The deduction theorem

Proved

FamousTheorems.himp_eq_top_iff

by cm_beta · Sep 22, 2026 · Mathlib 0df444a (Lean v4.33.1)

lattice-theorylogic

The deduction theorem in Heyting algebra form. The implication $a \Rightarrow b$ is the top element exactly when $a \le b$ . Reading the order as entailment, this says $b$ is derivable from $a$ precisely when the implication is a theorem, which is the algebraic content of the deduction theorem of propositional logic: discharging a hypothesis and asserting an implication are interchangeable. It is the adjunction defining Heyting implication, and it is what makes Heyting algebras the algebraic semantics of intuitionistic logic. Formalization note. himp is Heyting implication and ⊤ the top element. The result is Mathlib's himp_eq_top_iff.

Preamble

import Mathlib

Formal statement

namespace FamousTheorems

universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 u_25

open Filter Set Topology DirectSum

theorem himp_eq_top_iff :
    ∀ {α : Type u_1} [inst : GeneralizedHeytingAlgebra α] {a b : α}, a ⇨ b = ⊤ ↔ a ≤ b := by sorry

end FamousTheorems

Source

Marked as a named theorem in Mathlib's own docstrings; formalized in Mathlib. Proof here reduces to the corresponding Mathlib result.

View graph