QUICK_REFERENCE.txt

thm:
REFL `x`
gives `|- x = x`

TRANS `ASM1 |- a = b` `ASM2 |- b = c`
gives `ASM1+ASM2 |- a = c`

MK_COMB (`ASM1 |- f = g`, `ASM2 |- a = b`)
gives `ASM1+ASM2 |- (f a) = (g b)`

ABS `x` `ASM1{-x} |- a=b`
gives `ASM1 |- \x.a = \x.b`

BETA is useless; use BETA_CONV

ASSUME `a`
gives `a |- a`

EQ_MP `ASM1 |- a = b` `ASM2 |- a`
gives `ASM1+ASM2 |- b`

DEDUCT_ANTISYM_RULE `ASM1 |- a` `ASM2 |- b`
gives `(ASM1-{b})+(ASM2-{a}) |- a=b`

INST_TYPE instantiation theorem
gives a new theorem with type variables instantiated

INST instantiation theorem
gives a new theorem with variables instantiated

equal:
BETA_CONV `(\x. A) y`
gives `|- (\x. A) y = A[y/x]`

AP_TERM `f` `ASM1 |- a = b`
gives `ASM1 |- (f a) = (f b)`

AP_THM `ASM1 |- f = g` `a`
gives `ASM1 |- (f a) = (g a)`

SYM `ASM1 |- a = b`
gives `ASM1 |- b = a`

ALPHA `(\x.x)` `(\y.y)`
gives `|- (\x.x) = (\y.y)`

ALPHA_CONV `y` `(\x.x)`
gives `|- (\x.x) = (\y.y)`

GEN_ALPHA_CONV `y` `!x. P[x]`
gives `|- (!x. P[x]) = (!y. P[y])` (it looks inside one level of application)

MK_BINOP `(+)` (`ASM1 |- a = b`, `ASM2 |- c = d`)
gives `ASM1+ASM2 |- a + c = b + d`

A conversion takes a term and returns an equality theorem with the term on
the lhs (or it fails).

NO_CONV is a conversion which always fails.
ALL_CONV is a conversion which always succeeds without changing the term.
c1 THENC c2 rewrites with c1 then c2.
c1 ORELSEC c2 rewrites with c1; if it fails, it rewrites with c2 instead.
FIRST_CONV [c1;...;cn] rewrites with the first non-failing conversion in
           the list.
EVERY_CONV [c1;...;cn] rewrites with the first conversion, then the
           second, then ... then the last.
REPEATC c rewrites with c until it fails (and returns the result of the
        last successful rewrite).
CHANGED_CONV c rewrites with c, but fails if the result is alpha-equivalent
             to the original term.
TRY_CONV c rewrites with c; if it fails, it does not change the term
         (and does not fail).

RATOR_CONV c uses the conversion to rewrite the operator of a combination.
RAND_CONV c uses the conversion to rewrite the operand of a combination.
LAND_CONV c uses the conversion to rewrite the first argument of
          a binary function.
COMB2_CONV c1 c2 uses c1 to rewrite the operator and c2 to rewrite the
           operand of a combination.
COMB_CONV c rewrites both the operator and operand of a combination with c.
ABS_CONV c rewrites the body of an abstraction with c.
BINDER_CONV c rewrites the body of an abstraction or of a
            binder/abstraction combination.
SUB_CONV c either rewrites both parts of a combination, or the body
         of an abstraction, or does nothing (it never fails).
BINOP_CONV c rewrites both arguments of a binary function.
THENQC c1 c2 is like (c1 THENC c2) ORELSEC c1 ORELSEC c2
THENCQC c1 c2 is like (c1 THENC c2) ORELSEC c1
REPEATQC c rewrites with c one or more times, until it fails (returning
         the last succeeding rewrite; if the initial rewrite fails,
         then REPEATQC fails)
COMB2_QCONV c1 c2 is like
            (RATOR_CONV c1 THENC RAND_CONV c2) ORELSEC RATOR_CONV c1 ORELSEC RAND_CONV c2
COMB_QCONV c = COMB2_QCONV c c
SUB_QCONV c is like ABS_CONV c ORELSEC COMB_QCONV c
ONCE_DEPTH_QCONV c uses c to rewrite all maximal applicable terms
                 (terms which are not properly contained in another
                 applicable term).  Fails if c does not apply to any terms.
DEPTH_QCONV c repeatedly rewrites with c (in a single bottom-up sweep),
            failing if c does not apply anywhere.
REDEPTH_QCONV c rewrites with c (in a bottom-up sweep); after any
              successful rewrite, it starts over again at the leaves
              of the new term.  Fails if c does not apply anywhere.
TOP_DEPTH_QCONV c repeatedly rewrites with c top-to-bottom;
                recursively do something like:
                do (rewrite current until no change)
                   then rewrite children with TOP_DEPTH_QCONV
                until no change
TOP_SWEEP_QCONV c rewrite with c top-to-bottom; something like:
                (rewrite current until no change)
                then rewrite children with TOP_SWEEP_QCONV

ONCE_DEPTH_CONV, DEPTH_CONV, REDEPTH_CONV, TOP_DEPTH_CONV, TOP_SWEEP_CONV:
like the QCONV variants, except they never fail

SINGLE_DEPTH_CONV c rewrites maximal applicable subterms with c,
                  then rewrites parents of changed terms bottom-up

DEPTH_BINOP_CONV op c
                 For example, DEPTH_BINOP_CONV `+` c rewrites
                 x,y,z,w in `(x + ((y + z) + w))` (fails if any
                 of these rewrites fail)

PATH_CONV path c
          For example, PATH_CONV ["b";"l";"r";"r"] c rewrites the
          operand of the operand of the operator of the abstraction body
          with c.

SYM_CONV rewrites `a = b` to `b = a`

CONV_RULE c th uses c to rewrite the conclusion of the theorem
          (and return a new theorem).

SUBS_CONV ths is a conversion.  It takes its list of equality
          theorems and rewrites (anywhere in its argument term)
          any lhs to its corresponding rhs (matching with alpha-equivalence).

BETA_RULE takes a theorem, does all possible beta reductions, and
          returns the new theorem.
GSYM applies symmetry on all outermost equalities in the conclusion
     of a theorem.
SUBS applies SUBS_CONV to the conclusion of a theorem.

CACHE_CONV c is equivalent to c, except that it caches all
           conversion applications.

bool:
PINST tyin tmin th instantiates types in th according to tyin and terms
      according to tmin.

PROVE_HYP th1 th2
If the conclusion of th1 is a hypothesis of th2, then returns th2
except that this hypothesis is replaced by the hypotheses of th1; otherwise
returns th2 unchanged.

T_DEF: `T = ((\x:bool. x) = (\x:bool. x))`

TRUTH: `|- T`

EQT_ELIM `ASM1 |- a = T` gives `ASM1 |- a`

EQT_INTRO `ASM1 |- a` gives `ASM1 |- a = T`

AND_DEF: `(/\) = \t1 t2. (\f:bool->bool->bool. f t1 t2) = (\f. f T T)`
CONJ `ASM1 |- a` `ASM2 |- b` gives `ASM1+ASM2 |- a /\ b`
CONJUNCT1 `ASM |- a /\ b` gives `ASM1 |- a`
CONJUNCT2 `ASM |- a /\ b` gives `ASM1 |- b`
CONJ_PAIR th = (CONJUNCT1 th, CONJUNCT2 th)
CONJUNCTS th gives a list of theorems, one for each conjunct of the
          conclusion of th (no matter how they are associated)

IMP_DEF: `(==>) = \t1 t2. t1 /\ t2 = t1`
MP `ASM1 |- a ==> b` `ASM2 |- a` gives `ASM1+ASM2 |- b`
DISCH `a` `ASM,a |- b` gives `ASM |- a ==> b`
DISCH_ALL repeats DISCH until there are no more hypotheses.
UNDISCH `ASM |- a ==> b` gives `ASM,a |- b`
UNDISCH_ALL repeats UNDISCH until the conclusion is not an implication
IMP_ANTISYM_RULE `ASM1 |- a ==> b` `ASM2 |- b ==> a` gives `ASM1+ASM2 |- a = b`
ADD_ASSUM `a` `ASM |- b` gives `ASM,a |- b`
EQ_IMP_RULE `ASM |- (a:bool) = b` gives (`ASM |- a ==> b`, `ASM |- b ==> a`)
IMP_TRANS `ASM1 |- a ==> b` `ASM2 |- b ==> c` gives `ASM1+ASM2 |- a ==> c`

FORALL_DEF: `(!) = \P:A->bool. P = \x. T`
SPEC `a` `ASM |- !x.P[x]` gives `ASM |- P[a]`
SPECL [`a`;`b`;`c`] `ASM |- !x y z.P[x,y,z]` gives `ASM |- P[a,b,c]`
SPEC_VAR `ASM |- !x.P[x]` gives (`x17`, `ASM |- P[x17]`)
SPEC_ALL repeats SPEC_VAR until the conclusion is not a "forall", and returns
         the final theorem.
ISPEC is like SPEC, except that the specialized term may be an instance
      of the type of the quantified variable (in which case the theorem
      is type-instantiated first), rather than matching exactly.
ISPECL is like SPECL with type instantiations.
GEN `x` `ASM |- P[x]` gives `ASM |- !x. P[x]` (if x is not free in ASM)
GENL [`x`;`y`;`z`] `ASM |- P[x,y,z]` gives `ASM |- !x y z. P[x,y,z]`
GEN_ALL generalizes over all variables free in the conclusion but not
        in the assumptions

EXISTS_DEF: `(?) = \P:A->bool. !Q. (!x. P x ==> Q) ==> Q`
EXISTS (`?x. P[x]`,`a`) `ASM |- P[a]` gives `ASM |- ?x. P[x]`
SIMPLE_EXISTS `x` `ASM |- P[x]` gives `ASM |- ?x. P[x]`
CHOOSE (`x`,`ASM1 |- ?y.P[y]`) `ASM2,P[x] |- a` gives `ASM1+ASM2 |- a`
SIMPLE_CHOOSE `x` `ASM,P[x] |- a` gives `ASM,?x.P[x] |- a` (P[x] must be
              the first hypothesis)

OR_DEF: `(\/) = \t1 t2. !t. (t1 ==> t) ==> (t2 ==> t) ==> t`
DISJ1 `ASM |- a` `b` gives `ASM |- a \/ b`
DISJ2 `a` `ASM |- b` gives `ASM |- a \/ b`
DISJ_CASES `ASM1 |- a \/ b` `ASM2,a |- c` `ASM3,b |- c` gives
           `ASM1,ASM2,ASM3 |- c`
SIMPLE_DISJ_CASES `ASM1,a |- c` `ASM2,b |- c` gives
                  `ASM1,ASM2,a \/ b |- c` (a and b must be the first
                  hypotheses)

F_DEF: `F = !t:bool. t`
NOT_DEF: `(~) = \t. t ==> F`
NOT_ELIM `ASM |- ~a` gives `ASM |- a ==> F`
NOT_INTRO `ASM |- a ==> F` gives `ASM |- ~a`
EQF_INTRO `ASM |- ~a` gives `ASM |- a = F`
EQF_ELIM `ASM |- a = F` gives `ASM |- ~a`
NEG_DISCH `a` `ASM,a |- F` gives `ASM |- ~a` (if the conclusion of the
          initial theorem is not F, then NEG_DISCH acts like DISCH)
CONTR `a` `ASM |- F` gives `ASM |- a`

EXISTS_UNIQUE_DEF: `(?!) = \P:A->bool. ((?) P) /\ (!x y. ((P x) /\ (P y)) ==> (x = y))`
EXISTENCE `ASM |- ?!x. P[x]` gives `ASM |- ?x. P[x]`

drule:

MK_CONJ `ASM1 |- a = b` `ASM2 |- c = d` gives `ASM1,ASM2 |- a /\ c = b /\ d`
MK_DISJ `ASM1 |- a = b` `ASM2 |- c = d` gives `ASM1,ASM2 |- a \/ c = b \/ d`
MK_FORALL `x` `ASM |- a = b` gives `ASM |- (!x.a) = (!x.b)`
MK_EXISTS `x` `ASM |- a = b` gives `ASM |- (?x.a) = (?x.b)`

BETAS_CONV (n:int) is a conversion which rewrites with BETA_CONV n times.
(so BETAS_CONV 3 rewrites `(\x y z.P[x,y,z]) a b c` to `P[a,b,c]`)

INSTANTIATE takes an "instantiation" and a theorem and instantiates
the theorem.  (I haven't figured out yet what an "instantiation" is.)
INSTANTIATE_ALL has the same type as INSTANTIATE.

For the rest of the file, I will treat '@' as a metavariable for
an instantiation (written postfix).  For instance, I may mention
the terms `a ==> b`, `a@`, and `b@`; this means that
`a@ ==> b@` is an instantiation of `a ==> b`.

In some cases where I use this notation, universal quantification is
allowed; that is, where I mention terms `a` and `a@`, the implementation
would allow `!x. P[x]` and `P[b]`.

PART_MATCH partfn th tm matches (partfn (concl (SPEC_ALL th))) against tm
and instantiates the theorem appropriately.

MATCH_MP ith th is like MP, except it uses matching instead of
requiring an exact match.

HIGHER_REWRITE_CONV ths top
A conversion which finds the first largest (if top == true) or smallest
(if top == false) subterm which matches (using higher-order matching)
a lhs of a conclusion in ths, and rewrites it.

tactics:

t1 THEN t2
Apply t1, then apply t2 to all subgoals created.

t THENL [t1;t2;...;tn]
Apply t, then apply t1 to the first subgoal, ..., tn to the last subgoal
(there must be exactly n subgoals).

t1 ORELSE t2
Apply t1; if it fails, apply T2.

FAIL_TAC s
A tactic which always fails (with error message s).

NO_TAC
A tactic which always fails.

ALL_TAC
A tactic which does nothing (the identity tactic).

TRY t = tac ORELSE ALL_TAC

REPEAT t
Apply t, then apply it again to all subgoals, etc.; until it fails.

EVERY [t1;...;tn] = t1 THEN ... THEN tn

FIRST [t1;...;tn] = t1 ORELSE ... ORELSE tn

MAP_EVERY tf [x1;...;xn] = tf x1 THEN ... THEN tf xn

MAP_FIRST tf [x1;...;xn] = tf x1 ORELSE ... ORELSE tf xn

CHANGED_TAC t
Apply t; fail if the result is a single subgoal which is equal to the
original goal (warning: does not use alpha-equivalence!)

A "theorem tactic" is a function from theorems to tactics.
A "theorem tactical" is a function from theorem tactics to theorem tactics.
Equivalently: a theorem tactical is a function from theorem tactics
and theorems to tactics.

In other words, a theorem tactic takes a theorem and does something
to the current goal using that theorem.

A theorem tactical takes a theorem tactic and a theorem and does something
to the current goal.  Typically, it will preprocess the theorem somehow
before handing the result to the theorem tactic.  (In fact, the
tactical may apply the theorem tactic multiple times, sequentially
or in parallel (in different subgoals).)

The functions in this section manipulate theorem tacticals.
I will write out pseudo-definitions of these functions that pretend
that a theorem tactical is a function from theorems to theorems which
happens to side-effect the goal; remember that the actual type is
very different than this.

(thtc1 THEN_TCL thtc2) tht th
= tht (thtc1 (thtc2 th))

(thtc1 ORELSE_TCL thtc2) tht th
= (tht (thtc1 th)) ORELSE (tht (thtc2 th))

ALL_THEN = I
(all_then is the theorem tactical which does nothing to the theorem
before handing it to the theorem tactic)

NO_THEN
theorem tactical which always fails

REPEAT_TCL thtc
= (thtc THEN_TCL (REPEAT_TCL thtc)) ORELSE_TCL ALL_THEN

REPEAT_GTCL ???
I don't understand why REPEAT_GTCL is different than REPEAT_TCL.
Fortunately, REPEAT_GTCL is never used, so it can't be very important :-)

EVERY_TCL [thtc1;...;thtcn] = thtc1 THEN_TCL ... THEN_TCL thtcn
FIRST_TCL [thtc1;...;thtcn] = thtc1 ORELSE_TCL ... ORELSE_TCL thtcn

LABEL_TAC s th
Adds th as a new assumption, with label s.  (Assumes that any hypotheses
of th are also hypotheses of the goal.)

ASSUME_TAC = LABEL_TAC ""

USE_ASSUM s tht
Find the first assumption with label s (call this assumption th).
Applies tactic (tht th).

FIND_ASSUM tht tm
Find the first assumption whose conclusion is equal (not alpha-equivalent!)
to tm (call this assumption th).  Applies tactic (tht th).

