```(def-theorem ()
"forall(x:kk, not(x=o_kk) implies #(inv(x)))"
(theory fields)
(usages d-r-convergence)
(proof
(
direct-and-antecedent-inference-strategy
(instantiate-theorem multiplicative-inverse-for-fields ("x"))
)))
```

```(def-overloading + (h-o-real-arithmetic +) (fields +_kk))
```

```(def-overloading * (h-o-real-arithmetic *) (fields *_kk))
```

```(def-overloading - (h-o-real-arithmetic -) (fields -_kk))
```

```(def-theorem ()
"forall(x:kk,o_kk+x=x)"
(theory fields)
(proof
(
)))
```

```(def-theorem ()
"forall(x:kk,-x+x=o_kk)"
(theory fields)
(proof
(
)))
```

```
(fixed-theories h-o-real-arithmetic)
(source abelian-groups)
(target fields)
(sort-pairs
(gg kk))
(constant-pairs
(inv -_kk)
(mul +_kk)
(e o_kk)))
```

```
(def-theorem o_kk-times-is-o_kk
"forall(x:kk, x*o_kk=o_kk)"
(theory fields)
(proof
(
direct-inference
(cut-with-single-formula "x*o_kk=x*o_kk+x*o_kk")
(incorporate-antecedent "with(x:kk,x*o_kk=x*o_kk+x*o_kk);")
(apply-macete-with-minor-premises tr%right-trivial-cancellation-law-left)
simplify
(cut-with-single-formula "o_kk=o_kk+o_kk")
(backchain "o_kk=o_kk+o_kk;")
(apply-macete-with-minor-premises left-distributive-law-for-fields)
)))
```

```
(def-constant sub_kk
"lambda(x,y:kk,x+(-y))"
(theory fields))
```

```
(def-constant /_kk
"lambda(x,y:kk, if(y=o_kk,?kk,inv(y)*x))"
(theory fields))
```

```(def-overloading sub (h-o-real-arithmetic sub) (fields sub_kk))
```

```(def-overloading / (h-o-real-arithmetic /) (fields /_kk))
```

```(def-theorem ()
"forall(y,x:kk,x-y=x+-y)"
(proof
(
unfold-defined-constants
simplify
))
(theory fields))
```

```(def-theorem ()
"total_q{sub_kk,[kk,kk,kk]}"
(theory fields)
(proof
(
unfold-defined-constants
insistent-direct-inference-strategy
beta-reduce-repeatedly
))
(usages d-r-convergence))
```

```
(def-theorem division-characterization-fields
"forall(a,b:kk, not(b=o_kk) implies b*(a/b)=a)"
(theory fields)
(proof
(
(unfold-single-defined-constant (0) /_kk)
simplify
(force-substitution "b*(inv(b)*a)" "(b*inv(b))*a" (0))
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
(apply-macete-with-minor-premises multiplicative-identity-for-fields)
(apply-macete-with-minor-premises associative-law-for-multiplication-for-fields)
)))
```

```
(def-theorem non-o_kk-is-closed-under-inv
"forall(x:kk, not(x=o_kk) implies not(inv(x)=o_kk))"
(theory fields)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "i_kk=x*inv(x)")
(contrapose "with(x:kk,not(x=o_kk));")
(contrapose "with(x:kk,i_kk=x*inv(x));")
(backchain "with(x:kk,inv(x)=o_kk);")
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(contrapose "with(x:kk,not(x=o_kk));")
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
)))

```

```
(def-theorem  non-o_kk-is-closed-under-*_kk
"forall(x,y:kk, not(x=o_kk) and not(y=o_kk) implies not(x*y=o_kk))"
(theory fields)
(proof
(
direct-and-antecedent-inference-strategy
(contrapose "with(x:kk,not(x=o_kk));")
(cut-with-single-formula "x=x*y*inv(y)")
(backchain "with(y,x:kk,x=x*y*inv(y));")
(backchain "with(y,x:kk,x*y=o_kk);")
(apply-macete-with-minor-premises commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(apply-macete-with-minor-premises associative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
