kdrag.kernel

The kernel hold core proof datatypes and core inference rules. By and large, all proofs must flow through this module.

Module Attributes

defns

defn holds definitional axioms for function symbols.

Functions

`FreshVar`(prefix, sort)	Generate a fresh variable.
`Inductive`(name[, ctx, auto_fresh])	Declare datatypes with auto generated induction principles.
`andI`(pfs)	Prove an and from two kd.Proofs of its conjuncts.
`axiom`(thm[, fvs, by])	Assert an axiom.
`choose`(pf)	Convert a theorem of the form `\|= forall xs, exists y0 y1.
`compose`(ab, bc)	Compose two implications.
`decl_occurs`(f, body)	Check if symbol f appears in body.
`declare`(name, *sorts)	Reserve a name.
`define`(name, args, body)	Define a non recursive definition.
`define_fix`(name, args, retsort, fix_lam)	Define a recursive definition.
`define_simple`(name, args, body)	A simple non recursive definition is a conservative extension https://en.wikipedia.org/wiki/Conservative_extension
`ext`(domain, range_)
`forget`(ts, thm)	"Forget" a term using existentials.
`free_vars`(e)	Get an overapproximation of free variables in a term.
`fresh_const`(q[, prefixes])	Generate fresh constants of same sort as quantifier.
`generalize`(vs, pf)	Generalize a theorem with respect to a list of schema variables.
`herb`(thm[, prefixes])	Herbrandize a theorem.
`induct_inductive`(x, P)	Build a basic induction principle for an algebraic datatype
`instan`(ts, pf)	Instantiate a universally quantified formula.
`is_declared`(e)
`is_defined`(x)	Determined if expression head is in definitions.
`is_fresh_var`(v)	Check if a variable is a schema variable.
`is_proof`(p)
`make_unfolding`(pf)	Take an axiom of the form \|= forall x0 x1 ..., xn, f(x0, x1, ..., xn) = body and return a Defn object for f. Unfolding judgments can be used with unfold_defns.
`modus`(ab, a)	Modus ponens for implies and equality.
`obtain`(thm)	Skolemize an existential quantifier.
`open_binder`(lam)	Open a quantifier with fresh variables.
`prove`(thm[, by, admit, timeout, dump, ...])	Prove a theorem using a list of previously proved lemmas.
`register_definition`(defn)	Can register other unfoldings besides those generated by define.
`rename_vars2`(pf, vs[, patterns, no_patterns])	Rename bound variables and also attach triggers
`specialize`(ts, thm)	Instantiate a universally quantified formula forall xs, P(xs) -> P(ts) This is forall elimination
`subst`(t, eqs)	Substitute using equality proofs
`trigger`(pf[, patterns, no_patterns])	Add e-matching triggers to a quantified proof.
`unfold`(e, decls)	Unfold function definitions in an expression.
`unfold_defns`(e, defns)	Unfold definitions in an expression.
`weaken`(pf, fvs)

Classes

`Judgement`()	Judgements should be constructed by smart constructors.
`Proof`(fvs, thm, reason[, admit])	It is unlikely that users should be accessing the Proof constructor directly.
`Unfolding`(name, decl, args, body, ax, ...)	A record storing definition.

Exceptions

LemmaError

kdrag.kernel.FreshVar(prefix: str, sort: SortRef) → ExprRef

Generate a fresh variable. This is distinguished from FreshConst by the fact that it has freshness evidence. This is intended to be used for constants that represent arbitrary terms (implicitly universally quantified). For example, axioms like c_fresh = t should never be asserted about bare FreshVars as they imply a probably inconsistent axiom, whereas asserting such an axiom about FreshConst is ok, effectively defining a new rigid constant.

>>> FreshVar("x", smt.IntSort()).fresh_evidence
_FreshVarEvidence(v=x!...)

Parameters:

prefix (str)
sort (SortRef)

Return type:

ExprRef

kdrag.kernel.Inductive(name: str, ctx=None, auto_fresh=False) → Datatype

Declare datatypes with auto generated induction principles. Wrapper around z3.Datatype

>>> Nat = Inductive("Nat909")
>>> Nat.declare("zero")
>>> Nat.declare("succ", ("pred", Nat))
>>> Nat = Nat.create()
>>> Nat.succ(Nat.zero)
succ(zero)

Parameters:: name (str)
Return type:: Datatype

class kdrag.kernel.Judgement

Bases: object

Judgements should be constructed by smart constructors. Having an object of supertype judgement represents having shown some kind of truth. Judgements are the things that go above and below inference lines in a proof system. Don’t worry about it. It is just nice to have a name for the concept.

See: - https://en.wikipedia.org/wiki/Judgment_(mathematical_logic) - https://mathoverflow.net/questions/254518/what-exactly-is-a-judgement - https://ncatlab.org/nlab/show/judgment

exception kdrag.kernel.LemmaError

Bases: Exception

add_note(note, /): Add a note to the exception

args

with_traceback(tb, /): Set self.__traceback__ to tb and return self.

class kdrag.kernel.Proof(fvs: frozenset[ExprRef], thm: BoolRef, reason: list[Any], admit: bool = False)

Bases: Judgement

It is unlikely that users should be accessing the Proof constructor directly. This is not ironclad. If you really want the Proof constructor, I can’t stop you.

Parameters:

fvs (frozenset[ExprRef])
thm (BoolRef)
reason (list[Any])
admit (bool)

__call__(*args: ExprRef | Proof)

>>> x,y = smt.Ints("x y")
>>> p = prove(smt.ForAll([y], smt.ForAll([x], x >= x - 1)))
>>> p(x)
|= ForAll(x, x >= x - 1)
>>> p(x, 7)
|= 7 >= 7 - 1

>>> a,b,c = smt.Bools("a b c")
>>> ab = prove(smt.Implies(a,smt.Implies(a, a)))
>>> a = axiom(a)
>>> ab(a)
|= Implies(a, a)
>>> ab(a,a)
|= a

Parameters:: args (ExprRef | Proof)

admit: bool = False

forall(fresh_vars: list[ExprRef]) → Proof

Generalize a proof involved schematic variables generated by FreshVar

>>> x = FreshVar("x", smt.IntSort())
>>> prove(x + 1 > x).forall([x])
|= ForAll(x!..., x!... + 1 > x!...)

Parameters:: fresh_vars (list[ExprRef])
Return type:: Proof

fvs: frozenset[ExprRef]

reason: list[Any]

thm: BoolRef

class kdrag.kernel.Unfolding(name: str, decl: FuncDeclRef, args: list[ExprRef], body: ExprRef, ax: Proof, _subst_fun_body: ExprRef)

Bases: Judgement

A record storing definition. It is useful to record definitions as special axioms because we often must unfold them. It’s more a recognition of a fact in unfoldable form

Parameters:

name (str)
decl (FuncDeclRef)
args (list[ExprRef])
body (ExprRef)
ax (Proof)
_subst_fun_body (ExprRef)

args: list[ExprRef]

ax: Proof

body: ExprRef

decl: FuncDeclRef

name: str

kdrag.kernel.andI(pfs: Sequence[Proof]) → Proof

Prove an and from two kd.Proofs of its conjuncts.

>>> a, b = smt.Bools("a b")
>>> pa = kd.axiom(smt.Implies(True, a))
>>> pb = kd.axiom(smt.Implies(True, b))
>>> andI([pa, pb, pb])
|= Implies(True, And(a, b, b))

Parameters:: pfs (Sequence[Proof])
Return type:: Proof

kdrag.kernel.axiom(thm: BoolRef, fvs=frozenset({}), by=['axiom']) → Proof

Assert an axiom.

Axioms are necessary and useful. But you must use great care.

Parameters:

thm (BoolRef) – The axiom to assert.
by – A python object explaining why the axiom should exist. Often a string explaining the axiom.

Return type:

Proof

kdrag.kernel.choose(pf: Proof) → tuple[list[FuncDeclRef], Proof]

Convert a theorem of the form |= forall xs, exists y0 y1… , P(xs, y0, y1, …) to |= forall xs, P(xs, f0(xs), f1(xs), …).

>>> x,y,z = smt.Ints("x y z")
>>> pf = kd.prove(smt.ForAll([x], smt.Exists([y], y >= x)))
>>> choose(pf)
([f!...], |= ForAll(x!..., f!...(x!...) >= x!...))

Parameters:: pf (Proof)
Return type:: tuple[list[FuncDeclRef], Proof]

kdrag.kernel.compose(ab: Proof, bc: Proof) → Proof

Compose two implications. Useful for chaining implications.

>>> a,b,c = smt.Bools("a b c")
>>> ab = axiom(smt.Implies(a, b))
>>> bc = axiom(smt.Implies(b, c))
>>> compose(ab, bc)
|= Implies(a, c)

Parameters:

ab (Proof)
bc (Proof)

Return type:

Proof

kdrag.kernel.decl_occurs(f: FuncDeclRef, body: ExprRef) → bool

Check if symbol f appears in body.

>>> x = smt.Int("x")
>>> y = smt.Bool("y")
>>> f = smt.Function("f", smt.IntSort(), smt.BoolSort(), smt.IntSort())
>>> assert decl_occurs(f, 1 + f(x, y))
>>> assert not decl_occurs(f, x + y + 1)

Parameters:

f (FuncDeclRef)
body (ExprRef)

Return type:

bool

kdrag.kernel.declare(name: str, *sorts: SortRef) → FuncDeclRef

Reserve a name. This helps avoid accidental clashing of axioms over the same symbol. It also make serialization possible and easier.

>>> declare("reserved_example", smt.IntSort())
reserved_example
>>> declare("reserved_example", smt.IntSort())
Traceback (most recent call last):
    ...
ValueError: ('Declaration name already reserved: reserved_example', 'with sorts', [], Int)

Parameters:

name (str)
sorts (SortRef)

Return type:

FuncDeclRef

kdrag.kernel.define(name: str, args: list[ExprRef], body: ExprRef) → FuncDeclRef

Define a non recursive definition. Useful for shorthand and abstraction. Does not currently defend against ill formed definitions. TODO: Check for bad circularity, record dependencies

Parameters:

name (str) – The name of the term to define.
args (list[ExprRef]) – The arguments of the term.
defn – The definition of the term.
body (ExprRef)

Returns:

A tuple of the defined term and the proof of the definition.

Return type:

tuple[smt.FuncDeclRef, Proof]

kdrag.kernel.define_fix(name: str, args: list[ExprRef], retsort, fix_lam) → FuncDeclRef

Define a recursive definition.

Parameters:

name (str)
args (list[ExprRef])

Return type:

FuncDeclRef

kdrag.kernel.define_simple(name: str, args: list[ExprRef], body: ExprRef) → FuncDeclRef

A simple non recursive definition is a conservative extension https://en.wikipedia.org/wiki/Conservative_extension

>>> x = smt.Int("x")
>>> define_simple("mysucc9034", [x], x + 1)
mysucc9034
>>> f = smt.Function("f102", smt.IntSort(), smt.IntSort())
>>> define_simple("f102", [x], f(x))
Traceback (most recent call last):
    ...
ValueError: ('Symbol appears in it's own definition. This is not a simple definition and requires further checks for consistency', f102, f102(x))

Parameters:

name (str)
args (list[ExprRef])
body (ExprRef)

Return type:

FuncDeclRef

kdrag.kernel.defns: dict[FuncDeclRef, Unfolding] = {(Array (Array Sierpinski Bool) Bool).finite: Unfolding(name='(Array (Array Sierpinski Bool) Bool).finite', decl=(Array (Array Sierpinski Bool) Bool).finite, args=[A!1056], body=Exists(finwit!1055, ForAll(x!1054, A!1056[x!1054] == Contains(finwit!1055, Unit(x!1054)))), ax=|= ForAll(A, (Array (Array Sierpinski Bool) Bool).finite(A) == (Exists(finwit!1055, ForAll(x!1054, A[x!1054] == Contains(finwit!1055, Unit(x!1054)))))), _subst_fun_body=Exists(finwit!1055, ForAll(x!1054, Var(2)[x!1054] == Contains(finwit!1055, Unit(x!1054))))), (Array Bool Bool).choose: Unfolding(name='(Array Bool Bool).choose', decl=(Array Bool Bool).choose, args=[P!208], body=If(P!208[True], True, False), ax=|= ForAll(P, (Array Bool Bool).choose(P) == If(P[True], True, False)), _subst_fun_body=If(Var(0)[True], True, False)), (Array Bool Bool).finite: Unfolding(name='(Array Bool Bool).finite', decl=(Array Bool Bool).finite, args=[A!100], body=Exists(finwit!99, ForAll(x!98, A!100[x!98] == Contains(finwit!99, Unit(x!98)))), ax=|= ForAll(A, (Array Bool Bool).finite(A) == (Exists(finwit!99, ForAll(x!98, A[x!98] == Contains(finwit!99, Unit(x!98)))))), _subst_fun_body=Exists(finwit!99, ForAll(x!98, Var(2)[x!98] == Contains(finwit!99, Unit(x!98))))), (Array EReal Bool).finite: Unfolding(name='(Array EReal Bool).finite', decl=(Array EReal Bool).finite, args=[A!1672], body=Exists(finwit!1671, ForAll(x!1670, A!1672[x!1670] == Contains(finwit!1671, Unit(x!1670)))), ax=|= ForAll(A, (Array EReal Bool).finite(A) == (Exists(finwit!1671, ForAll(x!1670, A[x!1670] == Contains(finwit!1671, Unit(x!1670)))))), _subst_fun_body=Exists(finwit!1671, ForAll(x!1670, Var(2)[x!1670] == Contains(finwit!1671, Unit(x!1670))))), (Array Int Bool).choose: Unfolding(name='(Array Int Bool).choose', decl=(Array Int Bool).choose, args=[P!339], body=search(P!339, 0), ax=|= ForAll(P, (Array Int Bool).choose(P) == search(P, 0)), _subst_fun_body=search(Var(0), 0)), (Array Int Bool).finite: Unfolding(name='(Array Int Bool).finite', decl=(Array Int Bool).finite, args=[A!62], body=Exists(finwit!61, ForAll(x!60, A!62[x!60] == Contains(finwit!61, Unit(x!60)))), ax=|= ForAll(A, (Array Int Bool).finite(A) == (Exists(finwit!61, ForAll(x!60, A[x!60] == Contains(finwit!61, Unit(x!60)))))), _subst_fun_body=Exists(finwit!61, ForAll(x!60, Var(2)[x!60] == Contains(finwit!61, Unit(x!60))))), (Array Int C).add: Unfolding(name='(Array Int C).add', decl=(Array Int C).add, args=[a!1576, b!1577], body=Lambda(i, C.add(a!1576[i], b!1577[i])), ax=|= ForAll([a, b], (Array Int C).add(a, b) == (Lambda(i, C.add(a[i], b[i])))), _subst_fun_body=Lambda(i, C.add(Var(1)[i], Var(2)[i]))), (Array Int C).mul: Unfolding(name='(Array Int C).mul', decl=(Array Int C).mul, args=[a!1591, b!1592], body=Lambda(i, C.mul(a!1591[i], b!1592[i])), ax=|= ForAll([a, b], (Array Int C).mul(a, b) == (Lambda(i, C.mul(a[i], b[i])))), _subst_fun_body=Lambda(i, C.mul(Var(1)[i], Var(2)[i]))), (Array Int Real).add: Unfolding(name='(Array Int Real).add', decl=(Array Int Real).add, args=[a!1296, b!1297], body=Lambda(i, a!1296[i] + b!1297[i]), ax=|= ForAll([a, b], (Array Int Real).add(a, b) == (Lambda(i, a[i] + b[i]))), _subst_fun_body=Lambda(i, Var(1)[i] + Var(2)[i])), (Array Int Real).div_: Unfolding(name='(Array Int Real).div_', decl=(Array Int Real).div_, args=[a!1321, b!1322], body=Lambda(i, a!1321[i]/b!1322[i]), ax=|= ForAll([a, b], (Array Int Real).div_(a, b) == (Lambda(i, a[i]/b[i]))), _subst_fun_body=Lambda(i, Var(1)[i]/Var(2)[i])), (Array Int Real).mul: Unfolding(name='(Array Int Real).mul', decl=(Array Int Real).mul, args=[a!1308, b!1309], body=Lambda(i, a!1308[i]*b!1309[i]), ax=|= ForAll([a, b], (Array Int Real).mul(a, b) == (Lambda(i, a[i]*b[i]))), _subst_fun_body=Lambda(i, Var(1)[i]*Var(2)[i])), (Array Int Real).neg: Unfolding(name='(Array Int Real).neg', decl=(Array Int Real).neg, args=[a!1323], body=Lambda(i, -a!1323[i]), ax=|= ForAll(a, (Array Int Real).neg(a) == (Lambda(i, -a[i]))), _subst_fun_body=Lambda(i, -Var(1)[i])), (Array Int Real).sub: Unfolding(name='(Array Int Real).sub', decl=(Array Int Real).sub, args=[a!1306, b!1307], body=Lambda(i, a!1306[i] - b!1307[i]), ax=|= ForAll([a, b], (Array Int Real).sub(a, b) == (Lambda(i, a[i] - b[i]))), _subst_fun_body=Lambda(i, Var(1)[i] - Var(2)[i])), (Array Real Bool).finite: Unfolding(name='(Array Real Bool).finite', decl=(Array Real Bool).finite, args=[A!24], body=Exists(finwit!23, ForAll(x!22, A!24[x!22] == Contains(finwit!23, Unit(x!22)))), ax=|= ForAll(A, (Array Real Bool).finite(A) == (Exists(finwit!23, ForAll(x!22, A[x!22] == Contains(finwit!23, Unit(x!22)))))), _subst_fun_body=Exists(finwit!23, ForAll(x!22, Var(2)[x!22] == Contains(finwit!23, Unit(x!22))))), (Array Real Real).add: Unfolding(name='(Array Real Real).add', decl=(Array Real Real).add, args=[f!444, g!445], body=Lambda(x, f!444[x] + g!445[x]), ax=|= ForAll([f, g], (Array Real Real).add(f, g) == (Lambda(x, f[x] + g[x]))), _subst_fun_body=Lambda(x, Var(1)[x] + Var(2)[x])), (Array Real Real).div_: Unfolding(name='(Array Real Real).div_', decl=(Array Real Real).div_, args=[f!450, g!451], body=Lambda(x, f!450[x]/g!451[x]), ax=|= ForAll([f, g], (Array Real Real).div_(f, g) == (Lambda(x, f[x]/g[x]))), _subst_fun_body=Lambda(x, Var(1)[x]/Var(2)[x])), (Array Real Real).mul: Unfolding(name='(Array Real Real).mul', decl=(Array Real Real).mul, args=[f!448, g!449], body=Lambda(x, f!448[x]*g!449[x]), ax=|= ForAll([f, g], (Array Real Real).mul(f, g) == (Lambda(x, f[x]*g[x]))), _subst_fun_body=Lambda(x, Var(1)[x]*Var(2)[x])), (Array Real Real).sub: Unfolding(name='(Array Real Real).sub', decl=(Array Real Real).sub, args=[f!446, g!447], body=Lambda(x, f!446[x] - g!447[x]), ax=|= ForAll([f, g], (Array Real Real).sub(f, g) == (Lambda(x, f[x] - g[x]))), _subst_fun_body=Lambda(x, Var(1)[x] - Var(2)[x])), (Array Sierpinski Bool).finite: Unfolding(name='(Array Sierpinski Bool).finite', decl=(Array Sierpinski Bool).finite, args=[A!1000], body=Exists(finwit!999, ForAll(x!998, A!1000[x!998] == Contains(finwit!999, Unit(x!998)))), ax=|= ForAll(A, (Array Sierpinski Bool).finite(A) == (Exists(finwit!999, ForAll(x!998, A[x!998] == Contains(finwit!999, Unit(x!998)))))), _subst_fun_body=Exists(finwit!999, ForAll(x!998, Var(2)[x!998] == Contains(finwit!999, Unit(x!998))))), (Array Sierpinski Bool).open: Unfolding(name='(Array Sierpinski Bool).open', decl=(Array Sierpinski Bool).open, args=[A!1084], body=Or(A!1084 == K(Sierpinski, False), A!1084 == K(Sierpinski, True), A!1084 == Store(K(Sierpinski, False), S1, True)), ax=|= ForAll(A, (Array Sierpinski Bool).open(A) == Or(A == K(Sierpinski, False), A == K(Sierpinski, True), A == Store(K(Sierpinski, False), S1, True))), _subst_fun_body=Or(Var(0) == K(Sierpinski, False), Var(0) == K(Sierpinski, True), Var(0) == Store(K(Sierpinski, False), S1, True))), (Array Vec2 Bool).finite: Unfolding(name='(Array Vec2 Bool).finite', decl=(Array Vec2 Bool).finite, args=[A!1784], body=Exists(finwit!1783, ForAll(x!1782, A!1784[x!1782] == Contains(finwit!1783, Unit(x!1782)))), ax=|= ForAll(A, (Array Vec2 Bool).finite(A) == (Exists(finwit!1783, ForAll(x!1782, A[x!1782] == Contains(finwit!1783, Unit(x!1782)))))), _subst_fun_body=Exists(finwit!1783, ForAll(x!1782, Var(2)[x!1782] == Contains(finwit!1783, Unit(x!1782))))), Atom.fresh: Unfolding(name='Atom.fresh', decl=Atom.fresh, args=[x!1146, a!1147], body=a!1147 != x!1146, ax=|= ForAll([x, a], Atom.fresh(x, a) == a != x), _subst_fun_body=Var(1) != Var(0)), Atom.swap: Unfolding(name='Atom.swap', decl=Atom.swap, args=[x!1148, a!1149, b!1150], body=If(x!1148 == a!1149, b!1150, If(x!1148 == b!1150, a!1149, x!1148)), ax=|= ForAll([x, a, b], Atom.swap(x, a, b) == If(x == a, b, If(x == b, a, x))), _subst_fun_body=If(Var(0) == Var(1), Var(2), If(Var(0) == Var(2), Var(1), Var(0)))), BitVecN.to_int: Unfolding(name='BitVecN.to_int', decl=BitVecN.to_int, args=[x!189], body=If(Length(val(x!189)) == 0, 0, BV2Int(Nth(val(x!189), 0)) + 2* BitVecN.to_int(BitVecN(seq.extract(val(x!189), 1, Length(val(x!189)) - 1)))), ax=|= ForAll(x, BitVecN.to_int(x) == If(Length(val(x)) == 0, 0, BV2Int(Nth(val(x), 0)) + 2* BitVecN.to_int(BitVecN(seq.extract(val(x), 1, Length(val(x)) - 1))))), _subst_fun_body=If(Length(val(Var(0))) == 0, 0, BV2Int(Nth(val(Var(0)), 0)) + 2* BitVecN.to_int(BitVecN(seq.extract(val(Var(0)), 1, Length(val(Var(0))) - 1))))), C.add: Unfolding(name='C.add', decl=C.add, args=[z1!0, z2!1], body=C(re(z1!0) + re(z2!1), im(z1!0) + im(z2!1)), ax=|= ForAll([z1, z2], C.add(z1, z2) == C(re(z1) + re(z2), im(z1) + im(z2))), _subst_fun_body=C(re(Var(0)) + re(Var(1)), im(Var(0)) + im(Var(1)))), C.div_: Unfolding(name='C.div_', decl=C.div_, args=[z1!4, z2!5], body=C((re(z1!4)*re(z2!5) + im(z1!4)*im(z2!5))/ (re(z2!5)**2 + im(z2!5)**2), (im(z1!4)*re(z2!5) - re(z1!4)*im(z2!5))/ (re(z2!5)**2 + im(z2!5)**2)), ax=|= ForAll([z1, z2], C.div_(z1, z2) == C((re(z1)*re(z2) + im(z1)*im(z2))/ (re(z2)**2 + im(z2)**2), (im(z1)*re(z2) - re(z1)*im(z2))/ (re(z2)**2 + im(z2)**2))), _subst_fun_body=C((re(Var(0))*re(Var(1)) + im(Var(0))*im(Var(1)))/ (re(Var(1))**2 + im(Var(1))**2), (im(Var(0))*re(Var(1)) - re(Var(0))*im(Var(1)))/ (re(Var(1))**2 + im(Var(1))**2))), C.mul: Unfolding(name='C.mul', decl=C.mul, args=[z1!2, z2!3], body=C(re(z1!2)*re(z2!3) - im(z1!2)*im(z2!3), re(z1!2)*im(z2!3) + im(z1!2)*re(z2!3)), ax=|= ForAll([z1, z2], C.mul(z1, z2) == C(re(z1)*re(z2) - im(z1)*im(z2), re(z1)*im(z2) + im(z1)*re(z2))), _subst_fun_body=C(re(Var(0))*re(Var(1)) - im(Var(0))*im(Var(1)), re(Var(0))*im(Var(1)) + im(Var(0))*re(Var(1)))), C.norm2: Unfolding(name='C.norm2', decl=C.norm2, args=[z!1259], body=C.mul(z!1259, conj(z!1259)), ax=|= ForAll(z, C.norm2(z) == C.mul(z, conj(z))), _subst_fun_body=C.mul(Var(0), conj(Var(0)))), C.rezip: Unfolding(name='C.rezip', decl=C.rezip, args=[ab!1607], body=Lambda(i, C(_0(ab!1607)[i], _1(ab!1607)[i])), ax=|= ForAll(ab, C.rezip(ab) == (Lambda(i, C(_0(ab)[i], _1(ab)[i])))), _subst_fun_body=Lambda(i, C(_0(Var(1))[i], _1(Var(1))[i]))), C.unzip: Unfolding(name='C.unzip', decl=C.unzip, args=[a!1606], body=Tuple_Array_Array(Lambda(i, re(a!1606[i])), Lambda(i, im(a!1606[i]))), ax=|= ForAll(a, C.unzip(a) == Tuple_Array_Array(Lambda(i, re(a[i])), Lambda(i, im(a[i])))), _subst_fun_body=Tuple_Array_Array(Lambda(i, re(Var(1)[i])), Lambda(i, im(Var(1)[i])))), EPosReal.wf: Unfolding(name='EPosReal.wf', decl=EPosReal.wf, args=[x!1704], body=Implies(is(real, x!1704), val(x!1704) >= 0), ax=|= ForAll(x, EPosReal.wf(x) == Implies(is(real, x), val(x) >= 0)), _subst_fun_body=Implies(is(real, Var(0)), val(Var(0)) >= 0)), EReal.add: Unfolding(name='EReal.add', decl=EReal.add, args=[x!1631, y!1632], body=If(And(is(Real, x!1631), is(Real, y!1632)), Real(val(x!1631) + val(y!1632)), If(And(is(Inf, x!1631), Not(is(NegInf, y!1632))), Inf, If(And(Not(is(NegInf, x!1631)), is(Inf, y!1632)), Inf, If(And(is(NegInf, x!1631), Not(is(Inf, y!1632))), NegInf, If(And(Not(is(Inf, x!1631)), is(NegInf, y!1632)), NegInf, add_undef(x!1631, y!1632)))))), ax=|= ForAll([x, y], EReal.add(x, y) == If(And(is(Real, x), is(Real, y)), Real(val(x) + val(y)), If(And(is(Inf, x), Not(is(NegInf, y))), Inf, If(And(Not(is(NegInf, x)), is(Inf, y)), Inf, If(And(is(NegInf, x), Not(is(Inf, y))), NegInf, If(And(Not(is(Inf, x)), is(NegInf, y)), NegInf, add_undef(x, y))))))), _subst_fun_body=If(And(is(Real, Var(0)), is(Real, Var(1))), Real(val(Var(0)) + val(Var(1))), If(And(is(Inf, Var(0)), Not(is(NegInf, Var(1)))), Inf, If(And(Not(is(NegInf, Var(0))), is(Inf, Var(1))), Inf, If(And(is(NegInf, Var(0)), Not(is(Inf, Var(1)))), NegInf, If(And(Not(is(Inf, Var(0))), is(NegInf, Var(1))), NegInf, add_undef(Var(0), Var(1)))))))), EReal.le: Unfolding(name='EReal.le', decl=EReal.le, args=[x!1621, y!1622], body=If(x!1621 == y!1622, True, If(is(NegInf, x!1621), True, If(is(Inf, y!1622), True, If(is(NegInf, y!1622), False, If(is(Inf, x!1621), False, If(And(is(Real, x!1621), is(Real, y!1622)), val(x!1621) <= val(y!1622), unreachable!1620)))))), ax=|= ForAll([x, y], EReal.le(x, y) == If(x == y, True, If(is(NegInf, x), True, If(is(Inf, y), True, If(is(NegInf, y), False, If(is(Inf, x), False, If(And(is(Real, x), is(Real, y)), val(x) <= val(y), unreachable!1620))))))), _subst_fun_body=If(Var(0) == Var(1), True, If(is(NegInf, Var(0)), True, If(is(Inf, Var(1)), True, If(is(NegInf, Var(1)), False, If(is(Inf, Var(0)), False, If(And(is(Real, Var(0)), is(Real, Var(1))), val(Var(0)) <= val(Var(1)), unreachable!1620))))))), Interval.add: Unfolding(name='Interval.add', decl=Interval.add, args=[i!1909, j!1910], body=Interval(lo(i!1909) + lo(j!1910), hi(i!1909) + hi(j!1910)), ax=|= ForAll([i, j], Interval.add(i, j) == Interval(lo(i) + lo(j), hi(i) + hi(j))), _subst_fun_body=Interval(lo(Var(0)) + lo(Var(1)), hi(Var(0)) + hi(Var(1)))), Interval.mul: Unfolding(name='Interval.mul', decl=Interval.mul, args=[I!1923, J!1924], body=Interval(If(And(hi(I!1923)*hi(J!1924) <= lo(I!1923)*lo(J!1924), hi(I!1923)*hi(J!1924) <= hi(I!1923)*lo(J!1924), hi(I!1923)*hi(J!1924) <= lo(I!1923)*hi(J!1924)), hi(I!1923)*hi(J!1924), If(And(lo(I!1923)*lo(J!1924) <= hi(I!1923)*lo(J!1924), lo(I!1923)*lo(J!1924) <= lo(I!1923)*hi(J!1924)), lo(I!1923)*lo(J!1924), If(And(hi(I!1923)*lo(J!1924) <= lo(I!1923)*hi(J!1924)), hi(I!1923)*lo(J!1924), lo(I!1923)*hi(J!1924)))), If(And(hi(I!1923)*hi(J!1924) >= lo(I!1923)*lo(J!1924), hi(I!1923)*hi(J!1924) >= hi(I!1923)*lo(J!1924), hi(I!1923)*hi(J!1924) >= lo(I!1923)*hi(J!1924)), hi(I!1923)*hi(J!1924), If(And(lo(I!1923)*lo(J!1924) <= hi(I!1923)*lo(J!1924), lo(I!1923)*lo(J!1924) <= lo(I!1923)*hi(J!1924)), lo(I!1923)*lo(J!1924), If(And(hi(I!1923)*lo(J!1924) <= lo(I!1923)*hi(J!1924)), hi(I!1923)*lo(J!1924), lo(I!1923)*hi(J!1924))))), ax=|= ForAll([I, J], Interval.mul(I, J) == Interval(If(And(hi(I)*hi(J) <= lo(I)*lo(J), hi(I)*hi(J) <= hi(I)*lo(J), hi(I)*hi(J) <= lo(I)*hi(J)), hi(I)*hi(J), If(And(lo(I)*lo(J) <= hi(I)*lo(J), lo(I)*lo(J) <= lo(I)*hi(J)), lo(I)*lo(J), If(And(hi(I)*lo(J) <= lo(I)*hi(J)), hi(I)*lo(J), lo(I)*hi(J)))), If(And(hi(I)*hi(J) >= lo(I)*lo(J), hi(I)*hi(J) >= hi(I)*lo(J), hi(I)*hi(J) >= lo(I)*hi(J)), hi(I)*hi(J), If(And(lo(I)*lo(J) <= hi(I)*lo(J), lo(I)*lo(J) <= lo(I)*hi(J)), lo(I)*lo(J), If(And(hi(I)*lo(J) <= lo(I)*hi(J)), hi(I)*lo(J), lo(I)*hi(J)))))), _subst_fun_body=Interval(If(And(hi(Var(0))*hi(Var(1)) <= lo(Var(0))*lo(Var(1)), hi(Var(0))*hi(Var(1)) <= hi(Var(0))*lo(Var(1)), hi(Var(0))*hi(Var(1)) <= lo(Var(0))*hi(Var(1))), hi(Var(0))*hi(Var(1)), If(And(lo(Var(0))*lo(Var(1)) <= hi(Var(0))*lo(Var(1)), lo(Var(0))*lo(Var(1)) <= lo(Var(0))*hi(Var(1))), lo(Var(0))*lo(Var(1)), If(And(hi(Var(0))*lo(Var(1)) <= lo(Var(0))*hi(Var(1))), hi(Var(0))*lo(Var(1)), lo(Var(0))*hi(Var(1))))), If(And(hi(Var(0))*hi(Var(1)) >= lo(Var(0))*lo(Var(1)), hi(Var(0))*hi(Var(1)) >= hi(Var(0))*lo(Var(1)), hi(Var(0))*hi(Var(1)) >= lo(Var(0))*hi(Var(1))), hi(Var(0))*hi(Var(1)), If(And(lo(Var(0))*lo(Var(1)) <= hi(Var(0))*lo(Var(1)), lo(Var(0))*lo(Var(1)) <= lo(Var(0))*hi(Var(1))), lo(Var(0))*lo(Var(1)), If(And(hi(Var(0))*lo(Var(1)) <= lo(Var(0))*hi(Var(1))), hi(Var(0))*lo(Var(1)), lo(Var(0))*hi(Var(1))))))), Interval.setof: Unfolding(name='Interval.setof', decl=Interval.setof, args=[i!1890], body=Lambda(x, And(lo(i!1890) <= x, x <= hi(i!1890))), ax=|= ForAll(i, Interval.setof(i) == (Lambda(x, And(lo(i) <= x, x <= hi(i))))), _subst_fun_body=Lambda(x, And(lo(Var(1)) <= x, x <= hi(Var(1))))), Interval.sub: Unfolding(name='Interval.sub', decl=Interval.sub, args=[i!1916, j!1917], body=Interval(lo(i!1916) - hi(j!1917), hi(i!1916) - lo(j!1917)), ax=|= ForAll([i, j], Interval.sub(i, j) == Interval(lo(i) - hi(j), hi(i) - lo(j))), _subst_fun_body=Interval(lo(Var(0)) - hi(Var(1)), hi(Var(0)) - lo(Var(1)))), KnownBits_1.and_: Unfolding(name='KnownBits_1.and_', decl=KnownBits_1.and_, args=[x!742, y!743], body=KnownBits_1(ones(x!742) & ones(y!743), ~(~unknowns(x!742) & ~ones(x!742) | ~unknowns(y!743) & ~ones(y!743) | ones(x!742) & ones(y!743))), ax=|= ForAll([x, y], KnownBits_1.and_(x, y) == KnownBits_1(ones(x) & ones(y), ~(~unknowns(x) & ~ones(x) | ~unknowns(y) & ~ones(y) | ones(x) & ones(y)))), _subst_fun_body=KnownBits_1(ones(Var(0)) & ones(Var(1)), ~(~unknowns(Var(0)) & ~ones(Var(0)) | ~unknowns(Var(1)) & ~ones(Var(1)) | ones(Var(0)) & ones(Var(1))))), KnownBits_1.const: Unfolding(name='KnownBits_1.const', decl=KnownBits_1.const, args=[a!709], body=KnownBits_1(a!709, 0), ax=|= ForAll(a, KnownBits_1.const(a) == KnownBits_1(a, 0)), _subst_fun_body=KnownBits_1(Var(0), 0)), KnownBits_1.invert: Unfolding(name='KnownBits_1.invert', decl=KnownBits_1.invert, args=[x!733], body=KnownBits_1(~unknowns(x!733) & ~ones(x!733), unknowns(x!733)), ax=|= ForAll(x, KnownBits_1.invert(x) == KnownBits_1(~unknowns(x) & ~ones(x), unknowns(x))), _subst_fun_body=KnownBits_1(~unknowns(Var(0)) & ~ones(Var(0)), unknowns(Var(0)))), KnownBits_1.is_const: Unfolding(name='KnownBits_1.is_const', decl=KnownBits_1.is_const, args=[x!710], body=unknowns(x!710) == 0, ax=|= ForAll(x, KnownBits_1.is_const(x) == (unknowns(x) == 0)), _subst_fun_body=unknowns(Var(0)) == 0), KnownBits_1.or_: Unfolding(name='KnownBits_1.or_', decl=KnownBits_1.or_, args=[x!760, y!761], body=KnownBits_1(ones(x!760) | ones(y!761), ~(ones(x!760) | ones(y!761) | ~unknowns(x!760) & ~ones(x!760) & ~unknowns(y!761) & ~ones(y!761))), ax=|= ForAll([x, y], KnownBits_1.or_(x, y) == KnownBits_1(ones(x) | ones(y), ~(ones(x) | ones(y) | ~unknowns(x) & ~ones(x) & ~unknowns(y) & ~ones(y)))), _subst_fun_body=KnownBits_1(ones(Var(0)) | ones(Var(1)), ~(ones(Var(0)) | ones(Var(1)) | ~unknowns(Var(0)) & ~ones(Var(0)) & ~unknowns(Var(1)) & ~ones(Var(1))))), KnownBits_1.setof: Unfolding(name='KnownBits_1.setof', decl=KnownBits_1.setof, args=[x!707], body=Lambda(a, ~unknowns(x!707) & a == ones(x!707)), ax=|= ForAll(x, KnownBits_1.setof(x) == (Lambda(a, ~unknowns(x) & a == ones(x)))), _subst_fun_body=Lambda(a, ~unknowns(Var(1)) & a == ones(Var(1)))), KnownBits_1.wf: Unfolding(name='KnownBits_1.wf', decl=KnownBits_1.wf, args=[x!708], body=ones(x!708) & unknowns(x!708) == 0, ax=|= ForAll(x, KnownBits_1.wf(x) == (ones(x) & unknowns(x) == 0)), _subst_fun_body=ones(Var(0)) & unknowns(Var(0)) == 0), KnownBits_1.zeros: Unfolding(name='KnownBits_1.zeros', decl=KnownBits_1.zeros, args=[x!732], body=~unknowns(x!732) & ~ones(x!732), ax=|= ForAll(x, KnownBits_1.zeros(x) == ~unknowns(x) & ~ones(x)), _subst_fun_body=~unknowns(Var(0)) & ~ones(Var(0))), NDArray.add: Unfolding(name='NDArray.add', decl=NDArray.add, args=[u!2191, v!2192], body=If(shape(u!2191) == shape(v!2192), NDArray(shape(u!2191), Lambda(k, data(u!2191)[k] + data(v!2192)[k])), add_undef(u!2191, v!2192)), ax=|= ForAll([u, v], NDArray.add(u, v) == If(shape(u) == shape(v), NDArray(shape(u), Lambda(k, data(u)[k] + data(v)[k])), add_undef(u, v))), _subst_fun_body=If(shape(Var(0)) == shape(Var(1)), NDArray(shape(Var(0)), Lambda(k, data(Var(1))[k] + data(Var(2))[k])), add_undef(Var(0), Var(1)))), Nat.add: Unfolding(name='Nat.add', decl=Nat.add, args=[x!408, y!409], body=If(is(Z, x!408), y!409, S(Nat.add(pred(x!408), y!409))), ax=|= ForAll([x, y], Nat.add(x, y) == If(is(Z, x), y, S(Nat.add(pred(x), y)))), _subst_fun_body=If(is(Z, Var(0)), Var(1), S(Nat.add(pred(Var(0)), Var(1))))), PosEReal0.add: Unfolding(name='PosEReal0.add', decl=PosEReal0.add, args=[x!2134, y!2135], body=If(Or(is(Inf, x!2134), is(Inf, y!2135)), Inf, Fin(fin(x!2134) + fin(y!2135))), ax=|= ForAll([x, y], PosEReal0.add(x, y) == If(Or(is(Inf, x), is(Inf, y)), Inf, Fin(fin(x) + fin(y)))), _subst_fun_body=If(Or(is(Inf, Var(0)), is(Inf, Var(1))), Inf, Fin(fin(Var(0)) + fin(Var(1))))), PosEReal0.le: Unfolding(name='PosEReal0.le', decl=PosEReal0.le, args=[x!2146, y!2147], body=If(is(Inf, y!2147), True, If(is(Inf, x!2146), False, fin(x!2146) <= fin(y!2147))), ax=|= ForAll([x, y], PosEReal0.le(x, y) == If(is(Inf, y), True, If(is(Inf, x), False, fin(x) <= fin(y)))), _subst_fun_body=If(is(Inf, Var(1)), True, If(is(Inf, Var(0)), False, fin(Var(0)) <= fin(Var(1))))), PosEReal0.mul: Unfolding(name='PosEReal0.mul', decl=PosEReal0.mul, args=[x!2156, y!2157], body=If(Or(is(Inf, x!2156), is(Inf, y!2157)), Inf, Fin(fin(x!2156)*fin(y!2157))), ax=|= ForAll([x, y], PosEReal0.mul(x, y) == If(Or(is(Inf, x), is(Inf, y)), Inf, Fin(fin(x)*fin(y)))), _subst_fun_body=If(Or(is(Inf, Var(0)), is(Inf, Var(1))), Inf, Fin(fin(Var(0))*fin(Var(1))))), R.add: Unfolding(name='R.add', decl=R.add, args=[x!452, y!453], body=x!452 + y!453, ax=|= ForAll([x, y], R.add(x, y) == x + y), _subst_fun_body=Var(0) + Var(1)), R.dist: Unfolding(name='R.dist', decl=R.dist, args=[x!1240, y!1241], body=absR(x!1240 - y!1241), ax=|= ForAll([x!1218, y!1219], R.dist(x!1218, y!1219) == absR(x!1218 - y!1219)), _subst_fun_body=absR(Var(0) - Var(1))), R.max: Unfolding(name='R.max', decl=R.max, args=[x!542, y!543], body=If(x!542 >= y!543, x!542, y!543), ax=|= ForAll([x, y], R.max(x, y) == If(x >= y, x, y)), _subst_fun_body=If(Var(0) >= Var(1), Var(0), Var(1))), R.sqr: Unfolding(name='R.sqr', decl=R.sqr, args=[x!629], body=x!629*x!629, ax=|= ForAll(x, R.sqr(x) == x*x), _subst_fun_body=Var(0)*Var(0)), R.zero: Unfolding(name='R.zero', decl=R.zero, args=[], body=0, ax=|= R.zero == 0, _subst_fun_body=0), RFun.const: Unfolding(name='RFun.const', decl=RFun.const, args=[x!690], body=K(Real, x!690), ax=|= ForAll(x, RFun.const(x) == K(Real, x)), _subst_fun_body=K(Real, Var(0))), RFun.ident: Unfolding(name='RFun.ident', decl=RFun.ident, args=[], body=Lambda(x, x), ax=|= RFun.ident == (Lambda(x, x)), _subst_fun_body=Lambda(x, x)), RFun.smul: Unfolding(name='RFun.smul', decl=RFun.smul, args=[c!1705, f!1706], body=Lambda(x, mul(c!1705, f!1706[x])), ax=|= ForAll([c, f], RFun.smul(c, f) == (Lambda(x, mul(c, f[x])))), _subst_fun_body=Lambda(x, mul(Var(1), Var(2)[x]))), RSeq.const: Unfolding(name='RSeq.const', decl=RSeq.const, args=[x!1273], body=Lambda(i, x!1273), ax=|= ForAll(x, RSeq.const(x) == (Lambda(i, x))), _subst_fun_body=Lambda(i, Var(1))), RSeq.id: Unfolding(name='RSeq.id', decl=RSeq.id, args=[], body=Lambda(i, ToReal(i)), ax=|= RSeq.id == (Lambda(i, ToReal(i))), _subst_fun_body=Lambda(i, ToReal(i))), RSeq.rev: Unfolding(name='RSeq.rev', decl=RSeq.rev, args=[a!1281], body=Lambda(i, a!1281[-i]), ax=|= ForAll(a, RSeq.rev(a) == (Lambda(i, a[-i]))), _subst_fun_body=Lambda(i, Var(1)[-i])), RSeq.shift: Unfolding(name='RSeq.shift', decl=RSeq.shift, args=[a!1278, n!1279], body=Lambda(i, a!1278[i - n!1279]), ax=|= ForAll([a, n], RSeq.shift(a, n) == (Lambda(i, a[i - n]))), _subst_fun_body=Lambda(i, Var(1)[i - Var(2)])), RSeq.smul: Unfolding(name='RSeq.smul', decl=RSeq.smul, args=[x!1324, a!1325], body=Lambda(n, x!1324*a!1325[n]), ax=|= ForAll([x, a], RSeq.smul(x, a) == (Lambda(n, x*a[n]))), _subst_fun_body=Lambda(n, Var(1)*Var(2)[n])), RSeq.zero: Unfolding(name='RSeq.zero', decl=RSeq.zero, args=[], body=K(Int, 0), ax=|= RSeq.zero == K(Int, 0), _subst_fun_body=K(Int, 0)), RSet.has_lb: Unfolding(name='RSet.has_lb', decl=RSet.has_lb, args=[A!2115, M!2116], body=ForAll(x, Implies(A!2115[x], M!2116 <= x)), ax=|= ForAll([A, M], RSet.has_lb(A, M) == (ForAll(x, Implies(A[x], M <= x)))), _subst_fun_body=ForAll(x, Implies(Var(1)[x], Var(2) <= x))), RSet.inf: Unfolding(name='RSet.inf', decl=RSet.inf, args=[A!2117], body=RSet.sup(Lambda(x, RSet.has_lb(A!2117, x))), ax=|= ForAll(A, RSet.inf(A) == RSet.sup(Lambda(x, RSet.has_lb(A, x)))), _subst_fun_body=RSet.sup(Lambda(x, RSet.has_lb(Var(1), x)))), RSet.is_ub: Unfolding(name='RSet.is_ub', decl=RSet.is_ub, args=[A!2094], body=Exists(M, has_ub(A!2094, M)), ax=|= ForAll(A, RSet.is_ub(A) == (Exists(M, has_ub(A, M)))), _subst_fun_body=Exists(M, has_ub(Var(1), M))), RSet.sing: Unfolding(name='RSet.sing', decl=RSet.sing, args=[c!2113], body=Lambda(x, x == c!2113), ax=|= ForAll(c, RSet.sing(c) == (Lambda(x, x == c))), _subst_fun_body=Lambda(x, x == Var(1))), Real.sqrt: Unfolding(name='Real.sqrt', decl=Real.sqrt, args=[x!632], body=x!632**(1/2), ax=|= ForAll(x, Real.sqrt(x) == x**(1/2)), _subst_fun_body=Var(0)**(1/2)), T_Int.add: Unfolding(name='T_Int.add', decl=T_Int.add, args=[x!1151, y!1152], body=T_Int(Lambda(t, val(x!1151)[t] + val(y!1152)[t])), ax=|= ForAll([x, y], T_Int.add(x, y) == T_Int(Lambda(t, val(x)[t] + val(y)[t]))), _subst_fun_body=T_Int(Lambda(t, val(Var(1))[t] + val(Var(2))[t]))), T_Int.ge: Unfolding(name='T_Int.ge', decl=T_Int.ge, args=[x!1155, y!1156], body=T_Bool(Lambda(t, val(x!1155)[t] >= val(y!1156)[t])), ax=|= ForAll([x, y], T_Int.ge(x, y) == T_Bool(Lambda(t, val(x)[t] >= val(y)[t]))), _subst_fun_body=T_Bool(Lambda(t, val(Var(1))[t] >= val(Var(2))[t]))), T_Int.le: Unfolding(name='T_Int.le', decl=T_Int.le, args=[x!1157, y!1158], body=T_Bool(Lambda(t, val(x!1157)[t] <= val(y!1158)[t])), ax=|= ForAll([x, y], T_Int.le(x, y) == T_Bool(Lambda(t, val(x)[t] <= val(y)[t]))), _subst_fun_body=T_Bool(Lambda(t, val(Var(1))[t] <= val(Var(2))[t]))), T_Real.add: Unfolding(name='T_Real.add', decl=T_Real.add, args=[x!1153, y!1154], body=T_Real(Lambda(t, val(x!1153)[t] + val(y!1154)[t])), ax=|= ForAll([x, y], T_Real.add(x, y) == T_Real(Lambda(t, val(x)[t] + val(y)[t]))), _subst_fun_body=T_Real(Lambda(t, val(Var(1))[t] + val(Var(2))[t]))), Vec2.add: Unfolding(name='Vec2.add', decl=Vec2.add, args=[u!1736, v!1737], body=Vec2(x(u!1736) + x(v!1737), y(u!1736) + y(v!1737)), ax=|= ForAll([u, v], Vec2.add(u, v) == Vec2(x(u) + x(v), y(u) + y(v))), _subst_fun_body=Vec2(x(Var(0)) + x(Var(1)), y(Var(0)) + y(Var(1)))), Vec2.cross: Unfolding(name='Vec2.cross', decl=Vec2.cross, args=[u!1753, v!1754], body=x(u!1753)*y(v!1754) - y(u!1753)*x(v!1754), ax=|= ForAll([u, v], Vec2.cross(u, v) == x(u)*y(v) - y(u)*x(v)), _subst_fun_body=x(Var(0))*y(Var(1)) - y(Var(0))*x(Var(1))), Vec2.dist: Unfolding(name='Vec2.dist', decl=Vec2.dist, args=[u!1751, v!1752], body=Real.sqrt(Vec2.norm2(Vec2.sub(u!1751, v!1752))), ax=|= ForAll([u, v], Vec2.dist(u, v) == Real.sqrt(Vec2.norm2(Vec2.sub(u, v)))), _subst_fun_body=Real.sqrt(Vec2.norm2(Vec2.sub(Var(0), Var(1))))), Vec2.dot: Unfolding(name='Vec2.dot', decl=Vec2.dot, args=[u!1743, v!1744], body=x(u!1743)*x(v!1744) + y(u!1743)*y(v!1744), ax=|= ForAll([u, v], Vec2.dot(u, v) == x(u)*x(v) + y(u)*y(v)), _subst_fun_body=x(Var(0))*x(Var(1)) + y(Var(0))*y(Var(1))), Vec2.neg: Unfolding(name='Vec2.neg', decl=Vec2.neg, args=[u!1740], body=Vec2(-x(u!1740), -y(u!1740)), ax=|= ForAll(u, Vec2.neg(u) == Vec2(-x(u), -y(u))), _subst_fun_body=Vec2(-x(Var(0)), -y(Var(0)))), Vec2.norm: Unfolding(name='Vec2.norm', decl=Vec2.norm, args=[u!1746], body=Real.sqrt(Vec2.dot(u!1746, u!1746)), ax=|= ForAll(u, Vec2.norm(u) == Real.sqrt(Vec2.dot(u, u))), _subst_fun_body=Real.sqrt(Vec2.dot(Var(0), Var(0)))), Vec2.norm2: Unfolding(name='Vec2.norm2', decl=Vec2.norm2, args=[u!1745], body=Vec2.dot(u!1745, u!1745), ax=|= ForAll(u, Vec2.norm2(u) == Vec2.dot(u, u)), _subst_fun_body=Vec2.dot(Var(0), Var(0))), Vec2.smul: Unfolding(name='Vec2.smul', decl=Vec2.smul, args=[r!1741, u!1742], body=Vec2(r!1741*x(u!1742), r!1741*y(u!1742)), ax=|= ForAll([r, u], Vec2.smul(r, u) == Vec2(r*x(u), r*y(u))), _subst_fun_body=Vec2(Var(0)*x(Var(1)), Var(0)*y(Var(1)))), Vec2.sub: Unfolding(name='Vec2.sub', decl=Vec2.sub, args=[u!1738, v!1739], body=Vec2(x(u!1738) - x(v!1739), y(u!1738) - y(v!1739)), ax=|= ForAll([u, v], Vec2.sub(u, v) == Vec2(x(u) - x(v), y(u) - y(v))), _subst_fun_body=Vec2(x(Var(0)) - x(Var(1)), y(Var(0)) - y(Var(1)))), Vec3.add: Unfolding(name='Vec3.add', decl=Vec3.add, args=[u!2199, v!2200], body=Vec3(x(u!2199) + x(v!2200), y(u!2199) + y(v!2200), z(u!2199) + z(v!2200)), ax=|= ForAll([u, v], Vec3.add(u, v) == Vec3(x(u) + x(v), y(u) + y(v), z(u) + z(v))), _subst_fun_body=Vec3(x(Var(0)) + x(Var(1)), y(Var(0)) + y(Var(1)), z(Var(0)) + z(Var(1)))), Vec3.dist: Unfolding(name='Vec3.dist', decl=Vec3.dist, args=[u!2234, v!2235], body=Real.sqrt(Vec3.norm2(Vec3.sub(u!2234, v!2235))), ax=|= ForAll([u, v], Vec3.dist(u, v) == Real.sqrt(Vec3.norm2(Vec3.sub(u, v)))), _subst_fun_body=Real.sqrt(Vec3.norm2(Vec3.sub(Var(0), Var(1))))), Vec3.dot: Unfolding(name='Vec3.dot', decl=Vec3.dot, args=[u!2216, v!2217], body=x(u!2216)*x(v!2217) + y(u!2216)*y(v!2217) + z(u!2216)*z(v!2217), ax=|= ForAll([u, v], Vec3.dot(u, v) == x(u)*x(v) + y(u)*y(v) + z(u)*z(v)), _subst_fun_body=x(Var(0))*x(Var(1)) + y(Var(0))*y(Var(1)) + z(Var(0))*z(Var(1))), Vec3.neg: Unfolding(name='Vec3.neg', decl=Vec3.neg, args=[u!2205], body=Vec3(-x(u!2205), -y(u!2205), -z(u!2205)), ax=|= ForAll(u, Vec3.neg(u) == Vec3(-x(u), -y(u), -z(u))), _subst_fun_body=Vec3(-x(Var(0)), -y(Var(0)), -z(Var(0)))), Vec3.norm: Unfolding(name='Vec3.norm', decl=Vec3.norm, args=[u!2229], body=Real.sqrt(Vec3.norm2(u!2229)), ax=|= ForAll(u, Vec3.norm(u) == Real.sqrt(Vec3.norm2(u))), _subst_fun_body=Real.sqrt(Vec3.norm2(Var(0)))), Vec3.norm2: Unfolding(name='Vec3.norm2', decl=Vec3.norm2, args=[u!2225], body=Vec3.dot(u!2225, u!2225), ax=|= ForAll(u, Vec3.norm2(u) == Vec3.dot(u, u)), _subst_fun_body=Vec3.dot(Var(0), Var(0))), Vec3.smul: Unfolding(name='Vec3.smul', decl=Vec3.smul, args=[a!2206, u!2207], body=Vec3(a!2206*x(u!2207), a!2206*y(u!2207), a!2206*z(u!2207)), ax=|= ForAll([a, u], Vec3.smul(a, u) == Vec3(a*x(u), a*y(u), a*z(u))), _subst_fun_body=Vec3(Var(0)*x(Var(1)), Var(0)*y(Var(1)), Var(0)*z(Var(1)))), Vec3.sub: Unfolding(name='Vec3.sub', decl=Vec3.sub, args=[u!2203, v!2204], body=Vec3(x(u!2203) - x(v!2204), y(u!2203) - y(v!2204), z(u!2203) - z(v!2204)), ax=|= ForAll([u, v], Vec3.sub(u, v) == Vec3(x(u) - x(v), y(u) - y(v), z(u) - z(v))), _subst_fun_body=Vec3(x(Var(0)) - x(Var(1)), y(Var(0)) - y(Var(1)), z(Var(0)) - z(Var(1)))), ZF.elts: Unfolding(name='ZF.elts', decl=ZF.elts, args=[A!2265], body=Lambda(x, ∈(x, A!2265)), ax=|= ForAll(A, ZF.elts(A) == (Lambda(x, ∈(x, A)))), _subst_fun_body=Lambda(x, ∈(x, Var(1)))), ZF.sing: Unfolding(name='ZF.sing', decl=ZF.sing, args=[x!2338], body=upair(x!2338, x!2338), ax=|= ForAll(x, ZF.sing(x) == upair(x, x)), _subst_fun_body=upair(Var(0), Var(0))), ZFSet.le: Unfolding(name='ZFSet.le', decl=ZFSet.le, args=[A!2270, B!2271], body=ForAll(x, Implies(∈(x, A!2270), ∈(x, B!2271))), ax=|= ForAll([A, B], ZFSet.le(A, B) == (ForAll(x, Implies(∈(x, A), ∈(x, B))))), _subst_fun_body=ForAll(x, Implies(∈(x, Var(1)), ∈(x, Var(2))))), ZFSet.sub: Unfolding(name='ZFSet.sub', decl=ZFSet.sub, args=[A!2620, B!2621], body=sep(A!2620, Lambda(x, Not(∈(x, B!2621)))), ax=|= ForAll([A, B], ZFSet.sub(A, B) == sep(A, Lambda(x, Not(∈(x, B))))), _subst_fun_body=sep(Var(0), Lambda(x, Not(∈(x, Var(2)))))), ZSet.finite: Unfolding(name='ZSet.finite', decl=ZSet.finite, args=[A!352], body=And(ZSet.is_ub(A!352), ZSet.is_lb(A!352)), ax=|= ForAll(A, ZSet.finite(A) == And(ZSet.is_ub(A), ZSet.is_lb(A))), _subst_fun_body=And(ZSet.is_ub(Var(0)), ZSet.is_lb(Var(0)))), ZSet.has_lb: Unfolding(name='ZSet.has_lb', decl=ZSet.has_lb, args=[A!349, y!350], body=ForAll(x, Implies(A!349[x], y!350 <= x)), ax=|= ForAll([A, y], ZSet.has_lb(A, y) == (ForAll(x, Implies(A[x], y <= x)))), _subst_fun_body=ForAll(x, Implies(Var(1)[x], Var(2) <= x))), ZSet.has_ub: Unfolding(name='ZSet.has_ub', decl=ZSet.has_ub, args=[A!346, y!347], body=ForAll(x, Implies(A!346[x], x <= y!347)), ax=|= ForAll([A, y], ZSet.has_ub(A, y) == (ForAll(x, Implies(A[x], x <= y)))), _subst_fun_body=ForAll(x, Implies(Var(1)[x], x <= Var(2)))), ZSet.is_lb: Unfolding(name='ZSet.is_lb', decl=ZSet.is_lb, args=[A!351], body=Exists(x, ZSet.has_lb(A!351, x)), ax=|= ForAll(A, ZSet.is_lb(A) == (Exists(x, ZSet.has_lb(A, x)))), _subst_fun_body=Exists(x, ZSet.has_lb(Var(1), x))), ZSet.is_ub: Unfolding(name='ZSet.is_ub', decl=ZSet.is_ub, args=[A!348], body=Exists(x, ZSet.has_ub(A!348, x)), ax=|= ForAll(A, ZSet.is_ub(A) == (Exists(x, ZSet.has_ub(A, x)))), _subst_fun_body=Exists(x, ZSet.has_ub(Var(1), x))), abelian: Unfolding(name='abelian', decl=abelian, args=[], body=ForAll([x!808, y!809], mul(x!808, y!809) == mul(y!809, x!808)), ax=|= abelian == (ForAll([x!808, y!809], mul(x!808, y!809) == mul(y!809, x!808))), _subst_fun_body=ForAll([x!808, y!809], mul(x!808, y!809) == mul(y!809, x!808))), absR: Unfolding(name='absR', decl=absR, args=[x!475], body=If(x!475 >= 0, x!475, -x!475), ax=|= ForAll(x, absR(x) == If(x >= 0, x, -x)), _subst_fun_body=If(Var(0) >= 0, Var(0), -Var(0))), add_defined: Unfolding(name='add_defined', decl=add_defined, args=[x!1633, y!1634], body=Or(And(is(Real, x!1633), is(Real, y!1634)), And(is(Inf, x!1633), Not(is(NegInf, y!1634))), And(Not(is(NegInf, x!1633)), is(Inf, y!1634)), And(is(NegInf, x!1633), Not(is(Inf, y!1634))), And(Not(is(Inf, x!1633)), is(NegInf, y!1634))), ax=|= ForAll([x, y], add_defined(x, y) == Or(And(is(Real, x), is(Real, y)), And(is(Inf, x), Not(is(NegInf, y))), And(Not(is(NegInf, x)), is(Inf, y)), And(is(NegInf, x), Not(is(Inf, y))), And(Not(is(Inf, x)), is(NegInf, y)))), _subst_fun_body=Or(And(is(Real, Var(0)), is(Real, Var(1))), And(is(Inf, Var(0)), Not(is(NegInf, Var(1)))), And(Not(is(NegInf, Var(0))), is(Inf, Var(1))), And(is(NegInf, Var(0)), Not(is(Inf, Var(1)))), And(Not(is(Inf, Var(0))), is(NegInf, Var(1))))), aeq: Unfolding(name='aeq', decl=aeq, args=[x!518, y!519, eps!520], body=If(x!518 - y!519 > 0, x!518 - y!519, -(x!518 - y!519)) <= eps!520, ax=|= ForAll([x, y, eps], aeq(x, y, eps) == (If(x - y > 0, x - y, -(x - y)) <= eps)), _subst_fun_body=If(Var(0) - Var(1) > 0, Var(0) - Var(1), -(Var(0) - Var(1))) <= Var(2)), always: Unfolding(name='always', decl=always, args=[p!1175], body=T_Bool(Lambda(t!1173, ForAll(dt!1174, Implies(dt!1174 >= 0, val(p!1175)[t!1173 + dt!1174])))), ax=|= ForAll(p, always(p) == T_Bool(Lambda(t!1173, ForAll(dt!1174, Implies(dt!1174 >= 0, val(p)[t!1173 + dt!1174]))))), _subst_fun_body=T_Bool(Lambda(t!1173, ForAll(dt!1174, Implies(dt!1174 >= 0, val(Var(2))[t!1173 + dt!1174]))))), and1: Unfolding(name='and1', decl=and1, args=[x!783, y!784], body=not1(nand1(x!783, y!784)), ax=|= ForAll([x, y], and1(x, y) == not1(nand1(x, y))), _subst_fun_body=not1(nand1(Var(0), Var(1)))), and2: Unfolding(name='and2', decl=and2, args=[x!798, y!799], body=Concat(and1(Extract(1, 1, x!798), Extract(1, 1, y!799)), and1(Extract(0, 0, x!798), Extract(0, 0, y!799))), ax=|= ForAll([x, y], and2(x, y) == Concat(and1(Extract(1, 1, x), Extract(1, 1, y)), and1(Extract(0, 0, x), Extract(0, 0, y)))), _subst_fun_body=Concat(and1(Extract(1, 1, Var(0)), Extract(1, 1, Var(1))), and1(Extract(0, 0, Var(0)), Extract(0, 0, Var(1))))), ball: Unfolding(name='ball', decl=ball, args=[x!1223, r!1224], body=Lambda(y!1222, absR(y!1222 - x!1223) < r!1224), ax=|= ForAll([x!1218, r!1221], ball(x!1218, r!1221) == (Lambda(y!1222, absR(y!1222 - x!1218) < r!1221))), _subst_fun_body=Lambda(y!1222, absR(y!1222 - Var(1)) < Var(2))), bchoose: Unfolding(name='bchoose', decl=bchoose, args=[P!342, N!343], body=downsearch(P!342, N!343), ax=|= ForAll([P, N], bchoose(P, N) == downsearch(P, N)), _subst_fun_body=downsearch(Var(0), Var(1))), beq: Unfolding(name='beq', decl=beq, args=[p!1179, q!1180], body=T_Bool(Lambda(t!1178, val(p!1179)[t!1178] == val(q!1180)[t!1178])), ax=|= ForAll([p, q], beq(p, q) == T_Bool(Lambda(t!1178, val(p)[t!1178] == val(q)[t!1178]))), _subst_fun_body=T_Bool(Lambda(t!1178, val(Var(1))[t!1178] == val(Var(2))[t!1178]))), between: Unfolding(name='between', decl=between, args=[p!1857, q!1858, a!1859], body=And(collinear(p!1857, q!1858, a!1859), Vec2.dot(Vec2.sub(q!1858, p!1857), Vec2.sub(a!1859, p!1857)) >= 0, Vec2.dot(Vec2.sub(p!1857, q!1858), Vec2.sub(a!1859, q!1858)) >= 0), ax=|= ForAll([p, q, a], between(p, q, a) == And(collinear(p, q, a), Vec2.dot(Vec2.sub(q, p), Vec2.sub(a, p)) >= 0, Vec2.dot(Vec2.sub(p, q), Vec2.sub(a, q)) >= 0)), _subst_fun_body=And(collinear(Var(0), Var(1), Var(2)), Vec2.dot(Vec2.sub(Var(1), Var(0)), Vec2.sub(Var(2), Var(0))) >= 0, Vec2.dot(Vec2.sub(Var(0), Var(1)), Vec2.sub(Var(2), Var(1))) >= 0)), bexist: Unfolding(name='bexist', decl=bexist, args=[A!1099, n!1100], body=If(n!1100 < 0, False, Or(A!1099[n!1100], bexists(A!1099, n!1100 - 1))), ax=|= ForAll([A, n], bexist(A, n) == If(n < 0, False, Or(A[n], bexists(A, n - 1)))), _subst_fun_body=If(Var(1) < 0, False, Or(Var(0)[Var(1)], bexists(Var(0), Var(1) - 1)))), bexists: Unfolding(name='bexists', decl=bexists, args=[P!344, N!345], body=P!344[bchoose(P!344, N!345)], ax=|= ForAll([P, N], bexists(P, N) == P[bchoose(P, N)]), _subst_fun_body=Var(0)[bchoose(Var(0), Var(1))]), bforall: Unfolding(name='bforall', decl=bforall, args=[A!1101, n!1102], body=If(n!1102 < 0, True, And(A!1101[n!1102], bforall(A!1101, n!1102 - 1))), ax=|= ForAll([A, n], bforall(A, n) == If(n < 0, True, And(A[n], bforall(A, n - 1)))), _subst_fun_body=If(Var(1) < 0, True, And(Var(0)[Var(1)], bforall(Var(0), Var(1) - 1)))), biginter: Unfolding(name='biginter', decl=biginter, args=[a!2455], body=sep(pick(a!2455), Lambda(x, ForAll(b!2261, Implies(∈(b!2261, a!2455), ∈(x, b!2261))))), ax=|= ForAll(a!2260, biginter(a!2260) == sep(pick(a!2260), Lambda(x, ForAll(b!2261, Implies(∈(b!2261, a!2260), ∈(x, b!2261)))))), _subst_fun_body=sep(pick(Var(0)), Lambda(x, ForAll(b!2261, Implies(∈(b!2261, Var(2)), ∈(x, b!2261)))))), cauchy_mod: Unfolding(name='cauchy_mod', decl=cauchy_mod, args=[a!1525, mod!1526], body=ForAll(eps, Implies(eps > 0, ForAll([m, k], Implies(And(m > mod!1526[eps], k > mod!1526[eps]), absR(a!1525[m] - a!1525[k]) < eps)))), ax=|= ForAll([a, mod], cauchy_mod(a, mod) == (ForAll(eps, Implies(eps > 0, ForAll([m, k], Implies(And(m > mod[eps], k > mod[eps]), absR(a[m] - a[k]) < eps)))))), _subst_fun_body=ForAll(eps, Implies(eps > 0, ForAll([m, k], Implies(And(m > Var(4)[eps], k > Var(4)[eps]), absR(Var(3)[m] - Var(3)[k]) < eps))))), ceil: Unfolding(name='ceil', decl=ceil, args=[x!607], body=-floor(-x!607), ax=|= ForAll(x, ceil(x) == -floor(-x)), _subst_fun_body=-floor(-Var(0))), cinter: Unfolding(name='cinter', decl=cinter, args=[An!2131], body=Lambda(x, ForAll(n, An!2131[n][x])), ax=|= ForAll(An, cinter(An) == (Lambda(x, ForAll(n, An[n][x])))), _subst_fun_body=Lambda(x, ForAll(n, Var(2)[n][x]))), circle: Unfolding(name='circle', decl=circle, args=[c!1887, p!1888], body=Lambda(q, Vec2.norm2(Vec2.sub(p!1888, c!1887)) == Vec2.norm2(Vec2.sub(q, c!1887))), ax=|= ForAll([c, p], circle(c, p) == (Lambda(q, Vec2.norm2(Vec2.sub(p, c)) == Vec2.norm2(Vec2.sub(q, c))))), _subst_fun_body=Lambda(q, Vec2.norm2(Vec2.sub(Var(2), Var(1))) == Vec2.norm2(Vec2.sub(q, Var(1))))), closed: Unfolding(name='closed', decl=closed, args=[A!1096], body=(Array Sierpinski Bool).open(complement(A!1096)), ax=|= ForAll(A, closed(A) == (Array Sierpinski Bool).open(complement(A))), _subst_fun_body=(Array Sierpinski Bool).open(complement(Var(0)))), closed_int: Unfolding(name='closed_int', decl=closed_int, args=[x!2109, y!2110], body=Lambda(z, And(x!2109 <= z, z <= y!2110)), ax=|= ForAll([x, y], closed_int(x, y) == (Lambda(z, And(x <= z, z <= y)))), _subst_fun_body=Lambda(z, And(Var(1) <= z, z <= Var(2)))), collinear: Unfolding(name='collinear', decl=collinear, args=[p!1852, q!1853, a!1854], body=Vec2.cross(Vec2.sub(q!1853, p!1852), Vec2.sub(a!1854, p!1852)) == 0, ax=|= ForAll([p, q, a], collinear(p, q, a) == (Vec2.cross(Vec2.sub(q, p), Vec2.sub(a, p)) == 0)), _subst_fun_body=Vec2.cross(Vec2.sub(Var(1), Var(0)), Vec2.sub(Var(2), Var(0))) == 0), comp: Unfolding(name='comp', decl=comp, args=[f!688, g!689], body=Lambda(x, f!688[g!689[x]]), ax=|= ForAll([f, g], comp(f, g) == (Lambda(x, f[g[x]]))), _subst_fun_body=Lambda(x, Var(1)[Var(2)[x]])), cong_seg: Unfolding(name='cong_seg', decl=cong_seg, args=[p!1861, q!1862, a!1863, b!1864], body=Vec2.norm2(Vec2.sub(p!1861, q!1862)) == Vec2.norm2(Vec2.sub(a!1863, b!1864)), ax=|= ForAll([p, q, a, b], cong_seg(p, q, a, b) == (Vec2.norm2(Vec2.sub(p, q)) == Vec2.norm2(Vec2.sub(a, b)))), _subst_fun_body=Vec2.norm2(Vec2.sub(Var(0), Var(1))) == Vec2.norm2(Vec2.sub(Var(2), Var(3)))), conj: Unfolding(name='conj', decl=conj, args=[z!1249], body=C(re(z!1249), -im(z!1249)), ax=|= ForAll(z, conj(z) == C(re(z), -im(z))), _subst_fun_body=C(re(Var(0)), -im(Var(0)))), cont_at: Unfolding(name='cont_at', decl=cont_at, args=[f!696, x!697], body=ForAll(eps, Implies(eps > 0, Exists(delta, And(delta > 0, ForAll(y, Implies(absR(x!697 - y) < delta, absR(f!696[x!697] - f!696[y]) < eps)))))), ax=|= ForAll([f, x], cont_at(f, x) == (ForAll(eps, Implies(eps > 0, Exists(delta, And(delta > 0, ForAll(y, Implies(absR(x - y) < delta, absR(f[x] - f[y]) < eps)))))))), _subst_fun_body=ForAll(eps, Implies(eps > 0, Exists(delta, And(delta > 0, ForAll(y, Implies(absR(Var(4) - y) < delta, absR(Var(3)[Var(4)] - Var(3)[y]) < eps))))))), cross: Unfolding(name='cross', decl=cross, args=[u!2238, v!2239], body=Vec3(y(u!2238)*z(v!2239) - z(u!2238)*y(v!2239), z(u!2238)*x(v!2239) - x(u!2238)*z(v!2239), x(u!2238)*y(v!2239) - y(u!2238)*x(v!2239)), ax=|= ForAll([u, v], cross(u, v) == Vec3(y(u)*z(v) - z(u)*y(v), z(u)*x(v) - x(u)*z(v), x(u)*y(v) - y(u)*x(v))), _subst_fun_body=Vec3(y(Var(0))*z(Var(1)) - z(Var(0))*y(Var(1)), z(Var(0))*x(Var(1)) - x(Var(0))*z(Var(1)), x(Var(0))*y(Var(1)) - y(Var(0))*x(Var(1)))), cumsum: Unfolding(name='cumsum', decl=cumsum, args=[a!1340], body=Lambda(i, If(i == 0, a!1340[0], If(i > 0, cumsum(a!1340)[i - 1] + a!1340[i], -cumsum(a!1340)[i + 1] - a!1340[i + 1]))), ax=|= ForAll(a, cumsum(a) == (Lambda(i, If(i == 0, a[0], If(i > 0, cumsum(a)[i - 1] + a[i], -cumsum(a)[i + 1] - a[i + 1]))))), _subst_fun_body=Lambda(i, If(i == 0, Var(1)[0], If(i > 0, cumsum(Var(1))[i - 1] + Var(1)[i], -cumsum(Var(1))[i + 1] - Var(1)[i + 1])))), cunion: Unfolding(name='cunion', decl=cunion, args=[An!2132], body=Lambda(x, Exists(n, An!2132[n][x])), ax=|= ForAll(An, cunion(An) == (Lambda(x, Exists(n, An[n][x])))), _subst_fun_body=Lambda(x, Exists(n, Var(2)[n][x]))), decimate: Unfolding(name='decimate', decl=decimate, args=[a!1284, n!1285], body=Lambda(i, a!1284[i*n!1285]), ax=|= ForAll([a, n], decimate(a, n) == (Lambda(i, a[i*n]))), _subst_fun_body=Lambda(i, Var(1)[i*Var(2)])), delay: Unfolding(name='delay', decl=delay, args=[a!1280], body=RSeq.shift(a!1280, 1), ax=|= ForAll(a, delay(a) == RSeq.shift(a, 1)), _subst_fun_body=RSeq.shift(Var(0), 1)), diff: Unfolding(name='diff', decl=diff, args=[a!1295], body=Lambda(i, a!1295[i + 1] - a!1295[i]), ax=|= ForAll(a, diff(a) == (Lambda(i, a[i + 1] - a[i]))), _subst_fun_body=Lambda(i, Var(1)[i + 1] - Var(1)[i])), diff_at: Unfolding(name='diff_at', decl=diff_at, args=[f!694, x!695], body=Exists(y, has_diff_at(f!694, x!695, y)), ax=|= ForAll([f, x], diff_at(f, x) == (Exists(y, has_diff_at(f, x, y)))), _subst_fun_body=Exists(y, has_diff_at(Var(1), Var(2), y))), dilate: Unfolding(name='dilate', decl=dilate, args=[a!1282, n!1283], body=Lambda(i, a!1282[i/n!1283]), ax=|= ForAll([a, n], dilate(a, n) == (Lambda(i, a[i/n]))), _subst_fun_body=Lambda(i, Var(1)[i/Var(2)])), dir: Unfolding(name='dir', decl=dir, args=[p!1873], body=ray(Vec2(ToReal(0), ToReal(0)), p!1873), ax=|= ForAll(p, dir(p) == ray(Vec2(ToReal(0), ToReal(0)), p)), _subst_fun_body=ray(Vec2(ToReal(0), ToReal(0)), Var(0))), dom: Unfolding(name='dom', decl=dom, args=[R!2786], body=sep(bigunion(bigunion(R!2786)), Lambda(x, Exists(y, ∈(pair(x, y), R!2786)))), ax=|= ForAll(R, dom(R) == sep(bigunion(bigunion(R)), Lambda(x, Exists(y, ∈(pair(x, y), R))))), _subst_fun_body=sep(bigunion(bigunion(Var(0))), Lambda(x, Exists(y, ∈(pair(x, y), Var(2)))))), dommap: Unfolding(name='dommap', decl=dommap, args=[F!1276, a!1277], body=Lambda(i, a!1277[F!1276[i]]), ax=|= ForAll([F, a], dommap(F, a) == (Lambda(i, a[F[i]]))), _subst_fun_body=Lambda(i, Var(2)[Var(1)[i]])), double: Unfolding(name='double', decl=double, args=[n!407], body=If(is(Z, n!407), Z, S(S(double(pred(n!407))))), ax=|= ForAll(n, double(n) == If(is(Z, n), Z, S(S(double(pred(n)))))), _subst_fun_body=If(is(Z, Var(0)), Z, S(S(double(pred(Var(0))))))), downsearch: Unfolding(name='downsearch', decl=downsearch, args=[P!340, n!341], body=If(P!340[n!341], n!341, If(n!341 < 0, 0, downsearch(P!340, n!341 - 1))), ax=|= ForAll([P, n!314], downsearch(P, n!314) == If(P[n!314], n!314, If(n!314 < 0, 0, downsearch(P, n!314 - 1)))), _subst_fun_body=If(Var(0)[Var(1)], Var(1), If(Var(1) < 0, 0, downsearch(Var(0), Var(1) - 1)))), dvd: Unfolding(name='dvd', decl=dvd, args=[n!311, m!312], body=Exists(p, m!312 == n!311*p), ax=|= ForAll([n, m], dvd(n, m) == (Exists(p, m == n*p))), _subst_fun_body=Exists(p, Var(2) == Var(1)*p)), empty: Unfolding(name='empty', decl=empty, args=[], body=Interval(1, 0), ax=|= empty == Interval(1, 0), _subst_fun_body=Interval(1, 0)), even: Unfolding(name='even', decl=even, args=[x!284], body=Exists(y, x!284 == 2*y), ax=|= ForAll(x, even(x) == (Exists(y, x == 2*y))), _subst_fun_body=Exists(y, Var(1) == 2*y)), eventually: Unfolding(name='eventually', decl=eventually, args=[p!1172], body=T_Bool(Lambda(t!1170, Exists(dt!1171, And(dt!1171 >= 0, val(p!1172)[t!1170 + dt!1171])))), ax=|= ForAll(p, eventually(p) == T_Bool(Lambda(t!1170, Exists(dt!1171, And(dt!1171 >= 0, val(p)[t!1170 + dt!1171]))))), _subst_fun_body=T_Bool(Lambda(t!1170, Exists(dt!1171, And(dt!1171 >= 0, val(Var(2))[t!1170 + dt!1171]))))), expi: Unfolding(name='expi', decl=expi, args=[t!1260], body=C(Real.cos(t!1260), Real.sin(t!1260)), ax=|= ForAll(t, expi(t) == C(Real.cos(t), Real.sin(t))), _subst_fun_body=C(Real.cos(Var(0)), Real.sin(Var(0)))), eye: Unfolding(name='eye', decl=eye, args=[n!1482], body=Lambda([i, j], If(i == j, 1, 0)), ax=|= ForAll(n, eye(n) == (Lambda([i, j], If(i == j, 1, 0)))), _subst_fun_body=Lambda([i, j], If(i == j, 1, 0))), fact: Unfolding(name='fact', decl=fact, args=[n!313], body=If(n!313 <= 0, 1, n!313*fact(n!313 - 1)), ax=|= ForAll(n, fact(n) == If(n <= 0, 1, n*fact(n - 1))), _subst_fun_body=If(Var(0) <= 0, 1, Var(0)*fact(Var(0) - 1))), finsum: Unfolding(name='finsum', decl=finsum, args=[a!1436, n!1437], body=cumsum(a!1436)[n!1437], ax=|= ForAll([a, n], finsum(a, n) == cumsum(a)[n]), _subst_fun_body=cumsum(Var(0))[Var(1)]), floor: Unfolding(name='floor', decl=floor, args=[x!590], body=ToReal(ToInt(x!590)), ax=|= ForAll(x, floor(x) == ToReal(ToInt(x))), _subst_fun_body=ToReal(ToInt(Var(0)))), from_int: Unfolding(name='from_int', decl=from_int, args=[a!385], body=If(a!385 <= 0, Z, S(from_int(a!385 - 1))), ax=|= ForAll(a, from_int(a) == If(a <= 0, Z, S(from_int(a - 1)))), _subst_fun_body=If(Var(0) <= 0, Z, S(from_int(Var(0) - 1)))), from_nat: Unfolding(name='from_nat', decl=from_nat, args=[n!354], body=If(n!354%2 == 0, n!354/2, -(n!354 + 1)/2), ax=|= ForAll(n!314, from_nat(n!314) == If(n!314%2 == 0, n!314/2, -(n!314 + 1)/2)), _subst_fun_body=If(Var(0)%2 == 0, Var(0)/2, -(Var(0) + 1)/2)), fst: Unfolding(name='fst', decl=fst, args=[p!2658], body=pick(biginter(p!2658)), ax=|= ForAll(p!2264, fst(p!2264) == pick(biginter(p!2264))), _subst_fun_body=pick(biginter(Var(0)))), geom: Unfolding(name='geom', decl=geom, args=[x!1274], body=Lambda(i, pownat(x!1274, i)), ax=|= ForAll(x, geom(x) == (Lambda(i, pownat(x, i)))), _subst_fun_body=Lambda(i, pownat(Var(1), i))), has_bound: Unfolding(name='has_bound', decl=has_bound, args=[a!1519, M!1520], body=ForAll(n, absR(a!1519[n]) <= M!1520), ax=|= ForAll([a, M], has_bound(a, M) == (ForAll(n, absR(a[n]) <= M))), _subst_fun_body=ForAll(n, absR(Var(1)[n]) <= Var(2))), has_lim: Unfolding(name='has_lim', decl=has_lim, args=[a!1527, y!1528], body=ForAll(eps, Implies(eps > 0, Exists(N, ForAll(n, Implies(n > N, absR(a!1527[n] - y!1528) < eps))))), ax=|= ForAll([a, y], has_lim(a, y) == (ForAll(eps, Implies(eps > 0, Exists(N, ForAll(n, Implies(n > N, absR(a[n] - y) < eps))))))), _subst_fun_body=ForAll(eps, Implies(eps > 0, Exists(N, ForAll(n, Implies(n > N, absR(Var(3)[n] - Var(4)) < eps)))))), has_lim_at: Unfolding(name='has_lim_at', decl=has_lim_at, args=[f!691, p!692, L!693], body=ForAll(eps, Implies(0 < eps, Exists(delta, And(delta > 0, ForAll(x, Implies(And(0 < absR(x - p!692), absR(x - p!692) < delta), absR(f!691[x] - L!693) < eps)))))), ax=|= ForAll([f, p, L], has_lim_at(f, p, L) == (ForAll(eps, Implies(0 < eps, Exists(delta, And(delta > 0, ForAll(x, Implies(And(0 < absR(x - p), absR(x - p) < delta), absR(f[x] - L) < eps)))))))), _subst_fun_body=ForAll(eps, Implies(0 < eps, Exists(delta, And(delta > 0, ForAll(x, Implies(And(0 < absR(x - Var(4)), absR(x - Var(4)) < delta), absR(Var(3)[x] - Var(5)) < eps))))))), has_max: Unfolding(name='has_max', decl=has_max, args=[A!2105, x!2106], body=And(A!2105[x!2106], RSet.sup(A!2105) == x!2106), ax=|= ForAll([A, x], has_max(A, x) == And(A[x], RSet.sup(A) == x)), _subst_fun_body=And(Var(0)[Var(1)], RSet.sup(Var(0)) == Var(1))), has_sum: Unfolding(name='has_sum', decl=has_sum, args=[a!1572, y!1573], body=has_lim(cumsum(a!1572), y!1573), ax=|= ForAll([a, y], has_sum(a, y) == has_lim(cumsum(a), y)), _subst_fun_body=has_lim(cumsum(Var(0)), Var(1))), has_ub: Unfolding(name='has_ub', decl=has_ub, args=[A!2092, M!2093], body=ForAll(x, Implies(A!2092[x], x <= M!2093)), ax=|= ForAll([A, M], has_ub(A, M) == (ForAll(x, Implies(A[x], x <= M)))), _subst_fun_body=ForAll(x, Implies(Var(1)[x], x <= Var(2)))), iand: Unfolding(name='iand', decl=iand, args=[a!1106, b!1107], body=Prop(Lambda(w, And(val(a!1106)[w], val(b!1107)[w]))), ax=|= ForAll([a, b], iand(a, b) == Prop(Lambda(w, And(val(a)[w], val(b)[w])))), _subst_fun_body=Prop(Lambda(w, And(val(Var(1))[w], val(Var(2))[w])))), ieq: Unfolding(name='ieq', decl=ieq, args=[x!1206, y!1207], body=T_Bool(Lambda(t!1205, val(x!1206)[t!1205] == val(y!1207)[t!1205])), ax=|= ForAll([x, y], ieq(x, y) == T_Bool(Lambda(t!1205, val(x)[t!1205] == val(y)[t!1205]))), _subst_fun_body=T_Bool(Lambda(t!1205, val(Var(1))[t!1205] == val(Var(2))[t!1205]))), if_int: Unfolding(name='if_int', decl=if_int, args=[p!1214, x!1215, y!1216], body=T_Int(Lambda(t!1213, If(val(p!1214)[t!1213], val(x!1215)[t!1213], val(y!1216)[t!1213]))), ax=|= ForAll([p, x, y], if_int(p, x, y) == T_Int(Lambda(t!1213, If(val(p)[t!1213], val(x)[t!1213], val(y)[t!1213])))), _subst_fun_body=T_Int(Lambda(t!1213, If(val(Var(1))[t!1213], val(Var(2))[t!1213], val(Var(3))[t!1213])))), iimpl: Unfolding(name='iimpl', decl=iimpl, args=[a!1110, b!1111], body=Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(a!1110)[u], val(b!1111)[u]))))), ax=|= ForAll([a, b], iimpl(a, b) == Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(a)[u], val(b)[u])))))), _subst_fun_body=Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(Var(2))[u], val(Var(3))[u])))))), ind: Unfolding(name='ind', decl=ind, args=[S!1275], body=Lambda(i, If(S!1275[i], 1, 0)), ax=|= ForAll(S, ind(S) == (Lambda(i, If(S[i], 1, 0)))), _subst_fun_body=Lambda(i, If(Var(1)[i], 1, 0))), ineq: Unfolding(name='ineq', decl=ineq, args=[x!1209, y!1210], body=T_Bool(Lambda(t!1208, val(x!1209)[t!1208] != val(y!1210)[t!1208])), ax=|= ForAll([x, y], ineq(x, y) == T_Bool(Lambda(t!1208, val(x)[t!1208] != val(y)[t!1208]))), _subst_fun_body=T_Bool(Lambda(t!1208, val(Var(1))[t!1208] != val(Var(2))[t!1208]))), inext: Unfolding(name='inext', decl=inext, args=[x!1212], body=T_Int(Lambda(t!1211, val(x!1212)[t!1211 + 1])), ax=|= ForAll(x, inext(x) == T_Int(Lambda(t!1211, val(x)[t!1211 + 1]))), _subst_fun_body=T_Int(Lambda(t!1211, val(Var(1))[t!1211 + 1]))), inot: Unfolding(name='inot', decl=inot, args=[a!1112], body=Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(a!1112)[u], val(Prop(K(World, False)))[u]))))), ax=|= ForAll(a, inot(a) == Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(a)[u], val(Prop(K(World, False)))[u])))))), _subst_fun_body=Prop(Lambda(w, ForAll(u, Implies(acc(w, u), Implies(val(Var(2))[u], val(Prop(K(World, False)))[u])))))), inter: Unfolding(name='inter', decl=inter, args=[A!2473, B!2474], body=sep(A!2473, Lambda(x, ∈(x, B!2474))), ax=|= ForAll([A, B], inter(A, B) == sep(A, Lambda(x, ∈(x, B)))), _subst_fun_body=sep(Var(0), Lambda(x, ∈(x, Var(2))))), ior: Unfolding(name='ior', decl=ior, args=[a!1108, b!1109], body=Prop(Lambda(w, Or(val(a!1108)[w], val(b!1109)[w]))), ax=|= ForAll([a, b], ior(a, b) == Prop(Lambda(w, Or(val(a)[w], val(b)[w])))), _subst_fun_body=Prop(Lambda(w, Or(val(Var(1))[w], val(Var(2))[w])))), is_bounded: Unfolding(name='is_bounded', decl=is_bounded, args=[a!1521], body=Exists(M, has_bound(a!1521, M)), ax=|= ForAll(a, is_bounded(a) == (Exists(M, has_bound(a, M)))), _subst_fun_body=Exists(M, has_bound(Var(1), M))), is_cauchy: Unfolding(name='is_cauchy', decl=is_cauchy, args=[a!1524], body=ForAll(eps, Implies(eps > 0, Exists(N, ForAll([m, k], Implies(And(m > N, k > N), absR(a!1524[m] - a!1524[k]) < eps))))), ax=|= ForAll(a, is_cauchy(a) == (ForAll(eps, Implies(eps > 0, Exists(N, ForAll([m, k], Implies(And(m > N, k > N), absR(a[m] - a[k]) < eps))))))), _subst_fun_body=ForAll(eps, Implies(eps > 0, Exists(N, ForAll([m, k], Implies(And(m > N, k > N), absR(Var(4)[m] - Var(4)[k]) < eps)))))), is_circle: Unfolding(name='is_circle', decl=is_circle, args=[A!1889], body=Exists([c, q], ForAll(p, circle(c, q)[p] == A!1889[p])), ax=|= ForAll(A, is_circle(A) == (Exists([c, q], ForAll(p, circle(c, q)[p] == A[p])))), _subst_fun_body=Exists([c, q], ForAll(p, circle(c, q)[p] == Var(3)[p]))), is_cont: Unfolding(name='is_cont', decl=is_cont, args=[f!702], body=ForAll(x, cont_at(f!702, x)), ax=|= ForAll(f, is_cont(f) == (ForAll(x, cont_at(f, x)))), _subst_fun_body=ForAll(x, cont_at(Var(1), x))), is_conv: Unfolding(name='is_conv', decl=is_conv, args=[a!1529], body=Exists(y, has_lim(a!1529, y)), ax=|= ForAll(a, is_conv(a) == (Exists(y, has_lim(a, y)))), _subst_fun_body=Exists(y, has_lim(Var(1), y))), is_diff: Unfolding(name='is_diff', decl=is_diff, args=[f!701], body=ForAll(x, diff_at(f!701, x)), ax=|= ForAll(f, is_diff(f) == (ForAll(x, diff_at(f, x)))), _subst_fun_body=ForAll(x, diff_at(Var(1), x))), is_dir: Unfolding(name='is_dir', decl=is_dir, args=[A!1874], body=Exists(p, And(p != Vec2(ToReal(0), ToReal(0)), Vec2.norm2(p) == 1, dir(p) == A!1874)), ax=|= ForAll(A, is_dir(A) == (Exists(p, And(p != Vec2(ToReal(0), ToReal(0)), Vec2.norm2(p) == 1, dir(p) == A)))), _subst_fun_body=Exists(p, And(p != Vec2(ToReal(0), ToReal(0)), Vec2.norm2(p) == 1, dir(p) == Var(1)))), is_empty: Unfolding(name='is_empty', decl=is_empty, args=[I!1906], body=hi(i) > lo(i), ax=|= ForAll(I, is_empty(I) == (hi(i) > lo(i))), _subst_fun_body=hi(i) > lo(i)), is_idem: Unfolding(name='is_idem', decl=is_idem, args=[A!971], body=mul(A!971, A!971) == A!971, ax=|= ForAll(A, is_idem(A) == (mul(A, A) == A)), _subst_fun_body=mul(Var(0), Var(0)) == Var(0)), is_line: Unfolding(name='is_line', decl=is_line, args=[A!1816], body=Exists([p, q], And(p != q, ForAll(a, line(p, q)[a] == A!1816[a]))), ax=|= ForAll(A, is_line(A) == (Exists([p, q], And(p != q, ForAll(a, line(p, q)[a] == A[a]))))), _subst_fun_body=Exists([p, q], And(p != q, ForAll(a, line(p, q)[a] == Var(3)[a])))), is_linear: Unfolding(name='is_linear', decl=is_linear, args=[f!1707], body=And(ForAll([x, y], f!1707[R.add(x, y)] == R.add(f!1707[x], f!1707[y])), ForAll([x, c], f!1707[mul(c, x)] == mul(c, f!1707[x]))), ax=|= ForAll(f, is_linear(f) == And(ForAll([x, y], f[R.add(x, y)] == R.add(f[x], f[y])), ForAll([x, c], f[mul(c, x)] == mul(c, f[x])))), _subst_fun_body=And(ForAll([x, y], Var(2)[R.add(x, y)] == R.add(Var(2)[x], Var(2)[y])), ForAll([x, c], Var(2)[mul(c, x)] == mul(c, Var(2)[x])))), is_lub: Unfolding(name='is_lub', decl=is_lub, args=[A!1702, x!1703], body=And(upper_bound(A!1702, x!1703), ForAll(y, Implies(upper_bound(A!1702, y), EReal.le(x!1703, y)))), ax=|= ForAll([A, x], is_lub(A, x) == And(upper_bound(A, x), ForAll(y, Implies(upper_bound(A, y), EReal.le(x, y))))), _subst_fun_body=And(upper_bound(Var(0), Var(1)), ForAll(y, Implies(upper_bound(Var(1), y), EReal.le(Var(2), y))))), is_monotone: Unfolding(name='is_monotone', decl=is_monotone, args=[a!1522], body=ForAll([n, m], Implies(n <= m, a!1522[n] <= a!1522[m])), ax=|= ForAll(a, is_monotone(a) == (ForAll([n, m], Implies(n <= m, a[n] <= a[m])))), _subst_fun_body=ForAll([n, m], Implies(n <= m, Var(2)[n] <= Var(2)[m]))), is_nonneg: Unfolding(name='is_nonneg', decl=is_nonneg, args=[a!1523], body=ForAll(n, a!1523[n] >= 0), ax=|= ForAll(a, is_nonneg(a) == (ForAll(n, a[n] >= 0))), _subst_fun_body=ForAll(n, Var(1)[n] >= 0)), is_open: Unfolding(name='is_open', decl=is_open, args=[S!1225], body=ForAll(x!1218, Implies(S!1225[x!1218], Exists(r!1221, And(0 < r!1221, subset(ball(x!1218, r!1221), S!1225))))), ax=|= ForAll(S, is_open(S) == (ForAll(x!1218, Implies(S[x!1218], Exists(r!1221, And(0 < r!1221, subset(ball(x!1218, r!1221), S))))))), _subst_fun_body=ForAll(x!1218, Implies(Var(1)[x!1218], Exists(r!1221, And(0 < r!1221, subset(ball(x!1218, r!1221), Var(2))))))), is_orth: Unfolding(name='is_orth', decl=is_orth, args=[A!970], body=mul(A!970, trans(A!970)) == I, ax=|= ForAll(A, is_orth(A) == (mul(A, trans(A)) == I)), _subst_fun_body=mul(Var(0), trans(Var(0))) == I), is_pair: Unfolding(name='is_pair', decl=is_pair, args=[c!2657], body=Exists([a!2260, b!2261], pair(a!2260, b!2261) == c!2657), ax=|= ForAll(c!2262, is_pair(c!2262) == (Exists([a!2260, b!2261], pair(a!2260, b!2261) == c!2262))), _subst_fun_body=Exists([a!2260, b!2261], pair(a!2260, b!2261) == Var(2))), is_parallel: Unfolding(name='is_parallel', decl=is_parallel, args=[A!1848, B!1849], body=Vec2.cross(Vec2.sub(f!1844(A!1848), f!1845(A!1848)), Vec2.sub(f!1844(B!1849), f!1845(B!1849))) == 