POP_ASSUM tht
Call the first (most recently added) assumption th.  Removes th
from assumption list, and applies tactic (tht th).

ASSUM_LIST thlt
Applies tactic (thlt thl), where thl is the list of assumptions.

POP_ASSUM_LIST thlt
Applies tactic (thlt thl) after removing all assumptions, where thl
is the list of assumptions.

EVERY_ASSUM tht = ASSUM_LIST (MAP_EVERY tht)

(* This is not the actual definition; I think it is equivalent. *)
FIRST_ASSUM tht = ASSUM_LIST (MAP_FIRST tht)

RULE_ASSUM_TAC thth
Replaces every assumption with thth applied to that assumption.

ASM thlt thl
Applies tactic (thlt (asm @ thl)), where asm is the list of assumptions.

ACCEPT_TAC th
A tactic which solves the current goal, assuming the conclusion of th
is alpha-equivalent to the goal.

CONV_TAC c
Create tactic from a conversion. This allows the conversion to return
|- p rather than |- p = T on a term "p". It also eliminates any goals of
the form "T" automatically.

REFL_TAC
Accepts if the current goal is of the form `a = a`.

ABS_TAC
Converts goal `(\x. a) = (\x. b)` to `a = b`.

MK_COMB_TAC
Converts goal `f a = g b` to `f = g` and `a = b`.

AP_TERM_TAC
Converts goal `f a = f b` to `a = b`.

AP_THM_TAC
Converts goal `f a = g a` to `f = g`.

BINOP_TAC
Converts goal `f a b = f c d` to `a = c` and `b = d`.

SUBST1_TAC `|- a = b`
Converts goal `P[a]` to `P[b]`

SUBST_ALL_TAC `|- a = b`
Rewrites `a` to `b` in goal and all assumptions.

BETA_TAC
Does all possible beta-reductions in goal.

DISCH_TAC
Converts goal `a ==> b` to `b` and adds `a` as a new assumption.
(Treats goal `~a` as `a ==> F`.)

MP_TAC `|- a`
Converts goal `b` to `a ==> b`.

EQ_TAC
Converts goal `(a:bool) = b` to `a ==> b` and `b ==> a`.

UNDISCH_TAC `a`
Finds an assumption with a conclusion alpha-equivalent to `a`.
Removes this assumption, and converts goal `b` to `a ==> b`.

SPEC_TAC (`x`,`a`)
Converts goal `P[a]` to `!x. P[x]`

X_GEN_TAC `x`
Converts goal `!y. P[y]` to `P[x]`

GEN_TAC
Converts goal `!x. P[x]` to `P[x]`

EXISTS_TAC `a`
Converts goal `?x. P[x]` to `P[a]`

X_CHOOSE_TAC `x` `|- ?y. P[y]`
Adds a new assumption `P[x]`

CHOOSE_TAC `|- ?x. P[x]`
Adds a new assumption `P[x]`

CONJ_TAC
Converts goal `a /\ b` to `a` and `b`

DISJ1_TAC
Converts goal `a \/ b` to `a`

DISJ2_TAC
Converts goal `a \/ b` to `b`

DISJ_CASES_TAC `|- a \/ b`
Creates two subgoals.  Adds assumption `a` in one subgoal, `b` in the other.

CONTR_TAC `|- F`
Accepts the goal.

MATCH_ACCEPT_TAC `|- a`
First applies (REPEAT GEN_TAC), then accepts if the conclusion is an
instance of `a`

MATCH_MP_TAC `|- a ==> b`
Converts a goal `b@` into `a@`.

CONJUNCTS_THEN2 tht1 tht2 `|- a /\ b`
Applies tactic (tht1 `|- a`) THEN (tht2 `|- b`)

CONJUNCTS_THEN tht `|- a /\ b`
Applies tactic (tht `|- a`) THEN (tht `|- b`)

DISJ_CASES_THEN2 tht1 tht2 `|- a \/ b`
Generates two subgoals.  Applies (tht1 `|- a`) in one subgoal,
(tht2 `|- b`) in the other.

DISJ_CASES_THEN tht `|- a \/ b`
Generates two subgoals.  Applies (tht `|- a`) in one subgoal,
(tht `|- b`) in the other.

DISCH_THEN tht
Converts goal `a ==> b` to `b`, then applies tactic (tht `|- a`).
(Treats `~a` as `a ==> F`)

X_CHOOSE_THEN `x` tht `|- ?y. P[y]`
Applies tactic (tht `|- P[x]`)

CHOOSE_THEN tht `|- ?x. P[x]`
Applies tactic (tht `|- P[x]`)

STRIP_THM_THEN = FIRST_TCL [CONJUNCTS_THEN; DISJ_CASES_THEN; CHOOSE_THEN]
(That is, it acts like CONJUNCTS_THEN, DISJ_CASES_THEN, or CHOOSE_THEN,
depending on whether the theorem is a conjunction, disjunction, or
exists.)

ANTE_RES_THEN tht `|- a@`
For every assumption `|- a ==> b`, applies tactic (tht `|- b@`)

IMP_RES_THEN tht `|- a ==> b`
For every assumption `|- a@`, applies tactic (tht `|- b@`)

STRIP_ASSUME_TAC th
Starts with (REPEAT_TCL STRIP_THM_THEN) applied to th.
Call the resulting theorem(s) gth.  If gth is `F`, or equal to the
goal, then solve the goal; if gth is already an assumption, do nothing;
otherwise, add gth as an assumption.

STRUCT_CASES_TAC th
Starts with (REPEAT_TCL STRIP_THM_THEN) applied to th.
Call the resulting theorem(s) gth.  If gth is an equality, then use
it to rewrite the goal; otherwise, add gth as an assumption.

STRIP_GOAL_THEN tht
= FIRST [GEN_TAC; CONJ_TAC; DISCH_THEN tht]
(If the current goal is a forall, then generalize; if it is a conjunction,
prove the two cases separately; if it is an implication `a ==> b` then
convert to `b` and apply (tht `a`).)

STRIP_TAC = STRIP_GOAL_THEN STRIP_ASSUME_TAC

ABBREV_TAC `x = a`
Rewrites `a` to `x` in the goal and all assumptions, then adds `a = x`
as a new assumption.

EXPAND_TAC `x`
Finds the first assumption of the form `a = x`, rewrites `x` to `a` in
the goal, and beta-reduces the goal.

UNDISCH_THEN `a` tht
Finds an assumption `|- a`, removes the assumption, and applies the
tactic (tht `|- a`)

FIRST_X_ASSUM tht
Like FIRST_ASSUM, but removes the assumption successfully used by the
tactic.

SUBGOAL_THEN `a` tht
Creates two subgoals.  In the first, changes the goal to `a`.
In the second, applies tactic (tht `|- a`).

FREEZE_THEN tht `|- a`
Applies tactic (tht `|- a`) (while "freezing variables"?  Does this have
something to do with metavariables?  I don't understand this...) (???)

X_META_EXISTS_TAC `x`
Converts goal `?y. P[y]` to `P[x]`, where `x` is a metavariable.

META_EXISTS_TAC
Converts goal `?x. P[x]` to `P[x]`, where `x` is a metavariable.

META_SPEC_TAC `x` `!y. P[y]` adds a new assumption `P[x]`, where
`x` is a metavariable.

CHEAT_TAC
Introduce the goal as a new axiom, then solve the goal.

RECALL_ACCEPT_TAC f x = ACCEPT_TAC (f x)
As a side-effect, prints out the time taken to compute (f x); delays
this computation until it is required.

ANTS_TAC
Converts goal `(p ==> q) ==> r` to `p` and `q ==> r`

itab:
ITAUT_TAC
Intuitionistic theorem prover: understands
and,forall,implies,not,iff (boolean equality),or,exists,T,F;
applies a long list of rules dealing with the above types of terms
until it runs out of rules to apply or proves the theorem.
Either succeeds or leaves the goal state unchanged.

ITAUT `a` gives `|-a` if ITAUT can prove it.

CONTRAPOS `|- a ==> b` gives `|- ~b ==> ~a`

simp:
REWR_CONV `|- a = b`
A conversion which rewrites `a@` to `b@`

A conditional rewrite is a conversion which produces a theorem
of the form `|- P ==> (a = b)`.  This rewrites `a` to `b` under
the condition `P`.  Previous conversionals cannot deal with conditional
rewrites.

IMP_REWR_CONV `|- P ==> (a = b)`
A conversion which rewrites `a@` to `b@` under the condition `P@`.

ORDERED_REWR_CONV ord `|- a = b`
A conversion which rewrites `a@` to `b@` if (ord `a@` `b@`)
(otherwise it fails)

ORDERED_IMP_REWR_CONV ord `|- P ==> (a = b)`
A conversion which rewrites `a@` to `b@` under the condition `P@`
if (ord `a@` `b@`)

term_order
An ordering function which is AC-compatible for any binary operator.

net_of_thm rep `|- P ==> (a = b)` (or `|- a = b`) net
Adds a component to net which matches a term and returns a
conversion rewriting that term.
If rep is true and `a` appears in `b`, then it rewrites `(a = b)` to `T`.
If rep is true and the rewrite is permutative, then it uses an
ordered rewrite to rewrite `a` to `b`.
If neither of the above is true, then it uses an ordinary rewrite.
Conditional rewrites are entered as level 3; unconditional as level 1.

net_of_conv tm conv net
Adds a component to net which matches tm and returns conv.  Entered as level 2.

net_of_cong th net
Adds a component to net for a congruence rule.  (This is a rule of the
form `P1 ==> (P2 ==> (...(Pn ==> (a = b))))` ).  Entered at level 4.

mk_rewrites cf th rew_list
Prepends rewrites for th to new_list
I will describe the action in terms of a notional
add_rew function.
(add_rew l `|- !x. P[x]`) --> (add_rew l `|- P[x]`)
(add_rew l `|- P1 /\ P2`) --> (add_rew l `|- P1`), (add_rew l `|- P2`)
if cf, then (add_rew l `|- P1 ==> P2`) --> (add_rew (`P1`::l) `|- P2`)
(add_rew l `|- a = b`) adds a conditional rewrite with conditions l
         (if l is empty, this is an unconditional rewrite;
         if cf is false, then l is always empty)
(add_rew l `|- ~(a = b)`) --> (add_rew l `|- (a = b) = F`), (add_rew l `|- (b = a) = F`)
(add_rew l `|- ~a`) --> (add_rew l `|- a = F`)
If none of the above hold, (add_rew l `|- a`) --> (add_rew l `|- a = T`)

REWRITES_CONV net `a`
Looks up `a` in the net, and applies the resulting conversion to `a`.

There's a lot of generality in the simpset data structure which is not
used; I'm not going to try to understand the general-purpose operation,
but only the portions of it which are actually used in HOL Light.
Of the four fields in the simpset data structure, only the first ever
changes: the second is always basic_prover, the third is always the
empty list, and the fourth is always (mk_rewrites true).
The first field is a conversion net.

basic_prover strat ss lev `a`
Tries to prove `|- a`.  Succeeds if `a` = `T`, or if
(strat ss lev `a`) proves `|- a = T`.

(* None of the following are ever called.
ss_of_thms thms ss
Augments a simpset with a list of theorems (using the rewrite maker
from the simpset to add the theorems to the conversion net).

ss_of_conv keytm conv ss
Augments a simpset with a conversion (using (net_of_conv keytm conv)
to add the conversion to the conversion net).

ss_of_prover newprover ss
Replaces the prover in the simpset with newprover.

ss_of_provers newprovers ss
Prepends newprovers to the list of subprovers in the simpset.

ss_of_maker newmaker ss
Replaces the rewrite maker in the simpset with newmaker.
*)

AUGMENT_SIMPSET th ss
Uses the rewrite maker from the simpset (always (mk_rewrites true))
to create a list of rewrite theorems from th; use (net_of_thm true)
to add these theorems to the conversion net in the simpset.

Note: IMP_REWRITES_CONV and GEN_SUB_CONV have a different signature than
what I am documenting; if you ever want to call them directly,
look at the source.  I will first describe the meaning of these functions;
then I will describe their implementation.
(IMP_REWRITES_CONV strat ss lev) is a conversion which looks up the
term in the ss conversion net.  It tries unconditional rewrites before
conditional rewrites.  It will only accept a conditional rewrite if
(strat ss (lev-1)) can rewrite the condition to `T`.
(GEN_SUB_CONV strat ss lev) rewrites combinations and abstractions
in a depth-first manner by calling (strat ss lev) on subterms.  It
has special handling for if-then-else and implications; it adds
the condition to the simpset before simplifying the rest of the expression.
Both of these functions fail if they perform no rewrites.

IMP_REWRITES_CONV strat ss lev
Try to find a conversion for tm at level < 4 (i.e., not a
congruence rule) which is either unconditional or (if lev > 0) such that
the condition can be rewritten to `T` by (strat ss (lev-1)).

RUN_SUB_CONV -- called only by GEN_SUB_CONV; not documented.

GEN_SUB_CONV strat ss lev
Try to find a congruence rule for tm (i.e., a rule at level 4).
The conclusion of a congruence rule is of the form
`A[x1,...,xn] = A[x1',...,xn']`, so tm must be of the form `A[a1,...,an]`.
Each hypothesis of a congruence rule is either of the form
`xi = xi'` or `P[x1',...,x(i-1)'] ==> (xi = xi')`.
In the former case, use (strat ss lev) to rewrite `a1` to `a1'`;
in the latter case, use AUGMENT_SIMPSET to add
`P[a1',...,a(i-1)']` to ss (getting ss') and use (strat ss' lev) for
the rewrite.
Combine all of these rewrites to rewrite `A[a1,...,an]` to
`A[a1',...,an']`.
If this fails, then if tm is `f a`, then use (strat ss lev) to
rewrite `f` to `g` and `a` to `b` and return `|- f a = g b`.
If tm is `\x. a[x]`, then use (strat ss lev) to rewrite `a[x]` to `b[x]`
and return `|- (\x. a[x]) = (\x. b[x])`.
Throughout GEN_SUB_CONV (in both the congruence rule case, and the combination
case), if the call to strat fails to rewrite `a`, then it rewrites
`a` to `a`; but if all the calls to strat fail (so that GEN_SUB_CONV
would rewrite tm to tm), GEN_SUB_CONV fails instead.

Note that there are only two congruence rules used in HOL Light: one for
rewriting `if g then t else e` and one for rewriting `p ==> q`.

ONCE_DEPTH_SQCONV ss lev
= (IMP_REWRITES_CONV ONCE_DEPTH_SQCONV ss lev) ORELSEC
  (GEN_SUB_CONV ONCE_DEPTH_SQCONF ss lev)

DEPTH_SQCONV ss lev
= THENQC (GEN_SUB_CONV DEPTH_SQCONV ss lev)
         (IMP_REWRITES_CONV DEPTH_SQCONV ss lev)

REDEPTH_SQCONV ss lev
= REPEATQC (DEPTH_CONV ss lev)

TOP_DEPTH_SQCONV ss lev
= REPEATQC ((IMP_REWRITES_CONV TOP_DEPTH_SQCONV ss lev) ORELSEC
            (GEN_SUB_CONV TOP_DEPTH_SQCONV ss lev))

TOP_SWEEP_SQCONV ss lev
= THENQC (REPEATC (IMP_REWRITES_CONV TOP_SWEEP_SQCONV ss lev))
         (GEN_SUB_CONV TOP_SWEEP_SQCONV ss lev)

Basic rewrites:
There is a global set of "basic rewrites".  These are canonicalized with
(mk_rewrites false) (so there is no handling of conditional rewrites),
and added to a basic conversion net.

set_basic_rewrites thl: sets the "basic rewrite" set to thl
extend_basic_rewrites thl: adds thl to the "basic rewrite" set
basic_rewrites (): retrieves the (canonicalized) "basic rewrite" set
basic_net (): retrieves the conversion net for the "basic rewrite" set

There is also a set of basic congruences; since HOL has only two congruence
rules, I won't bother documenting set_basic_congs, extend_basic_congs,
basic_congs.

GENERAL_REWRITE_CONV rep cnvl net thl
Canonicalizes thl with (mk_rewrites false), adds these conversions
to net with (net_of_thm rep) (giving final_net), and then rewrites with
(cnvl (REWRITES_CONV final_net)).