(apply-macete-with-minor-premises commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises multiplicative-identity-for-fields)
)))
```

```(def-theorem ()
"forsome(x:kk,not(x=o_kk))"
lemma
(theory fields)
(proof
(
(instantiate-existential ("i_kk"))
simplify
)))
```

```(def-theorem ()
"forall(x,y:kk,
not(x=o_kk) implies y=o_kk or not(x*y=o_kk))"
lemma
(theory fields)
(proof
(
(apply-macete-with-minor-premises non-o_kk-is-closed-under-*_kk)
simplify
)))
```

```(def-theorem ()
"forall(x:kk,not(x=o_kk) implies inv(x)*x=i_kk)"
lemma
(theory fields)
(proof
(
(apply-macete-with-minor-premises commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
)))
```

```(def-theorem ()
"forall(x,y,z:kk,
not(x=o_kk) and not(y=o_kk) and not(z=o_kk)
implies
not(x *_kk y=o_kk)
and
(not(y *_kk z=o_kk) and x *_kk y *_kk z=x *_kk (y *_kk z)))"
lemma
(theory fields)
(proof
(
(apply-macete-with-minor-premises non-o_kk-is-closed-under-*_kk)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises associative-law-for-multiplication-for-fields)
)))
```

```(def-theorem ()
"forall(x,y:kk,not(x=o_kk) and not(y=o_kk) implies x*y=y*x)"
lemma
(theory fields)
(proof
(
direct-and-antecedent-inference-strategy
(assume-theorem commutative-law-for-multiplication-for-fields)
(backchain "forall(y,x:kk,x*y=y*x);")
)))
```

```(def-theorem ()
"forall(x:kk,not(x=o_kk) implies x*i_kk=x)"
lemma
(theory fields)
(proof
(
(apply-macete-with-minor-premises commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises multiplicative-identity-for-fields)
)))
```

```(def-theorem ()
"forall(x:kk,
not(x=o_kk)
implies
not(inv(x)=o_kk) and inv(x) *_kk x=i_kk)"
lemma
(theory fields)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises non-o_kk-is-closed-under-inv)
(apply-macete-with-minor-premises commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
)))
```

```(def-theorem ()
"forall(x:kk,
not(x=o_kk)
implies
not(inv(x)=o_kk) and x *_kk inv(x)=i_kk)"
lemma
(theory fields)
(proof
(
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises non-o_kk-is-closed-under-inv)
(apply-macete-with-minor-premises multiplicative-inverse-for-fields)
)))
```

```(def-theorem ()
"forall(x:kk,x=o_kk or not(inv(x)=o_kk))"
lemma
(theory fields)
(proof
(
(apply-macete-with-minor-premises non-o_kk-is-closed-under-inv)
direct-and-antecedent-inference-strategy
)))
```

```
(def-translation groups-to-multiplicative-kk
(source abelian-groups)
(target fields)
(fixed-theories h-o-real-arithmetic)
(sort-pairs
(gg (pred "lambda(x:kk, not(x=o_kk))")))
(constant-pairs
(e i_kk)
(inv "lambda(x:kk, if(x=o_kk,?kk,inv(x)))")
(mul "lambda(x,y:kk,if(x=o_kk or y=o_kk,?kk,x*y))"))
(theory-interpretation-check using-simplification))
```

```(def-transported-symbols (multiple)
(renamer groups-multiplicative-kk-renamer)
(translation groups-to-multiplicative-kk)
)
```

```
(def-constant ^_kk
"lambda(x:kk,n:zz, if(not(x=o_kk),pexp(n,x),0<n,o_kk,?kk))"
(theory fields))
```

```(def-overloading ^ (h-o-real-arithmetic ^) (fields ^_kk))
```

```(def-theorem ()
"forall(y,x:kk, x/y==x*y^[-1])"
(theory fields)
(proof
(
