Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
Datasets:
statement stringlengths 5 10.7k | proof stringlengths 0 9.25k | type stringclasses 4
values | symbolic_name stringlengths 1 80 | library stringclasses 103
values | filename stringclasses 746
values | imports listlengths 0 49 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
refl-right-neutral : {A : Type} {a b : A} (p : a ＝ b) → p ∙ refl ＝ p | refl-right-neutral refl = refl | function | refl-right-neutral | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/CommutativeLoopSpaces.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices-streamlined"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl-left-neutral : {A : Type} {a b : A} (p : a ＝ b) → refl ∙ p ＝ p | refl-left-neutral refl = refl | function | refl-left-neutral | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/CommutativeLoopSpaces.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices-streamlined"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_＝⟨_⟩_ : {X : Type} (x : X) {y z : X} → x ＝ y → y ＝ z → x ＝ z | function | _＝⟨_⟩_ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
_∎ : {X : Type} (x : X) → x ＝ x | _∎ _ = refl | function | _∎ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl-left-neutral : {X : Type} {x y : X} (p : x ＝ y) → refl ∙ p ＝ p | refl-left-neutral refl = refl | function | refl-left-neutral | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl-right-neutral : {X : Type} {x y : X} (p : x ＝ y) → p ∙ refl ＝ p | refl-right-neutral refl = refl | function | refl-right-neutral | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
majorities-only-act-on-sets : (M : Type) (m : M → M → M → M)
→ ((a b : M) → m b a a ＝ a)
→ ((a b : M) → m a b a ＝ a)
→ ((a b : M) → m a a b ＝ a)
→ (m₀ : M) → (p : m₀ ＝ m₀) → p ＝ refl | majorities-only-act-on-sets M m eq₀ eq₁ eq₂ m₀ =
Ω-trivial
M
m₀
(m m₀ m₀ m₀)
f
idem₁
side₁-is-p
side₀-is-refl
side₂-is-refl
(λ p q r s → sym (m'-is-homo p r refl refl q s))
where
open Ω-trivial-criterion
open type-with-majority M m eq₀ eq₁ eq₂ m₀
f : (m₀ ＝ m₀) → (m₀ ＝ m₀) → m m₀ m₀ m₀ ... | function | majorities-only-act-on-sets | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [
"Type",
"refl",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_ ＝⟨ p ⟩ q = p ∙ q | function | _ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority-streamlined.lagda | [
"Agda.Primitive",
"AlgebraicStructuresForcingSethood.Semilattices",
"AlgebraicStructuresForcingSethood.Majority"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
ap₃ : {A B C D : Type} (f : A → B → C → D) {a₁ a₂ : A} {b₁ b₂ : B} {c₁ c₂ : C}
→ a₁ ＝ a₂
→ b₁ ＝ b₂
→ c₁ ＝ c₂
→ f a₁ b₁ c₁ ＝ f a₂ b₂ c₂ | ap₃ f refl refl refl = refl | function | ap₃ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Majority.lagda | [
"AlgebraicStructuresForcingSethood.Semilattices",
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_＝_ {A : Type} (a : A) : A → Type where
refl : a ＝ a | data | _＝_ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
_∙_ : {A : Type} {a b c : A} → a ＝ b → b ＝ c → a ＝ c | function | _∙_ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
sym : {A : Type} {a b : A} → a ＝ b → b ＝ a | sym refl = refl | function | sym | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap : {A B : Type} {a b : A} (f : A → B) → a ＝ b → f a ＝ f b | ap f refl = refl | function | ap | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap₂ : {A B C : Type} (f : A → B → C) {a₁ a₂ : A} {b₁ b₂ : B}
→ a₁ ＝ a₂
→ b₁ ＝ b₂