GEN_REWRITE_CONV cnvl thl
= GENERAL_REWRITE_CONV false cnvl empty_net thl

PURE_REWRITE_CONV thl
= GENERAL_REWRITE_CONV true TOP_DEPTH_CONV empty_net thl

REWRITE_CONV thl
= GENERAL_REWRITE_CONV true TOP_DEPTH_CONV (basic_net()) thl

PURE_ONCE_REWRITE_CONV thl
= GENERAL_REWRITE_CONV false ONCE_DEPTH_CONV empty_net thl

ONCE_REWRITE_CONV thl
= GENERAL_REWRITE_CONV false ONCE_DEPTH_CONV (basic_net()) thl

LIMITED_REWRITE_CONV n thl
Rewrite n times with
(GEN_REWRITE_CONV ONCE_DEPTH_CONV ths THENC
 GENERAL_REWRITE_CONV true TOP_DEPTH_CONV (basic_net()) [])

GEN_REWRITE_RULE cnvl thl
= CONV_RULE(GEN_REWRITE_CONV cnvl thl)

PURE_REWRITE_RULE thl
= CONV_RULE(PURE_REWRITE_CONV thl)

REWRITE_RULE thl
= CONV_RULE(REWRITE_CONV thl)

PURE_ONCE_REWRITE_RULE thl
= CONV_RULE(PURE_ONCE_REWRITE_CONV thl)

ONCE_REWRITE_RULE thl
= CONV_RULE(ONCE_REWRITE_CONV thl)

PURE_ASM_REWRITE_RULE, ASM_REWRITE_RULE,
PURE_ONCE_ASM_REWRITE_RULE, ONCE_ASM_REWRITE_RULE
As non-"ASM_", but adds theorem hypotheses to the rewrite list.

GEN_REWRITE_TAC, PURE_REWRITE_TAC, REWRITE_TAC,
PURE_ONCE_REWRITE_TAC, ONCE_REWRITE_TAC
As "_RULE", but use CONV_TAC instead of CONV_RULE.

PURE_ASM_REWRITE_TAC, ASM_REWRITE_TAC,
PURE_ONCE_ASM_REWRITE_TAC, ONCE_ASM_REWRITE_TAC
As non-"ASM_", but adds current assumptions to the rewrite list.

GEN_SIMPLIFY_CONV strat ss lev thl
Use AUGMENT_SIMPSET to add thl to ss (giving ss'); then do
TRY_CONV (strat ss' lev)

ONCE_SIMPLIFY_CONV ss
= GEN_SIMPLIFY_CONV ONCE_DEPTH_SQCONV ss 1

SIMPLIFY_CONV ss
= GEN_SIMPLIFY_CONV TOP_DEPTH_SQCONV ss 3

empty_ss: an empty simpset (no conversions in the conversion net)

basic_ss thl
Canonicalizes thl with (mk_rewrites true), then adds the rewrites
to (basic_net()) with (net_of_thm true).  Also adds the basic congruences.
Returns the resulting simpset.

SIMP_CONV
= SIMPLIFY_CONV (basic_ss [])

PURE_SIMP_CONV
= SIMPLIFY_CONV empty_ss

ONCE_SIMP_CONV
= ONCE_SIMPLIFY_CONV (basic_ss []) thl

SIMP_RULE, PURE_SIMP_RULE, ONCE_SIMP_RULE
As "_CONV", but calls CONV_RULE

SIMP_TAC, PURE_SIMP_TAC, ONCE_SIMP_TAC
As "_CONV", but calls CONV_TAC

ASM_SIMP_TAC, PURE_ASM_SIMP_TAC, ONCE_ASM_SIMP_TAC
As non-"ASM_", but adds current assumptions to the rewrite list

theorems:

val ( EQ_REFL ) : thm = |- !x. x = x
val ( EQ_REFL_T ) : thm = |- !x. (x = x) = T
val ( EQ_SYM ) : thm = |- !x y. (x = y) ==> (y = x)
val ( EQ_SYM_EQ ) : thm = |- !x y. (x = y) = y = x
val ( EQ_TRANS ) : thm = |- !x y z. (x = y) /\ (y = z) ==> (x = z)
val ( REFL_CLAUSE ) : thm = |- !x. (x = x) = T

AC thl `a = b`
Returns `|- a = b` or fails; succeeds if ordered rewriting using thl
can rewrite `a = b` to `T` or `c = c`

val ( BETA_THM ) : thm = |- !f y. (\x. f x) y = f y
val ( ABS_SIMP ) : thm = |- !t1 t2. (\x. t1) t2 = t1
val ( CONJ_ASSOC ) : thm = |- !t1 t2 t3. t1 /\ t2 /\ t3 = (t1 /\ t2) /\ t3
val ( CONJ_SYM ) : thm = |- !t1 t2. t1 /\ t2 = t2 /\ t1
val ( CONJ_ACI ) : thm =
  |- (p /\ q = q /\ p) /\
     ((p /\ q) /\ r = p /\ q /\ r) /\
     (p /\ q /\ r = q /\ p /\ r) /\
     (p /\ p = p) /\
     (p /\ p /\ q = p /\ q)
val ( DISJ_ASSOC ) : thm = |- !t1 t2 t3. t1 \/ t2 \/ t3 = (t1 \/ t2) \/ t3
val ( DISJ_SYM ) : thm = |- !t1 t2. t1 \/ t2 = t2 \/ t1
val ( DISJ_ACI ) : thm =
  |- (p \/ q = q \/ p) /\
     ((p \/ q) \/ r = p \/ q \/ r) /\
     (p \/ q \/ r = q \/ p \/ r) /\
     (p \/ p = p) /\
     (p \/ p \/ q = p \/ q)
val ( FORALL_SIMP ) : thm = |- !t. (!x. t) = t
val ( EXISTS_SIMP ) : thm = |- !t. (?x. t) = t
val ( EQ_CLAUSES ) : thm =
  |- !t. ((T = t) = t) /\ ((t = T) = t) /\ ((F = t) = ~t) /\ ((t = F) = ~t)
val ( NOT_CLAUSES_WEAK ) : thm = |- (~T = F) /\ (~F = T)
val ( AND_CLAUSES ) : thm =
  |- !t. (T /\ t = t) /\
         (t /\ T = t) /\
         (F /\ t = F) /\
         (t /\ F = F) /\
         (t /\ t = t)
val ( OR_CLAUSES ) : thm =
  |- !t. (T \/ t = T) /\
         (t \/ T = T) /\
         (F \/ t = t) /\
         (t \/ F = t) /\
         (t \/ t = t)
val ( IMP_CLAUSES ) : thm =
  |- !t. (T ==> t = t) /\
         (t ==> T = T) /\
         (F ==> t = T) /\
         (t ==> t = T) /\
         (t ==> F = ~t)

REFL_CLAUSE, EQ_CLAUSES, NOT_CLAUSES_WEAK, AND_CLAUSES, OR_CLAUSES,
IMP_CLAUSES, FORALL_SIMP, EXISTS_SIMP, BETA_THM, and `((x = x) ==> p) = p`
are all "basic rewrites".

val ( EXISTS_UNIQUE_THM ) : thm =
  |- !P. (?!x. P x) = (?x. P x) /\ (!x x'. P x /\ P x' ==> (x = x'))
val ( EXISTS_REFL ) : thm = |- !a. ?x. x = a
val ( EXISTS_UNIQUE_REFL ) : thm = |- !a. ?!x. x = a
val ( EXISTS_UNIQUE_ALT ) : thm = |- !P. (?!x. P x) = (?x. !y. P y = x = y)
val ( UNWIND_THM1 ) : thm = |- !P a. (?x. (a = x) /\ P x) = P a
val ( UNWIND_THM2 ) : thm = |- !P a. (?x. (x = a) /\ P x) = P a
val ( SWAP_FORALL_THM ) : thm = |- !P. (!x y. P x y) = (!y x. P x y)
val ( SWAP_EXISTS_THM ) : thm = |- !P. (?x y. P x y) = (?y x. P x y)
val ( FORALL_AND_THM ) : thm =
  |- !P Q. (!x. P x /\ Q x) = (!x. P x) /\ (!x. Q x)
val ( AND_FORALL_THM ) : thm =
  |- !P Q. (!x. P x) /\ (!x. Q x) = (!x. P x /\ Q x)
val ( LEFT_AND_FORALL_THM ) : thm = |- !P Q. (!x. P x) /\ Q = (!x. P x /\ Q)
val ( RIGHT_AND_FORALL_THM ) : thm = |- !P Q. P /\ (!x. Q x) = (!x. P /\ Q x)
val ( EXISTS_OR_THM ) : thm =
  |- !P Q. (?x. P x \/ Q x) = (?x. P x) \/ (?x. Q x)
val ( OR_EXISTS_THM ) : thm =
  |- !P Q. (?x. P x) \/ (?x. Q x) = (?x. P x \/ Q x)
val ( LEFT_OR_EXISTS_THM ) : thm = |- !P Q. (?x. P x) \/ Q = (?x. P x \/ Q)
val ( RIGHT_OR_EXISTS_THM ) : thm = |- !P Q. P \/ (?x. Q x) = (?x. P \/ Q x)
val ( LEFT_EXISTS_AND_THM ) : thm = |- !P Q. (?x. P x /\ Q) = (?x. P x) /\ Q
val ( RIGHT_EXISTS_AND_THM ) : thm = |- !P Q. (?x. P /\ Q x) = P /\ (?x. Q x)
val ( TRIV_EXISTS_AND_THM ) : thm =
  |- !P Q. (?x. P /\ Q) = (?x. P) /\ (?x. Q)
val ( LEFT_AND_EXISTS_THM ) : thm = |- !P Q. (?x. P x) /\ Q = (?x. P x /\ Q)
val ( RIGHT_AND_EXISTS_THM ) : thm = |- !P Q. P /\ (?x. Q x) = (?x. P /\ Q x)
val ( TRIV_AND_EXISTS_THM ) : thm =
  |- !P Q. (?x. P) /\ (?x. Q) = (?x. P /\ Q)
val ( TRIV_FORALL_OR_THM ) : thm = |- !P Q. (!x. P \/ Q) = (!x. P) \/ (!x. Q)
val ( TRIV_OR_FORALL_THM ) : thm = |- !P Q. (!x. P) \/ (!x. Q) = (!x. P \/ Q)
val ( RIGHT_IMP_FORALL_THM ) : thm =
  |- !P Q. P ==> (!x. Q x) = (!x. P ==> Q x)
val ( RIGHT_FORALL_IMP_THM ) : thm =
  |- !P Q. (!x. P ==> Q x) = P ==> (!x. Q x)
val ( LEFT_IMP_EXISTS_THM ) : thm =
  |- !P Q. (?x. P x) ==> Q = (!x. P x ==> Q)
val ( LEFT_FORALL_IMP_THM ) : thm =
  |- !P Q. (!x. P x ==> Q) = (?x. P x) ==> Q
val ( TRIV_FORALL_IMP_THM ) : thm =
  |- !P Q. (!x. P ==> Q) = (?x. P) ==> (!x. Q)
val ( TRIV_EXISTS_IMP_THM ) : thm =
  |- !P Q. (?x. P ==> Q) = (!x. P) ==> (?x. Q)

ind-defs:

RIGHT_BETAS [`x`;`y`] `|- f = \x y. A[x,y]` gives
`|- f x y = A[x,y]`

HALF_BETA_EXPAND [`x`;`y`] `|- f = \x y. A[x,y]` gives
`|- !x y. f x y = A[x,y]`

SIMPLE_DISJ_PAIR `P \/ Q |- R` gives
(`P |- R`, `Q |- R`)

FORALL_IMPS_CONV
Rewrites `!x y. P[x,y] ==> Q` to `(?x y. P[x,y]) ==> Q`

AND_IMPS_CONV
Rewrites:
(*    (!x1..xn. P1[xs] ==> Q[xs]) /\ ... /\ (!x1..xn. Pm[xs] ==> Q[xs])      *)
(* -> (!x1..xn. P1[xs] \/ ... \/ Pm[xs] ==> Q[xs])                           *)

EXISTS_EQUATION `x = a` `ASM,x = a |- P[x]`
gives `ASM |- ?x. P[x]`

val ( MONO_AND ) : thm = |- (A ==> B) /\ (C ==> D) ==> A /\ C ==> B /\ D
val ( MONO_OR ) : thm = |- (A ==> B) /\ (C ==> D) ==> A \/ C ==> B \/ D
val ( MONO_IMP ) : thm = |- (B ==> A) /\ (C ==> D) ==> (A ==> C) ==> B ==> D
val ( MONO_NOT ) : thm = |- (B ==> A) ==> ~A ==> ~B
val ( MONO_FORALL ) : thm = |- (!x. P x ==> Q x) ==> (!x. P x) ==> (!x. Q x)
val ( MONO_EXISTS ) : thm = |- (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)

BACKCHAIN_TAC:
"Simplified version of MATCH_MP_TAC to avoid quantifier troubles."

MONO_ABS_TAC:
(*   ?- (\x. P[x]) x1 .. xn ==> (\y. Q[y]) x1 .. xn                          *)
(* ==================================================                        *)
(*     ?- !x1. P[x1] x2 .. xn ==> Q[x1] x2 .. xn                             *)

mono_tactics:
A global variable holding a set of tactics for proving monotonicity.

APPLY_MONOTAC
Prove `a ==> a` automatically; otherwise, select a tactic from
mono_tactics and apply it.

MONO_STEP_TAC
= REPEAT GEN_TAC THEN APPLY_MONOTAC

MONO_TAC
= REPEAT MONO_STEP_TAC THEN ASM_REWRITE_TAC[]

class:

This is an axiom:
val ( ETA_AX ) : thm = |- !t. (\x. t x) = t

ETA_CONV
Rewrites `(\x. f x)` to `f`

val ( EQ_EXT ) : thm = |- !f g. (!x. f x = g x) ==> (f = g)
val ( FUN_EQ_THM ) : thm = |- !f g. (f = g) = (!x. f x = g x)

EXT `|- (!x. f x = g x)` gives `|- f = x`

This is an axiom:
val ( SELECT_AX ) : thm = |- !P x. P x ==> P ((@) P)

val ( EXISTS_THM ) : thm = |- (?) = (\P. P ((@) P))

SELECT_INTRO `|- P a` gives `|- P ((@) P)`

SELECT_RULE `|- ?x. P[x]` gives `|- P[ @x. P[x] ]`

SELECT_ELIM `|- P ((@) P)` (`x`, `P x |- a`) gives `|- a`

SELECT_CONV rewrites `P[ @x. P[x] ]` to `?x. P[x]`

val ( SELECT_REFL ) : thm = |- !x. (@y. y = x) = x
val ( SELECT_UNIQUE ) : thm = |- !P x. (!y. P y = y = x) ==> ((@) P = x)

SELECT_REFL is a "basic rewrite".

new_specification ["foo";"bar";"baz"] `?x y z.P[x,y,z]`
Defines "foo", "bar", and "baz" as new constants, and returns a theorem
`|- P[foo,bar,baz]`

simple_new_specification th
Like new_specification, but gets the new constant names from the
bound variable names in the theorem.

new_type_definition "newtype" ("newtypeABS","newtypeREP") `|- ?x. P[x]`
Creates a new type "newtype", which is isomorphic to a subset
of the type of the bound variable `x` ("oldtype").  Creates new constants
"newtypeABS", which maps from oldtype to newtype, and "newtypeREP",
which maps from newtype to oldtype.
Returns a theorem
`|- (!(a:newtype). newtypeABS (newtypeREP a) = a) /\
    (!(r:oldtype). P[r] = (newtypeREP (newtypeABS r) = r))`

val ( EXCLUDED_MIDDLE ) : thm = |- !t. t \/ ~t
val ( BOOL_CASES_AX ) : thm = |- !t. (t = T) \/ (t = F)

BOOL_CASES_TAC `a`
Creates two subgoals.  In the first, rewrite `a` to `T` within the goal;
in the second, rewrite `a` to `F`.

ASM_CASES_TAC `a`
Creates two subgoals.  In the first, add `a` as an assumption; in the second,
add `~a` as an assumption.

TAUT tm
Tries to prove the term.  Starts with (REPEAT GEN_TAC), then rewrites with
the basic rewrites and does case splits on free boolean variables
using BOOL_CASES_TAC.

val ( DE_MORGAN_THM ) : thm =
  |- !t1 t2. (~(t1 /\ t2) = ~t1 \/ ~t2) /\ (~(t1 \/ t2) = ~t1 /\ ~t2)
val ( NOT_CLAUSES ) : thm = |- (!t. ~~t = t) /\ (~T = F) /\ (~F = T)
val ( NOT_IMP ) : thm = |- !t1 t2. ~(t1 ==> t2) = t1 /\ ~t2
val ( CONTRAPOS_THM ) : thm = |- !t1 t2. ~t1 ==> ~t2 = t2 ==> t1

NOT_CLAUSES is a "basic rewrite".

