```(def-theory-ensemble-instances
metric-spaces
(target-theories fixed-interval-theory h-o-real-arithmetic)
force-under-quick-load
(permutations (0) (0 1))
(theory-interpretation-check using-simplification)
(sorts (pp ii rr))
(constants (dist "lambda(x,y:ii, abs(x-y))" "lambda(x,y:rr, abs(x-y))"))
(special-renamings
(continuous%at%point ii%continuous%at%point)
(continuous ii%continuous)))
```

```(def-theory-ensemble-instances
metric-spaces
(target-theories fixed-interval-theory)
force-under-quick-load
(permutations (0))
(theory-interpretation-check using-simplification)
(sorts (pp ii))
(constants (dist "lambda(x,y:ii, abs(x-y))")))
```

```(def-theory-ensemble-overloadings metric-spaces (1 2))
```

```
(def-theorem sup-property-of-continuity
Remark: This entry is multiply defined. See:  [1] [2]
"forall(f:[ii,rr],x,a:rr,s:sets[ii],
total_q{f,[ii,rr]}
and
ii%continuous(f)
and
#(sup(s),ii)
and
forall(x:rr,x in s implies f(x)<=a)
implies
f(sup(s))<=a)"
(theory fixed-interval-theory)
(proof
(
(apply-macete-with-minor-premises tr%eps-delta-characterization-of-continuity)
(unfold-single-defined-constant (0) ii%continuous%at%point)
direct-and-antecedent-inference-strategy
(instantiate-universal-antecedent "with(p:prop,forall(x:ii,forall(eps:rr,p)));"
("sup(s)"))
(contrapose "with(p:prop,
forall(eps:rr,0<eps implies forsome(delta:rr,p)));")
(instantiate-existential ("(f(sup(s))-a)/2"))
simplify
(cut-with-single-formula "forsome(x:ii, x in s and sup(s)-delta_\$0<x)")
(antecedent-inference "with(p:prop,forsome(x:ii,p));")
(instantiate-existential ("x"))
(apply-macete-with-minor-premises absolute-value-non-negative)
simplify
(apply-macete-with-minor-premises tr%minorizes-property-of-prec%sup)
direct-inference
(instantiate-existential ("x"))
simplify
(cut-with-single-formula "f(x)<=a")
(move-to-sibling 1)
simplify
(apply-macete-with-minor-premises absolute-value-non-negative)
simplify
(cut-with-single-formula "forsome(x:rr,x in s and sup(s)-delta_\$0<x)")
(antecedent-inference "with(p:prop,forsome(x:rr,p));")
(instantiate-existential ("with(x:rr, x)"))
(apply-macete-with-minor-premises sup-minus-epsilon-corollary)
direct-and-antecedent-inference-strategy
)))
```

```
(def-theorem subset-of-interval-characterization
"forall(s:sets[rr],a,b:ii,
forall(x:rr,x in s implies a<=x and x<=b)
implies
#(s,sets[ii]))"
(theory fixed-interval-theory)
(proof
(
direct-and-antecedent-inference-strategy
sort-definedness
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises interval-characterization)
(instantiate-existential ("b" "a"))
simplify
simplify
)))
```

```
(def-theorem inf-lemma
"forall(x,y,z:rr, forall(y_0:rr, y<y_0 implies x<=y_0*z) implies x<=y*z)"
lemma
(theory h-o-real-arithmetic)
(proof
(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "z<=0 or 0<z")
(block
(script-comment "node added by `cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,p or p);")
(block
(script-comment "node added by `antecedent-inference' at (0)")
(cut-with-single-formula "x<=(y+1)*z and (y+1)*z<=y*z")
simplify
(block
(script-comment "node added by `cut-with-single-formula' at (1)")
direct-and-antecedent-inference-strategy
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (0)")
(backchain "with(p:prop,forall(y_0:rr,p));")
simplify)
simplify))
(block
(script-comment "node added by `antecedent-inference' at (1)")
(cut-with-single-formula "y<x/z or x/z<=y")
(block
(script-comment "node added by `cut-with-single-formula' at (0)")
(antecedent-inference "with(p:prop,p or p);")
(block
(script-comment "node added by `antecedent-inference' at (0)")
(instantiate-universal-antecedent "with(p:prop,forall(y_0:rr,p));"
("[1/2]*(y+x/z)"))
(block
(script-comment "node added by `instantiate-universal-antecedent' at (0 0 0)")
(contrapose "with(p:prop,not(p));")
simplify)
(block
(script-comment "node added by `instantiate-universal-antecedent' at (0 0 1)")
(incorporate-antecedent "with(r,x:rr,x<=r);")
