-theorem increasing_to_injective: ∀f:nat → nat.
- increasing f → injective nat nat f.
-#f #incr #n #m cases(decidable_le n m)
- [#lenm cases(le_to_or_lt_eq … lenm) //
- #lenm #eqf @False_ind @(absurd … eqf) @lt_to_not_eq
- @increasing_to_monotonic //
- |#nlenm #eqf @False_ind @(absurd … eqf) @sym_not_eq
- @lt_to_not_eq @increasing_to_monotonic /2/
+theorem increasing_to_injective: ∀f:\ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 → \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6.
+ \ 5a href="cic:/matita/arithmetics/nat/increasing.def(2)"\ 6increasing\ 5/a\ 6 f → \ 5a href="cic:/matita/basics/relations/injective.def(1)"\ 6injective\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 \ 5a href="cic:/matita/arithmetics/nat/nat.ind(1,0,0)"\ 6nat\ 5/a\ 6 f.
+#f #incr #n #m cases(\ 5a href="cic:/matita/arithmetics/nat/decidable_le.def(6)"\ 6decidable_le\ 5/a\ 6 n m)
+ [#lenm cases(\ 5a href="cic:/matita/arithmetics/nat/le_to_or_lt_eq.def(5)"\ 6le_to_or_lt_eq\ 5/a\ 6 … lenm) //
+ #lenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6
+ @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 //
+ |#nlenm #eqf @\ 5a href="cic:/matita/basics/logic/False_ind.fix(0,1,1)"\ 6False_ind\ 5/a\ 6 @(\ 5a href="cic:/matita/basics/logic/absurd.def(2)"\ 6absurd\ 5/a\ 6 … eqf) @\ 5a href="cic:/matita/basics/logic/sym_not_eq.def(4)"\ 6sym_not_eq\ 5/a\ 6
+ @\ 5a href="cic:/matita/arithmetics/nat/lt_to_not_eq.def(7)"\ 6lt_to_not_eq\ 5/a\ 6 @\ 5a href="cic:/matita/arithmetics/nat/increasing_to_monotonic.def(4)"\ 6increasing_to_monotonic\ 5/a\ 6 /2/