→ f a₁ b₁ ＝ f a₂ b₂ | ap₂ f refl refl = refl | function | ap₂ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
∙refl : {A : Type} {a b : A} (p : a ＝ b) → p ∙ refl ＝ p | ∙refl refl = refl | function | ∙refl | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl∙ : {A : Type} {a b : A} (p : a ＝ b) → refl ∙ p ＝ p | refl∙ refl = refl | function | refl∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
∙-cancel : {A : Type} {a b c : A} (p : a ＝ b) (q₁ q₂ : b ＝ c)
→ p ∙ q₁ ＝ p ∙ q₂ → q₁ ＝ q₂ | ∙-cancel refl refl q₂ h = refl ＝⟨ h ⟩
refl ∙ q₂ ＝⟨ refl∙ q₂ ⟩
q₂ ∎ | function | ∙-cancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl",
"refl∙"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr : {A : Type} {a b x y : A} → a ＝ x → b ＝ y → a ＝ b → x ＝ y | eq-congr refl refl p = p | function | eq-congr | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-refl : {A : Type} {a x : A} (h : a ＝ x) → eq-congr h h refl ＝ refl | eq-congr-refl refl = refl | function | eq-congr-refl | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-nat : {A : Type} {a b x y : A}
(ha : a ＝ a) (hb : b ＝ b) (hax : a ＝ x) (hby : b ＝ y)
(p : a ＝ b)
→ eq-congr hax hby (eq-congr ha hb p)
＝ eq-congr
(eq-congr hax hax ha)
(eq-congr hby hby hb)
(eq-congr hax ... | eq-congr-nat ha hb refl refl p = refl | function | eq-congr-nat | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-nat' : {A : Type} {a b x y : A}
(hab : a ＝ b) (hax : a ＝ x) (hby : b ＝ y)
(p : a ＝ a)
→ eq-congr hby hby (eq-congr hab hab p)
＝ eq-congr
(eq-congr hax hby hab)
(eq-congr hax hby hab)
(eq-co... | eq-congr-nat' refl refl refl p = refl | function | eq-congr-nat' | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-∙ : {A : Type} {a b c x y z : A}
{h₁ : a ＝ x} {h₂ : b ＝ y} {h₃ : c ＝ z}
(p : a ＝ b) (q : b ＝ c)
→ eq-congr h₁ h₃ (p ∙ q) ＝ eq-congr h₁ h₂ p ∙ eq-congr h₂ h₃ q | eq-congr-∙ {h₁ = refl} {h₂ = refl} {h₃ = refl} p q = refl | function | eq-congr-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
congr-∙ : {A : Type} {a b u v x y : A}
(l₁ : a ＝ u) (l₂ : u ＝ x) (r₁ : b ＝ v) (r₂ : v ＝ y) (p : a ＝ b)
→ eq-congr (l₁ ∙ l₂) (r₁ ∙ r₂) p ＝ eq-congr l₂ r₂ (eq-congr l₁ r₁ p) | congr-∙ refl refl refl refl p = refl | function | congr-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"l₂",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
conjugate-loop : {A : Type} {a b : A} → a ＝ b → a ＝ a → b ＝ b | conjugate-loop p = eq-congr p p | function | conjugate-loop | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
conjugate-loop' : {A : Type} {a b : A} → a ＝ b → a ＝ a → b ＝ b | conjugate-loop' p q = sym p ∙ q ∙ p | function | conjugate-loop' | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
conjugate-loops-agree : {A : Type} {a b x y : A} (p : a ＝ x) (q : a ＝ a)
→ conjugate-loop p q ＝ conjugate-loop' p q | conjugate-loops-agree refl q =
q ＝⟨ sym (refl∙ q) ⟩
refl ∙ q ＝⟨ ap (refl ∙_) (sym (∙refl q)) ⟩
refl ∙ q ∙ refl ∎ | function | conjugate-loops-agree | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [
"Type",
"ap",
"conjugate-loop",
"conjugate-loop'",
"refl",
"refl∙",
"sym",
"∙refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl ∙ refl = refl | function | refl | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices-streamlined.lagda | [
"Agda.Primitive"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
ap₂ : {A B C : Type} (f : A → B → C) {a₁ a₂ : A} {b₁ b₂ : B}
→ a₁ ＝ a₂ → b₁ ＝ b₂ → f a₁ b₁ ＝ f a₂ b₂ | ap₂ f refl refl = refl | function | ap₂ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-sym : {A : Type} {a b x y : A}
{hax : a ＝ x} {hby : b ＝ y}
{p : a ＝ b} {q : x ＝ y}