CCONTR `a` `~a |- F` gives `|- a`

CONTRAPOS_CONV rewrites `a ==> b` to `~b ==> ~a`

val ( NOT_EXISTS_THM ) : thm = |- !P. ~(?x. P x) = (!x. ~P x)
val ( EXISTS_NOT_THM ) : thm = |- !P. (?x. ~P x) = ~(!x. P x)
val ( NOT_FORALL_THM ) : thm = |- !P. ~(!x. P x) = (?x. ~P x)
val ( FORALL_NOT_THM ) : thm = |- !P. (!x. ~P x) = ~(?x. P x)
val ( LEFT_FORALL_OR_THM ) : thm = |- !P Q. (!x. P x \/ Q) = (!x. P x) \/ Q
val ( RIGHT_FORALL_OR_THM ) : thm = |- !P Q. (!x. P \/ Q x) = P \/ (!x. Q x)
val ( LEFT_OR_FORALL_THM ) : thm = |- !P Q. (!x. P x) \/ Q = (!x. P x \/ Q)
val ( RIGHT_OR_FORALL_THM ) : thm = |- !P Q. P \/ (!x. Q x) = (!x. P \/ Q x)
val ( LEFT_IMP_FORALL_THM ) : thm =
  |- !P Q. (!x. P x) ==> Q = (?x. P x ==> Q)
val ( LEFT_EXISTS_IMP_THM ) : thm =
  |- !P Q. (?x. P x ==> Q) = (!x. P x) ==> Q
val ( RIGHT_IMP_EXISTS_THM ) : thm =
  |- !P Q. P ==> (?x. Q x) = (?x. P ==> Q x)
val ( RIGHT_EXISTS_IMP_THM ) : thm =
  |- !P Q. (?x. P ==> Q x) = P ==> (?x. Q x)

This is the definition of COND, also known as if-then-else.  That is,
if the parser sees "if c then t else e", it transforms it into
"COND c t e".  (Actually, it does the same for "c => t | e".)
val ( COND_DEF ) : thm =
  |- COND = (\t t1 t2. @x. ((t = T) ==> (x = t1)) /\ ((t = F) ==> (x = t2)))

val ( COND_CLAUSES ) : thm =
  |- !t1 t2. ((if T then t1 else t2) = t1) /\ ((if F then t1 else t2) = t2)

COND_CLAUSES is a "basic rewrite".

val ( COND_EXPAND ) : thm =
  |- !b t1 t2. (if b then t1 else t2) = (~b \/ t1) /\ (b \/ t2)
val ( COND_ID ) : thm = |- !b t. (if b then t else t) = t
val ( COND_RAND ) : thm =
  |- !b f x y. f (if b then x else y) = (if b then f x else f y)
val ( COND_RATOR ) : thm =
  |- !b f g x. (if b then f else g) x = (if b then f x else g x)
val ( COND_ABS ) : thm =
  |- !b f g. (\x. if b then f x else g x) = (if b then f else g)
val ( MONO_COND ) : thm =
  |- (A ==> B) /\ (C ==> D) ==> (if b then A else C) ==> (if b then B else D)

MONO_COND will automatically be used by MONO_TAC to prove
monotonicity for COND expressions.

val ( COND_ELIM_THM ) : thm =
  |- P (if c then x else y) = (c ==> P x) /\ (~c ==> P y)

COND_ELIM_CONV rewrites `P[if c then x else y]` to
`(c ==> P[x]) /\ (~c ==> P[y])`

COND_CASES_TAC does a case split on a COND expression in the goal.
If the goal is `P[if c then x else y]`, then in one subgoal,
it adds an assumption `c` and changes the goal to `P[x]` (and then
rewrites `c` to `T` in the goal).
In the other subgoal, it adds an assumption `~c`, changes the goal to
`P[y]`, and rewrites `c` to `F`.  (If `c` has the form `~d`, then instead,
it adds an assumption `d` and rewrites `d` to `T`.)

val ( SKOLEM_THM ) : thm = |- !P. (!x. ?y. P x y) = (?y. !x. P x (y x))
val ( UNIQUE_SKOLEM_ALT ) : thm =
  |- !P. (!x. ?!y. P x y) = (?f. !x y. P x y = f x = y)
val ( UNIQUE_SKOLEM_THM ) : thm =
  |- !P. (!x. ?!y. P x y) = (?!f. !x. P x (f x))

The theorem `|- (if x = x then y else z) = y` is a "basic rewrite".

This is the definition of `(o)`, the function composition operator.
val o_DEF : thm = |- !f g. f o g = (\x. f (g x))

This is the definition of `I`, the identity function.
val ( I_DEF ) : thm = |- I = (\x. x)

val o_THM : thm = |- !f g x. (f o g) x = f (g x)
val o_ASSOC : thm = |- !f g h. f o g o h = (f o g) o h
val ( I_THM ) : thm = |- !x. I x = x
val ( I_O_ID ) : thm = |- !f. (I o f = f) /\ (f o I = f)
val ( EXISTS_ONE_REP ) : thm = |- ?b. b

This defines a new type "one", with one element; and functions that
map between bool and one (where the single element of "one" maps to `T`).
val one_tydef : thm =
  |- (!a. one_ABS (one_REP a) = a) /\ (!r. r = one_REP (one_ABS r) = r)

This is the definition of "one", the name of the sole value of type "one".
val one_DEF : thm = |- one = (@x. T)

val one : thm = |- !v. v = one
val one_axiom : thm = |- !f g. f = g
val one_Axiom : thm = |- !e. ?!fn. fn one = e

canon:

PRESIMP_CONV is a conversion which simplifies literal `T` and `F` as the
arguments of not,and,or,implies,equality, etc.; and moves connectives
out of quantifiers whenever possible.

NNF_CONV and NNFC_CONV move negations inward past and,or,implies,forall,
exists,exists_unique,iff.  These conversions also eliminate iff
and implication.  (There are two possible ways to eliminate
iff; NNF_CONV uses the method which makes DNF smaller, and NNFC_CONV uses
the method which makes CNF smaller.)

SKOLEM_CONV moves existential quantifiers outward past or,and,forall.
(skolemizing when an existential moves past a universal).  Does not handle
existentials inside negations.

PRENEX_CONV moves universal quantifiers outward past or,and.

MINISCOPE_FORALL moves universal quantifiers inward past or,and,forall.
(For instance, `!x y. P x y \/ Q y` would become `!y. (!x. P x y) \/ Q y`.)

PROP_CNF_CONV uses DeMorgan's laws and associativity to move to
conjunctive normal form.

PROP_DNF_CONV uses DeMorgan's laws and associativity to move to
disjunctive normal form.

REFUTE_THEN tht changes the goal from `p` to `F` and runs tht on the
assumption `|- ~p`.

SPLIT_TAC lev
Starts by replacing any assumption `|- a /\ b` with separate assumptions
`|- a` and `|- b`.  Then, up to lev levels deep, it removes an assumption
`|- a \/ b`, splits into two subgoals, adds assumption `|- a` in one
subgoal, and adds assumption `|- b` in the other.

EQ_ABS_CONV rewrites `f = \x y z. A[x,y,z]` to `!x y z. f x y z = A[x,y,z]`.
It never fails.

DELAMB_CONV repeatedly rewrites `(\x. s x) = t` to `!x. s x = t x`
(and similarly if the lambda is on the right) and applies beta reduction
until no more progress can be made.  It never fails.

I'm not documenting GEN_FOL_CONV.
FOL_CONV takes a term and makes sure that for every constant, whenever
it is used, it is always applied to the same number of arguments.  It
does this by finding the minimal number of arguments for each constant;
anywhere it is applied to more arguments, the application
`f x1 ... xn` is changed to `I (f x1 ... x(n-1)) xn` (this process
is repeated, if necessary).

ASM_FOL_TAC
Uses the technique of FOL_CONV, but applies throughout the entire goalstate
(goal and assumptions).

meson:

A general first-order theorem prover.

ASM_MESON_TAC thl
Tries to prove the goal using meson; uses the assumptions and the theorems
in thl.

MESON_TAC thl
Tries to prove the goal using meson; ignores assumptions.

GEN_MESON_TAC min max step thl
Like MESON_TAC but with explicit start (min), finish (max) and step (step) for
the iterative deepening.

quot:

define_quotient_type "newtype" ("newtypeABS","newtypeREP") `\x y. P[x,y]`
Defines a new type "newtype" with a bijection to a subset of "oldtype->bool"
(where "oldtype" is the type of `x`).  This subset is defined by the
predicate `\s. ?x. s = \y. P[x,y]`.  (If `P[x,y]` is symmetric,
reflexive, and transitive, then "newtype" is isomorphic to the
quotient of "oldtype" by P.)
Along with the new type, it also defines new constants "newtypeABS"
and "newtypeREP", and returns theorems:
(`|- newtypeABS (newtypeREP a) = a`,
 `|- (?x. r = (\x y. P[x,y]) x) = newtypeREP (newtypeABS r) = r`)

lift_function:
(* Given a welldefinedness theorem for a curried function f, of the form:    *)
(*                                                                           *)
(* |- !x1 x1' .. xn xn'. (x1 == x1') /\ ... /\ (xn == xn')                   *)
(*                       ==> (f x1 .. xn == f x1' .. f nx')                  *)
(*                                                                           *)
(* where each "==" is either equality or some fixed binary relation R, a     *)
(* new operator called "opname" is introduced which lifts "f" up to the      *)
(* R-equivalence classes. Two theorems are returned: the actual definition   *)
(* and a useful consequence for lifting theorems.                            *)
(*                                                                           *)
(* The function also needs the second (more complicated) type bijection, and *)
(* the reflexivity and transitivity (not symmetry!) of the equivalence       *)
(* relation. The use also gives a name for the new function.                 *)

lift_theorem:
(* Lifts a theorem. This can be done by higher order rewriting alone.        *)
(*                                                                           *)
(* NB! All and only the first order variables must be bound by quantifiers.  *)

recursion.ml:

Functions for defining recursive functions. Mostly subsumed by "define", but
sometimes useful for efficiency or other reasons.

pair.ml:

This is the definition of LET.
val ( LET_DEF ) : thm = |- !f x. LET f x = f x

This is the definition of LET_END.
val ( LET_END_DEF ) : thm = |- !t. LET_END t = t

This is the definition of GABS.
val ( GABS_DEF ) : thm = |- !P. GABS P = (@) P

This is the definition of GEQ.
val ( GEQ_DEF ) : thm = |- !a b. GEQ a b = a = b

LET, LET_END, GABS, and GEQ are syntactic markers used by the
parser and printer, so that you can read in a let expression, or
an extended lambda expression, and print it back out.

The term `let x = 3 and y = 4 in x+y` would be parsed as
`LET (\x y. LET_END (x+y)) 3 4`.

This is the definition of mk_pair.
val mk_pair_def : thm = |- !x y. mk_pair x y = (\a b. (a = x) /\ (b = y))

val ( PAIR_EXISTS_THM ) : thm = |- ?x a b. x = mk_pair a b

This defines a new type (a,b)prod, which is in bijection with a subset
of the type `a->b->bool` (the range of the function mk_pair).
val prod_tybij : thm =
  |- (!a. ABS_prod (REP_prod a) = a) /\
     (!r. (?a b. r = mk_pair a b) = REP_prod (ABS_prod r) = r)

val ( REP_ABS_PAIR ) : thm =
  |- !x y. REP_prod (ABS_prod (mk_pair x y)) = mk_pair x y

This defines the infix "," (comma) operator.
val ( COMMA_DEF ) : thm = |- !x y. x,y = ABS_prod (mk_pair x y)

This defines the FST function.
val ( FST_DEF ) : thm = |- !p. FST p = (@x. ?y. p = x,y)

This defines the SND function.
val ( SND_DEF ) : thm = |- !p. SND p = (@y. ?x. p = x,y)

val ( PAIR_EQ ) : thm = |- !x y a b. (x,y = a,b) = (x = a) /\ (y = b)
val ( PAIR_SURJECTIVE ) : thm = |- !p. ?x y. p = x,y
val ( FST ) : thm = |- !x y. FST (x,y) = x
val ( SND ) : thm = |- !x y. SND (x,y) = y
val ( PAIR ) : thm = |- !x. FST x,SND x = x

FST, SND, and PAIR are "basic rewrites".

This defines the CURRY function.
val ( CURRY_DEF ) : thm = |- !f x y. CURRY f x y = f (x,y)

This defines the UNCURRY function.
val ( UNCURRY_DEF ) : thm = |- !f x y. UNCURRY f (x,y) = f x y

This defines the PASSOC function.
val ( PASSOC_DEF ) : thm = |- !f x y z. PASSOC f (x,y,z) = f ((x,y),z)

Let me describe extended lambda expressions.  The "variable"
for a lambda expression can actually be an arbitrary expression;
`(\P[x,y]. Q[x,y])` is taken to mean:
`@f. !x y. (f (P[x,y])) = Q[x,y]`
This makes perfect sense for pairs; for instance,
`\(x,y).x` is equal to `FST`.  It also makes sense in other situations:
`\(x+&1).x` is equal to `\x.x-&1`.  Other cases are stranger:
`\(x*2).x` is a function which divides even numbers by 2, but its
behavior on odd numbers is undefined.
Extended binders of all sorts are defined by extension.
`!(x*2).P[x]` means `(!) (\(x*2).P[x])`; the value of this expression
may not be determined, since it may depend on the (unspecified) behavior
of the lambda term on odd arguments.
`!(x,y).P[x,y]` is perfectly meaningful, although not necessarily useful;
it means the same thing as `!x y.P[x,y]`.

`let (x,y) = (1,2) in x+y` also has the expected meaning.

I said above that
`(\P[x,y]. Q[x,y])` is taken to mean:
`@f. !x y. (f (P[x,y])) = Q[x,y]`
However, to allow these functions to be printed back out in the same
syntax they were read in, special markers are used. GABS is defined
to equal `(@)`, and GEQ is defined to equal `(=)`, so the internal
form of the above is actually `GABS (\f. !x y. GEQ (f (P[x,y])) Q[x,y])`.

PAIRED_BETA_CONV handles beta reduction of extended lambda expressions
whose arguments are pairs.

GEN_PAIR_TAC converts a goal of the form `!x. P[x]` (where `x` has a
pair type) to `P[(FST x, SND x)]`; but a comment says that rewriting with
FORALL_PAIR_THM (below) is better.

val ( FORALL_PAIR_THM ) : thm = |- (!p. P p) = (!p1 p2. P (p1,p2))
val ( EXISTS_PAIR_THM ) : thm = |- (?p. P p) = (?p1 p2. P (p1,p2))

let_CONV reduces a "let" expression.  (Again, it handles extended
definitions, if they only use pairs.)

LET_TAC
replaces "let x = t in p[x]" in goal with "p[x]" given a new
hypothesis "t = x".

GEN_BETA_CONV reduces generalized beta-redexes such as
`(\(x,y). x + y) (1,2)`

num.ml:

We define a new type, "ind"; we are going to axiomatize that "ind" has
an infinite number of elements.

