adantl ex ) EFABABGZACG
ZHIZJZKZFACUPUOHIZJZKZUSEVBFADADGZUOHIZJZKZVBUSADVDUSAVCUPUOUSADLZABUQVCV
CUPHIAUOUPABLZACLZMZVGAANUOUPVCHUOVCHIZAOZVHVIVKAANUOVCHVLVHVGPQRSAUOUPUS
VHABUQVJUCUDUAUEVBVFHIUSANUBZNFURAUFZANBUQVJTUGVMNVAVEFEVNANCUTAUPUOVIVHM
ZTANCDUTVDVOAANUPUOVCHUPVCHIZVLVIVHVPAUPVCVIVGMQRUHUIUJUKULUMUN $.
$( [10-Oct-2014] $)
$}
${
$d x A $. $d x y al $.
ax11.1 $e |- A : bool $.
$( Axiom of Variable Substitution. It
is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105,
from which it can be proved by cases. $)
ax11 $p |- T. |= [ [ x : al = y : al ] ==> [ ( ! \ y : al . A ) ==>
( ! \ x : al . [ [ x : al = y : al ] ==> A ] ) ] ] $=
( kt tv ke kbr tal kl kc tim kct hb wl wc id a1i ex ht wal eta ceq2 mpbir
wv wim weqi wov simpr ax-cb1 ax-cb2 simpl wct beta eqtri mpbi wtru adantl
alrimiv ax5 mpd syl ) FABGZACGZHIZJACDKZLZJABVFDMIZKZLZMIFVFNZVHVKVHVLVKVH
JABVGVDLZKZLZVKVHVOVHVHAOUAZOJVGAUBZAOCDEPZQZRVOVHHIZVHVSVPOVNVGJFVQAOBVM
AOVGVDVRABUFZQZPZAOBVGVRUCUDZSUEVOVOVKVPOJVJVQAOBVIOOOVFDMUGAVDVEWAACUFUH
ZEUIZPQZVOVPOJVNVQWCQZRVOVKMIZVOWHFJABVMVIMIZKLWIOOOVOVKMUGVOVFVHNZVHVOWK
VFVHWEVSUJZVTWKVHWKWLUKWDSUEULWGUIABWJFFVMVIVMFVIVMVFDVMDVMVFNZVMVFWBWEUM
OVMVGVELZDWMWBAOVDVEVGWMVRWAVMVFWBWEUJUDWNDHIWMVMVFWBWEUNAOCDEUOSUPUQTURU
STUTABVMVIWBWFVAVBSVBVCFVFURWEUNUSTT $.
$( [10-Oct-2014] $)
$}
${
$d p x z $. $d p y z $. $d p z al $.
$( Axiom of Quantifier Introduction. Axiom scheme C9' in [Megill]
p. 448 (p. 16 of the preprint). $)
ax12 $p |- T. |=
[ ( ~ ( ! \ z : al . [ z : al = x : al ] ) ) ==>
[ ( ~ ( ! \ z : al . [ z : al = y : al ] ) ) ==>
[ [ x : al = y : al ] ==> ( ! \ z : al . [ x : al = y : al ] ) ] ] ] $=
( vp kt tne tal tv ke kbr kl kc tim wv weqi hb wnot wl wc ax-17 isfree ht
wal adantr ex ) FGHADADIZABIZJKZLZMZMZGHADUGACIZJKZLZMZMZUHUMJKZHADURLMNK
ZNKZFULUTFUQUSFUQUSADEURAUHUMABOZACOZPZAQDURAEIVCAEOUAUBQQGUPRAQUCZQHUOAU
DZAQDUNAUGUMADOZVBPSTTUEUFQQGUKRVDQHUJVEAQDUIAUGUHVFVAPSTTUEUF $.
$( [10-Oct-2014] $)
$}
${
ax13.1 $e |- A : al $.
ax13.2 $e |- B : al $.
ax13.3 $e |- C : ( al -> bool ) $.
$( Axiom of Equality. Axiom scheme C12' in [Megill] p. 448
(p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of
system S2 of [Tarski] p. 77. $)
ax13 $p |- T. |= [ [ A = B ] ==> [ ( C A ) ==> ( C B ) ] ] $=
( kt ke kbr kc tim kct wtru weqi wct hb wc simpr ex ceq2 adantr mpbi ) HB
CIJZDBKZDCKZLJHUDMZUEUFUEUFUGUEMUGUEHUDNABCEFOZPAQDBGERZSUGUEUEUFIJAQBCDU
GGEHUDNUHSUAUIUBUCTT $.
$( [10-Oct-2014] $)
$}
${
ax14.1 $e |- A : ( al -> bool ) $.
ax14.2 $e |- B : ( al -> bool ) $.
ax14.3 $e |- C : al $.