(unfold-single-defined-constant (0) /_kk)
direct-and-antecedent-inference-strategy
(case-split ("y=o_kk"))
simplify
(unfold-single-defined-constant (0) ^_kk)
simplify
simplify
(assume-theorem commutative-law-for-multiplication-for-fields)
(backchain "forall(y,x:kk,x*y=y*x);")
(cut-with-single-formula "inv(y)=y^[-1]")
(unfold-single-defined-constant (0) ^_kk)
simplify
(instantiate-transported-theorem
multiple-commutes-with-negation-corollary groups-to-multiplicative-kk ("y"))
simplify
)))
```

```
(def-theorem o_kk-exponent
"forall(m:zz, #(o_kk^m,kk) implies o_kk^m=o_kk)"
(theory fields)
(proof
(
unfold-defined-constants
simplify
)))
```

```(def-theorem ()
"forall(n:zz, #(i_kk^n,kk) implies i_kk^n=i_kk)"
(theory fields)
(proof
(
unfold-defined-constants
simplify
direct-and-antecedent-inference-strategy
(instantiate-transported-theorem
multiple-of-group-identity groups-to-multiplicative-kk ("n"))
)))

```

```(def-theorem ()
"forall(x:kk,#(x^0,kk) implies x^0=i_kk)"
(theory fields)
(proof
(
unfold-defined-constants
simplify
direct-inference
direct-and-antecedent-inference-strategy
(instantiate-transported-theorem
0-multiple-is-group-identity groups-to-multiplicative-kk ("x"))
)))
```

```(def-theorem ()
"forall(x:kk,x^1=x)"
(theory fields)
(proof
(
(unfold-single-defined-constant (0) ^_kk)
direct-inference
(case-split ("x=o_kk"))
simplify
simplify
(instantiate-transported-theorem 1-multiple-is-element groups-to-multiplicative-kk ("x"))
)))
```

```(def-theorem ()
"forall(n,m:zz, x:kk ,#((x^m)^n,kk) implies (x^m)^n=x^(m*n))"
(theory fields)
(proof
(
direct-inference
(case-split ("x=o_kk"))
(block
(script-comment "`case-split' at (1)")
(backchain-repeatedly ("with(x:kk,x=o_kk);")) direct-inference
(cut-with-single-formula "0<m and 0<n and 0<m*n")
(block
(script-comment "`cut-with-single-formula' at (0)")
(apply-macete-with-minor-premises o_kk-exponent)
(block
(script-comment "`apply-macete-with-minor-premises' at (0)")
(apply-macete-with-minor-premises o_kk-exponent)
(unfold-single-defined-constant (0) ^_kk) simplify
)
(block
(script-comment "`apply-macete-with-minor-premises' at (1)")
(unfold-single-defined-constant (0) ^_kk) simplify
) )
(block
(script-comment "`cut-with-single-formula' at (1)")
(cut-with-single-formula "#(o_kk^m) and o_kk^m=o_kk")
(block
(script-comment "`cut-with-single-formula' at (0)")
(contrapose "with(k:kk,#(k,kk));") (backchain "with(p:prop,p and p);")
(contrapose "with(p:prop,p or p or p);")
(antecedent-inference "with(p:prop,p and p);")
(incorporate-antecedent "with(k:kk,#(k));")
(incorporate-antecedent "with(k:kk,#(k,kk));")
(unfold-single-defined-constant (0 1) ^_kk) simplify
)
(apply-macete-with-minor-premises o_kk-exponent)
) )
(block
(script-comment "`case-split' at (2)")
(cut-with-single-formula "not(pexp(m,x)=o_kk)")
(block
(script-comment "`cut-with-single-formula' at (0)")
(apply-macete-with-minor-premises commutative-law-for-multiplication)
(move-to-ancestor 1) unfold-defined-constants simplify
beta-reduce-with-minor-premises
(block
(script-comment "`beta-reduce-with-minor-premises' at (0)") simplify
(instantiate-transported-theorem multiple-is-homogenuous