This is the definition of ONE_ONE.
val ( ONE_ONE ) : thm =
  |- !f. ONE_ONE f = (!x1 x2. (f x1 = f x2) ==> (x1 = x2))

This is the definition of ONTO.
val ( ONTO ) : thm = |- !f. ONTO f = (!y. ?x. y = f x)

This is an axiom!  (Note that `f` has type `ind->ind`.)
val ( INFINITY_AX ) : thm = |- ?f. ONE_ONE f /\ ~ONTO f

val ( IND_SUC_0_EXISTS ) : thm =
  |- ?f z. (!x1 x2. (f x1 = f x2) = x1 = x2) /\ (!x. ~(f x = z))

Now we define constants IND_SUC and IND_0, using new_specification with
IND_SUC_0_EXISTS.  These are the two theorems returned.
val ( IND_SUC_INJ ) : thm = |- !x1 x2. (IND_SUC x1 = IND_SUC x2) = x1 = x2
val ( IND_SUC_0 ) : thm = |- !x. ~(IND_SUC x = IND_0)

Now define the natural number representations as a subset of ind, using the
inductive definition
   `NUM_REP IND_0 /\
    (!i. NUM_REP i ==> NUM_REP (IND_SUC i))`
This returns the following theorems:
val ( NUM_REP_RULES ) : thm =
  |- NUM_REP IND_0 /\ (!i. NUM_REP i ==> NUM_REP (IND_SUC i))
val ( NUM_REP_INDUCT ) : thm =
  |- !NUM_REP'. NUM_REP' IND_0 /\ (!i. NUM_REP' i ==> NUM_REP' (IND_SUC i))
                ==> (!a. NUM_REP a ==> NUM_REP' a)
val ( NUM_REP_CASES ) : thm =
  |- !a. NUM_REP a = (a = IND_0) \/ (?i. (a = IND_SUC i) /\ NUM_REP i)

Now we define the type "num", which has a bijection with the NUM_REP
subset of ind.  This gives the following theorems:
val num_tydef : thm * thm =
  (|- mk_num (dest_num a) = a, |- NUM_REP r = dest_num (mk_num r) = r)

This is the definition of _0.
val ( ZERO_DEF ) : thm = |- _0 = mk_num IND_0

This is the definition of SUC.
val ( SUC_DEF ) : thm = |- !n. SUC n = mk_num (IND_SUC (dest_num n))

val ( NOT_SUC ) : thm = |- !n. ~(SUC n = _0)
val ( SUC_INJ ) : thm = |- !m n. (SUC m = SUC n) = m = n
val num_INDUCTION : thm =
  |- !P. P _0 /\ (!n. P n ==> P (SUC n)) ==> (!n. P n)
val num_Axiom : thm =
  |- !e f. ?!fn. (fn _0 = e) /\ (!n. fn (SUC n) = f (fn n) n)

This is the definition of NUMERAL.
The HOL Light parser reads decimal numbers and turns them internally
into terms starting with NUMERAL; then the printer turns them back
into decimal numbers.
val ( NUMERAL ) : thm = |- !n. NUMERAL n = n

Now we can use the decimal number 0, rather than the symbol _0; we prove
new versions of the above four theorems, replacing _0 with 0.  (We give
them the same names as the old theorems, which are now inaccessible.)

val ( NOT_SUC ) : thm = |- !n. ~(SUC n = 0)
val ( SUC_INJ ) : thm = |- !m n. (SUC m = SUC n) = m = n
val num_INDUCTION : thm = |- !P. P 0 /\ (!n. P n ==> P (SUC n)) ==> (!n. P n)
val num_Axiom : thm =
  |- !e f. ?!fn. (fn 0 = e) /\ (!n. fn (SUC n) = f (fn n) n)

INDUCT_TAC takes a goal `!x. P[x]` and creates subgoals `P[0]` and
`P[SUC n]` (the latter with an assumption `P[n]`).

val num_RECURSION : thm =
  |- !e f. ?fn. (fn 0 = e) /\ (!n. fn (SUC n) = f (fn n) n)

val num_CASES : thm = |- !m. (m = 0) \/ (?n. m = SUC n)

arith.ml:

This is the definition of PRE (predecessor).
val ( PRE ) : thm = |- (PRE 0 = 0) /\ (!n. PRE (SUC n) = n)

This is the definition of (+).
val ( ADD ) : thm = |- (!n. 0 + n = n) /\ (!m n. SUC m + n = SUC (m + n))

val ( ADD_0 ) : thm = |- !m. m + 0 = m
val ( ADD_SUC ) : thm = |- !m n. m + SUC n = SUC (m + n)
val ( ADD_CLAUSES ) : thm =
  |- (!n. 0 + n = n) /\
     (!m. m + 0 = m) /\
     (!m n. SUC m + n = SUC (m + n)) /\
     (!m n. m + SUC n = SUC (m + n))
val ( ADD_SYM ) : thm = |- !m n. m + n = n + m
val ( ADD_ASSOC ) : thm = |- !m n p. m + n + p = (m + n) + p
val ( ADD_AC ) : thm =
  |- (m + n = n + m) /\ ((m + n) + p = m + n + p) /\ (m + n + p = n + m + p)
val ( ADD_EQ_0 ) : thm = |- !m n. (m + n = 0) = (m = 0) /\ (n = 0)
val ( EQ_ADD_LCANCEL ) : thm = |- !m n p. (m + n = m + p) = n = p
val ( EQ_ADD_RCANCEL ) : thm = |- !m n p. (m + p = n + p) = m = n
val ( EQ_ADD_LCANCEL_0 ) : thm = |- !m n. (m + n = m) = n = 0
val ( EQ_ADD_RCANCEL_0 ) : thm = |- !m n. (m + n = n) = m = 0

These are the definitions of BIT0 and BIT1.  The HOL Light parser
parses decimal numbers into terms made of 0, BIT0, and BIT1, surrounded
by NUMERAL.
val ( BIT0 ) : thm = |- !n. BIT0 n = n + n
val ( BIT1 ) : thm = |- !n. BIT1 n = SUC (n + n)

val ( BIT0_THM ) : thm = |- !n. NUMERAL (BIT0 n) = NUMERAL n + NUMERAL n
val ( BIT1_THM ) : thm =
  |- !n. NUMERAL (BIT1 n) = SUC (NUMERAL n + NUMERAL n)
val ( ONE ) : thm = |- 1 = SUC 0
val ( TWO ) : thm = |- 2 = SUC 1
val ( ADD1 ) : thm = |- !m. SUC m = m + 1

This is the definition of (*)
val ( MULT ) : thm = |- (!n. 0 * n = 0) /\ (!m n. SUC m * n = m * n + n)

val ( MULT_0 ) : thm = |- !m. m * 0 = 0
val ( MULT_SUC ) : thm = |- !m n. m * SUC n = m + m * n
val ( MULT_CLAUSES ) : thm =
  |- (!n. 0 * n = 0) /\
     (!m. m * 0 = 0) /\
     (!n. 1 * n = n) /\
     (!m. m * 1 = m) /\
     (!m n. SUC m * n = m * n + n) /\
     (!m n. m * SUC n = m + m * n)
val ( MULT_SYM ) : thm = |- !m n. m * n = n * m
val ( LEFT_ADD_DISTRIB ) : thm = |- !m n p. m * (n + p) = m * n + m * p
val ( RIGHT_ADD_DISTRIB ) : thm = |- !m n p. (m + n) * p = m * p + n * p
val ( MULT_ASSOC ) : thm = |- !m n p. m * n * p = (m * n) * p
val ( MULT_AC ) : thm =
  |- (m * n = n * m) /\ ((m * n) * p = m * n * p) /\ (m * n * p = n * m * p)
val ( MULT_EQ_0 ) : thm = |- !m n. (m * n = 0) = (m = 0) \/ (n = 0)
val ( EQ_MULT_LCANCEL ) : thm =
  |- !m n p. (m * n = m * p) = (m = 0) \/ (n = p)
val ( EQ_MULT_RCANCEL ) : thm =
  |- !m n p. (m * p = n * p) = (m = n) \/ (p = 0)
val ( MULT_2 ) : thm = |- !n. 2 * n = n + n
val ( MULT_EQ_1 ) : thm = |- !m n. (m * n = 1) = (m = 1) /\ (n = 1)

This is the definition of EXP (exponentiation).
val ( EXP ) : thm = |- (!m. m EXP 0 = 1) /\ (!m n. m EXP SUC n = m * m EXP n)

val ( EXP_EQ_0 ) : thm = |- !m n. (m EXP n = 0) = (m = 0) /\ ~(n = 0)
val ( EXP_ADD ) : thm = |- !m n p. m EXP (n + p) = m EXP n * m EXP p
val ( EXP_ONE ) : thm = |- !n. 1 EXP n = 1
val ( EXP_1 ) : thm = |- !n. n EXP 1 = n
val ( EXP_2 ) : thm = |- !n. n EXP 2 = n * n
val ( MULT_EXP ) : thm = |- !p m n. (m * n) EXP p = m EXP p * n EXP p
val ( EXP_MULT ) : thm = |- !m n p. m EXP (n * p) = m EXP n EXP p

This is the definition of (<=)
val ( LE ) : thm =
  |- (!m. m <= 0 = m = 0) /\ (!m n. m <= SUC n = (m = SUC n) \/ m <= n)

This is the definition of (<)
val ( LT ) : thm = |- (!m. m < 0 = F) /\ (!m n. m < SUC n = (m = n) \/ m < n)

This is the definition of (>=)
val ( GE ) : thm = |- !n m. m >= n = n <= m

This is the definition of (>)
val ( GT ) : thm = |- !n m. m > n = n < m

val ( LE_SUC_LT ) : thm = |- !m n. SUC m <= n = m < n
val ( LT_SUC_LE ) : thm = |- !m n. m < SUC n = m <= n
val ( LE_SUC ) : thm = |- !m n. SUC m <= SUC n = m <= n
val ( LT_SUC ) : thm = |- !m n. SUC m < SUC n = m < n
val ( LE_0 ) : thm = |- !n. 0 <= n
val ( LT_0 ) : thm = |- !n. 0 < SUC n
val ( LE_REFL ) : thm = |- !n. n <= n
val ( LT_REFL ) : thm = |- !n. ~(n < n)
val ( LE_ANTISYM ) : thm = |- !m n. m <= n /\ n <= m = m = n
val ( LT_ANTISYM ) : thm = |- !m n. ~(m < n /\ n < m)
val ( LET_ANTISYM ) : thm = |- !m n. ~(m <= n /\ n < m)
val ( LTE_ANTISYM ) : thm = |- !m n. ~(m < n /\ n <= m)
val ( LE_TRANS ) : thm = |- !m n p. m <= n /\ n <= p ==> m <= p
val ( LT_TRANS ) : thm = |- !m n p. m < n /\ n < p ==> m < p
val ( LET_TRANS ) : thm = |- !m n p. m <= n /\ n < p ==> m < p
val ( LTE_TRANS ) : thm = |- !m n p. m < n /\ n <= p ==> m < p
val ( LE_CASES ) : thm = |- !m n. m <= n \/ n <= m
val ( LT_CASES ) : thm = |- !m n. m < n \/ n < m \/ (m = n)
val ( LET_CASES ) : thm = |- !m n. m <= n \/ n < m
val ( LTE_CASES ) : thm = |- !m n. m < n \/ n <= m
val ( LE_LT ) : thm = |- !m n. m <= n = m < n \/ (m = n)
val ( LT_LE ) : thm = |- !m n. m < n = m <= n /\ ~(m = n)
val ( NOT_LE ) : thm = |- !m n. ~(m <= n) = n < m
val ( NOT_LT ) : thm = |- !m n. ~(m < n) = n <= m
val ( LT_IMP_LE ) : thm = |- !m n. m < n ==> m <= n
val ( EQ_IMP_LE ) : thm = |- !m n. (m = n) ==> m <= n
val ( LE_EXISTS ) : thm = |- !m n. m <= n = (?d. n = m + d)
val ( LT_EXISTS ) : thm = |- !m n. m < n = (?d. n = m + SUC d)
val ( LE_ADD ) : thm = |- !m n. m <= m + n
val ( LE_ADDR ) : thm = |- !m n. n <= m + n
val ( LT_ADD ) : thm = |- !m n. m < m + n = 0 < n
val ( LT_ADDR ) : thm = |- !m n. n < m + n = 0 < m
val ( LE_ADD_LCANCEL ) : thm = |- !m n p. m + n <= m + p = n <= p
val ( LE_ADD_RCANCEL ) : thm = |- !m n p. m + p <= n + p = m <= n
val ( LT_ADD_LCANCEL ) : thm = |- !m n p. m + n < m + p = n < p
val ( LT_ADD_RCANCEL ) : thm = |- !m n p. m + p < n + p = m < n
val ( LE_ADD2 ) : thm = |- !m n p q. m <= p /\ n <= q ==> m + n <= p + q
val ( LET_ADD2 ) : thm = |- !m n p q. m <= p /\ n < q ==> m + n < p + q
val ( LTE_ADD2 ) : thm = |- !m n p q. m < p /\ n <= q ==> m + n < p + q
val ( LT_ADD2 ) : thm = |- !m n p q. m < p /\ n < q ==> m + n < p + q
val ( LT_MULT ) : thm = |- !m n. 0 < m * n = 0 < m /\ 0 < n
val ( LE_MULT2 ) : thm = |- !m n p q. m <= n /\ p <= q ==> m * p <= n * q
val ( LT_LMULT ) : thm = |- !m n p. ~(m = 0) /\ n < p ==> m * n < m * p
val ( LE_MULT_LCANCEL ) : thm = |- !m n p. m * n <= m * p = (m = 0) \/ n <= p
val ( LE_MULT_RCANCEL ) : thm = |- !m n p. m * p <= n * p = m <= n \/ (p = 0)
val ( LT_MULT_LCANCEL ) : thm = |- !m n p. m * n < m * p = ~(m = 0) /\ n < p
val ( LT_MULT_RCANCEL ) : thm = |- !m n p. m * p < n * p = m < n /\ ~(p = 0)
val ( EQ_SUC ) : thm = |- !m n. (SUC m = SUC n) = m = n
val ( LT_MULT2 ) : thm = |- !m n p q. m < n /\ p < q ==> m * p < n * q
val ( LE_SQUARE_REFL ) : thm = |- !n. n <= n * n
val num_WF : thm = |- !P. (!n. (!m. m < n ==> P m) ==> P n) ==> (!n. P n)
val num_WOP : thm = |- !P. (?n. P n) = (?n. P n /\ (!m. m < n ==> ~P m))
val num_MAX : thm =
  |- !P. (?x. P x) /\ (?M. !x. P x ==> x <= M) =
         (?m. P m /\ (!x. P x ==> x <= m))

These are the definitions of EVEN and ODD.
val ( EVEN ) : thm = |- (EVEN 0 = T) /\ (!n. EVEN (SUC n) = ~EVEN n)
val ( ODD ) : thm = |- (ODD 0 = F) /\ (!n. ODD (SUC n) = ~ODD n)

val ( NOT_EVEN ) : thm = |- !n. ~EVEN n = ODD n
val ( NOT_ODD ) : thm = |- !n. ~ODD n = EVEN n
val ( EVEN_OR_ODD ) : thm = |- !n. EVEN n \/ ODD n
val ( EVEN_AND_ODD ) : thm = |- !n. ~(EVEN n /\ ODD n)
val ( EVEN_ADD ) : thm = |- !m n. EVEN (m + n) = EVEN m = EVEN n
val ( EVEN_MULT ) : thm = |- !m n. EVEN (m * n) = EVEN m \/ EVEN n
val ( EVEN_EXP ) : thm = |- !m n. EVEN (m EXP n) = EVEN m /\ ~(n = 0)
val ( ODD_ADD ) : thm = |- !m n. ODD (m + n) = ~(ODD m = ODD n)
val ( ODD_MULT ) : thm = |- !m n. ODD (m * n) = ODD m /\ ODD n
val ( ODD_EXP ) : thm = |- !m n. ODD (m EXP n) = ODD m \/ (n = 0)
val ( EVEN_DOUBLE ) : thm = |- !n. EVEN (2 * n)
val ( ODD_DOUBLE ) : thm = |- !n. ODD (SUC (2 * n))
val ( EVEN_EXISTS_LEMMA ) : thm =
  |- !n. (EVEN n ==> (?m. n = 2 * m)) /\ (~EVEN n ==> (?m. n = SUC (2 * m)))
val ( EVEN_EXISTS ) : thm = |- !n. EVEN n = (?m. n = 2 * m)
val ( ODD_EXISTS ) : thm = |- !n. ODD n = (?m. n = SUC (2 * m))

This is the definition of (-).  Note that according to this
definition, if a<b, then a-b = 0.
val ( SUB ) : thm = |- (!m. m - 0 = m) /\ (!m n. m - SUC n = PRE (m - n))

val ( SUB_0 ) : thm = |- !m. (0 - m = 0) /\ (m - 0 = m)
val ( SUB_PRESUC ) : thm = |- !m n. PRE (SUC m - n) = m - n
val ( SUB_SUC ) : thm = |- !m n. SUC m - SUC n = m - n
val ( SUB_REFL ) : thm = |- !n. n - n = 0
val ( ADD_SUB ) : thm = |- !m n. (m + n) - n = m
val ( ADD_SUB2 ) : thm = |- !m n. (m + n) - m = n
val ( SUB_EQ_0 ) : thm = |- !m n. (m - n = 0) = m <= n
val ( ADD_SUBR2 ) : thm = |- !m n. m - (m + n) = 0
val ( ADD_SUBR ) : thm = |- !m n. n - (m + n) = 0
val ( SUB_ADD ) : thm = |- !m n. n <= m ==> (m - n + n = m)
val ( SUB_ADD_LCANCEL ) : thm = |- !m n p. (m + n) - (m + p) = n - p
val ( SUB_ADD_RCANCEL ) : thm = |- !m n p. (m + p) - (n + p) = m - n
val ( LEFT_SUB_DISTRIB ) : thm = |- !m n p. m * (n - p) = m * n - m * p
val ( RIGHT_SUB_DISTRIB ) : thm = |- !m n p. (m - n) * p = m * p - n * p