(apply-macete-with-minor-premises fractional-expression-manipulation)
simplify))
(block
(script-comment "node added by `antecedent-inference' at (1)")
(incorporate-antecedent "with(y,r:rr,r<=y);")
(apply-macete-with-minor-premises fractional-expression-manipulation)))
simplify))
simplify
)))
```

```
(def-constant ii%locally%lipschitz%bound
"lambda(phi:[ii,pp], r:rr, 0<r and
forall(x,z:ii, x<z implies forsome(y:ii, x<y and y<=z and dist(phi(y),phi(x))<=r*(y-x))))"
(theory mappings-from-an-interval))
```

```
(def-theorem lipschitz-on-interval-characterization
"forall(phi:[ii,pp], r:rr, 0<r and forall(x,y:ii, y<=x
implies
dist(phi(x),phi(y))<=r*(x-y))
implies
lipschitz%bound(phi,r))"
lemma
(proof
(
(while (progresses? unfold-directly-defined-constants) (skip))
direct-and-antecedent-inference-strategy
(case-split ("0<=x_\$0-y_\$0"))
(let-script instantiate-antecedents 2
(
(instantiate-universal-antecedent
"with(p:prop,forall(x,y:ii,p));"
(\$1 \$2))
(simplify-antecedent "with(p:prop,not(p));")
simplify))
(\$instantiate-antecedents "x_\$0" "y_\$0")
(\$instantiate-antecedents "y_\$0" "x_\$0")
))
(theory mappings-from-an-interval))
```

```
(def-theorem continuity-of-dist-expression
"forall(phi:[ii,pp],y:ii,a,b:rr,
total_q{phi,[ii,pp]}
and
continuous(phi)
implies
continuous(lambda(z:ii,a+b*z+dist(phi(z),phi(y)))))"
(theory mappings-from-an-interval)
(proof
(

direct-and-antecedent-inference-strategy
(incorporate-antecedent "with(phi:[ii,pp],continuous(phi));")
(apply-macete-with-minor-premises tr%eps-delta-characterization-of-continuity)
(move-to-sibling 1)
simplify-insistently
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises polynomial-continuity-macete)
(incorporate-antecedent "with(phi:[ii,pp],forall(x:ii,continuous%at%point(phi,x)));")
unfold-defined-constants
direct-and-antecedent-inference-strategy
(instantiate-existential ("eps"))
(instantiate-universal-antecedent "with(p:prop,forall(x:ii,forall(eps:rr,p)));"
("x"))
(instantiate-universal-antecedent "with(p:pp,r:rr,
forall(eps:rr,
0<eps
implies
forsome(delta:rr,
0<delta
and
forall(y:ii,abs(r)<=delta implies dist(p,p)<=eps))));"
("eps_\$0"))
(instantiate-existential ("delta"))
(cut-with-single-formula
"abs(dist(phi(x),phi(y))-dist(phi(y_\$0),phi(y)))<=dist(phi(x),phi(y_\$0))")
(move-to-sibling 1)
(apply-macete-with-minor-premises dist-continuity-lemma)
(instantiate-universal-antecedent "with(eps_\$0,delta,r:rr,
forall(y:ii,r<=delta implies r<=eps_\$0));"
("y_\$0"))
simplify
)))
```

```
(def-theorem locally-lipschitz-implies-lipschitz
"forall(phi:[ii,pp],r:rr,
total_q{phi,[ii,pp]}
and
continuous(phi)
and
ii%locally%lipschitz%bound(phi,r)
implies
lipschitz%bound(phi,r))"
lemma
(proof
(
(apply-macete-with-minor-premises lipschitz-on-interval-characterization)
(apply-macete-with-minor-premises tr%eps-delta-characterization-of-continuity)
(unfold-single-defined-constant (0) ii%locally%lipschitz%bound)
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forall(z:rr, y<=z and z<=x implies lambda(x:rr,dist(phi(x),phi(y))-r*(x-y)<=0)(z))")
(block
(script-comment "node added by `cut-with-single-formula' at (0)")
(instantiate-universal-antecedent "with(p:prop,forall(z:rr,p implies p));"
("x"))
(simplify-antecedent "with(p:prop,not(p));")
simplify)
(block
(script-comment "node added by `cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises tr%induction-principle-for-cpos)
beta-reduce-repeatedly
direct-and-antecedent-inference-strategy
simplify
(move-to-ancestor 2)
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (1 0 0)")
simplify-insistently
direct-and-antecedent-inference-strategy
(cut-with-single-formula "#(sup(t_\$0)) and y<=sup(t_\$0) and sup(t_\$0)<=x ")
(move-to-sibling 1)
(block
(script-comment "node added by `cut-with-single-formula' at (1)")
direct-and-antecedent-inference-strategy
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (0)")
(apply-macete-with-minor-premises tr%prec%sup-defined)