→ eq-congr hax hby p ＝ q
→ p ＝ eq-congr (sym hax) (sym hby) q | eq-congr-sym {hax = refl} {hby = refl} refl = refl | function | eq-congr-sym | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-congr-sq : {A : Type} {a b x y : A}
(p : a ＝ b) (q : a ＝ x) (r : b ＝ y)
→ q ∙ eq-congr q r p ＝ p ∙ r | eq-congr-sq refl refl refl = refl | function | eq-congr-sq | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
iter-congr : {A : Type} {a b u v x y : A}
(l₁ : a ＝ u) (l₂ : u ＝ x) (r₁ : b ＝ v) (r₂ : v ＝ y) (p : a ＝ b)
→ eq-congr (l₁ ∙ l₂) (r₁ ∙ r₂) p ＝ eq-congr l₂ r₂ (eq-congr l₁ r₁ p) | iter-congr refl refl refl refl p = refl | function | iter-congr | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"l₂",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
∙-cancel : {A : Type} {a b c : A} (p : a ＝ b) (q₁ q₂ : b ＝ c)
→ p ∙ q₁ ＝ p ∙ q₂ → q₁ ＝ q₂ | ∙-cancel refl q₁ q₂ h = eq-congr (refl∙ q₁) (refl∙ q₂) h | function | ∙-cancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"eq-congr",
"refl",
"refl∙"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
assoc₄ : {A : Type} {a b c d e : A}
{p : a ＝ b} {q : b ＝ c} {r : c ＝ d} {s : d ＝ e}
→ (p ∙ q) ∙ (r ∙ s) ＝ p ∙ (q ∙ r) ∙ s | assoc₄ {p = refl} {q = refl} {r = refl} {s = refl} = refl | function | assoc₄ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
comm₂ : {A : Type} {a : A} {p q : a ＝ a} (h : p ∙ q ＝ q ∙ p)
→ (p ∙ p) ∙ (q ∙ q) ＝ (q ∙ q) ∙ (p ∙ p) | comm₂ {p = p} {q = q} h =
eq-congr
(sym assoc₄)
(sym assoc₄)
(eq-congr
(ap (λ x → p ∙ x ∙ q) (sym h))
(ap (λ x → q ∙ x ∙ p) h)
(eq-congr assoc₄ assoc₄ (ap (λ x → x ∙ x) h))) | function | comm₂ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/Semilattices.lagda | [
"Agda.Primitive"
] | [
"Type",
"ap",
"assoc₄",
"eq-congr",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
sym : {A : Type} {a b : A} → a ＝ b → b ＝ a | sym = _⁻¹ -- I will use sym for inverting paths | function | sym | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"_⁻¹"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl∙ : {A : Type} {a b : A} (q : a ＝ b) → refl ∙ q ＝ q | refl∙ refl = refl | function | refl∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
∙-cancel : {A : Type} {a b c : A} (q : a ＝ b) {p p' : b ＝ c}
→ q ∙ p ＝ q ∙ p'
→ p ＝ p' | ∙-cancel refl {p} {p'} h = sym (refl∙ p) ∙ h ∙ (refl∙ p') | function | ∙-cancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl",
"refl∙",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap₃-homo : {A B C D : Type} (f : A → B → C → D)
{a₁ a₂ a₃ : A} {b₁ b₂ b₃ : B} {c₁ c₂ c₃ : C}
(pa : a₁ ＝ a₂) (qa : a₂ ＝ a₃)
(pb : b₁ ＝ b₂) (qb : b₂ ＝ b₃)
(pc : c₁ ＝ c₂) (qc : c₂ ＝ c₃)
→ ap₃ f pa pb pc ∙ ap₃ f qa qb qc ＝ ap₃ f (pa ∙ qa) (pb ∙ qb) (pc ∙ qc) | ap₃-homo f refl refl refl refl refl refl = refl | function | ap₃-homo | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"ap₃",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong : {A : Type} {a a' b b' : A} → a ＝ a' → b ＝ b' → a ＝ b → a' ＝ b' | eq-cong refl refl p = p | function | eq-cong | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-∙ : {A : Type} {a a' b b' c c' : A}
→ {q : a ＝ a'}
→ {q' : b ＝ b'}
→ {q'' : c ＝ c'}
→ (p : a ＝ b)
→ (r : b ＝ c)
→ eq-cong q q'' (p ∙ r) ＝ eq-cong q q' p ∙ eq-cong q' q'' r | eq-cong-∙ {q = refl} {q' = refl} {q'' = refl} p r = refl | function | eq-cong-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-refl : {A : Type} {a a' : A} (q : a ＝ a') → eq-cong q q refl ＝ refl | eq-cong-refl refl = refl | function | eq-cong-refl | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-sq : {A : Type} {a a' b b' : A} (h₁ : a ＝ a') (h₂ : b ＝ b') (p : a ＝ b)