0, ax=|= ForAll([A, B], is_parallel(A, B) == (Vec2.cross(Vec2.sub(f!1844(A), f!1845(A)), Vec2.sub(f!1844(B), f!1845(B))) == 0)), _subst_fun_body=Vec2.cross(Vec2.sub(f!1844(Var(0)), f!1845(Var(0))), Vec2.sub(f!1844(Var(1)), f!1845(Var(1)))) == 0), is_perp: Unfolding(name='is_perp', decl=is_perp, args=[A!1846, B!1847], body=Vec2.dot(Vec2.sub(f!1844(A!1846), f!1845(A!1846)), Vec2.sub(f!1844(B!1847), f!1845(B!1847))) == 0, ax=|= ForAll([A, B], is_perp(A, B) == (Vec2.dot(Vec2.sub(f!1844(A), f!1845(A)), Vec2.sub(f!1844(B), f!1845(B))) == 0)), _subst_fun_body=Vec2.dot(Vec2.sub(f!1844(Var(0)), f!1845(Var(0))), Vec2.sub(f!1844(Var(1)), f!1845(Var(1)))) == 0), is_point: Unfolding(name='is_point', decl=is_point, args=[A!1813], body=Exists(p, point(p) == A!1813), ax=|= ForAll(A, is_point(A) == (Exists(p, point(p) == A))), _subst_fun_body=Exists(p, point(p) == Var(1))), is_rel: Unfolding(name='is_rel', decl=is_rel, args=[R!2785], body=ForAll(p!2264, Implies(∈(p!2264, R!2785), is_pair(p!2264))), ax=|= ForAll(R, is_rel(R) == (ForAll(p!2264, Implies(∈(p!2264, R), is_pair(p!2264))))), _subst_fun_body=ForAll(p!2264, Implies(∈(p!2264, Var(1)), is_pair(p!2264)))), is_set: Unfolding(name='is_set', decl=is_set, args=[P!2268], body=Exists(A, reflects(P!2268, A)), ax=|= ForAll(P, is_set(P) == (Exists(A, reflects(P, A)))), _subst_fun_body=Exists(A, reflects(Var(1), A))), is_summable: Unfolding(name='is_summable', decl=is_summable, args=[a!1574], body=Exists(y, has_sum(a!1574, y)), ax=|= ForAll(a, is_summable(a) == (Exists(y, has_sum(a, y)))), _subst_fun_body=Exists(y, has_sum(Var(1), y))), is_symm: Unfolding(name='is_symm', decl=is_symm, args=[A!969], body=trans(A!969) == A!969, ax=|= ForAll(A, is_symm(A) == (trans(A) == A)), _subst_fun_body=trans(Var(0)) == Var(0)), iterate: Unfolding(name='iterate', decl=iterate, args=[f!1286, x!1287], body=Lambda(i, If(i <= 0, x!1287, f!1286[iterate(f!1286, x!1287)[ToReal(i - 1)]])), ax=|= ForAll([f, x], iterate(f, x) == (Lambda(i, If(i <= 0, x, f[iterate(f, x)[ToReal(i - 1)]])))), _subst_fun_body=Lambda(i, If(i <= 0, Var(2), Var(1)[iterate(Var(1), Var(2))[ToReal(i - 1)]]))), join: Unfolding(name='join', decl=join, args=[i!1900, j!1901], body=Interval(min(lo(i!1900), lo(j!1901)), R.max(hi(i!1900), hi(j!1901))), ax=|= ForAll([i, j], join(i, j) == Interval(min(lo(i), lo(j)), R.max(hi(i), hi(j)))), _subst_fun_body=Interval(min(lo(Var(0)), lo(Var(1))), R.max(hi(Var(0)), hi(Var(1))))), krondelta: Unfolding(name='krondelta', decl=krondelta, args=[n!1291], body=Lambda(i, If(i == n!1291, 1, 0)), ax=|= ForAll(n, krondelta(n) == (Lambda(i, If(i == n, 1, 0)))), _subst_fun_body=Lambda(i, If(i == Var(1), 1, 0))), lim: Unfolding(name='lim', decl=lim, args=[A!1531], body=f!1530(A!1531), ax=|= ForAll(A!1270, lim(A!1270) == f!1530(A!1270)), _subst_fun_body=f!1530(Var(0))), line: Unfolding(name='line', decl=line, args=[p!1814, q!1815], body=Lambda(a, Vec2.cross(Vec2.sub(a, p!1814), Vec2.sub(q!1815, p!1814)) == 0), ax=|= ForAll([p, q], line(p, q) == (Lambda(a, Vec2.cross(Vec2.sub(a, p), Vec2.sub(q, p)) == 0))), _subst_fun_body=Lambda(a, Vec2.cross(Vec2.sub(a, Var(1)), Vec2.sub(Var(2), Var(1))) == 0)), max: Unfolding(name='max', decl=max, args=[a!1391], body=Lambda(i, If(i == 0, a!1391[0], If(i > 0, R.max(max(a!1391)[i - 1], a!1391[i]), R.max(max(a!1391)[i + 1], a!1391[i])))), ax=|= ForAll(a, max(a) == (Lambda(i, If(i == 0, a[0], If(i > 0, R.max(max(a)[i - 1], a[i]), R.max(max(a)[i + 1], a[i])))))), _subst_fun_body=Lambda(i, If(i == 0, Var(1)[0], If(i > 0, R.max(max(Var(1))[i - 1], Var(1)[i]), R.max(max(Var(1))[i + 1], Var(1)[i]))))), meet: Unfolding(name='meet', decl=meet, args=[i!1891, j!1892], body=Interval(R.max(lo(i!1891), lo(j!1892)), min(hi(i!1891), hi(j!1892))), ax=|= ForAll([i, j], meet(i, j) == Interval(R.max(lo(i), lo(j)), min(hi(i), hi(j)))), _subst_fun_body=Interval(R.max(lo(Var(0)), lo(Var(1))), min(hi(Var(0)), hi(Var(1))))), mid: Unfolding(name='mid', decl=mid, args=[i!1908], body=(lo(i!1908) + hi(i!1908))/2, ax=|= ForAll(i, mid(i) == (lo(i) + hi(i))/2), _subst_fun_body=(lo(Var(0)) + hi(Var(0)))/2), midp: Unfolding(name='midp', decl=midp, args=[p!1850, q!1851], body=Vec2.smul(1/2, Vec2.add(p!1850, q!1851)), ax=|= ForAll([p, q], midp(p, q) == Vec2.smul(1/2, Vec2.add(p, q))), _subst_fun_body=Vec2.smul(1/2, Vec2.add(Var(0), Var(1)))), min: Unfolding(name='min', decl=min, args=[x!561, y!562], body=If(x!561 <= y!562, x!561, y!562), ax=|= ForAll([x, y], min(x, y) == If(x <= y, x, y)), _subst_fun_body=If(Var(0) <= Var(1), Var(0), Var(1))), mu: Unfolding(name='mu', decl=mu, args=[A!1105], body=mu_iter(A!1105, 0), ax=|= ForAll(A, mu(A) == mu_iter(A, 0)), _subst_fun_body=mu_iter(Var(0), 0)), mu_iter: Unfolding(name='mu_iter', decl=mu_iter, args=[A!1103, n!1104], body=If(A!1103[n!1104], n!1104, mu_iter(A!1103, n!1104 + 1)), ax=|= ForAll([A, n], mu_iter(A, n) == If(A[n], n, mu_iter(A, n + 1))), _subst_fun_body=If(Var(0)[Var(1)], Var(1), mu_iter(Var(0), Var(1) + 1))), mul: Unfolding(name='mul', decl=mul, args=[x!463, y!464], body=x!463*y!464, ax=|= ForAll([x, y], mul(x, y) == x*y), _subst_fun_body=Var(0)*Var(1)), nand1: Unfolding(name='nand1', decl=nand1, args=[x!778, y!779], body=~(x!778 & y!779), ax=|= ForAll([x, y], nand1(x, y) == ~(x & y)), _subst_fun_body=~(Var(0) & Var(1))), nand2: Unfolding(name='nand2', decl=nand2, args=[x!793, y!794], body=Concat(nand1(Extract(1, 1, x!793), Extract(1, 1, y!794)), nand1(Extract(0, 0, x!793), Extract(0, 0, y!794))), ax=|= ForAll([x, y], nand2(x, y) == Concat(nand1(Extract(1, 1, x), Extract(1, 1, y)), nand1(Extract(0, 0, x), Extract(0, 0, y)))), _subst_fun_body=Concat(nand1(Extract(1, 1, Var(0)), Extract(1, 1, Var(1))), nand1(Extract(0, 0, Var(0)), Extract(0, 0, Var(1))))), nand2step: Unfolding(name='nand2step', decl=nand2step, args=[st!120], body=If(rom(st!120)[pc(st!120)] >> 15 == 0, Nand2State(pc(st!120) + 1, rom(st!120), ram(st!120), D(st!120), rom(st!120)[pc(st!120)]), Nand2State(If(If(rom(st!120)[pc(st!120)] & 7 == 0, False, If(rom(st!120)[pc(st!120)] & 7 == 1, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 42, 0, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 63, 1, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 58, 65535, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 12, D(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 48, A(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 112, rom(st!120)[A(st!120)], If(rom(st!120)[pc(st!120)] >> 6 & 127 == 13, ~D(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 49, ~A(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 113, ~rom(st!120)[A(st!120)], If(rom(...)[pc(...)] >> 6 & 127 == 15, -D(st!120), If(...[...] >> 6 & 127 == 51, -A(st!120), If(... >> ... & 127 == 115, -rom(...)[A(...)], If(... & ... == 31, D(...) + 1, If(... == ..., ... + ..., If(..., ..., ...))))))))))))))) > 0, If(rom(st!120)[pc(st!120)] & 7 == 2, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 42, 0, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 63, 1, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 58, 65535, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 12, D(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 48, A(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 112, rom(st!120)[A(st!120)], If(rom(st!120)[pc(st!120)] >> 6 & 127 == 13, ~D(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 49, ~A(st!120), If(rom(...)[pc(...)] >> 6 & 127 == 113, ~rom(st!120)[A(st!120)], If(...[...] >> 6 & 127 == 15, -D(st!120), If(... >> ... & 127 == 51, -A(st!120), If(... & ... == 115, -...[...], If(... == ..., ... + ..., If(..., ..., ...)))))))))))))) == 0, If(rom(st!120)[pc(st!120)] & 7 == 3, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 42, 0, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 63, 1, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 58, 65535, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 12, D(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 48, A(st!120), If(rom(st!120)[pc(st!120)] >> 6 & 127 == 112, rom(st!120)[A(st!120)], If(rom(st!120)[pc(st!120)] >> 6 & 127 == 13, ~D(st!120), If(rom(...)[pc(...)] >> 6 & 127 == 49, ~A(st!120), If(...[...] >> 6 & 127 == 113, ~rom(st!120)[A(st!120)], If(... >> ... & 127 == 15, -D(st!120), If(... & ... == 51, -A(...), If(... == ..., -..., If(..., ..., ...))))))))))))) >= 0, If(rom(st!120)[pc(st!120)] & 7 == 4, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 42, 0, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 63, 1, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 58, 65535, If(rom(st!120)[pc(st!120)] >> 6 & 127 == 12, ..., ax=|= ForAll(st, nand2step(st) == If(rom(st)[pc(st)] >> 15 == 0, Nand2State(pc(st) + 1, rom(st), ram(st), D(st), rom(st)[pc(st)]), Nand2State(If(If(rom(st)[pc(st)] & 7 == 0, False, If(rom(st)[pc(st)] & 7 == 1, If(rom(st)[pc(st)] >> 6 & 127 == 42, 0, If(rom(st)[pc(st)] >> 6 & 127 == 63, 1, If(rom(st)[pc(st)] >> 6 & 127 == 58, 65535, If(rom(st)[pc(st)] >> 6 & 127 == 12, D(st), If(rom(st)[pc(st)] >> 6 & 127 == 48, A(st), If(rom(st)[pc(st)] >> 6 & 127 == 112, rom(st)[A(st)], If(rom(st)[pc(st)] >> 6 & 127 == 13, ~D(st), If(rom(...)[pc(...)] >> 6 & 127 == 49, ~A(st), If(...[...] >> 6 & 127 == 113, ~rom(st)[A(st)], If(... >> ... & 127 == 15, -D(st), If(... & ... == 51, -A(...), If(... == ..., -..., If(..., ..., ...))))))))))))) > 0, If(rom(st)[pc(st)] & 7 == 2, If(rom(st)[pc(st)] >> 6 & 127 == 42, 0, If(rom(st)[pc(st)] >> 6 & 127 == 63, 1, If(rom(st)[pc(st)] >> 6 & 127 == 58, 65535, If(rom(st)[pc(st)] >> 6 & 127 == 12, D(st), If(rom(st)[pc(st)] >> 6 & 127 == 48, A(st), If(rom(st)[pc(st)] >> 6 & 127 == 112, rom(st)[A(st)], If(rom(...)[pc(...)] >> 6 & 127 == 13, ~D(st), If(...[...] >> 6 & 127 == 49, ~A(st), If(... >> ... & 127 == 113, ~rom(...)[A(...)], If(... & ... == 15, -D(...), If(... == ..., -..., If(..., ..., ...)))))))))))) == 0, If(rom(st)[pc(st)] & 7 == 3, If(rom(st)[pc(st)] >> 6 & 127 == 42, 0, If(rom(st)[pc(st)] >> 6 & 127 == 63, 1, If(rom(st)[pc(st)] >> 6 & 127 == 58, 65535, If(rom(st)[pc(st)] >> 6 & 127 == 12, D(st), If(rom(st)[pc(st)] >> 6 & 127 == 48, A(st), If(rom(...)[pc(...)] >> 6 & 127 == 112, rom(st)[A(st)], If(...[...] >> 6 & 127 == 13, ~D(st), If(... >> ... & 127 == 49, ~A(st), If(... & ... == 113, ~...[...], If(... == ..., -..., If(..., ..., ...))))))))))) >= 0, If(rom(st)[pc(st)] & 7 == 4, If(rom(st)[pc(st)] >> 6 & 127 == 42, 0, If(rom(st)[pc(st)] >> 6 & 127 == 63, 1, If(rom(st)[pc(st)] >> 6 & 127 == 58, 65535, If(rom(st)[pc(st)] >> 6 & 127 == 12, D(st), If(rom(...)[pc(...)] >> 6 & 127 == 48, A(st), If(...[...] >> 6 & 127 == 112, rom(st)[A(st)], If(... >> ... & 127 == 13, ~D(st), If(... & ... == 49, ~A(...), If(... == ..., ~..., If(..., ..., ...)))))))))) < 0, If(rom(st)[pc(st)] & 7 == 5, If(rom(st)[pc(st)] >> 6 & 127 == 42, 0, If(rom(st)[pc(st)] >> 6 & ..., _subst_fun_body=If(rom(Var(0))[pc(Var(0))] >> 15 == 0, Nand2State(pc(Var(0)) + 1, rom(Var(0)), ram(Var(0)), D(Var(0)), rom(Var(0))[pc(Var(0))]), Nand2State(If(If(rom(Var(0))[pc(Var(0))] & 7 == 0, False, If(rom(Var(0))[pc(Var(0))] & 7 == 1, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 42, 0, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 63, 1, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 58, 65535, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 12, D(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 48, A(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 112, rom(Var(0))[A(Var(0))], If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 13, ~D(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 49, ~A(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 113, ~rom(Var(0))[A(Var(0))], If(rom(...)[pc(...)] >> 6 & 127 == 15, -D(Var(0)), If(...[...] >> 6 & 127 == 51, -A(Var(0)), If(... >> ... & 127 == 115, -rom(...)[A(...)], If(... & ... == 31, D(...) + 1, If(... == ..., ... + ..., If(..., ..., ...))))))))))))))) > 0, If(rom(Var(0))[pc(Var(0))] & 7 == 2, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 42, 0, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 63, 1, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 58, 65535, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 12, D(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 48, A(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 112, rom(Var(0))[A(Var(0))], If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 13, ~D(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 49, ~A(Var(0)), If(rom(...)[pc(...)] >> 6 & 127 == 113, ~rom(Var(0))[A(Var(0))], If(...[...] >> 6 & 127 == 15, -D(Var(0)), If(... >> ... & 127 == 51, -A(Var(0)), If(... & ... == 115, -...[...], If(... == ..., ... + ..., If(..., ..., ...)))))))))))))) == 0, If(rom(Var(0))[pc(Var(0))] & 7 == 3, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 42, 0, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 63, 1, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 58, 65535, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 12, D(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 48, A(Var(0)), If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 112, rom(Var(0))[A(Var(0))], If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 13, ~D(Var(0)), If(rom(...)[pc(...)] >> 6 & 127 == 49, ~A(Var(0)), If(...