k4t assumes that precedence is transitive (transitivity of ≺), but the axiom is framed in terms of our tense operators.
Add the proof of temporal modal logic axiom TG4:

(TG4) Gp → GGp

And the corollaries

(TF4) FFp → Fp
(TP4) PPp → Pp
(TH4) Hp  → HHp

Checklist

• TG4
• TH4
• TF4
• TP4

Inductive Temporal Logic (Zt)

Zt include both forward induction and backward induction. It will include both no end and no beginning as well. Basically Zt is the same as #11 (Nt) , but i t has ne beginning (so isn't well ordered, in the same way that integers are like natural numbers. The backward induction claim that all moments must have an immediate successor.

(TANE) Gp → Fp
(TANB) Hp → Pp
(TAFI) (Fp ∧ G(p → Fp)) → GFp
(TABI) (Pp ∧ H(p → Pp)) → HPp 

Well-Ordered Temporal Logic (Nt)

Nt is going to include everything from #10 (Lt), but add on three new axioms, No End, Forward Induction (implies backwards discreteness), and Well Ordering. It creates a set of times instants that is isomporphically similar to ℕ.

Add the proofs of temporal modal logic axioms:

No End and No Beginning

(TANE) Gp → Fp
(TANB) Hp → Pp

Forward Induction

(TAFI) (Fp ∧ G(p → Fp)) → GFp

Well Ordering

(TAWO)  H(Hp → p) → H p

Checklist

• TANE
• TANB
• TAFI
• TAWO

⚠️ this PR will need #7 to be completed because of Well Ordering proof

NNPP define the equivalence: □p ≡ ¬◇¬p.

We defined NNPP like that for it's first part:
(eval (Globally p) t valuation) -> not (eval (Not (Future p)) t valuation)

which is not correct, the correct definition is
(eval (Globally p) t valuation) -> not (eval (Future (Not p)) t valuation)

Complete Temporal Logic (Rt)

Make a time frame Dedeking complete, which means that any set of instants that is defined by an instant that all instants in the set are before must be able to be defined by an instant that all instant in the set are before which is a instant in the original timeline. Neither can Dedekind completeness be expressed, stating that every non-empty set of instants which has an upper bound has a least upper bound.

completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.

Rt will need the axiom of #13 and the temporal completeness axiom

(TAC) A(Hp  → F(Hp)) → (Hp → Gp)

A stand for: forall instant

We have to look if we can use coqlang module to export only part of Proof.
This will allow user to Require import only the axiomatic system they will need.

For now, we only use Futur operators. An enhancement could be to add the Past operators:

M,t ⊨ Hφ iff M,t ⊨φ for all time instant  t′ such that t′≺t;
M,t ⊨ Pϕ iff M,h,t′ ⊨ϕ for some t' ∈ h such that t'≺t.

It is my hope that the proof for Past theorem will almost be the same as the proof we wrote for Future theorem.

Overview

The binary temporal operators S (“Since”) and U (“Until”) means:

• φ S ψ “φ has been true since a time when ψ was true”
• φ U ψ “φ will be true until a time when ψ is true”

The formal semantics of S and U in temporal models can be given as:

M,t ⊨ φ S ψ  iff M,s ⊨ ψ for some s such that s ⪯ t and M,u ⊨ φ for every u such that s ⪯ u ≺t.
M,t ⊨ φ U ψ iff M, s ⊨ ψ for some s such that t ⪯ s and M,u ⊨ φ for every u such that t ⪯ u ≺ s

The Axioms of Since and Until

We can define Fp and Pp in terms of since and until:

Fp = True U p
Pp = True S p
Xp = ⊥ U p
Bp = ⊥ S p

NOTE: we should do this by adding a definition and not a proof

We can add this proofs:

G(p → q) → p U x →q U x
G(p → q) → x U p →x U q
p ∧ x U q → x U (q  ∧ x S p)
p U q  ⟺ (p ∧ p U q) U q

⚠️ this will need #7

Lt will include all of #5 (Kt) , as well as #8 (transitivity) and axiom L that add linearity.
Add the proofs of temporal modal logic axioms:

(TAL) FPp ∨ PFp →Pp ∨ p ∨ Fp

checklist

• TAL

We introduced the next time operator X(p) for our linear and forward discrete temporal systems.
We have to add proofs for axioms:

(TKX) X(p → q) → (Xp →Xq)
(TNX) ~Xp ≡ X~p 

Checklist

• TKX
• TNX

The minimal temporal logic Kt axiomatizing all valid formulae of TL.
To achieve this we have to:

Add the proof of temporal modal logic axiom TAKG:

Whatever will always follow from what always will be, always will be
KG = G(x→y)→(Gx→Gy)

Add the proof of temporal modal logic axiom TAKH:

Whatever has always followed from what always has been, always has been
`KH = H(x→y)→(Hx→Hy)

Add the proof of temporal axiom TAGP:

“What is, will always have been”
p →GP(p)

Add the proof of temporal axiom TAHF:

“What is, has always been going to be”
p →GP(p)

Checklist

• TAKG
• TAKH
• TAGP
• TAHF

S4t assumes that temporal frame are reflexive. S4t includes all of the axioms of #8 (K4t), but adds an equivalent to the modal alethic Axiom T as well.

Add the proofs of temporal modal logic axioms:

(TGT) Gp → p

And the corollaries

(THT) Hp → p
(TPT) p  → Fp
(TFT) p  → Fp

Checklist

• TGT
• THT
• TPT
• TFT

This issue can need #7 to be completed.

Dense Temporal Logic (Qt)

Make the time frame dense, which means that between any two instants of time there is another instant. It is infinitely divisible. This is in direct contradiction to forward and backward induction.
This system, which we call Qt can be mapped onto the rational numbers.

Add the proofs of temporal modal logic axioms:

(TADG) GGp → Gp
(TADF) Fp → FFp

Add temporal logic necessitation rule TNR :

if  ⊢ p then ⊢ Ap

Where A(p) is true at all times past and present.