→ h₁ ∙ eq-cong h₁ h₂ p ＝ p ∙ h₂ | eq-cong-sq refl refl p = (refl∙ p) | function | eq-cong-sq | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/SimpleCaseOfWillard.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl",
"refl∙"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
sym : {A : Type} {a b : A} → (a ＝ b) → (b ＝ a) | sym = _⁻¹ | function | sym | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"_⁻¹"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
sym-lcancel : {A : Type} {a b : A} (p : a ＝ b)
→ sym p ∙ p ＝ refl | sym-lcancel refl = refl | function | sym-lcancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
sym-rcancel : {A : Type} {a b : A} (p : a ＝ b)
→ p ∙ sym p ＝ refl | sym-rcancel refl = refl | function | sym-rcancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
∙-assoc : {A : Type} {a b c d : A} (p : a ＝ b) (q : b ＝ c) (r : c ＝ d)
→ p ∙ (q ∙ r) ＝ (p ∙ q) ∙ r | ∙-assoc p refl refl = refl | function | ∙-assoc | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
refl' : {A : Type} → (a : A) → a ＝ a | refl' a = refl | function | refl' | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
lap-∙ : {A : Type} {a b c : A} {p p' : a ＝ b} (q : b ＝ c)
→ (p ＝ p') → (p ∙ q ＝ p' ∙ q) | lap-∙ refl h = h | function | lap-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
rap-∙ : {A : Type} {a b c : A} {p p' : b ＝ c} (q : a ＝ b)
→ (p ＝ p') → (q ∙ p ＝ q ∙ p') | rap-∙ q refl = refl | function | rap-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
map-∙ : {A : Type} {a b c d : A}
→ (p : a ＝ b)
→ {q q' : b ＝ c}
→ (q ＝ q')
→ (r : c ＝ d)
→ (p ∙ q ∙ r ＝ p ∙ q' ∙ r) | map-∙ p refl refl = refl | function | map-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap₃-homo : {A B C D : Type}
(f : A → B → C → D)
{a₁ a₂ a₃ : A} {b₁ b₂ b₃ : B} {c₁ c₂ c₃ : C}
(pa : a₁ ＝ a₂) (qa : a₂ ＝ a₃)
(pb : b₁ ＝ b₂) (qb : b₂ ＝ b₃)
(pc : c₁ ＝ c₂) (qc : c₂ ＝ c₃)
→ ((ap₃ f) pa pb pc) ∙ ((ap₃ f) qa qb qc)
＝ (ap₃ f) (pa ∙ qa) ... | ap₃-homo f {a₁ = a} {b₁ = b} {c₁ = c} refl refl refl refl refl refl = refl | function | ap₃-homo | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"ap₃",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap₃-homo' : {A B C D : Type}
(f : A → B → C → D)
{a₁ a₂ a₃ : A} {b₁ b₂ b₃ : B} {c₁ c₂ c₃ : C}
(pa : a₁ ＝ a₂) (qa : a₂ ＝ a₃) {ra : a₁ ＝ a₃}
(pb : b₁ ＝ b₂) (qb : b₂ ＝ b₃) {rb : b₁ ＝ b₃}
(pc : c₁ ＝ c₂) (qc : c₂ ＝ c₃) {rc : c₁ ＝ c₃}
(a^ : ra ＝ pa... | ap₃-homo' f pa qa pb qb pc qc a^ b^ c^ =
ap₃ f _ _ _ ＝⟨ ap₃ (ap₃ f) a^ b^ c^ ⟩
ap₃ f (pa ∙ qa) (pb ∙ qb) (pc ∙ qc) ＝⟨ sym (ap₃-homo f pa qa pb qb pc qc) ⟩
ap₃ f pa pb pc ∙ ap₃ f qa qb qc ∎ | function | ap₃-homo' | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"ap₃",
"ap₃-homo",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
ap₃-sym : {A B C D : Type}
(f : A → B → C → D)
{a a' : A} {b b' : B} {c c' : C}
(p : a ＝ a')
(q : b ＝ b')
(r : c ＝ c')
→ sym (ap₃ f p q r) ＝ ap₃ f (sym p) (sym q) (sym r) | ap₃-sym f refl refl refl = refl | function | ap₃-sym | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"ap₃",
"refl",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong : {A : Type} {a a' b b' : A}
→ (a ＝ a') → (b ＝ b') → (a ＝ b) → (a' ＝ b') | eq-cong refl refl p = p | function | eq-cong | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-∙ : {A : Type} {a a' b b' c c' : A}
{q : a ＝ a'} {q' : b ＝ b'} {q'' : c ＝ c'}
(p : a ＝ b) (r : b ＝ c)