[...] >> 6 & 127 == 113, ~rom(Var(0))[A(Var(0))], If(... >> ... & 127 == 15, -D(Var(0)), If(... & ... == 51, -A(...), If(... == ..., -..., If(..., ..., ...))))))))))))) >= 0, If(rom(Var(0))[pc(Var(0))] & 7 == 4, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 42, 0, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 63, 1, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 58, 65535, If(rom(Var(0))[pc(Var(0))] >> 6 & 127 == 12, ...), natpair: Unfolding(name='natpair', decl=natpair, args=[n!361, m!362], body=((n!361 + m!362)*(n!361 + m!362 + 1))/2 + m!362, ax=|= ForAll([n!314, m!324], natpair(n!314, m!324) == ((n!314 + m!324)*(n!314 + m!324 + 1))/2 + m!324), _subst_fun_body=((Var(0) + Var(1))*(Var(0) + Var(1) + 1))/2 + Var(1)), natpair_to_tuple: Unfolding(name='natpair_to_tuple', decl=natpair_to_tuple, args=[k!371], body=If(k!371 < 0, Tuple_Int_Int(-1, -1), If(k!371 == 0, Tuple_Int_Int(0, 0), If(_0(natpair_to_tuple(k!371 - 1)) == 0, Tuple_Int_Int(_1(natpair_to_tuple(k!371 - 1)) + 1, 0), Tuple_Int_Int(_0(natpair_to_tuple(k!371 - 1)) - 1, _1(natpair_to_tuple(k!371 - 1)) + 1)))), ax=|= ForAll(k!315, natpair_to_tuple(k!315) == If(k!315 < 0, Tuple_Int_Int(-1, -1), If(k!315 == 0, Tuple_Int_Int(0, 0), If(_0(natpair_to_tuple(k!315 - 1)) == 0, Tuple_Int_Int(_1(natpair_to_tuple(k!315 - 1)) + 1, 0), Tuple_Int_Int(_0(natpair_to_tuple(k!315 - 1)) - 1, _1(natpair_to_tuple(k!315 - 1)) + 1))))), _subst_fun_body=If(Var(0) < 0, Tuple_Int_Int(-1, -1), If(Var(0) == 0, Tuple_Int_Int(0, 0), If(_0(natpair_to_tuple(Var(0) - 1)) == 0, Tuple_Int_Int(_1(natpair_to_tuple(Var(0) - 1)) + 1, 0), Tuple_Int_Int(_0(natpair_to_tuple(Var(0) - 1)) - 1, _1(natpair_to_tuple(Var(0) - 1)) + 1))))), nested: Unfolding(name='nested', decl=nested, args=[An!2133], body=ForAll(n, subset(An!2133[n + 1], An!2133[n])), ax=|= ForAll(An, nested(An) == (ForAll(n, subset(An[n + 1], An[n])))), _subst_fun_body=ForAll(n, subset(Var(1)[n + 1], Var(1)[n]))), next: Unfolding(name='next', decl=next, args=[p!1177], body=T_Bool(Lambda(t!1176, val(p!1177)[t!1176 + 1])), ax=|= ForAll(p, next(p) == T_Bool(Lambda(t!1176, val(p)[t!1176 + 1]))), _subst_fun_body=T_Bool(Lambda(t!1176, val(Var(1))[t!1176 + 1]))), nonneg: Unfolding(name='nonneg', decl=nonneg, args=[x!539], body=absR(x!539) == x!539, ax=|= ForAll(x, nonneg(x) == (absR(x) == x)), _subst_fun_body=absR(Var(0)) == Var(0)), not1: Unfolding(name='not1', decl=not1, args=[x!780], body=nand1(x!780, x!780), ax=|= ForAll(x, not1(x) == nand1(x, x)), _subst_fun_body=nand1(Var(0), Var(0))), not2: Unfolding(name='not2', decl=not2, args=[x!795], body=Concat(not1(Extract(1, 1, x!795)), not1(Extract(0, 0, x!795))), ax=|= ForAll(x, not2(x) == Concat(not1(Extract(1, 1, x)), not1(Extract(0, 0, x)))), _subst_fun_body=Concat(not1(Extract(1, 1, Var(0))), not1(Extract(0, 0, Var(0))))), odd: Unfolding(name='odd', decl=odd, args=[x!285], body=Exists(y, x!285 == 2*y + 1), ax=|= ForAll(x, odd(x) == (Exists(y, x == 2*y + 1))), _subst_fun_body=Exists(y, Var(1) == 2*y + 1)), one: Unfolding(name='one', decl=one, args=[], body=1, ax=|= one == 1, _subst_fun_body=1), ones: Unfolding(name='ones', decl=ones, args=[n!2190], body=NDArray(Unit(n!2190), K(Int, 1)), ax=|= ForAll(n, ones(n) == NDArray(Unit(n), K(Int, 1))), _subst_fun_body=NDArray(Unit(Var(0)), K(Int, 1))), open_int: Unfolding(name='open_int', decl=open_int, args=[x!2107, y!2108], body=Lambda(z, And(x!2107 < z, z < y!2108)), ax=|= ForAll([x, y], open_int(x, y) == (Lambda(z, And(x < z, z < y)))), _subst_fun_body=Lambda(z, And(Var(1) < z, z < Var(2)))), or1: Unfolding(name='or1', decl=or1, args=[x!788, y!789], body=nand1(not1(x!788), not1(y!789)), ax=|= ForAll([x, y], or1(x, y) == nand1(not1(x), not1(y))), _subst_fun_body=nand1(not1(Var(0)), not1(Var(1)))), or2: Unfolding(name='or2', decl=or2, args=[x!803, y!804], body=Concat(or1(Extract(1, 1, x!803), Extract(1, 1, y!804)), or1(Extract(0, 0, x!803), Extract(0, 0, y!804))), ax=|= ForAll([x, y], or2(x, y) == Concat(or1(Extract(1, 1, x), Extract(1, 1, y)), or1(Extract(0, 0, x), Extract(0, 0, y)))), _subst_fun_body=Concat(or1(Extract(1, 1, Var(0)), Extract(1, 1, Var(1))), or1(Extract(0, 0, Var(0)), Extract(0, 0, Var(1))))), pair: Unfolding(name='pair', decl=pair, args=[n!383], body=from_nat(natpair(to_nat(n!383), to_nat(m!324))), ax=|= ForAll(n!314, pair(n!314) == from_nat(natpair(to_nat(n!314), to_nat(m!324)))), _subst_fun_body=from_nat(natpair(to_nat(Var(0)), to_nat(m!324)))), pair: Unfolding(name='pair', decl=pair, args=[a!2655, b!2656], body=upair(ZF.sing(a!2655), upair(a!2655, b!2656)), ax=|= ForAll([a!2260, b!2261], pair(a!2260, b!2261) == upair(ZF.sing(a!2260), upair(a!2260, b!2261))), _subst_fun_body=upair(ZF.sing(Var(0)), upair(Var(0), Var(1)))), perp: Unfolding(name='perp', decl=perp, args=[u!2236, v!2237], body=Vec3.dot(u!2236, v!2237) == 0, ax=|= ForAll([u, v], perp(u, v) == (Vec3.dot(u, v) == 0)), _subst_fun_body=Vec3.dot(Var(0), Var(1)) == 0), pick: Unfolding(name='pick', decl=pick, args=[a!2285], body=f!2284(a!2285), ax=|= ForAll(a!2260, pick(a!2260) == f!2284(a!2260)), _subst_fun_body=f!2284(Var(0))), point: Unfolding(name='point', decl=point, args=[p!1812], body=Store(K(Vec2, False), p!1812, True), ax=|= ForAll(p, point(p) == Store(K(Vec2, False), p, True)), _subst_fun_body=Store(K(Vec2, False), Var(0), True)), pos: Unfolding(name='pos', decl=pos, args=[a!1294], body=Lambda(i, If(i >= 0, a!1294[i], 0)), ax=|= ForAll(a, pos(a) == (Lambda(i, If(i >= 0, a[i], 0)))), _subst_fun_body=Lambda(i, If(i >= 0, Var(1)[i], 0))), pow: Unfolding(name='pow', decl=pow, args=[x!613, y!614], body=x!613**y!614, ax=|= ForAll([x, y], pow(x, y) == x**y), _subst_fun_body=Var(0)**Var(1)), powers: Unfolding(name='powers', decl=powers, args=[x!1575], body=Lambda(i, If(i == 0, 1, If(i < 0, powers(x!1575)[i + 1]/x!1575, x!1575*powers(x!1575)[i - 1]))), ax=|= ForAll(x, powers(x) == (Lambda(i, If(i == 0, 1, If(i < 0, powers(x)[i + 1]/x, x*powers(x)[i - 1]))))), _subst_fun_body=Lambda(i, If(i == 0, 1, If(i < 0, powers(Var(1))[i + 1]/Var(1), Var(1)*powers(Var(1))[i - 1])))), pownat: Unfolding(name='pownat', decl=pownat, args=[x!622, n!623], body=If(n!623 == 0, 1, If(n!623 > 0, x!622*pownat(x!622, n!623 - 1), pownat(x!622, n!623 + 1)/x!622)), ax=|= ForAll([x, n], pownat(x, n) == If(n == 0, 1, If(n > 0, x*pownat(x, n - 1), pownat(x, n + 1)/x))), _subst_fun_body=If(Var(1) == 0, 1, If(Var(1) > 0, Var(0)*pownat(Var(0), Var(1) - 1), pownat(Var(0), Var(1) + 1)/Var(0)))), prime: Unfolding(name='prime', decl=prime, args=[n!310], body=And(n!310 > 1, Not(Exists([p, q], And(p > 1, q > 1, n!310 == p*q)))), ax=|= ForAll(n, prime(n) == And(n > 1, Not(Exists([p, q], And(p > 1, q > 1, n == p*q))))), _subst_fun_body=And(Var(0) > 1, Not(Exists([p, q], And(p > 1, q > 1, Var(2) == p*q))))), prod: Unfolding(name='prod', decl=prod, args=[A!2712, B!2713], body=sep(pow(pow(union(A!2712, B!2713))), Lambda(p!2264, Exists([x, y], And(And(∈(x, A!2712), ∈(y, B!2713)), p!2264 == pair(x, y))))), ax=|= ForAll([A, B], prod(A, B) == sep(pow(pow(union(A, B))), Lambda(p!2264, Exists([x, y], And(And(∈(x, A), ∈(y, B)), p!2264 == pair(x, y)))))), _subst_fun_body=sep(pow(pow(union(Var(0), Var(1)))), Lambda(p!2264, Exists([x, y], And(And(∈(x, Var(3)), ∈(y, Var(4))), p!2264 == pair(x, y)))))), ran: Unfolding(name='ran', decl=ran, args=[R!2787], body=sep(bigunion(bigunion(R!2787)), Lambda(y, Exists(x, ∈(pair(x, y), R!2787)))), ax=|= ForAll(R, ran(R) == sep(bigunion(bigunion(R)), Lambda(y, Exists(x, ∈(pair(x, y), R))))), _subst_fun_body=sep(bigunion(bigunion(Var(0))), Lambda(y, Exists(x, ∈(pair(x, y), Var(2)))))), ray: Unfolding(name='ray', decl=ray, args=[p!1871, q!1872], body=Lambda(a, And(collinear(p!1871, q!1872, a), Vec2.dot(Vec2.sub(q!1872, p!1871), Vec2.sub(a, p!1871)) >= 0)), ax=|= ForAll([p, q], ray(p, q) == (Lambda(a, And(collinear(p, q, a), Vec2.dot(Vec2.sub(q, p), Vec2.sub(a, p)) >= 0)))), _subst_fun_body=Lambda(a, And(collinear(Var(1), Var(2), a), Vec2.dot(Vec2.sub(Var(2), Var(1)), Vec2.sub(a, Var(1))) >= 0))), reflects: Unfolding(name='reflects', decl=reflects, args=[P!2266, A!2267], body=ForAll(x, ∈(x, A!2267) == P!2266[x]), ax=|= ForAll([P, A], reflects(P, A) == (ForAll(x, ∈(x, A) == P[x]))), _subst_fun_body=ForAll(x, ∈(x, Var(2)) == Var(1)[x])), rseq.abs: Unfolding(name='rseq.abs', decl=rseq.abs, args=[a!1485], body=Map(absR, a!1485), ax=|= ForAll(a, rseq.abs(a) == Map(absR, a)), _subst_fun_body=Map(absR, Var(0))), rseq.cos: Unfolding(name='rseq.cos', decl=rseq.cos, args=[a!1484], body=Map(Real.cos, a!1484), ax=|= ForAll(a, rseq.cos(a) == Map(Real.cos, a)), _subst_fun_body=Map(Real.cos, Var(0))), rseq.dot: Unfolding(name='rseq.dot', decl=rseq.dot, args=[n!1450, a!1451, b!1452], body=finsum((Array Int Real).mul(a!1451, b!1452), n!1450), ax=|= ForAll([n, a, b], rseq.dot(n, a, b) == finsum((Array Int Real).mul(a, b), n)), _subst_fun_body=finsum((Array Int Real).mul(Var(1), Var(2)), Var(0))), rseq.outer: Unfolding(name='rseq.outer', decl=rseq.outer, args=[a!1480, b!1481], body=Lambda([i, j], a!1480[i]*b!1481[j]), ax=|= ForAll([a, b], rseq.outer(a, b) == (Lambda([i, j], a[i]*b[j]))), _subst_fun_body=Lambda([i, j], Var(2)[i]*Var(3)[j])), rseq.sin: Unfolding(name='rseq.sin', decl=rseq.sin, args=[a!1483], body=Map(Real.sin, a!1483), ax=|= ForAll(a, rseq.sin(a) == Map(Real.sin, a)), _subst_fun_body=Map(Real.sin, Var(0))), safe_pred: Unfolding(name='safe_pred', decl=safe_pred, args=[n!384], body=If(is(Z, n!384), Z, pred(n!384)), ax=|= ForAll(n, safe_pred(n) == If(is(Z, n), Z, pred(n))), _subst_fun_body=If(is(Z, Var(0)), Z, pred(Var(0)))), search: Unfolding(name='search', decl=search, args=[P!337, n!338], body=If(P!337[n!338], n!338, If(P!337[-n!338], -n!338, If(P!337[n!338 + 1], 1, 0))), ax=|= ForAll([P, n!314], search(P, n!314) == If(P[n!314], n!314, If(P[-n!314], -n!314, If(P[n!314 + 1], 1, 0)))), _subst_fun_body=If(Var(0)[Var(1)], Var(1), If(Var(0)[-Var(1)], -Var(1), If(Var(0)[Var(1) + 1], 1, 0)))), sgn: Unfolding(name='sgn', decl=sgn, args=[x!476], body=If(x!476 > 0, 1, If(x!476 < 0, -1, 0)), ax=|= ForAll(x, sgn(x) == If(x > 0, 1, If(x < 0, -1, 0))), _subst_fun_body=If(Var(0) > 0, 1, If(Var(0) < 0, -1, 0))), shift: Unfolding(name='shift', decl=shift, args=[A!2111, c!2112], body=Lambda(x, A!2111[x - c!2112]), ax=|= ForAll([A, c], shift(A, c) == (Lambda(x, A[x - c]))), _subst_fun_body=Lambda(x, Var(1)[x - Var(2)])), snd: Unfolding(name='snd', decl=snd, args=[p!2659], body=If(ZF.sing(ZF.sing(fst(p!2659))) == p!2659, fst(p!2659), pick(ZFSet.sub(bigunion(p!2659), ZF.sing(fst(p!2659))))), ax=|= ForAll(p!2264, snd(p!2264) == If(ZF.sing(ZF.sing(fst(p!2264))) == p!2264, fst(p!2264), pick(ZFSet.sub(bigunion(p!2264), ZF.sing(fst(p!2264)))))), _subst_fun_body=If(ZF.sing(ZF.sing(fst(Var(0)))) == Var(0), fst(Var(0)), pick(ZFSet.sub(bigunion(Var(0)), ZF.sing(fst(Var(0))))))), sub: Unfolding(name='sub', decl=sub, args=[x!459, y!460], body=x!459 - y!460, ax=|= ForAll([x, y], sub(x, y) == x - y), _subst_fun_body=Var(0) - Var(1)), tan: Unfolding(name='tan', decl=tan, args=[x!687], body=Real.sin(x!687)/Real.cos(x!687), ax=|= ForAll(x, tan(x) == Real.sin(x)/Real.cos(x)), _subst_fun_body=Real.sin(Var(0))/Real.cos(Var(0))), tand: Unfolding(name='tand', decl=tand, args=[p!1162, q!1163], body=T_Bool(Lambda(t!1161, And(val(p!1162)[t!1161], val(q!1163)[t!1161]))), ax=|= ForAll([p, q], tand(p, q) == T_Bool(Lambda(t!1161, And(val(p)[t!1161], val(q)[t!1161])))), _subst_fun_body=T_Bool(Lambda(t!1161, And(val(Var(1))[t!1161], val(Var(2))[t!1161])))), temp.valid: Unfolding(name='temp.valid', decl=temp.valid, args=[p!1181], body=val(p!1181)[0], ax=|= ForAll(p, temp.valid(p) == val(p)[0]), _subst_fun_body=val(Var(0))[0]), timpl: Unfolding(name='timpl', decl=timpl, args=[p!1168, q!1169], body=T_Bool(Lambda(t!1167, Implies(val(p!1168)[t!1167], val(q!1169)[t!1167]))), ax=|= ForAll([p, q], timpl(p, q) == T_Bool(Lambda(t!1167, Implies(val(p)[t!1167], val(q)[t!1167])))), _subst_fun_body=T_Bool(Lambda(t!1167, Implies(val(Var(1))[t!1167], val(Var(2))[t!1167])))), tint: Unfolding(name='tint', decl=tint, args=[x!1217], body=T_Int(K(Int, x!1217)), ax=|= ForAll(x, tint(x) == T_Int(K(Int, x))), _subst_fun_body=T_Int(K(Int, Var(0)))), tnot: Unfolding(name='tnot', decl=tnot, args=[p!1160], body=T_Bool(Lambda(t!1159, Not(val(p!1160)[t!1159]))), ax=|= ForAll(p, tnot(p) == T_Bool(Lambda(t!1159, Not(val(p)[t!1159])))), _subst_fun_body=T_Bool(Lambda(t!1159, Not(val(Var(1))[t!1159])))), to_int: Unfolding(name='to_int', decl=to_int, args=[n!386], body=If(is(Z, n!386), 0, 1 + to_int(pred(n!386))), ax=|= ForAll(n, to_int(n) == If(is(Z, n), 0, 1 + to_int(pred(n)))), _subst_fun_body=If(is(Z, Var(0)), 0, 1 + to_int(pred(Var(0))))), to_nat: Unfolding(name='to_nat', decl=to_nat, args=[n!353], body=If(n!353 >= 0, n!353*2, n!353*-2 - 1), ax=|= ForAll(n!314, to_nat(n!314) == If(n!314 >= 0, n!314*2, n!314*-2 - 1)), _subst_fun_body=If(Var(0) >= 0, Var(0)*2, Var(0)*-2 - 1)), tor: Unfolding(name='tor', decl=tor, args=[p!1165, q!1166], body=T_Bool(Lambda(t!1164, Or(val(p!1165)[t!1164], val(q!1166)[t!1164]))), ax=|= ForAll([p, q], tor(p, q) == T_Bool(Lambda(t!1164, Or(val(p)[t!1164], val(q)[t!1164])))), _subst_fun_body=T_Bool(Lambda(t!1164, Or(val(Var(1))[t!1164], val(Var(2))[t!1164])))), union: Unfolding(name='union', decl=union, args=[A!2361, B!2362], body=bigunion(upair(A!2361, B!2362)), ax=|= ForAll([A, B], union(A, B) == bigunion(upair(A, B))), _subst_fun_body=bigunion(upair(Var(0), Var(1)))), upper_bound: Unfolding(name='upper_bound', decl=upper_bound, args=[A!1700, x!1701], body=ForAll(y, Implies(A!1700[y], EReal.le(y, x!1701))), ax=|= ForAll([A, x], upper_bound(A, x) == (ForAll(y, Implies(A[y], EReal.le(y, x))))), _subst_fun_body=ForAll(y, Implies(Var(1)[y], EReal.le(y, Var(2))))), valid: Unfolding(name='valid', decl=valid, args=[a!1113], body=ForAll(w, val(a!1113)[w]), ax=|= ForAll(a, valid(a) == (ForAll(w, val(a)[w]))), _subst_fun_body=ForAll(w, val(Var(1))[w])), where: Unfolding(name='where', decl=where, args=[mask!1288, a!1289, b!1290], body=Lambda(i, If(mask!1288[i], a!1289[i], b!1290[i])), ax=|= ForAll([mask, a, b], where(mask, a, b) == (Lambda(i, If(mask[i], a[i], b[i])))), _subst_fun_body=Lambda(i, If(Var(1)[i], Var(2)[i], Var(3)[i]))), width: Unfolding(name='width', decl=width, args=[i!1907], body=hi(i!1907) - lo(i!1907), ax=|= ForAll(i, width(i) == hi(i) - lo(i)), _subst_fun_body=hi(Var(0)) - lo(Var(0))), zero: Unfolding(name='zero', decl=zero, args=[n!2189], body=NDArray(Unit(n!2189), K(Int, 0)), ax=|= ForAll(n, zero(n) == NDArray(Unit(n), K(Int, 0))), _subst_fun_body=NDArray(Unit(Var(0)), K(Int, 0))), zf.false: Unfolding(name='zf.false', decl=zf.false, args=[], body=∅, ax=|= zf.false == ∅, _subst_fun_body=∅), zf.succ: Unfolding(name='zf.succ', decl=zf.succ, args=[x!2788], body=union(x!2788, ZF.sing(x!2788)), ax=|= ForAll(x, zf.succ(x) == union(x, ZF.sing(x))), _subst_fun_body=union(Var(0), ZF.sing(Var(0)))), zf.true: Unfolding(name='zf.true', decl=zf.true, args=[], body=ZF.sing(∅), ax=|= zf.true == ZF.sing(∅), _subst_fun_body=ZF.sing(∅)), zf.zero: Unfolding(name='zf.zero', decl=zf.zero, args=[], body=∅, ax=|= zf.zero == ∅, _subst_fun_body=∅)}: defn holds definitional axioms for function symbols.