This is the definition of FACT (factorial).
val ( FACT ) : thm = |- (FACT 0 = 1) /\ (!n. FACT (SUC n) = SUC n * FACT n)

val ( FACT_LT ) : thm = |- !n. 0 < FACT n
val ( FACT_LE ) : thm = |- !n. 1 <= FACT n
val ( FACT_MONO ) : thm = |- !m n. m <= n ==> FACT m <= FACT n
val ( DIVMOD_EXIST ) : thm =
  |- !m n. ~(n = 0) ==> (?q r. (m = q * n + r) /\ r < n)

This is the simultaneous definition, using new_specification, of DIV
and MOD.
val ( DIVISION ) : thm =
  |- !m n. ~(n = 0) ==> (m = m DIV n * n + m MOD n) /\ m MOD n < n

val ( DIVMOD_UNIQ_LEMMA ) : thm =
  |- !m n q1 r1 q2 r2.
         ((m = q1 * n + r1) /\ r1 < n) /\ (m = q2 * n + r2) /\ r2 < n
         ==> (q1 = q2) /\ (r1 = r2)
val ( DIVMOD_UNIQ ) : thm =
  |- !m n q r. (m = q * n + r) /\ r < n ==> (m DIV n = q) /\ (m MOD n = r)
val ( MOD_UNIQ ) : thm =
  |- !m n q r. (m = q * n + r) /\ r < n ==> (m MOD n = r)
val ( DIV_UNIQ ) : thm =
  |- !m n q r. (m = q * n + r) /\ r < n ==> (m DIV n = q)
val ( MOD_MULT ) : thm = |- !m n. ~(m = 0) ==> ((m * n) MOD m = 0)
val ( DIV_MULT ) : thm = |- !m n. ~(m = 0) ==> ((m * n) DIV m = n)
val ( DIV_DIV ) : thm =
  |- !m n p. ~(n * p = 0) ==> (m DIV n DIV p = m DIV (n * p))
val ( MOD_LT ) : thm = |- !m n. m < n ==> (m MOD n = m)
val ( MOD_EQ ) : thm = |- !m n p q. (m = n + q * p) ==> (m MOD p = n MOD p)
val ( DIV_MOD ) : thm =
  |- !m n p. ~(n * p = 0) ==> ((m DIV n) MOD p = (m MOD (n * p)) DIV n)
val ( DIV_1 ) : thm = |- !n. n DIV 1 = n
val ( EXP_LT_0 ) : thm = |- !n x. 0 < x EXP n = ~(x = 0) \/ (n = 0)
val ( DIV_LE ) : thm = |- !m n. ~(n = 0) ==> m DIV n <= m
val ( DIV_MUL_LE ) : thm = |- !m n. n * m DIV n <= m
val ( DIV_0 ) : thm = |- !n. ~(n = 0) ==> (0 DIV n = 0)
val ( MOD_0 ) : thm = |- !n. ~(n = 0) ==> (0 MOD n = 0)
val ( DIV_LT ) : thm = |- !m n. m < n ==> (m DIV n = 0)
val ( MOD_MOD ) : thm =
  |- !m n p. ~(n * p = 0) ==> (m MOD (n * p) MOD n = m MOD n)
val ( MOD_MOD_REFL ) : thm = |- !m n. ~(n = 0) ==> (m MOD n MOD n = m MOD n)
val ( DIV_MULT2 ) : thm =
  |- !m n p. ~(m * p = 0) ==> ((m * n) DIV (m * p) = n DIV p)
val ( MOD_MULT2 ) : thm =
  |- !m n p. ~(m * p = 0) ==> ((m * n) MOD (m * p) = m * n MOD p)
val ( MOD_1 ) : thm = |- !n. n MOD 1 = 0
val ( MOD_EXISTS ) : thm =
  |- !m n. (?q. m = n * q) = (if n = 0 then m = 0 else m MOD n = 0)
val ( LT_EXP ) : thm =
  |- !x m n.
         x EXP m < x EXP n =
         2 <= x /\ m < n \/ (x = 0) /\ ~(m = 0) /\ (n = 0)
val ( LE_EXP ) : thm =
  |- !x m n.
         x EXP m <= x EXP n =
         (if x = 0 then (m = 0) ==> (n = 0) else (x = 1) \/ m <= n)
val ( DIV_MONO ) : thm = |- !m n p. ~(p = 0) /\ m <= n ==> m DIV p <= n DIV p
val ( DIV_MONO_LT ) : thm =
  |- !m n p. ~(p = 0) /\ m + p <= n ==> m DIV p < n DIV p
val ( LE_LDIV ) : thm = |- !a b n. ~(a = 0) /\ b <= a * n ==> b DIV a <= n
val ( LE_RDIV_EQ ) : thm =
  |- !a b n. ~(a = 0) ==> (n <= b DIV a = a * n <= b)
val ( LE_LDIV_EQ ) : thm =
  |- !a b n. ~(a = 0) ==> (b DIV a <= n = b < a * (n + 1))
val ( DIV_EQ_0 ) : thm = |- !m n. ~(n = 0) ==> ((m DIV n = 0) = m < n)
val ( MOD_EQ_0 ) : thm =
  |- !m n. ~(n = 0) ==> ((m MOD n = 0) = (?q. m = q * n))
val ( EVEN_MOD ) : thm = |- !n. EVEN n = n MOD 2 = 0
val ( ODD_MOD ) : thm = |- !n. ODD n = n MOD 2 = 1
val ( MOD_MULT_RMOD ) : thm =
  |- !m n p. ~(n = 0) ==> ((m * p MOD n) MOD n = (m * p) MOD n)
val ( MOD_MULT_LMOD ) : thm =
  |- !m n p. ~(n = 0) ==> ((m MOD n * p) MOD n = (m * p) MOD n)
val ( MOD_MULT_MOD2 ) : thm =
  |- !m n p. ~(n = 0) ==> ((m MOD n * p MOD n) MOD n = (m * p) MOD n)
val ( MOD_EXP_MOD ) : thm =
  |- !m n p. ~(n = 0) ==> ((m MOD n) EXP p MOD n = m EXP p MOD n)
val ( MOD_MULT_ADD ) : thm = |- !m n p. (m * n + p) MOD n = p MOD n
val ( MOD_ADD_MOD ) : thm =
  |- !a b n. ~(n = 0) ==> ((a MOD n + b MOD n) MOD n = (a + b) MOD n)
val ( DIV_ADD_MOD ) : thm =
  |- !a b n.
         ~(n = 0)
         ==> (((a + b) MOD n = a MOD n + b MOD n) =
              (a + b) DIV n =
              a DIV n + b DIV n)
val ( DIV_REFL ) : thm = |- !n. ~(n = 0) ==> (n DIV n = 1)
val ( MOD_LE ) : thm = |- !m n. ~(n = 0) ==> m MOD n <= m
val ( DIV_MONO2 ) : thm =
  |- !m n p. ~(p = 0) /\ p <= m ==> n DIV m <= n DIV p
val ( SUB_ELIM_THM ) : thm =
  |- P (a - b) = (!d. ((b = a + d) ==> P 0) /\ ((a = b + d) ==> P d))

SUB_ELIM_CONV rewrites with SUB_ELIM_THM (directly above).

SUB_ELIM_TAC takes a goal `P[a-b]` and creates two subgoals.  In the
first, the goal is changed to `P[0]`, `b` is rewritten to `a+d`
throughout the goal state, and an assumption `b = a+d` is added.
In the second, the goal is changed to `P[d]`, `a` is rewritten
to `b+d` throughout the goal state, and an assumption `a=b+d` is added.

val ( PRE_ELIM_THM ) : thm =
  |- P (PRE n) = (!m. ((n = 0) ==> P 0) /\ ((n = SUC m) ==> P m))

PRE_ELIM_CONV rewrites with PRE_ELIM_THM (directly above).

PRE_ELIM_TAC takes a goal `P[PRE n]` and creates two subgoals.  In the
first, the goal is changed to `P[0]`, `n` is rewritten to `0` throughout
the goal state, and an assumption `n=0` is added.  In the second, the
goal is changed to `P[m]`, `n` is rewritten to `SUC m` throughout
the goal state, and an assumption `n = SUC m` is added.

val ( DIVMOD_ELIM_THM ) : thm =
  |- ~(n = 0)
     ==> (P (m DIV n) (m MOD n) = (!q r. (m = q * n + r) /\ r < n ==> P q r))

DIVMOD_ELIM_TAC finds either a subterm `m DIV n` or `m MOD n` in
the current goal.  It uses these to view the goal as `P[m DIV n,m MOD n]`.
It then changes the goal to
`~(n = 0) /\
  ((m = q*n + r) /\ r < n ==> P[q;r]`

NUM_CANCEL_CONV rewrites `(c+(b+a)) = (b+d)+a` to `c=d`

LE_IMP `|- a <= b` gives (GEN_ALL `|- b <= p ==> a <= p`)

wf.ml:

This is the definition of WF.
val ( WF ) : thm =
  |- !(<<). WF (<<) = (!P. (?x. P x) ==> (?x. P x /\ (!y. y << x ==> ~P y)))

val ( WF_EQ ) : thm =
  |- WF (<<) = (!P. (?x. P x) = (?x. P x /\ (!y. y << x ==> ~P y)))
val ( WF_IND ) : thm =
  |- WF (<<) = (!P. (!x. (!y. y << x ==> P y) ==> P x) ==> (!x. P x))
val ( WF_DCHAIN ) : thm = |- WF (<<) = ~(?s. !n. s (SUC n) << s n)
val ( WF_UREC ) : thm =
  |- WF (<<)
     ==> (!H. (!f g x. (!z. z << x ==> (f z = g z)) ==> (H f x = H g x))
              ==> (!f g. (!x. f x = H f x) /\ (!x. g x = H g x) ==> (f = g)))
val ( WF_UREC_WF ) : thm =
  |- (!H. (!f g x. (!z. z << x ==> (f z = g z)) ==> (H f x = H g x))
          ==> (!f g. (!x. f x = H f x) /\ (!x. g x = H g x) ==> (f = g)))
     ==> WF (<<)
val ( WF_REC ) : thm =
  |- WF (<<)
     ==> (!H. (!f g x. (!z. z << x ==> (f z = g z)) ==> (H f x = H g x))
              ==> (?f. !x. f x = H f x))
val ( WF_REC_WF ) : thm =
  |- (!H. (!f g x. (!z. z << x ==> (f z = g z)) ==> (H f x = H g x))
          ==> (?f. !x. f x = H f x))
     ==> WF (<<)
val ( WF_EREC ) : thm =
  |- WF (<<)
     ==> (!H. (!f g x. (!z. z << x ==> (f z = g z)) ==> (H f x = H g x))
              ==> (?!f. !x. f x = H f x))
val ( WF_SUBSET ) : thm =
  |- (!x y. x << y ==> x <<< y) /\ WF (<<<) ==> WF (<<)
val ( WF_MEASURE_GEN ) : thm = |- !m. WF (<<) ==> WF (\x x'. m x << m x')
val ( WF_LEX_DEPENDENT ) : thm =
  |- !R S.
         WF R /\ (!a. WF (S a))
         ==> WF (\(r1,s1). \(r2,s2). R r1 r2 \/ (r1 = r2) /\ S r1 s1 s2)
val ( WF_LEX ) : thm =
  |- !R S.
         WF R /\ WF S
         ==> WF (\(r1,s1). \(r2,s2). R r1 r2 \/ (r1 = r2) /\ S s1 s2)
val ( WF_POINTWISE ) : thm =
  |- WF (<<) /\ WF (<<<) ==> WF (\(x1,y1). \(x2,y2). x1 << x2 /\ y1 <<< y2)
val ( WF_num ) : thm = |- WF (<)
val ( WF_REC_num ) : thm =
  |- !H. (!f g n. (!m. m < n ==> (f m = g m)) ==> (H f n = H g n))
         ==> (?f. !n. f n = H f n)

This is the definition of measure.
val measure : thm = |- !m. measure m = (\x y. m x < m y)

val ( WF_MEASURE ) : thm = |- !m. WF (measure m)
val ( WF_REFL ) : thm = |- !x. WF (<<) ==> ~(x << x)
val ( WF_REC_TAIL ) : thm =
  |- !P g h. ?f. !x. f x = (if P x then f (g x) else h x)

WF_INDUCT_THEN
WF_INDUCT_TAC
Perform wellfounded induction over a nominated measure function. Sometimes
avoids explicit "!n t. size(t) = n ==> ..." goal.

calc_num.ml:

mangle takes a theorem and rewrites its conclusion to elide NUMERAL.