→ eq-cong q q'' (p ∙ r) ＝ eq-cong q q' p ∙ eq-cong q' q'' r | eq-cong-∙ {q = refl} {q' = refl} {q'' = refl} p r = refl | function | eq-cong-∙ | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-refl : {A : Type} {a a' : A} (q : a ＝ a')
→ eq-cong q q refl ＝ refl | eq-cong-refl refl = refl | function | eq-cong-refl | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-ap : {A B C D : Type}
(f : A → B → C → D)
{a a' a'' a''' : A} {b b' b'' b''' : B} {c c' c'' c''' : C}
(qa : a' ＝ a) (qa' : a'' ＝ a''') (pa : a' ＝ a'')
(qb : b' ＝ b) (qb' : b'' ＝ b''') (pb : b' ＝ b'')
(qc : c' ＝ c) (qc' : c'' ＝ c''') (pc : c' ＝ c''... | eq-cong-ap f refl refl pa refl refl pb refl refl pc = refl | function | eq-cong-ap | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"ap₃",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eq-cong-cancel : {A : Type} {a a' b b' : A} {p q : a ＝ a'}
→ (h₁ : a ＝ b)
→ (h₂ : a' ＝ b')
→ eq-cong h₁ h₂ p ＝ eq-cong h₁ h₂ q
→ p ＝ q | eq-cong-cancel refl refl h = h | function | eq-cong-cancel | AlgebraicStructuresForcingSethood | source/AlgebraicStructuresForcingSethood/WeakNearUnanimity.lagda | [
"MLTT.Spartan"
] | [
"Type",
"eq-cong",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-prop-valued
is-irreflexive
is-symmetric
is-strongly-cotransitive
is-tight
is-strong-apartness
: {X : 𝓤 ̇ } → (X → X → 𝓥 ̇ ) → 𝓤 ⊔ 𝓥 ̇ | is-prop-valued _♯_ = ∀ x y → is-prop (x ♯ y) | function | is-prop-valued | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"is-irreflexive",
"is-prop",
"is-strong-apartness",
"is-strongly-cotransitive",
"is-symmetric",
"is-tight"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Strong-Apartness : 𝓤 ̇ → (𝓥 : Universe) → 𝓥 ⁺ ⊔ 𝓤 ̇ | Strong-Apartness X 𝓥 = Σ _♯_ ꞉ (X → X → 𝓥 ̇ ) , is-strong-apartness _♯_ | function | Strong-Apartness | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"is-strong-apartness"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
double-negation-of-equality-gives-negation-of-apartness
: {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ )
→ is-irreflexive _♯_
→ ¬¬ (x ＝ y)
→ ¬ (x ♯ y) | double-negation-of-equality-gives-negation-of-apartness x y _♯_ i
= contrapositive f
where
f : x ♯ y → ¬ (x ＝ y)
f a refl = i y a | function | double-negation-of-equality-gives-negation-of-apartness | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"contrapositive",
"is-irreflexive",
"refl",
"¬¬"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
tight-types-are-¬¬-separated' : {X : 𝓤 ̇ } (_♯_ : X → X → 𝓥 ̇ )
→ is-irreflexive _♯_
→ is-tight _♯_
→ is-¬¬-separated X | tight-types-are-¬¬-separated' _♯_ i t = f
where
f : ∀ x y → ¬¬ (x ＝ y) → x ＝ y
f x y φ = t x y (double-negation-of-equality-gives-negation-of-apartness
x y _♯_ i φ) | function | tight-types-are-¬¬-separated' | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"double-negation-of-equality-gives-negation-of-apartness",
"is-irreflexive",
"is-tight",
"is-¬¬-separated",
"¬¬"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
tight-types-are-sets' : {X : 𝓤 ̇ } (_♯_ : X → X → 𝓥 ̇ )
→ funext 𝓤 𝓤₀
→ is-irreflexive _♯_
→ is-tight _♯_
→ is-set X | tight-types-are-sets' _♯_ fe i t =
¬¬-separated-types-are-sets fe (tight-types-are-¬¬-separated' _♯_ i t) | function | tight-types-are-sets' | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"fe",
"funext",
"is-irreflexive",
"is-set",
"is-tight",
"tight-types-are-¬¬-separated'",
"¬¬-separated-types-are-sets"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-irreflexive _♯_ = ∀ x → ¬ (x ♯ x) | function | is-irreflexive | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