direct-and-antecedent-inference-strategy
auto-instantiate-existential
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (1 0)")
(unfold-single-defined-constant (0)

majorizes)
(instantiate-existential ("x"))
simplify))
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (1)")
(apply-macete-with-minor-premises tr%minorizes-property-of-prec%sup)
direct-and-antecedent-inference-strategy
(instantiate-existential ("y"))
simplify)
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (2)")
(apply-macete-with-minor-premises tr%lub-property-of-prec%sup)
direct-and-antecedent-inference-strategy
(unfold-single-defined-constant (0) majorizes)
direct-and-antecedent-inference-strategy
simplify))
(block
(script-comment "node added by `cut-with-single-formula' at (0)")
(cut-with-single-formula "lambda(z:ii,r*y+([-1]*r)*z+dist(phi(z),phi(y)))(sup(t_\$0))<=0")
simplify
(block
(script-comment "node added by `cut-with-single-formula' at (1)")
(apply-macete-with-minor-premises sup-property-of-continuity)
(block
(script-comment "node added by `apply-macete-with-minor-premises' at (0)")
direct-and-antecedent-inference-strategy
simplify-insistently
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (1)")
(apply-macete-with-minor-premises continuity-of-dist-expression)
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises tr%eps-delta-characterization-of-continuity))
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (2)")
(apply-macete-with-minor-premises interval-characterization)
(instantiate-existential ("x" "y")))
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (3 0 0)")
beta-reduce-with-minor-premises
(move-to-sibling 1)
(block
(script-comment "node added by `beta-reduce-with-minor-premises' at (1)")
(apply-macete-with-minor-premises interval-characterization)
(instantiate-existential ("x" "y"))
simplify
simplify)
simplify))
(block
(script-comment "node added by `apply-macete-with-minor-premises' at (1)")
(apply-macete-with-minor-premises subset-of-interval-characterization)
(instantiate-existential ("x" "y"))
simplify
simplify))))
(block
(script-comment "node added by `direct-and-antecedent-inference-strategy' at (1 1 0
0 0)")
(instantiate-universal-antecedent "with(p:prop,forall(x,z:ii,p));"
("y_\$1" "z_\$0"))
(simplify-antecedent "with(z_\$0,y_\$1:rr,not(y_\$1<z_\$0));")
(block
(script-comment "node added by `instantiate-universal-antecedent' at (0 0 1 0 0)")
(instantiate-existential ("y_\$0"))
(block
(script-comment "node added by `instantiate-existential' at (0 0)")
(cut-with-single-formula "dist(phi(y_\$0),phi(y))<=dist(phi(y_\$0),phi(y_\$1))+dist(phi(y_\$1),phi(y))")
(move-to-sibling 1)
(apply-macete-with-minor-premises triangle-inequality-for-distance)
simplify)
simplify
simplify)
(block
(script-comment "node added by `instantiate-universal-antecedent' at (1 2)")
(apply-macete-with-minor-premises interval-characterization)
(instantiate-existential ("x" "y"))
simplify)))
))
(theory mappings-from-an-interval))
```

```
(def-theorem locally-lipschitz-implies-lipschitz-plus
"forall(phi:[ii,pp], r_0:rr, 0<r_0 and total_q{phi,[ii,pp]} and continuous(phi) and
forall(r:rr,r_0<r implies ii%locally%lipschitz%bound(phi,r))
implies
lipschitz%bound(phi,r_0))"
(theory mappings-from-an-interval)
(usages transportable-macete)
(proof

(
direct-and-antecedent-inference-strategy
(cut-with-single-formula "forall(r:rr,r_0<r implies lipschitz%bound(phi,r))")
(incorporate-antecedent "with(phi:[ii,pp],r_0:rr,
forall(r:rr,r_0<r implies lipschitz%bound(phi,r)));")
unfold-defined-constants
unfold-defined-constants
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises inf-lemma)
direct-and-antecedent-inference-strategy
(backchain "with(p:prop, forall(r:rr, r_0<r implies 0<r and p))")
direct-and-antecedent-inference-strategy
(apply-macete-with-minor-premises locally-lipschitz-implies-lipschitz)
direct-and-antecedent-inference-strategy
(backchain "with(p:prop,  forall(r:rr, r_0<r implies p))")
)))
```