groups-to-multiplicative-kk ("n" "m" "x")
)
direct-inference
(apply-macete-with-minor-premises commutative-law-for-multiplication)
)
(instantiate-transported-theorem multiple-totality
groups-to-multiplicative-kk ("m" "x")
) )
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk ("x" "m")
) )
)))
```

```(def-theorem ()
"forall(n,m:zz, x:kk ,((#(x^m,kk) and #(x^n,kk)) iff #((x^m)^n,kk)))"
(theory fields)
(proof
(
direct-inference
unfold-defined-constants
(case-split ("x=o_kk"))
simplify
(block
(script-comment "`case-split' at (2)")
simplify
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("x" "m"))
(cut-with-single-formula "not pexp(m,x)=o_kk")
simplify
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("pexp(m,x)" "n"))
direct-and-antecedent-inference-strategy
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("x" "n")))
)))
```

```(def-theorem ()
"forall(m:zz ,y,x:kk,(#(x^m*y^m,kk) or #((x*y)^m,kk))
implies
x^m*y^m=(x*y)^m)"
(theory fields)
(proof
(
direct-inference
(case-split ("x=o_kk"))
(block
(script-comment "`case-split' at (1)")
simplify
(apply-macete-with-minor-premises
commutative-law-for-multiplication-for-fields)
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(unfold-single-defined-constant (1 2) ^_kk)
simplify
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(unfold-single-defined-constant (0) ^_kk)
(case-split ("y=o_kk"))
simplify
(block
(script-comment "`case-split' at (2)")
simplify
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("y" "m"))))
(block
(script-comment "`case-split' at (2)")
(case-split ("y=o_kk"))
(block
(script-comment "`case-split' at (1)")
simplify
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(unfold-single-defined-constant (1 2) ^_kk)
simplify
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
(unfold-single-defined-constant (0) ^_kk)
simplify
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("x" "m")))
(block
(script-comment "`case-split' at (2)")
direct-inference
(unfold-single-defined-constant-globally ^_kk)
(cut-with-single-formula "not(x*y=o_kk)")
(block
(script-comment "`cut-with-single-formula' at (0)")
simplify
groups-to-multiplicative-kk
("m" "x" "y"))
(contrapose "with(k:kk,k=k);")
(cut-with-single-formula "not pexp(m,x)=o_kk and not pexp(m,y)=o_kk")
(block
(script-comment "`cut-with-single-formula' at (0)")
simplify
beta-reduce-with-minor-premises
simplify
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("y" "m"))
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("x" "m")))
(block
(script-comment "`cut-with-single-formula' at (1)")
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("x" "m"))
direct-inference
(instantiate-transported-theorem multiple-totality-with-range
groups-to-multiplicative-kk
("y" "m"))))
(block
(script-comment "`cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises non-o_kk-is-closed-under-*_kk)
direct-inference)))
)))
```

```(def-theorem ()
"forall(n,m:zz, x:kk,#(x^m*x^n,kk) implies x^(m+n)=x^m*x^n)"
(theory fields)
(proof
(
direct-inference
(case-split ("x=o_kk"))
unfold-defined-constants
simplify
(apply-macete-with-minor-premises o_kk-times-is-o_kk)
unfold-defined-constants