val ( ARITH_ZERO ) : thm = |- (NUMERAL 0 = 0) /\ (BIT0 _0 = _0)
val ( ARITH_SUC ) : thm =
  |- (!n. SUC (NUMERAL n) = NUMERAL (SUC n)) /\
     (SUC _0 = BIT1 _0) /\
     (!n. SUC (BIT0 n) = BIT1 n) /\
     (!n. SUC (BIT1 n) = BIT0 (SUC n))
val ( ARITH_PRE ) : thm =
  |- (!n. PRE (NUMERAL n) = NUMERAL (PRE n)) /\
     (PRE _0 = _0) /\
     (!n. PRE (BIT0 n) = (if n = _0 then _0 else BIT1 (PRE n))) /\
     (!n. PRE (BIT1 n) = BIT0 n)
val ( ARITH_ADD ) : thm =
  |- (!m n. NUMERAL m + NUMERAL n = NUMERAL (m + n)) /\
     (_0 + _0 = _0) /\
     (!n. _0 + BIT0 n = BIT0 n) /\
     (!n. _0 + BIT1 n = BIT1 n) /\
     (!n. BIT0 n + _0 = BIT0 n) /\
     (!n. BIT1 n + _0 = BIT1 n) /\
     (!m n. BIT0 m + BIT0 n = BIT0 (m + n)) /\
     (!m n. BIT0 m + BIT1 n = BIT1 (m + n)) /\
     (!m n. BIT1 m + BIT0 n = BIT1 (m + n)) /\
     (!m n. BIT1 m + BIT1 n = BIT0 (SUC (m + n)))
val ( ARITH_MULT ) : thm =
  |- (!m n. NUMERAL m * NUMERAL n = NUMERAL (m * n)) /\
     (_0 * _0 = _0) /\
     (!n. _0 * BIT0 n = _0) /\
     (!n. _0 * BIT1 n = _0) /\
     (!n. BIT0 n * _0 = _0) /\
     (!n. BIT1 n * _0 = _0) /\
     (!m n. BIT0 m * BIT0 n = BIT0 (BIT0 (m * n))) /\
     (!m n. BIT0 m * BIT1 n = BIT0 m + BIT0 (BIT0 (m * n))) /\
     (!m n. BIT1 m * BIT0 n = BIT0 n + BIT0 (BIT0 (m * n))) /\
     (!m n. BIT1 m * BIT1 n = BIT1 m + BIT0 n + BIT0 (BIT0 (m * n)))
val ( ARITH_EXP ) : thm =
  |- (!m n. NUMERAL m EXP NUMERAL n = NUMERAL (m EXP n)) /\
     (_0 EXP _0 = BIT1 _0) /\
     (!m. BIT0 m EXP _0 = BIT1 _0) /\
     (!m. BIT1 m EXP _0 = BIT1 _0) /\
     (!n. _0 EXP BIT0 n = _0 EXP n * _0 EXP n) /\
     (!m n. BIT0 m EXP BIT0 n = BIT0 m EXP n * BIT0 m EXP n) /\
     (!m n. BIT1 m EXP BIT0 n = BIT1 m EXP n * BIT1 m EXP n) /\
     (!n. _0 EXP BIT1 n = _0) /\
     (!m n. BIT0 m EXP BIT1 n = BIT0 m * BIT0 m EXP n * BIT0 m EXP n) /\
     (!m n. BIT1 m EXP BIT1 n = BIT1 m * BIT1 m EXP n * BIT1 m EXP n)
val ( ARITH_EVEN ) : thm =
  |- (!n. EVEN (NUMERAL n) = EVEN n) /\
     (EVEN _0 = T) /\
     (!n. EVEN (BIT0 n) = T) /\
     (!n. EVEN (BIT1 n) = F)
val ( ARITH_ODD ) : thm =
  |- (!n. ODD (NUMERAL n) = ODD n) /\
     (ODD _0 = F) /\
     (!n. ODD (BIT0 n) = F) /\
     (!n. ODD (BIT1 n) = T)
val ( ARITH_LE ) : thm =
  |- (!m n. NUMERAL m <= NUMERAL n = m <= n) /\
     (_0 <= _0 = T) /\
     (!n. BIT0 n <= _0 = n = _0) /\
     (!n. BIT1 n <= _0 = F) /\
     (!n. _0 <= BIT0 n = T) /\
     (!n. _0 <= BIT1 n = T) /\
     (!m n. BIT0 m <= BIT0 n = m <= n) /\
     (!m n. BIT0 m <= BIT1 n = m <= n) /\
     (!m n. BIT1 m <= BIT0 n = m < n) /\
     (!m n. BIT1 m <= BIT1 n = m <= n)
val ( ARITH_LT ) : thm =
  |- (!m n. NUMERAL m < NUMERAL n = m < n) /\
     (_0 < _0 = F) /\
     (!n. BIT0 n < _0 = F) /\
     (!n. BIT1 n < _0 = F) /\
     (!n. _0 < BIT0 n = _0 < n) /\
     (!n. _0 < BIT1 n = T) /\
     (!m n. BIT0 m < BIT0 n = m < n) /\
     (!m n. BIT0 m < BIT1 n = m <= n) /\
     (!m n. BIT1 m < BIT0 n = m < n) /\
     (!m n. BIT1 m < BIT1 n = m < n)
val ( ARITH_GE ) : thm =
  |- (!m n. NUMERAL n >= NUMERAL m = n >= m) /\
     _0 >= _0 /\
     (!n. _0 >= BIT0 n = n = _0) /\
     (!n. ~(_0 >= BIT1 n)) /\
     (!n. BIT0 n >= _0) /\
     (!n. BIT1 n >= _0) /\
     (!m n. BIT0 n >= BIT0 m = n >= m) /\
     (!m n. BIT1 n >= BIT0 m = n >= m) /\
     (!m n. BIT0 n >= BIT1 m = n > m) /\
     (!m n. BIT1 n >= BIT1 m = n >= m)
val ( ARITH_GT ) : thm =
  |- (!m n. NUMERAL n > NUMERAL m = n > m) /\
     ~(_0 > _0) /\
     (!n. ~(_0 > BIT0 n)) /\
     (!n. ~(_0 > BIT1 n)) /\
     (!n. BIT0 n > _0 = n > _0) /\
     (!n. BIT1 n > _0) /\
     (!m n. BIT0 n > BIT0 m = n > m) /\
     (!m n. BIT1 n > BIT0 m = n >= m) /\
     (!m n. BIT0 n > BIT1 m = n > m) /\
     (!m n. BIT1 n > BIT1 m = n > m)
val ( ARITH_EQ ) : thm =
  |- (!m n. (NUMERAL m = NUMERAL n) = m = n) /\
     ((_0 = _0) = T) /\
     (!n. (BIT0 n = _0) = n = _0) /\
     (!n. (BIT1 n = _0) = F) /\
     (!n. (_0 = BIT0 n) = _0 = n) /\
     (!n. (_0 = BIT1 n) = F) /\
     (!m n. (BIT0 m = BIT0 n) = m = n) /\
     (!m n. (BIT0 m = BIT1 n) = F) /\
     (!m n. (BIT1 m = BIT0 n) = F) /\
     (!m n. (BIT1 m = BIT1 n) = m = n)
val ( ARITH_SUB ) : thm =
  |- (!m n. NUMERAL m - NUMERAL n = NUMERAL (m - n)) /\
     (_0 - _0 = _0) /\
     (!n. _0 - BIT0 n = _0) /\
     (!n. _0 - BIT1 n = _0) /\
     (!n. BIT0 n - _0 = BIT0 n) /\
     (!n. BIT1 n - _0 = BIT1 n) /\
     (!m n. BIT0 m - BIT0 n = BIT0 (m - n)) /\
     (!m n. BIT0 m - BIT1 n = PRE (BIT0 (m - n))) /\
     (!m n. BIT1 m - BIT0 n = (if n <= m then BIT1 (m - n) else _0)) /\
     (!m n. BIT1 m - BIT1 n = BIT0 (m - n))
val ( ARITH ) : thm =
  |- ((NUMERAL 0 = 0) /\ (BIT0 _0 = _0)) /\
     ((!n. SUC (NUMERAL n) = NUMERAL (SUC n)) /\
      (SUC _0 = BIT1 _0) /\
      (!n. SUC (BIT0 n) = BIT1 n) /\
      (!n. SUC (BIT1 n) = BIT0 (SUC n))) /\
     ((!n. PRE (NUMERAL n) = NUMERAL (PRE n)) /\
      (PRE _0 = _0) /\
      (!n. PRE (BIT0 n) = (if n = _0 then _0 else BIT1 (PRE n))) /\
      (!n. PRE (BIT1 n) = BIT0 n)) /\
     ((!m n. NUMERAL m + NUMERAL n = NUMERAL (m + n)) /\
      (_0 + _0 = _0) /\
      (!n. _0 + BIT0 n = BIT0 n) /\
      (!n. _0 + BIT1 n = BIT1 n) /\
      (!n. BIT0 n + _0 = BIT0 n) /\
      (!n. BIT1 n + _0 = BIT1 n) /\
      (!m n. BIT0 m + BIT0 n = BIT0 (m + n)) /\
      (!m n. BIT0 m + BIT1 n = BIT1 (m + n)) /\
      (!m n. BIT1 m + BIT0 n = BIT1 (m + n)) /\
      (!m n. BIT1 m + BIT1 n = BIT0 (SUC (m + n)))) /\
     ((!m n. NUMERAL m * NUMERAL n = NUMERAL (m * n)) /\
      (_0 * _0 = _0) /\
      (!n. _0 * BIT0 n = _0) /\
      (!n. _0 * BIT1 n = _0) /\
      (!n. BIT0 n * _0 = _0) /\
      (!n. BIT1 n * _0 = _0) /\
      (!m n. BIT0 m * BIT0 n = BIT0 (BIT0 (m * n))) /\
      (!m n. BIT0 m * BIT1 n = BIT0 m + BIT0 (BIT0 (m * n))) /\
      (!m n. BIT1 m * BIT0 n = BIT0 n + BIT0 (BIT0 (m * n))) /\
      (!m n. BIT1 m * BIT1 n = BIT1 m + BIT0 n + BIT0 (BIT0 (m * n)))) /\
     ((!m n. NUMERAL m EXP NUMERAL n = NUMERAL (m EXP n)) /\
      (_0 EXP _0 = BIT1 _0) /\
      (!m. BIT0 m EXP _0 = BIT1 _0) /\
      (!m. BIT1 m EXP _0 = BIT1 _0) /\
      (!n. _0 EXP BIT0 n = _0 EXP n * _0 EXP n) /\
      (!m n. BIT0 m EXP BIT0 n = BIT0 m EXP n * BIT0 m EXP n) /\
      (!m n. BIT1 m EXP BIT0 n = BIT1 m EXP n * BIT1 m EXP n) /\
      (!n. _0 EXP BIT1 n = _0) /\
      (!m n. BIT0 m EXP BIT1 n = BIT0 m * BIT0 m EXP n * BIT0 m EXP n) /\
      (!m n. BIT1 m EXP BIT1 n = BIT1 m * BIT1 m EXP n * BIT1 m EXP n)) /\
     ((!n. EVEN (NUMERAL n) = EVEN n) /\
      (EVEN _0 = T) /\
      (!n. EVEN (BIT0 n) = T) /\
      (!n. EVEN (BIT1 n) = F)) /\
     ((!n. ODD (NUMERAL n) = ODD n) /\
      (ODD _0 = F) /\
      (!n. ODD (BIT0 n) = F) /\
      (!n. ODD (BIT1 n) = T)) /\
     ((!m n. (NUMERAL m = NUMERAL n) = m = n) /\
      ((_0 = _0) = T) /\
      (!n. (BIT0 n = _0) = n = _0) /\
      (!n. (BIT1 n = _0) = F) /\
      (!n. (_0 = BIT0 n) = _0 = n) /\
      (!n. (_0 = BIT1 n) = F) /\
      (!m n. (BIT0 m = BIT0 n) = m = n) /\
      (!m n. (BIT0 m = BIT1 n) = F) /\
      (!m n. (BIT1 m = BIT0 n) = F) /\
      (!m n. (BIT1 m = BIT1 n) = m = n)) /\
     ((!m n. NUMERAL m <= NUMERAL n = m <= n) /\
      (_0 <= _0 = T) /\
      (!n. BIT0 n <= _0 = n = _0) /\
      (!n. BIT1 n <= _0 = F) /\
      (!n. _0 <= BIT0 n = T) /\
      (!n. _0 <= BIT1 n = T) /\
      (!m n. BIT0 m <= BIT0 n = m <= n) /\
      (!m n. BIT0 m <= BIT1 n = m <= n) /\
      (!m n. BIT1 m <= BIT0 n = m < n) /\
      (!m n. BIT1 m <= BIT1 n = m <= n)) /\
     ((!m n. NUMERAL m < NUMERAL n = m < n) /\
      (_0 < _0 = F) /\
      (!n. BIT0 n < _0 = F) /\
      (!n. BIT1 n < _0 = F) /\
      (!n. _0 < BIT0 n = _0 < n) /\
      (!n. _0 < BIT1 n = T) /\
      (!m n. BIT0 m < BIT0 n = m < n) /\
      (!m n. BIT0 m < BIT1 n = m <= n) /\
      (!m n. BIT1 m < BIT0 n = m < n) /\
      (!m n. BIT1 m < BIT1 n = m < n)) /\
     ((!m n. NUMERAL n >= NUMERAL m = n >= m) /\
      _0 >= _0 /\
      (!n. _0 >= BIT0 n = n = _0) /\
      (!n. ~(_0 >= BIT1 n)) /\
      (!n. BIT0 n >= _0) /\
      (!n. BIT1 n >= _0) /\
      (!m n. BIT0 n >= BIT0 m = n >= m) /\
      (!m n. BIT1 n >= BIT0 m = n >= m) /\
      (!m n. BIT0 n >= BIT1 m = n > m) /\
      (!m n. BIT1 n >= BIT1 m = n >= m)) /\
     ((!m n. NUMERAL n > NUMERAL m = n > m) /\
      ~(_0 > _0) /\
      (!n. ~(_0 > BIT0 n)) /\
      (!n. ~(_0 > BIT1 n)) /\
      (!n. BIT0 n > _0 = n > _0) /\
      (!n. BIT1 n > _0) /\
      (!m n. BIT0 n > BIT0 m = n > m) /\
      (!m n. BIT1 n > BIT0 m = n >= m) /\
      (!m n. BIT0 n > BIT1 m = n > m) /\
      (!m n. BIT1 n > BIT1 m = n > m)) /\
     (!m n. NUMERAL m - NUMERAL n = NUMERAL (m - n)) /\
     (_0 - _0 = _0) /\
     (!n. _0 - BIT0 n = _0) /\
     (!n. _0 - BIT1 n = _0) /\
     (!n. BIT0 n - _0 = BIT0 n) /\
     (!n. BIT1 n - _0 = BIT1 n) /\
     (!m n. BIT0 m - BIT0 n = BIT0 (m - n)) /\
     (!m n. BIT0 m - BIT1 n = PRE (BIT0 (m - n))) /\
     (!m n. BIT1 m - BIT0 n = (if n <= m then BIT1 (m - n) else _0)) /\
     (!m n. BIT1 m - BIT1 n = BIT0 (m - n))

NUM_SUC_CONV rewrites a term of the form `SUC k`, where k is a numeral.
NUM_ADD_CONV rewrites a term of the form `j + k`, where j and k are numerals.
NUM_SUC_CONV' and NUM_ADD_CONV' are the same, except that they expect
numerals without the NUMERAL tag.

NUM_PRE_CONV rewrites a term of the form `PRE k`, where k is a numeral.

NUM_REL_CONV rewrites a term of the form `j OP k`, where j and k
are numerals, and op is (<), (<=), (>), (>=), (=).
NUM_REL_CONV' is the same, except that it expects numerals without
the NUMERAL tag and does not deal with (>) or (>=).

NUM_EQ_CONV, NUM_LE_CONV, NUM_LT_CONV, NUM_GE_CONV, NUM_GT_CONV
Like NUM_REL_CONV, but only works if the op matches the conversion
name.  (You could always use NUM_REL_CONV instead of these, unless
you deliberately want to fail on the wrong op.  These will be
very slightly more efficient.)

NUM_EVEN_CONV, NUM_ODD_CONV
rewrite terms of the form `EVEN k` and `ODD k` respectively, where k
is a numeral.

NUM_SUB_CONV rewrites a term of the form `j - k`, where j and k
are numerals.

NUM_MULT_CONV rewrites a term of the form `j * k`, where j and k
are numerals.
NUM_MULT_CONV' is the same, except that it expects numerals without
the NUMERAL tag.

NUM_EXP_CONV rewrites a term of the form `j EXP k`, where j and k
are numerals.

NUM_DIV_CONV and NUM_MOD_CONV rewrite terms of the forms `j DIV k`
and `j MOD k` respectively, where j and k are numerals.
NUM_DIVMOD_CONV (j:Num.num) (k:Num.num) produces a theorem
`|- (J DIV K = JDIVK) /\ (J MOD K = JMODK)`, where J and K
are the numerals for j and k, and JDIVK and JMODK are the appropriate
numerals.  (NUM_DIV_CONV and NUM_MID_CONV each call this function;
if you need to compute both DIV and MOD, it's about twice as
fast to do both together with NUM_DIVMOD_CONV.)

NUM_FACT_CONV rewrites terms of the form `FACT k` where k is a numeral.

NUM_RED_CONV reduces `SUC j`, `PRE j`, `FACT j`, `j < k`, `j <= k`,
`j > k`, `j >= k`, `j = k`, `EVEN j`, `ODD j`, `j + k`, `j - k`,
`j * k`, `j EXP k`, `j DIV k`, or `j MOD k` (where j and k are numerals).

NUM_REDUCE_CONV reduces the above numeral expressions depth-first
throughout the expression.

NUM_REDUCE_TAC reduces numeral expressions in the goal.

num_CONV rewrites `j` to `SUC (J-1)`, where J-1 is the numeral for j - 1.