$( Axiom of Equality. Axiom scheme C12' in [Megill] p. 448
(p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of
system S2 of [Tarski] p. 77. $)
ax14 $p |- T. |= [ [ A = B ] ==> [ ( A C ) ==> ( B C ) ] ] $=
( kt ke kbr kc tim kct wtru hb ht weqi wct simpr ex wc ceq1 adantr mpbi )
HBCIJZBDKZCDKZLJHUEMZUFUGUFUGUHUFMUHUFHUENAOPBCEFQZRAOBDEGUAZSUHUFUFUGIJA
ODBUHCEGHUENUISUBUJUCUDTT $.
$( [10-Oct-2014] $)
$}
${
$d x y A $. $d x y al $.
ax17.1 $e |- A : bool $.
$( Axiom to quantify a variable over a formula in which it does not occur.
Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as
Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of
[Monk2] p. 113. $)
ax17 $p |- T. |= [ A ==> ( ! \ x : al . A ) ] $=
( vy hb tv wv ax-17 isfree ) ABECDAFBCAEGDAEHIJ $.
$( [10-Oct-2014] $)
$}
${
$d x y A $. $d x y B $. $d x y al $.
axext.1 $e |- A : ( al -> bool ) $.
axext.2 $e |- B : ( al -> bool ) $.
$( Axiom of Extensionality. An axiom of Zermelo-Fraenkel set theory. It
states that two sets are identical if they contain the same elements.
Axiom Ext of [BellMachover] p. 461. $)
axext $p |- T. |=
[ ( ! \ x : al . [ ( A x : al ) = ( B x : al ) ] ) ==> [ A = B ] ] $=
( vy kt tal tv kc ke kbr kl hb ht wv wc wl eta weqi ax4 wal ax-17 ax-hbl1
hbc leqf ax-cb1 a1i 3eqtr3i wtru adantl ex ) HIABCABJZKZDUNKZLMZNZKZCDLMZ
USHUTAOPZABUONZABUPNZUSCDAOBUOAOCUNEABQZRZSAOBGUOUPUSVEABUQOUOUPVEAODUNFV
DRUAZUBAVAOBURAGJZIHAUCZAOBUQVFSAGQZAVAOPBIVGVHVIUDAAOBUQVGVFVIUEUFUGZVBC
LMUSVBVCLMUSVJUHZAOBCETUIVCDLMUSVKAOBDFTUIUJUKULUM $.
$( [10-Oct-2014] $)
$}
${
$d f A $. $d f y B $. $d f y al $. $d f x y be $. $d f y z be $.
axrep.1 $e |- A : bool $.
axrep.2 $e |- B : ( al -> bool ) $.
$( Axiom of Replacement. An axiom scheme of Zermelo-Fraenkel set theory.
Axiom 5 of [TakeutiZaring] p. 19. $)
axrep $p |- T. |= [ ( ! \ x : al . ( ? \ y : be . ( ! \ z : be .
[ ( ! \ y : be . A ) ==> [ z : be = y : be ] ] ) ) ) ==>
( ? \ y : ( be -> bool ) . ( ! \ z : be . [ ( y : ( be -> bool ) z : be ) =
( ? \ x : al . [ ( B x : al ) /\ ( ! \ y : be . A ) ] ) ] ) ) ] $=
( kt tal tex kl kc ke kbr hb ht wl wc vf hga tim tan wal wex wim weqi wov
tv wv wtru wan eqid alrimiv a1i ax-cb1 weq id ceq1 beta eqtri oveq1 ax-17
ax-hbl1 hbov leqf ceq2 hbl1 hbc hbl clf mpbir ax4e syl adantl ex ) JKACLB
DKBEKBDFMZNZBEUJZBDUJZOPZUCPZMZNZMZNZMZNZLBQRZDKBEWJDUJZVTNZLACGACUJZNZVS
UDPZMZNZOPZMZNZMZNZWIJXBWIXABEWQMZNZXBKBEWQWQOPZMZNZXDWIXGWIAQRZQKWHAUEAQ
CWGWJQLWFBUFZBQDWEWJQKWDBUEZBQEWCQQQVSWBUCUGWJQKVRXJBQDFHSZTZBVTWABEUKZBD
UKUHUISTSTSTBEXEJQWQJULXHQLWPAUFZAQCWOQQQWNVSUDUMAQGWMIACUKTZXLUIZSZTZUNU
OUPZXDXGOPWIXGWIXSUQWJQDUAWTXGXCWJQKWSXJBQEWRQWLWQBQWKVTWJDUKZXMTZXRUHZSZ
TZBQEWQXRSZWJQWSXFKWKXCOPZXJYCBQEUAWRXEYFYBQQQWLWQWQOYFQURZYAXRQWLXCVTNZW
QYFYABQVTWKYFXCXTXMYFWJWKXCXTYEUHZUSUTYHWQOPYFYIBQEWQXRVAUPVBVCBWJWJQEWKB
UAUJZXCOJWJURXTBUAUKZYEBQQQRRZEOYJYGYKVDBWJEWKYJXTYKVDBBQEWQYJXRYKVEVFVGV
HWJWJQDXFWJUAUJZKJXJBQEXEQWQWQXRXRUHZSWJUAUKZWJWJQRZDKYMXJYOVDZWJBQDEXEYM
JYNYOWJQQQDWQYMWQOJYGXRYOXRWJUBUBQRRDOYMUBURYOVDWJXHQDWPYMLJXNXQYOWJYPDLY
MXIYOVDWJAQDCWOYMJXPYOWJQQQDWNYMVSUDJUMXOYOXLWJYLDUDYMUMYOVDWJQDWNYMXOYOV
DWJWJQDVRYMKJXJXKYOYQWJBQDFYMJHYOULVIVJVFVKVJZYRVFVKVJWJBQDEWQYMJXRYOYRVK
VLUPVMWJXCXAWJQDWTYDSYEVNVOULVPVQ $.