is-symmetric _♯_ = ∀ x y → x ♯ y → y ♯ x | function | is-symmetric | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
is-strongly-cotransitive _♯_ = ∀ x y z → x ♯ y → (x ♯ z) + (y ♯ z) | function | is-strongly-cotransitive | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
is-tight _♯_ = ∀ x y → ¬ (x ♯ y) → x ＝ y | function | is-tight | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
is-strong-apartness _♯_ = is-prop-valued _♯_
× is-irreflexive _♯_
× is-symmetric _♯_
× is-strongly-cotransitive _♯_ | function | is-strong-apartness | Apartness | source/Apartness/Definition.lagda | [
"MLTT.Spartan",
"UF.DiscreteAndSeparated",
"UF.FunExt",
"UF.Lower-FunExt",
"UF.NotNotStablePropositions",
"UF.PropTrunc",
"UF.Sets",
"UF.Sets-Properties",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_",
"is-irreflexive",
"is-prop-valued",
"is-strongly-cotransitive",
"is-symmetric"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
is-strongly-extensional : ∀ {𝓣} {X : 𝓤 ̇ } {Y : 𝓥 ̇ }
→ (X → X → 𝓦 ̇ ) → (Y → Y → 𝓣 ̇ ) → (X → Y) → 𝓤 ⊔ 𝓦 ⊔ 𝓣 ̇ | is-strongly-extensional _♯_ _♯'_ f = ∀ x x' → f x ♯' f x' → x ♯ x' | function | is-strongly-extensional | Apartness | source/Apartness/Morphisms.lagda | [
"Apartness.Definition",
"MLTT.Spartan",
"UF.FunExt",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
being-strongly-extensional-is-prop : Fun-Ext
→ {X : 𝓤 ̇ } {Y : 𝓥 ̇ }
→ (_♯_ : X → X → 𝓦 ̇ )
→ (_♯'_ : Y → Y → 𝓣 ̇ )
→ is-prop-valued _♯_
→ (f... | being-strongly-extensional-is-prop fe _♯_ _♯'_ ♯p f =
Π₃-is-prop fe (λ x x' a → ♯p x x') | function | being-strongly-extensional-is-prop | Apartness | source/Apartness/Morphisms.lagda | [
"Apartness.Definition",
"MLTT.Spartan",
"UF.FunExt",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [
"Fun-Ext",
"_♯_",
"fe",
"is-prop",
"is-prop-valued",
"is-strongly-extensional",
"Π₃-is-prop"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
preserves : ∀ {𝓣} {X : 𝓤 ̇ } {Y : 𝓥 ̇ }
→ (X → X → 𝓦 ̇ ) → (Y → Y → 𝓣 ̇ ) → (X → Y) → 𝓤 ⊔ 𝓦 ⊔ 𝓣 ̇ | preserves R S f = ∀ {x x'} → R x x' → S (f x) (f x') | function | preserves | Apartness | source/Apartness/Morphisms.lagda | [
"Apartness.Definition",
"MLTT.Spartan",
"UF.FunExt",
"UF.Subsingletons",
"UF.Subsingletons-FunExt"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
elements-that-are-not-apart-have-the-same-apartness-class
: {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ )
→ is-apartness _♯_
→ ¬ (x ♯ y)
→ ((z : X) → x ♯ z ↔ y ♯ z) | elements-that-are-not-apart-have-the-same-apartness-class
{𝓤} {𝓥} {X} x y _♯_ (p , _ , s , c) = g
where
g : ¬ (x ♯ y) → (z : X) → x ♯ z ↔ y ♯ z
g n z = g₁ , g₂
where
g₁ : x ♯ z → y ♯ z
g₁ a = s z y (left-fails-gives-right-holds (p z y) b n)
where
b : (x ♯ y) ∨ (z ♯ y)
b = c x z y a
... | function | elements-that-are-not-apart-have-the-same-apartness-class | Apartness | source/Apartness/Negation.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan"
] | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
elements-with-the-same-apartness-class-are-not-apart
: {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ )
→ is-irreflexive _♯_
→ ((z : X) → x ♯ z ↔ y ♯ z)
→ ¬ (x ♯ y) | elements-with-the-same-apartness-class-are-not-apart
{𝓤} {𝓥} {X} x y _♯_ i = f
where
f : ((z : X) → x ♯ z ↔ y ♯ z) → ¬ (x ♯ y)
f φ a = i y (pr₁(φ y) a) | function | elements-with-the-same-apartness-class-are-not-apart | Apartness | source/Apartness/Negation.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan"
] | [
"_♯_",
"is-irreflexive",
"pr₁"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