kdrag.kernel.ext(domain: Sequence[SortRef], range_: SortRef) → Proof

>>> ext([smt.IntSort()], smt.IntSort())
|= ForAll([f, g], (f == g) == (ForAll(x0, f[x0] == g[x0])))
>>> ext([smt.IntSort(), smt.RealSort()], smt.IntSort())
|= ForAll([f, g],
       (f == g) ==
       (ForAll([x0, x1],
               Select(f, x0, x1) == Select(g, x0, x1))))

Parameters:

domain (Sequence[SortRef])
range_ (SortRef)

Return type:

Proof

kdrag.kernel.forget(ts: Sequence[ExprRef], thm: QuantifierRef) → Proof

“Forget” a term using existentials. This is existential introduction. P(ts) -> exists xs, P(xs) thm is an existential formula, and ts are terms to substitute those variables with. forget easily follows. https://en.wikipedia.org/wiki/Existential_generalization

Parameters:

ts (Sequence[ExprRef])
thm (QuantifierRef)

Return type:

Proof

kdrag.kernel.free_vars(e: ExprRef) → set[ExprRef]

Get an overapproximation of free variables in a term. This is used for skolemization.

>>> x,y = smt.Ints("x y")
>>> z = FreshVar("z", smt.IntSort())
>>> assert z in free_vars(x + y + z)

Parameters:: e (ExprRef)
Return type:: set[ExprRef]

kdrag.kernel.fresh_const(q: QuantifierRef, prefixes=None) → list[ExprRef]

Generate fresh constants of same sort as quantifier.

Parameters:: q (QuantifierRef)
Return type:: list[ExprRef]

kdrag.kernel.generalize(vs: list[ExprRef], pf: Proof) → Proof

Generalize a theorem with respect to a list of schema variables. This introduces a universal quantifier for schema variables.

>>> x = FreshVar("x", smt.IntSort())
>>> y = FreshVar("y", smt.IntSort())
>>> generalize([x, y], prove(x == x))
|= ForAll([x!..., y!...], x!... == x!...)

Parameters:

vs (list[ExprRef])
pf (Proof)

Return type:

Proof

kdrag.kernel.herb(thm: QuantifierRef, prefixes=None) → tuple[list[ExprRef], Proof]

Herbrandize a theorem. It is sufficient to prove a theorem for fresh consts to prove a universal. Note: Perhaps lambdaized form is better? Return vars and lamda that could receive |= P[vars]

>>> x = smt.Int("x")
>>> herb(smt.ForAll([x], x >= x))
([x!...], |=  Implies(x!... >= x!..., ForAll(x, x >= x)))

Parameters:: thm (QuantifierRef)
Return type:: tuple[list[ExprRef], Proof]

kdrag.kernel.induct_inductive(x: DatatypeRef, P: QuantifierRef) → Proof

Build a basic induction principle for an algebraic datatype

Parameters:

x (DatatypeRef)
P (QuantifierRef)

Return type:

Proof

kdrag.kernel.instan(ts: Sequence[ExprRef], pf: Proof) → Proof

Instantiate a universally quantified formula. This is forall elimination

Parameters:

ts (Sequence[ExprRef])
pf (Proof)

Return type:

Proof

kdrag.kernel.is_declared(e: ExprRef | FuncDeclRef) → bool

Parameters:: e (ExprRef | FuncDeclRef)
Return type:: bool

kdrag.kernel.is_defined(x: ExprRef) → bool

Determined if expression head is in definitions.

Parameters:: x (ExprRef)
Return type:: bool

kdrag.kernel.is_fresh_var(v: ExprRef) → bool

Check if a variable is a schema variable. Schema variables are generated by FreshVar and have a _FreshVarEvidence attribute.

>>> is_fresh_var(FreshVar("x", smt.IntSort()))
True

Parameters:: v (ExprRef)
Return type:: bool

kdrag.kernel.is_proof(p: Proof) → bool

Parameters:: p (Proof)
Return type:: bool

kdrag.kernel.make_unfolding(pf: Proof) → Unfolding

Take an axiom of the form |= forall x0 x1 …, xn, f(x0, x1, …, xn) = body and return a Defn object for f. Unfolding judgments can be used with unfold_defns.

>>> x,y,z = smt.Ints("x y z")
>>> pf = kd.prove(smt.ForAll([x,y], x + y == y + x + 0)) # has shape of definition, but not "definitional"
>>> defn = make_unfolding(pf)
>>> unfold_defns(z + 42, [defn])
(42 + z + 0, |= z + 42 == 42 + z + 0)

Parameters:: pf (Proof)
Return type:: Unfolding

kdrag.kernel.modus(ab: Proof, a: Proof) → Proof

Modus ponens for implies and equality.

>>> a,b = smt.Bools("a b")
>>> ab = axiom(smt.Implies(a, b))
>>> a = axiom(a)
>>> modus(ab, a)
|= b
>>> ab1 = axiom(smt.Eq(a.thm, b))
>>> modus(ab1, a)
|= b

Parameters:

ab (Proof)
a (Proof)

Return type:

Proof

kdrag.kernel.obtain(thm: QuantifierRef) → tuple[list[ExprRef], Proof]

Skolemize an existential quantifier. exists xs, P(xs) -> P(cs) for fresh cs https://en.wikipedia.org/wiki/Existential_instantiation

>>> x = smt.Int("x")
>>> obtain(smt.Exists([x], x >= 0))
([x!...], |=  Implies(Exists(x, x >= 0), x!... >= 0))
>>> y = FreshVar("y", smt.IntSort())
>>> obtain(smt.Exists([x], x >= y))
([f!...(y!...)], |=  Implies(Exists(x, x >= y!...), f!...(y!...) >= y!...))

Parameters:: thm (QuantifierRef)
Return type:: tuple[list[ExprRef], Proof]

kdrag.kernel.open_binder(lam: QuantifierRef) → tuple[list[ExprRef], ExprRef]

Open a quantifier with fresh variables. This achieves the locally nameless representation https://chargueraud.org/research/2009/ln/main.pdf where it is harder to go wrong.

The variables are schema variables which when used in a proof may be generalized

>>> x = smt.Int("x")
>>> open_binder(smt.ForAll([x], x > 0))
([x!...], x!... > 0)

Parameters:: lam (QuantifierRef)
Return type:: tuple[list[ExprRef], ExprRef]

kdrag.kernel.prove(thm: BoolRef, by: Proof | Iterable[Proof] = [], admit=False, timeout=1000, dump=False, solver=None, fvs=frozenset({})) → Proof

Prove a theorem using a list of previously proved lemmas.

In essence prove(Implies(by, thm)).

Parameters:

thm (smt.BoolRef) – The theorem to prove.
thm – The theorem to prove.
by (list[Proof]) – A list of previously proved lemmas.
admit (bool) – If True, admit the theorem without proof.

Returns:

A proof object of thm

Return type:

Proof

>>> prove(smt.BoolVal(True))
|= True
>>> prove(smt.RealVal(1) >= smt.RealVal(0))
|= 1 >= 0

kdrag.kernel.register_definition(defn: Unfolding)

Can register other unfoldings besides those generated by define. This may be useful to get unfolding for objects constructed in roundabout ays.

Parameters:: defn (Unfolding)

kdrag.kernel.rename_vars2(pf: Proof, vs: list[ExprRef], patterns=[], no_patterns=[]) → Proof

Rename bound variables and also attach triggers

>>> x,y = smt.Ints("x y")
>>> f = smt.Function("f", smt.IntSort(), smt.IntSort())
>>> rename_vars2(kd.prove(smt.ForAll([x, y], x + 1 > x)), [y,x], patterns=[x+y])
|= ForAll([y, x], y + 1 > y)
>>> rename_vars2(kd.prove(smt.Exists([x], x + 1 > y)), [y])
Traceback (most recent call last):
    ...
ValueError: ('Cannot rename vars to ones that already occur in term', [y], Exists(x, x + 1 > y))

Parameters:

pf (Proof)
vs (list[ExprRef])

Return type:

Proof

kdrag.kernel.specialize(ts: Sequence[ExprRef], thm: BoolRef) → Proof

Instantiate a universally quantified formula forall xs, P(xs) -> P(ts) This is forall elimination

Parameters:

ts (Sequence[ExprRef])
thm (BoolRef)

Return type:

Proof

kdrag.kernel.subst(t: ExprRef, eqs: Sequence[Proof]) → tuple[ExprRef, Proof]

Substitute using equality proofs

>>> x, y = smt.Ints("x y")
>>> eq = kd.prove(x == ((x + 1) - 1))
>>> subst(x + 3, [eq])
(x + 1 - 1 + 3, |= x + 3 == x + 1 - 1 + 3)

Parameters:

t (ExprRef)
eqs (Sequence[Proof])

Return type:

tuple[ExprRef, Proof]

kdrag.kernel.trigger(pf: Proof, patterns: list[ExprRef] = [], no_patterns: list[ExprRef] = []) → Proof

Add e-matching triggers to a quantified proof. Logically speaking, does nothing.

>>> x,y = smt.Ints("x y")
>>> p = kd.prove(smt.ForAll([x,y], y > 0, x + y > x))
>>> p.thm.num_patterns()
0
>>> p2 = trigger(p, patterns=[x + y])
>>> p2
|= ForAll([x, y], Implies(y > 0, x + y > x))
>>> p2.thm.pattern(0)
Var(1) + Var(0)

Parameters:

pf (Proof)
patterns (list[ExprRef])
no_patterns (list[ExprRef])

Return type:

Proof

kdrag.kernel.unfold(e: ExprRef, decls: Sequence[FuncDeclRef]) → tuple[ExprRef, Proof]

Unfold function definitions in an expression.

>>> x,y = smt.Ints("x y")
>>> f = define("f", [x,y], x + 2*y)
>>> g = define("g", [x,y], f(x,y) + 1)
>>> unfold(f(42,13) + g(7,8), [f])
(42 + 2*13 + g(7, 8), |= f(42, 13) + g(7, 8) == 42 + 2*13 + g(7, 8))
>>> unfold(f(42,13) + g(7,8), [f,g])
(42 + 2*13 + f(7, 8) + 1, |= f(42, 13) + g(7, 8) == 42 + 2*13 + f(7, 8) + 1)

Parameters:

e (ExprRef)
decls (Sequence[FuncDeclRef])

Return type:

tuple[ExprRef, Proof]

kdrag.kernel.unfold_defns(e: ExprRef, defns: Sequence[Unfolding]) → tuple[ExprRef, Proof]

Unfold definitions in an expression.

Parameters:

e (ExprRef)
defns (Sequence[Unfolding])

Return type:

tuple[ExprRef, Proof]

kdrag.kernel.weaken(pf: Proof, fvs: Sequence[ExprRef]) → Proof

Parameters:

pf (Proof)
fvs (Sequence[ExprRef])

Return type:

Proof