```(def-theorem ()
"forall(s:[zz,prop],
forall(t:zz,0<t implies s(t))
iff
(s(1) and forall(t:zz,0<t implies (s(t) implies s(1+t)))))"
(theory h-o-real-arithmetic)
(proof
(
direct-inference
direct-inference
direct-and-antecedent-inference-strategy
(backchain "with(s:[zz,prop],forall(t:zz,0<t implies s(t)));")
simplify
(backchain "with(s:[zz,prop],forall(t:zz,0<t implies s(t)));")
simplify
(cut-with-single-formula "forall(t:zz,1<=t implies s(t))")
direct-and-antecedent-inference-strategy
(backchain "with(s:[zz,prop],forall(t:zz,1<=t implies s(t)));")
simplify
(induction integer-inductor ("t"))
)))
```

```(def-theorem ()
"forall(x:rr,#(x,qq) iff forsome(a,b:zz,x=b^[-1]*a))"
(theory h-o-real-arithmetic)
(proof
(

direct-and-antecedent-inference-strategy
(instantiate-theorem zz-quotient-field ("x"))
(instantiate-existential ("a" "b"))
(incorporate-antecedent "with(b,a:zz,x:rr,x=a/b);")
simplify
(backchain "with(a,b:zz,x:rr,x=b^[-1]*a);")
sort-definedness
(contrapose "with(a,b:zz,x:rr,x=b^[-1]*a);")
simplify
)))
```

```(def-theorem ()
"forall(x:rr,#(x,qq) iff forsome(a,b:zz,x=a*b^[-1]))"
(theory h-o-real-arithmetic)
(proof
(
direct-and-antecedent-inference-strategy
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 0)")
(instantiate-theorem zz-quotient-field ("x")) (instantiate-existential ("a" "b"))
(incorporate-antecedent "with(r,x:rr,x=r);") simplify
)
(block
(script-comment "`direct-and-antecedent-inference-strategy' at (0 1 0)")
(backchain "with(p:prop,p);") sort-definedness (contrapose "with(p:prop,p);") simplify
)
)))
```

```(def-theorem ()
"sub=lambda(x,y:rr,x+[-1]*y)"
(theory h-o-real-arithmetic)
(proof
(

extensionality
simplify
)))
```

```(def-theorem ()
"<= = lambda(x,y:rr,x=y or x<y)"
(theory h-o-real-arithmetic)
(proof
(
extensionality
simplify
direct-and-antecedent-inference-strategy
simplify
simplify
simplify
)))

```

```(def-theorem ()
"^
=lambda(r:rr,n:zz,
if(n<0, 1/r^(-n), 2<=n, r^n, 1=n, r, not(r=0), 1, ?rr))"
(theory h-o-real-arithmetic)
(proof
(
extensionality
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(case-split ("x_1<0"))
(block
(script-comment "node added by `case-split' at (1)")
simplify
(case-split ("x_0=0"))
simplify
simplify)
(block
(script-comment "node added by `case-split' at (2)")
(case-split ("1=x_1"))
simplify
(block
(script-comment "node added by `case-split' at (2)")
(case-split ("2<=x_1"))
simplify
(block
(script-comment "node added by `case-split' at (2)")
(cut-with-single-formula "x_1=0")
(block
(script-comment "node added by `cut-with-single-formula' at (0)")
simplify
(case-split ("x_0=0"))
simplify
simplify)
simplify)))
)))
```

```(def-theorem ()
"forall(h_0:[rr,zz,rr],
forall(u_0:rr,u_1:zz,
#(if(u_1<0,
h_0(u_0,[-1]*u_1)^[-1],
2<=u_1,
h_0(u_0,[-1]+u_1)*u_0,
1=u_1,
u_0,
not(u_0=0),
1,
?rr))
implies
if(u_1<0,
h_0(u_0,[-1]*u_1)^[-1],
2<=u_1,
h_0(u_0,[-1]+u_1)*u_0,
1=u_1,
u_0,
not(u_0=0),
1,
?rr)
=h_0(u_0,u_1))
implies
forall(u_0:rr,u_1:zz,
#(u_0^u_1) implies u_0^u_1=h_0(u_0,u_1)))"
(theory h-o-real-arithmetic)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "2<=u_1 or 1=u_1 or 0=u_1 or u_1<0")
(antecedent-inference "with(u_1:zz,2<=u_1 or 1=u_1 or 0=u_1 or u_1<0);")
(induction trivial-integer-inductor ("u_1"))
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
simplify
(cut-with-single-formula "u_0^2=h_0(u_0,1)*u_0")
direct-inference
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
simplify
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
simplify
(backchain-backwards "with(h_0:[rr,zz,rr],t:zz,u_0:rr,u_0^t=h_0(u_0,t));")
simplify
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
simplify
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
simplify
(contrapose "with(u_1:zz,u_0:rr,#(u_0^u_1));")
simplify
(cut-with-single-formula "forall(u_1:zz,1<=u_1 implies u_0^u_1=h_0(u_0,u_1))")
(force-substitution "u_0^u_1" "(u_0^(-u_1))^[-1]" (0))
(backchain "with(h_0:[rr,zz,rr],u_0:rr,
forall(u_1:zz,1<=u_1 implies u_0^u_1=h_0(u_0,u_1)));")
simplify
(force-substitution "h_0(u_0,[-1]*u_1)^[-1]=h_0(u_0,u_1)"
"h_0(u_0,u_1)=h_0(u_0,[-1]*u_1)^[-1]" (0))
(backchain-backwards "with(p,q:prop,forall(u_0:rr,u_1:zz,p implies q))")
(cut-with-single-formula "not(u_0=0)")
simplify
(backchain-backwards "with(h_0:[rr,zz,rr],u_0:rr,
forall(u_1:zz,1<=u_1 implies u_0^u_1=h_0(u_0,u_1)));")
simplify
(contrapose "with(u_1:zz,u_0:rr,#(u_0^u_1));")
simplify
simplify
(cut-with-single-formula "not(u_0=0)")
simplify
(weaken (0 1))
direct-and-antecedent-inference-strategy
(cut-with-single-formula "1=u_1 or 2<=u_1")
(antecedent-inference "with(u_1:zz,1=u_1 or 2<=u_1);")
(weaken (1))
(cut-with-single-formula "#(u_0^u_1)")
simplify
(weaken (1))
(cut-with-single-formula "#(u_0^u_1)")
simplify
simplify
simplify
)))
```

```
(def-translation complete-ordered-field-interpretable
(source complete-ordered-field)
(target  h-o-real-arithmetic)
(constant-pairs
(^ ^)
(sub sub)
(<= <=))
(theory-interpretation-check using-simplification))
```