$( [10-Oct-2014] $)
$}
${
$d x y $. $d y A $. $d p y z al $.
axpow.1 $e |- A : ( al -> bool ) $.
$( Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. $)
axpow $p |- T. |=
( ? \ y : ( ( al -> bool ) -> bool ) . ( ! \ z : ( al -> bool ) .
[ ( ! \ x : al . [ ( z : ( al -> bool ) x : al ) ==> ( A x : al ) ] ) ==>
( y : ( ( al -> bool ) -> bool ) z : ( al -> bool ) ) ] ) ) $=
( vp kt tal hb tv kc tim kbr kl wtru wim wv wc wl ht tex wal wov simpl ex
alrimiv ke weqi id ceq1 eqid cl a1i eqtri oveq2 leq ceq2 cla4ev syl ) HIA
JUAZDIABVADKZABKZLZEVCLZMNZOZLZHMNZOZLZUBVAJUAZCIVADVHVLCKZVBLZMNZOZLZOLV
ADVIHHVHHHVHPVAJIVGAUCAJBVFJJJVDVEMQAJVBVCVADRZABRZSAJEVCFVSSUDTSZUEUFUGV
LCVQVAGHOZVKVLJIVPVAUCZVAJDVOJJJVHVNMQVTVAJVMVBVLCRZVRSZUDZTZSVAJGHPTZVLJ
VPVJIVMWAUHNZWBWFVAJDVOVIWHWEJJJVHVNMWHHQVTWDJVNWAVBLZHWHWDVAJVBVMWHWAWCV
RWHVLVMWAWCWGUIZUJUKWIHUHNWHWJVAJGHHVBPVRJHVAGKZVBUHNVAWKVBVAGRVRUIPULUMU
NUOUPUQURUSUT $.
$( [10-Oct-2014] $)
$}
${
$d x y $. $d p y z $. $d y A $. $d y z al $. $d p y z al $.
axun.1 $e |- A : ( ( al -> bool ) -> bool ) $.
$( Axiom of Union. An axiom of Zermelo-Fraenkel set theory. $)
axun $p |- T. |=
( ? \ y : ( al -> bool ) . ( ! \ z : al . [ ( ? \ x : ( al -> bool ) .
[ ( x : ( al -> bool ) z : al ) /\ ( A x : ( al -> bool ) ) ] ) ==>
( y : ( al -> bool ) z : al ) ] ) ) $=
( vp kt tal tex hb tv kc kbr kl tim wtru wv wc wl tan kct wex wan wov wct
ht trud ex alrimiv wal wim ke weqi id ceq1 a17i eqtri leq ceq2 cla4ev syl
oveq2 ) HIADJAKUGZBVDBLZADLZMZEVEMZUANZOZMZHPNZOZMZJVDCIADVKVDCLZVFMZPNZO
ZMZOMADVLHHVKHHVKUBHVKQVDKUGKJVJVDUCVDKBVIKKKVGVHUAUDAKVEVFVDBRZADRZSVDKE
VEFVTSUETSZUFUHUIUJVDCVSAGHOZVNVDKIVRAUKZAKDVQKKKVKVPPULWBAKVOVFVDCRZWASZ
UEZTZSAKGHQTZVDKVRVMIVOWCUMNZWDWHAKDVQVLWJWGKKKVKVPPWJHULWBWFKVPWCVFMHWJW
FAKVFVOWJWCWEWAWJVDVOWCWEWIUNZUOUPAKGHVFWJQWAWKUQURVCUSUTVAVB $.
$( [10-Oct-2014] $)
$}
$( ax-reg, ax-inf, and ax-ac are "true" in the broad strokes in HOL, but
the set.mm versions of these axioms are not well-typed and so cannot be
expressed. $)
$( $t
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section uses C-style comment delimiters. Todo: Allow multiple
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$)
$( 456789012345 (79-character line to adjust text window width) 567890123456 $)