simplify
(instantiate-transported-theorem
(cut-with-single-formula "not(pexp(m,x)=o_kk) and not(pexp(n,x)=o_kk)")
(simplify-antecedent "with(x,y:kk, x=y)")
(weaken (0))
direct-inference
(instantiate-transported-theorem
multiple-totality-with-range groups-to-multiplicative-kk ("x" "m"))
(instantiate-transported-theorem
multiple-totality-with-range groups-to-multiplicative-kk ("x" "n"))
)))
```

```(def-algebraic-processor field-algebraic-processor
(language fields)
(base ((operations
(+ +_kk)
(* *_kk)
(- -_kk)
(^ ^_kk)
(sub sub_kk)
(zero o_kk)
(unit i_kk))
commutes))
(exponent rr-algebraic-processor))

```

```(def-theory-processors fields
(algebraic-simplifier (field-algebraic-processor ^_kk sub_kk *_kk +_kk -_kk))
(algebraic-term-comparator field-algebraic-processor))
```

```(def-theorem ()
"forall(x:kk,y:zz,(0<y or not(x=o_kk)) implies #(x^y))"
(proof
(
direct-and-antecedent-inference-strategy
unfold-defined-constants
simplify
unfold-defined-constants
simplify
))
(theory fields)
(usages d-r-convergence))
```

```(def-theorem ()
"forall(x:kk, y:zz, x=o_kk and not(0<y) implies not(#(x^y)))"
(proof
(
direct-and-antecedent-inference-strategy
unfold-defined-constants
simplify
))
(theory fields)
(usages d-r-convergence))
```

```(def-transported-symbols (multiple)
)
```

```(def-overloading * (fields *_ext))
```

```
(def-theorem *_ext-totality
"total_q{*_ext,[zz,kk,kk]}"
(proof
(
(apply-macete-with-minor-premises tr%multiple-totality)
))
(theory fields)
(usages d-r-convergence))
```

```
(def-recursive-constant sum_kk
"lambda(sigma:[zz,zz,[zz,kk],kk],lambda(m,n:zz,f:[zz,kk],
if(m<=n,sigma(m,n-1,f)+f(n),o_kk)))"
(definition-name sum_kk)
(theory fields))
```

```
(def-recursive-constant prod_kk
"lambda(pi:[zz,zz,[zz,kk],kk],lambda(m,n:zz,f:[zz,kk],
if(m<=n,pi(m,n-1,f)*f(n),i_kk)))"
(definition-name  prod_kk)
(theory fields))
```

```(def-overloading sum (h-o-real-arithmetic sum) (fields sum_kk))
```

```(def-overloading prod (h-o-real-arithmetic prod) (fields prod_kk))
```

```
(def-theorem sum_kk-definedness
"forall(m,n:zz,f:[zz,kk],forall(k:zz,m<=k and k<=n implies #(f(k)))
implies
#(sum(m,n,f)))"
(theory fields)

(proof

(

direct-inference
(case-split ("m<=n"))
(induction integer-inductor ())
direct-and-antecedent-inference-strategy
(prove-by-logic-and-simplification 3)
(prove-by-logic-and-simplification 3)
(unfold-single-defined-constant (0) sum_kk)
simplify

))
(usages d-r-convergence))
```

```
(fixed-theories h-o-real-arithmetic)
(source commutative-monoid-theory)
(target fields)
(sort-pairs
(uu kk))
(constant-pairs
(** +_kk)
(e o_kk))
(theory-interpretation-check using-simplification))
```

```
(def-theorem external-multiplication-conversion
"forall(m:zz,x:kk,m*x=m*i_kk*x)"
(theory fields)
(proof
(
direct-inference
(case-split ("0<=m"))
(induction integer-inductor ("m"))
(unfold-single-defined-constant (0 1) *_ext)
simplify
(cut-with-single-formula "forall(m:zz,x:kk,0<=m implies m*x=m*i_kk*x);")
(backchain-backwards "forall(m:zz,x:kk,0<=m implies m*x=m*i_kk*x);")
simplify
(weaken (0))
direct-and-antecedent-inference-strategy)))
```

```(def-theorem ()
"forall(x:kk,m,n:zz, (m+n)*x=m*x+n*x)"