EXPAND_CASES_CONV rewrites `!n. n < 5 ==> P[n]` to
`P[0] /\ P[1] /\ P[2] /\ P[3] /\ P[4]`

ind-types.ml:

val ( INJ_INVERSE2 ) : thm =
  |- !P. (!x1 y1 x2 y2. (P x1 y1 = P x2 y2) = (x1 = x2) /\ (y1 = y2))
         ==> (?X Y. !x y. (X (P x y) = x) /\ (Y (P x y) = y))

This is the definition of NUMPAIR.
val ( NUMPAIR ) : thm = |- !x y. NUMPAIR x y = 2 EXP x * (2 * y + 1)

val ( NUMPAIR_INJ_LEMMA ) : thm =
  |- !x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) ==> (x1 = x2)
val ( NUMPAIR_INJ ) : thm =
  |- !x1 y1 x2 y2. (NUMPAIR x1 y1 = NUMPAIR x2 y2) = (x1 = x2) /\ (y1 = y2)

This is the simultaneous definition of NUMFST and NUMSND (using
new_specification).
val ( NUMPAIR_DEST ) : thm =
  |- !x y. (NUMFST (NUMPAIR x y) = x) /\ (NUMSND (NUMPAIR x y) = y)

This is the definition of NUMSUM.
val ( NUMSUM ) : thm =
  |- !b x. NUMSUM b x = (if b then SUC (2 * x) else 2 * x)

val ( NUMSUM_INJ ) : thm =
  |- !b1 x1 b2 x2. (NUMSUM b1 x1 = NUMSUM b2 x2) = (b1 = b2) /\ (x1 = x2)

This is the simultaneous definition of NUMLEFT and NUMRIGHT
(using new_specification).
val ( NUMSUM_DEST ) : thm =
  |- !x y. (NUMLEFT (NUMSUM x y) = x) /\ (NUMRIGHT (NUMSUM x y) = y)

This is the definition of INJN.
val ( INJN ) : thm = |- !m. INJN m = (\n a. n = m)

val ( INJN_INJ ) : thm = |- !n1 n2. (INJN n1 = INJN n2) = n1 = n2

This is the definition of INJA.
val ( INJA ) : thm = |- !a. INJA a = (\n b. b = a)

val ( INJA_INJ ) : thm = |- !a1 a2. (INJA a1 = INJA a2) = a1 = a2

This is the definition of INJF.
val ( INJF ) : thm = |- !f. INJF f = (\n. f (NUMFST n) (NUMSND n))

val ( INJF_INJ ) : thm = |- !f1 f2. (INJF f1 = INJF f2) = f1 = f2

This is the definition of INJP.
val ( INJP ) : thm =
  |- !f1 f2.
         INJP f1 f2 =
         (\n a. if NUMLEFT n then f1 (NUMRIGHT n) a else f2 (NUMRIGHT n) a)

val ( INJP_INJ ) : thm =
  |- !f1 f1' f2 f2'. (INJP f1 f2 = INJP f1' f2') = (f1 = f1') /\ (f2 = f2')

These are the definitions of ZCONSTR and ZBOT.
val ( ZCONSTR ) : thm =
  |- !c i r. ZCONSTR c i r = INJP (INJN (SUC c)) (INJP (INJA i) (INJF r))
val ( ZBOT ) : thm = |- ZBOT = INJP (INJN 0) (@z. T)

val ( ZCONSTR_ZBOT ) : thm = |- !c i r. ~(ZCONSTR c i r = ZBOT)

This is the definition (via new_inductive_definition) of
ZRECSPACE_ZBOT.
val ( ZRECSPACE_RULES ) : thm =
  |- ZRECSPACE ZBOT /\
     (!c i r. (!n. ZRECSPACE (r n)) ==> ZRECSPACE (ZCONSTR c i r))
val ( ZRECSPACE_INDUCT ) : thm =
  |- !ZRECSPACE'. ZRECSPACE' ZBOT /\
                  (!c i r.
                       (!n. ZRECSPACE' (r n)) ==> ZRECSPACE' (ZCONSTR c i r))
                  ==> (!a. ZRECSPACE a ==> ZRECSPACE' a)
val ( ZRECSPACE_CASES ) : thm =
  |- !a. ZRECSPACE a =
         (a = ZBOT) \/ (?c i r. (a = ZCONSTR c i r) /\ (!n. ZRECSPACE (r n)))

This is the definition of a new type, "(A)recspace"; and its in and out
functions, "_mk_rec" and "_dest_rec".
val recspace_tydef : thm * thm =
  (|- _mk_rec (_dest_rec a) = a, |- ZRECSPACE r = _dest_rec (_mk_rec r) = r)

These are the definitions of BOTTOM and CONSTR.
val ( BOTTOM ) : thm = |- BOTTOM = _mk_rec ZBOT
val ( CONSTR ) : thm =
  |- !c i r. CONSTR c i r = _mk_rec (ZCONSTR c i (\n. _dest_rec (r n)))

val ( MK_REC_INJ ) : thm =
  |- !x y. (_mk_rec x = _mk_rec y) ==> ZRECSPACE x /\ ZRECSPACE y ==> (x = y)
val ( DEST_REC_INJ ) : thm = |- !x y. (_dest_rec x = _dest_rec y) = x = y
val ( CONSTR_BOT ) : thm = |- !c i r. ~(CONSTR c i r = BOTTOM)
val ( CONSTR_INJ ) : thm =
  |- !c1 i1 r1 c2 i2 r2.
         (CONSTR c1 i1 r1 = CONSTR c2 i2 r2) =
         (c1 = c2) /\ (i1 = i2) /\ (r1 = r2)
val ( CONSTR_IND ) : thm =
  |- !P. P BOTTOM /\ (!c i r. (!n. P (r n)) ==> P (CONSTR c i r))
         ==> (!x. P x)
val ( CONSTR_REC ) : thm =
  |- !Fn. ?f. !c i r. f (CONSTR c i r) = Fn c i r (\n. f (r n))

This is the definition of FCONS.
val ( FCONS ) : thm =
  |- (!a f. FCONS a f 0 = a) /\ (!a f n. FCONS a f (SUC n) = f n)

val ( FCONS_UNDO ) : thm = |- !f. f = FCONS (f 0) (f o SUC)

This is the definition of FNIL.
val ( FNIL ) : thm = |- !n. FNIL n = (@x. T)

sucivate 5 = `SUC (SUC (SUC (SUC (SUC 0))))`
(for example)

SCRUB_EQUATION `x = a` `x = a |- P[x]` gives `|- P[a]`

list.ml:

LIST_INDUCT_TAC takes a goal of the form `!l. P[l]` and creates two
subgoals: `P[ [] ]` and `P[CONS h t]`.  The latter subgoal has a new
assumption `P[t]`.

Definitions of list-related functions:
val ( HD ) : thm = |- HD (CONS h t) = h
val ( TL ) : thm = |- TL (CONS h t) = t
val ( APPEND ) : thm =
  |- (!l. APPEND [] l = l) /\
     (!h t l. APPEND (CONS h t) l = CONS h (APPEND t l))
val ( REVERSE ) : thm =
  |- (REVERSE [] = []) /\ (REVERSE (CONS x l) = APPEND (REVERSE l) [x])
val ( LENGTH ) : thm =
  |- (LENGTH [] = 0) /\ (!h t. LENGTH (CONS h t) = SUC (LENGTH t))
val ( MAP ) : thm =
  |- (!f. MAP f [] = []) /\ (!f h t. MAP f (CONS h t) = CONS (f h) (MAP f t))
val ( LAST ) : thm = |- LAST (CONS h t) = (if t = [] then h else LAST t)
val ( REPLICATE ) : thm =
  |- (REPLICATE 0 x = []) /\ (REPLICATE (SUC n) x = CONS x (REPLICATE n x))
val ( NULL ) : thm = |- (NULL [] = T) /\ (NULL (CONS h t) = F)
val ( ALL ) : thm = |- (ALL P [] = T) /\ (ALL P (CONS h t) = P h /\ ALL P t)
val ( EX ) : thm = |- (EX P [] = F) /\ (EX P (CONS h t) = P h \/ EX P t)
val ( ITLIST ) : thm =
  |- (ITLIST f [] b = b) /\ (ITLIST f (CONS h t) b = f h (ITLIST f t b))
val ( MEM ) : thm =
  |- (MEM x [] = F) /\ (MEM x (CONS h t) = (x = h) \/ MEM x t)
val ( ALL2_DEF ) : thm =
  |- (ALL2 P [] l2 = l2 = []) /\
     (ALL2 P (CONS h1 t1) l2 =
      (if l2 = [] then F else P h1 (HD l2) /\ ALL2 P t1 (TL l2)))
val ( MAP2_DEF ) : thm =
  |- (MAP2 f [] l = []) /\
     (MAP2 f (CONS h1 t1) l = CONS (f h1 (HD l)) (MAP2 f t1 (TL l)))
val ( EL ) : thm = |- (EL 0 l = HD l) /\ (EL (SUC n) l = EL n (TL l))
val ( FILTER ) : thm =
  |- (FILTER P [] = []) /\
     (FILTER P (CONS h t) =
      (if P h then CONS h (FILTER P t) else FILTER P t))
val ( ASSOC ) : thm =
  |- ASSOC a (CONS h t) = (if FST h = a then SND h else ASSOC a t)
val ( ITLIST2_DEF ) : thm =
  |- (ITLIST2 f [] l2 b = b) /\
     (ITLIST2 f (CONS h1 t1) l2 b = f h1 (HD l2) (ITLIST2 f t1 (TL l2) b))
val ( ZIP_DEF ) : thm =
  |- (ZIP [] l2 = []) /\
     (ZIP (CONS h1 t1) l2 = CONS (h1,HD l2) (ZIP t1 (TL l2)))

More convenient forms of some of the above:
val ( ALL2 ) : thm =
  |- (ALL2 P [] [] = T) /\
     (ALL2 P (CONS h1 t1) [] = F) /\
     (ALL2 P [] (CONS h2 t2) = F) /\
     (ALL2 P (CONS h1 t1) (CONS h2 t2) = P h1 h2 /\ ALL2 P t1 t2)
val ( MAP2 ) : thm =
  |- (MAP2 f [] [] = []) /\
     (MAP2 f (CONS h1 t1) (CONS h2 t2) = CONS (f h1 h2) (MAP2 f t1 t2))
val ( ITLIST2 ) : thm =
  |- (ITLIST2 f [] [] b = b) /\
     (ITLIST2 f (CONS h1 t1) (CONS h2 t2) b = f h1 h2 (ITLIST2 f t1 t2 b))
val ( ZIP ) : thm =
  |- (ZIP [] [] = []) /\
     (ZIP (CONS h1 t1) (CONS h2 t2) = CONS (h1,h2) (ZIP t1 t2))

val ( NOT_CONS_NIL ) : thm = |- !h t. ~(CONS h t = [])
val ( LAST_CLAUSES ) : thm =
  |- (LAST [h] = h) /\ (LAST (CONS h (CONS k t)) = LAST (CONS k t))
val ( APPEND_NIL ) : thm = |- !l. APPEND l [] = l
val ( APPEND_ASSOC ) : thm =
  |- !l m n. APPEND l (APPEND m n) = APPEND (APPEND l m) n
val ( REVERSE_APPEND ) : thm =
  |- !l m. REVERSE (APPEND l m) = APPEND (REVERSE m) (REVERSE l)
val ( REVERSE_REVERSE ) : thm = |- !l. REVERSE (REVERSE l) = l
val ( CONS_11 ) : thm =
  |- !h1 h2 t1 t2. (CONS h1 t1 = CONS h2 t2) = (h1 = h2) /\ (t1 = t2)
val list_CASES : thm = |- !l. (l = []) \/ (?h t. l = CONS h t)
val ( LENGTH_APPEND ) : thm =
  |- !l m. LENGTH (APPEND l m) = LENGTH l + LENGTH m
val ( MAP_APPEND ) : thm =
  |- !f l1 l2. MAP f (APPEND l1 l2) = APPEND (MAP f l1) (MAP f l2)
val ( LENGTH_MAP ) : thm = |- !l f. LENGTH (MAP f l) = LENGTH l
val ( LENGTH_EQ_NIL ) : thm = |- !l. (LENGTH l = 0) = l = []
val ( MAP_o ) : thm = |- !f g l. MAP (g o f) l = MAP g (MAP f l)
val ( MAP_EQ ) : thm =
  |- !f g l. ALL (\x. f x = g x) l ==> (MAP f l = MAP g l)
val ( ALL_IMP ) : thm =
  |- !P Q l. (!x. MEM x l /\ P x ==> Q x) /\ ALL P l ==> ALL Q l
val ( NOT_EX ) : thm = |- !P l. ~EX P l = ALL (\x. ~P x) l
val ( NOT_ALL ) : thm = |- !P l. ~ALL P l = EX (\x. ~P x) l
val ( ALL_MAP ) : thm = |- !P f l. ALL P (MAP f l) = ALL (P o f) l
val ( ALL_T ) : thm = |- !l. ALL (\x. T) l
val ( MAP_EQ_ALL2 ) : thm =
  |- !l m. ALL2 (\x y. f x = f y) l m ==> (MAP f l = MAP f m)
val ( ALL2_MAP ) : thm =
  |- !P f l. ALL2 P (MAP f l) l = ALL (\a. P (f a) a) l
val ( MAP_EQ_DEGEN ) : thm = |- !l f. ALL (\x. f x = x) l ==> (MAP f l = l)
val ( ALL2_AND_RIGHT ) : thm =
  |- !l m P Q. ALL2 (\x y. P x /\ Q x y) l m = ALL P l /\ ALL2 Q l m
val ( ITLIST_EXTRA ) : thm =
  |- !l. ITLIST f (APPEND l [a]) b = ITLIST f l (f a b)
val ( ALL_MP ) : thm =
  |- !P Q l. ALL (\x. P x ==> Q x) l /\ ALL P l ==> ALL Q l
val ( FORALL_ALL ) : thm = |- !l. (!x. ALL (P x) l) = ALL (\a. !x. P x a) l
val ( AND_ALL ) : thm = |- !l. ALL P l /\ ALL Q l = ALL (\x. P x /\ Q x) l
val ( EX_IMP ) : thm =
  |- !P Q l. (!x. MEM x l /\ P x ==> Q x) /\ EX P l ==> EX Q l
val ( ALL_MEM ) : thm = |- !P l. (!x. MEM x l ==> P x) = ALL P l
val ( LENGTH_REPLICATE ) : thm = |- !n x. LENGTH (REPLICATE n x) = n
val ( EX_MAP ) : thm = |- !P f l. EX P (MAP f l) = EX (P o f) l
val ( EXISTS_EX ) : thm = |- !P l. (?x. EX (P x) l) = EX (\s. ?x. P x s) l
val ( FORALL_ALL ) : thm = |- !P l. (!x. ALL (P x) l) = ALL (\s. !x. P x s) l
val ( MEM_APPEND ) : thm =
  |- !x l1 l2. MEM x (APPEND l1 l2) = MEM x l1 \/ MEM x l2
val ( MEM_MAP ) : thm =
  |- !f y l. MEM y (MAP f l) = (?x. MEM x l /\ (y = f x))
val ( FILTER_APPEND ) : thm =
  |- !P l1 l2. FILTER P (APPEND l1 l2) = APPEND (FILTER P l1) (FILTER P l2)
val ( FILTER_MAP ) : thm =
  |- !P f l. FILTER P (MAP f l) = MAP f (FILTER (P o f) l)
val ( MEM_FILTER ) : thm = |- !P l x. MEM x (FILTER P l) = P x /\ MEM x l
val ( EX_MEM ) : thm = |- !P l. EX P l = (?x. P x /\ MEM x l)
val ( MAP_FST_ZIP ) : thm =
  |- !l1 l2. (LENGTH l1 = LENGTH l2) ==> (MAP FST (ZIP l1 l2) = l1)
val ( MAP_SND_ZIP ) : thm =
  |- !l1 l2. (LENGTH l1 = LENGTH l2) ==> (MAP SND (ZIP l1 l2) = l2)
val ( MEM_ASSOC ) : thm = |- !l x. MEM (x,ASSOC x l) l = MEM x (MAP FST l)
val ( ALL_APPEND ) : thm =
  |- !P l1 l2. ALL P (APPEND l1 l2) = ALL P l1 /\ ALL P l2
val ( MEM_EL ) : thm = |- !l n. n < LENGTH l ==> MEM (EL n l) l
val ( ALL2_MAP2 ) : thm =
  |- !l m. ALL2 P (MAP f l) (MAP g m) = ALL2 (\x y. P (f x) (g y)) l m
val ( AND_ALL2 ) : thm =
  |- !P Q l m. ALL2 P l m /\ ALL2 Q l m = ALL2 (\x y. P x y /\ Q x y) l m
val ( ALL2_ALL ) : thm = |- !P l. ALL2 P l l = ALL (\x. P x x) l
val ( APPEND_EQ_NIL ) : thm =
  |- !l m. (APPEND l m = []) = (l = []) /\ (m = [])

val ( MONO_ALL ) : thm = |- (!x. P x ==> Q x) ==> ALL P l ==> ALL Q l
val ( MONO_ALL2 ) : thm =
  |- (!x y. P x y ==> Q x y) ==> ALL2 P l l' ==> ALL2 Q l l'
MONO_ALL and MONO_ALL2 will automatically be used by MONO_TAC.

LIST_CONV conv uses conv to rewrite every member of a (literal) list.

realax.ml:
The operators +,-,*,<,<=,>,>= are overloaded; there are versions for
num, real, int.  If an expression does not make it clear which type is
used, they default to the current "prioritized" type.

pioritize_num() sets operator overloading to default to the type num.

This is the definition of dist.
val dist : thm = |- !n m. dist (m,n) = m - n + n - m

val ( DIST_REFL ) : thm = |- !n. dist (n,n) = 0
val ( DIST_LZERO ) : thm = |- !n. dist (0,n) = n
val ( DIST_RZERO ) : thm = |- !n. dist (n,0) = n
val ( DIST_SYM ) : thm = |- !m n. dist (m,n) = dist (n,m)
val ( DIST_LADD ) : thm = |- !m p n. dist (m + n,m + p) = dist (n,p)
val ( DIST_RADD ) : thm = |- !m p n. dist (m + p,n + p) = dist (m,n)
val ( DIST_LADD_0 ) : thm = |- !m n. dist (m + n,m) = n
val ( DIST_RADD_0 ) : thm = |- !m n. dist (m,m + n) = n
val ( DIST_LMUL ) : thm = |- !m n p. m * dist (n,p) = dist (m * n,m * p)
val ( DIST_RMUL ) : thm = |- !m n p. dist (m,n) * p = dist (m * p,n * p)
val ( DIST_EQ_0 ) : thm = |- !m n. (dist (m,n) = 0) = m = n
val ( DIST_ELIM_THM ) : thm =
  |- P (dist (x,y)) = (!d. ((x = y + d) ==> P d) /\ ((y = x + d) ==> P d))