has-two-points-apart : {X : 𝓤 ̇ } → Apartness X 𝓥 → 𝓥 ⊔ 𝓤 ̇ | has-two-points-apart {𝓤} {𝓥} {X} (_♯_ , α) = Σ (x , y) ꞉ X × X , (x ♯ y) | function | has-two-points-apart | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"_♯_"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Nontrivial-Apartness : 𝓤 ̇ → (𝓥 : Universe) → 𝓥 ⁺ ⊔ 𝓤 ̇ | Nontrivial-Apartness X 𝓥 = Σ a ꞉ Apartness X 𝓥 , has-two-points-apart a | function | Nontrivial-Apartness | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"has-two-points-apart"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness
: funext 𝓤 𝓤₀
→ {X : 𝓤 ̇ }
→ has-two-distinct-points X
→ typal-WEM 𝓤
→ Nontrivial-Apartness X 𝓤 | WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness
{𝓤} fe {X} htdp wem = γ
where
s : (x y z : X) → x ≠ y → (x ≠ z) + (y ≠ z)
s x y z d =
Cases (wem (x ≠ z))
(λ (a : ¬ (x ≠ z)) → inr (λ {refl → a d}))
(λ (b : ¬¬ (x ≠ z)) → inl (three-negations-imply-one b))
c : is-cotransitive _≠... | function | WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"Cases",
"Nontrivial-Apartness",
"_≠_",
"fe",
"funext",
"has-two-distinct-points",
"inl",
"inr",
"negations-are-props",
"refl",
"three-negations-imply-one",
"typal-WEM",
"¬¬",
"≠-is-irrefl",
"≠-sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness⁺
: funext 𝓤 𝓤₀
→ {X : 𝓤 ⁺ ̇ }
→ is-locally-small X
→ has-two-distinct-points X
→ typal-WEM 𝓤
→ Nontrivial-Apartness X 𝓤 | WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness⁺
{𝓤} fe {X} ls ((x₀ , x₁) , d) wem = γ
where
_♯_ : X → X → 𝓤 ̇
x ♯ y = x ≠⟦ ls ⟧ y
s : (x y z : X) → x ♯ y → (x ♯ z) + (y ♯ z)
s x y z a = Cases (wem (x ♯ z)) (inr ∘ f) (inl ∘ g)
where
f : ¬ (x ♯ z) → y ♯ z
f = contrapositive
... | function | WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness⁺ | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"Cases",
"Nontrivial-Apartness",
"_♯_",
"contrapositive",
"fe",
"funext",
"has-two-distinct-points",
"inl",
"inr",
"is-locally-small",
"negations-are-props",
"three-negations-imply-one",
"transport",
"typal-WEM",
"x₀",
"¬¬",
"≠-gives-≠⟦",
"≠⟦",
"＝⟦"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
WEM-gives-non-trivial-apartness-on-universe
: funext (𝓤 ⁺) 𝓤₀
→ typal-WEM (𝓤 ⁺)
→ Nontrivial-Apartness (𝓤 ̇ ) (𝓤 ⁺) | WEM-gives-non-trivial-apartness-on-universe fe =
WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness
fe
universe-has-two-distinct-points | function | WEM-gives-non-trivial-apartness-on-universe | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"Nontrivial-Apartness",
"WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness",
"fe",
"funext",
"typal-WEM",
"universe-has-two-distinct-points"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
EM-gives-tight-apartness-is-≠ : DNE 𝓥
→ (X : 𝓤 ̇ )
→ ((_♯_ , _ , _) : Tight-Apartness X 𝓥)
→ ((x y : X) → x ♯ y ↔ x ≠ y) | EM-gives-tight-apartness-is-≠ dne X (_♯_ , ♯-is-apartness , ♯-is-tight) x y = III
where
I : x ♯ y → x ≠ y
I = not-equal-if-apart _♯_ ♯-is-apartness
II : x ≠ y → x ♯ y
II ν = dne (x ♯ y)
(apartness-is-prop-valued _♯_ ♯-is-apartness x y)
(contrapositive (♯-is-tight x y) ν)
III : x ♯ y... | function | EM-gives-tight-apartness-is-≠ | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"DNE",
"_♯_",
"contrapositive",
"dne",
"♯-is-apartness",
"♯-is-tight"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
At-Most-One-Tight-Apartness : (X : 𝓤 ̇ ) (𝓥 : Universe) → (𝓥 ⁺ ⊔ 𝓤) ̇ | At-Most-One-Tight-Apartness X 𝓥 = is-prop (Tight-Apartness X 𝓥) | function | At-Most-One-Tight-Apartness | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"is-prop"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