(theory  fields)
(usages rewrite)
(proof
(
simplify
)))
```

```(def-theorem ()
"forall(x:kk,0*x=o_kk)"
(theory  fields)
(usages rewrite)
(proof
(
unfold-defined-constants
simplify
)))
```

```(def-theorem ()
"forall(x:kk,1*x=x)"
(theory  fields)
(usages rewrite)
(proof
(
unfold-defined-constants
simplify
)))
```

```
(def-theorem uniqueness-of-exponentiation
"forall(f:[kk,zz,kk],
(forall(x:kk, n:zz, not(x=o_kk) and 0<=n implies f(x,n+1)=f(x,n)*x)
and forall(x:kk, not(x=o_kk)  implies f(x,0)=i_kk)
and forall(x:kk, n:zz, not(x=o_kk) and 0<=n implies f(x,-n)=(f(x,n))^[-1])
and forall(n:zz,1<=n implies f(o_kk,n)=o_kk)
and forall(n:zz, n<=0 implies not(#(f(o_kk,n)))))
iff
f=^_kk)"
lemma
(theory fields)
(proof
(
direct-inference
direct-inference
(antecedent-inference-strategy (0))
extensionality
direct-and-antecedent-inference-strategy
(case-split ("1<=x_1"))
(induction trivial-integer-inductor ("x_1"))
beta-reduce-repeatedly
(instantiate-universal-antecedent "with(f:[kk,zz,kk],
forall(x:kk,n:zz,
not(x=o_kk) and 0<=n implies f(x,n+1)=f(x,n)*x));" ("x_0" "0"))
simplify
(simplify-antecedent "not(0<=0);")
(incorporate-antecedent "with(x_0:kk,f:[kk,zz,kk],f(x_0,0+1)=f(x_0,0)*x_0);")
simplify
(case-split ("x_0=o_kk"))
simplify
simplify
(case-split ("x_0=o_kk"))
simplify
(instantiate-universal-antecedent "with(f:[kk,zz,kk],
forall(x:kk,n:zz,
not(x=o_kk) and 0<=n implies f(x,n+1)=f(x,n)*x));" ("x_0" "t"))
(simplify-antecedent "with(t:zz,not(0<=t));")
(incorporate-antecedent "with(t:zz,x_0:kk,f:[kk,zz,kk],f(x_0,t+1)=f(x_0,t)*x_0);")
simplify
(case-split ("x_0=o_kk"))
simplify
(case-split ("x_1=0"))
simplify
(force-substitution "x_1" "-(-x_1)" (0))
(backchain "with(f:[kk,zz,kk],
forall(x:kk,n:zz,
not(x=o_kk) and 0<=n implies f(x,-n)=f(x,n)^[-1]));")
direct-and-antecedent-inference-strategy
simplify
(cut-with-single-formula "forall(x_1:zz, 1<=x_1 implies f(x_0,x_1)==x_0^x_1)")
(backchain "with(x_0:kk,f:[kk,zz,kk],
forall(x_1:zz,1<=x_1 implies f(x_0,x_1)==x_0^x_1));")
simplify
simplify
simplify
)))
```

```
(def-theorem uniqueness-of-exponentiation-corollary
"forall(f:[kk,zz,kk],
(forall(x:kk, n:zz, not(x=o_kk) and 0<=n implies f(x,n+1)=f(x,n)*x)
and forall(x:kk, not(x=o_kk)  implies f(x,0)=i_kk)
and forall(x:kk, n:zz, not(x=o_kk) and 0<=n implies f(x,-n)*f(x,n)=i_kk)
and forall(n:zz,1<=n implies f(o_kk,n)=o_kk)
and forall(n:zz, n<=0 implies not(#(f(o_kk,n)))))
implies
f=^_kk)"
lemma
(theory fields)
(usages transportable-macete)
(proof
(
direct-and-antecedent-inference-strategy
(instantiate-theorem uniqueness-of-exponentiation ("f"))
(contrapose "with(n:zz,x:kk,f:[kk,zz,kk],not(f(x,-n)=f(x,n)^[-1]));")
(instantiate-universal-antecedent "with(f:[kk,zz,kk],
forall(x:kk,n:zz,
not(x=o_kk) and 0<=n implies f(x,-n)*f(x,n)=i_kk));" ("x" "n"))
(cut-with-single-formula "not(f(x,n)=o_kk) and #(f(x,n)) and #(f(x,-n))")
(force-substitution "f(x,n)^[-1]" "f(x,n)^[-1]*(f(x,-n)*f(x,n))" (0))
(weaken (1))
simplify
simplify
direct-and-antecedent-inference-strategy
(contrapose "with(n:zz,x:kk,f:[kk,zz,kk],f(x,-n)*f(x,n)=i_kk);")
(case-split ("#(f(x,-n))"))
simplify
(contrapose "with(n:zz,x:kk,f:[kk,zz,kk],not(#(f(x,-n))));")
)))
```