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https://github.com/kenny2github/language-tnt

Typographic Number Theory syntax highlighting in Atom
https://github.com/kenny2github/language-tnt

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Typographic Number Theory syntax highlighting in Atom

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# language-tnt



Typographic Number Theory support in Atom.



## Summary of TNT

### Rules of Well-Formedness

**Numerals**

> 0 is a numeral.


> A numeral preceded by S is also a numeral.


> *Examples*: `0` `S0` `SS0` `SSS0` `SSSS0` `SSSSS0`



**Variables**

> `a` is a variable. If we're not being austere, so are `b`, `c`, `d`, and `e`.


> A variable followed by a prime is also a variable.


> *Examples*: `a` `b′` `c′′` `d′′′` `e′′′′`


> *Other symbols permitted instead of `′`*: `'`



**Terms**

> All numerals and variables are terms.


> A term preceded by `S` is also a term.


> If `s` and `t` are terms, then so are `(s+t)` and `(s⋅t)`.


> *Examples*: `0` `b` `(S0⋅(SS0+c))` `S(Sa⋅(Sb⋅Sc))`


> *Other symbols permitted instead of `⋅`*: `*`, `.`



The above rules tell how to make *parts* of well-formed formulas; the remaining rules tell how to make *complete* well-formed formulas.



**Atoms**

> If `s` and `t` are terms, then `s=t` is an atom.


> *Examples*: `S0=0` `(SS0+SS0)=SSSS0` `S(b+c)=((c⋅d)⋅e)`


> If an atom contains a variable `u`, then `u` is *free* in it. Thus there are four free variables in the last example.



**Negations**

> A well-formed formula preceded by a tilde is well-formed.


> *Examples*: `~S0=0` `~∃b:(b+b)=S0` `~<0=0⊃S0=0>` `~b=S0`


> *Other symbols permitted instead of `~`*: `!`



**Compounds**

> If `x` and `y` are well-formed formulas, and provided that no variable which is free in one is quantified in the other, then the following are all well-formed formulas:


> ``, ``, ``


> *Examples*: `<0=0∧~0=0>` `` ``


> *Other symbols permitted instead of `∧`*: `&`, `^`


> *Other symbols permitted instead of `∨`*: `|`, `V`, `v`


> *Other symbols permitted instead of `⊃`*: `→` (Alt+26 on Windows), `]` (less obvious as "implies")



**Quantifications**

> If `u` is a variable, and `x` is a well-formed formula in which `u` is free, then the following strings are well-formed formulas:


> `∃u:x` and `∀u:x`


> *Examples*: `∀b:` `∀c:~∃b:(b+b)=c` `~∃c:Sc=d`


> *Other symbols permitted instead of `∀`*: `A`


> *Other symbols permitted instead of `∃`*: `E`



### Rules of Propositional Calculus

**Joining Rule**: `joining`

> If `x` and `y` are theorems, then `` is a theorem.



**Separation Rule**: `separation`

> If `` is a theorem, then both `x` and `y` are theorems.



**Double-Tilde Rule**: `double-tilde`

> The string `~~` can be deleted from any theorem. It can also be inserted into any theorem, provided that the resulting string is itself well-formed



**Fantasy Rule**: `fantasy rule`

> If `y` can be derived when `x` is assumed to be a theorem, then `` is a theorem.



**Carry-Over Rule**: `carry over line`*`n`*

> Inside a fantasy, any theorem from the "reality" one level higher can be brought in and used.



**Rule of Detachment**: `detachment`

> If `x` and `` are both theorems, then `y` is a theorem.



**Contrapositive Rule**: `contrapositive`

> `` and `<~y⊃~x>` are interchangeable.



**De Morgan's Rule**: `De Morgan`

> `<~x∧~y>` and `~` are interchangeable.



**Switcheroo Rule**: `switcheroo`

> `` and `<~x⊃y>` are interchangeable.



### Rules of TNT

**Rule of Specification**: `specification`

> Suppose `u` is a variable which occurs inside the string `x`. If the string `∀u:x` is a theorem, then so is `x`, and so are any strings made from `x` by replacing `u`, wherever it occurs, by one and the same term.


> (*Restriction*: The term which replaces `u` must not contain any variable that is quantified in `x`.)



**Rule of Generalization**: `generalization`

> Suppose `x` is a theorem in which `u`, a variable, occurs free. Then `∀u:x` is a theorem.


> (*Restriction*: No generalization is allowed in a fantasy on any variable which appeared free in the fantasy's premise.)



**Rule of Interchange**: `interchange`

> Suppose `u` is a variable. Then the strings `∀u:~` and `~∃u:` are interchangeable anywhere inside any theorem.



**Rule of Existence**: `existence`

> Suppose a term (which may contain variables as long as they are free) appears once, or multiply, in a theorem. Then any (or several, or all) of the appearances of the term may be replaced by a variable which otherwise does not occur in the theorem, and the corresponding existential quantifier must be placed in front.



**Rule of Symmetry** (in Equality); `symmetry`

> If `r=s` is a theorem, then so is `s=r`



**Rule of Transitivity** (in Equality): `transitivity`

> If `r=s` and `s=t` are theorems, then so is `r=t`



**Add S** (a Rule of Successorship): `add S`

> If `r=t` is a theorem, then `Sr=St` is a theorem.



**Drop S** (a Rule of Successorship): `drop S`

> If `Sr=St` is theorem, then `r=t` is a theorem.



**Rule of Induction**: `induction`

> Let `X{u}` represent a well-formed formula in which the variable `u` is free, and `X{x/u}` represent the same string, with each appearance of `u` replaced by `x`.


> If both `∀u:` and `X{0/u}` are theorems, then `∀u:X{u}` is also a theorem.



### Examples

Check the [examples folder](https://github.com/Kenny2github/language-tnt/tree/master/examples) - `ascii` contains examples using only ASCII-compatible symbols, `unicode` contains the same examples using the proper (intended) symbols.