At-Most-One-Tight-Apartness-on-Cantor-gives-WLPO-implies-LPO
: Fun-Ext
→ At-Most-One-Tight-Apartness 𝟚ᴺ 𝓤₀
→ WLPO-variation₂
→ LPO-variation | At-Most-One-Tight-Apartness-on-Cantor-gives-WLPO-implies-LPO fe hyp wlpo = VI
where
_♯_ = _♯[𝟚ᴺ]_
has-root : 𝟚ᴺ → 𝓤₀ ̇
has-root α = Σ n ꞉ ℕ , α n ＝ ₀
P⁺ : (α : 𝟚ᴺ) → Σ b ꞉ 𝟚 , (b ＝ ₀ ↔ ¬¬ (has-root α))
× (b ＝ ₁ ↔ ¬ (has-root α))
P⁺ α = boolean-value' (wlpo α)
P : 𝟚ᴺ ... | function | At-Most-One-Tight-Apartness-on-Cantor-gives-WLPO-implies-LPO | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"At-Most-One-Tight-Apartness",
"Cantor-is-totally-separated",
"Fun-Ext",
"Idtofun",
"LPO-variation",
"WLPO-variation₂",
"_♯_",
"ap",
"apartness-criterion",
"apartness-criterion-converse",
"boolean-value'",
"eq",
"fe",
"happly",
"has-root",
"id",
"inl",
"inr",
"is-decidable",
"l... | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
≠-is-apartness-on-discrete-type : funext 𝓤 𝓤₀
→ {X : 𝓤 ̇ }
→ is-discrete X
→ is-apartness _≠_ | ≠-is-apartness-on-discrete-type fe {X} X-discrete =
(λ x y → negations-are-props fe)
, ≠-is-irrefl
, (λ x y → ≠-sym)
, (λ x y z a → I x y z a (X-discrete x z))
where
I : (x y z : X) → x ≠ y
→ (x ＝ z) + ¬ (x ＝ z)
→ (x ≠ z) ∨ (y ≠ z)
I x y z a (inl refl) = ∣ inr (≠-sym a) ∣
I x y z a (inr ν) ... | function | ≠-is-apartness-on-discrete-type | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"_≠_",
"fe",
"funext",
"inl",
"inr",
"is-discrete",
"negations-are-props",
"refl",
"≠-is-irrefl",
"≠-sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
≠-is-tight-on-discrete-type : {X : 𝓤 ̇ }
→ is-discrete X
→ is-tight _≠_ | ≠-is-tight-on-discrete-type = discrete-is-¬¬-separated | function | ≠-is-tight-on-discrete-type | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"_≠_",
"discrete-is-¬¬-separated",
"is-discrete",
"is-tight"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE
: funext 𝓤 𝓤₀
→ (X : 𝓤 ̇ )
→ has-two-distinct-points X
→ is-discrete X
→ At-Most-One-Tight-Apartness X 𝓤
→ DNE 𝓤 | At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE
{𝓤} fe X ((x₀ , x₁) , x₀-is-not-x₁) X-discrete hyp P P-is-prop = II
where
_♯_ : X → X → 𝓤 ̇
x ♯ y = P × (x ≠ y)
pv : is-prop-valued _♯_
pv x y = ×-is-prop P-is-prop (negations-are-props fe)
ir : is-irreflexive _♯_
ir... | function | At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"At-Most-One-Tight-Apartness",
"DNE",
"Idtofun",
"_≠_",
"_♯_",
"ap",
"discrete-is-¬¬-separated",
"eq",
"fe",
"funext",
"happly",
"has-two-distinct-points",
"inl",
"inr",
"is-discrete",
"is-irreflexive",
"is-prop-valued",
"is-symmetric",
"is-tight",
"negations-are-props",
"pr₁... | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
At-Most-One-Tight-Apartness-on-ℕ-gives-DNE
: funext 𝓤₀ 𝓤₀
→ At-Most-One-Tight-Apartness ℕ 𝓤₀
→ DNE 𝓤₀ | At-Most-One-Tight-Apartness-on-ℕ-gives-DNE fe =
At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE
fe ℕ ((0 , 1) , zero-not-positive 0) ℕ-is-discrete | function | At-Most-One-Tight-Apartness-on-ℕ-gives-DNE | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"At-Most-One-Tight-Apartness",
"At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE",
"DNE",
"fe",
"funext",
"zero-not-positive",
"ℕ-is-discrete"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Exactly-One-Tight-Apartness-on-type-with-two-points-apart-gives-DNE
: {X : 𝓤 ̇ }
((_♯_ , a , _) : Tight-Apartness X 𝓤)
→ has-two-points-apart (_♯_ , a)
→ At-Most-One-Tight-Apartness X 𝓤
→ DNE 𝓤 | Exactly-One-Tight-Apartness-on-type-with-two-points-apart-gives-DNE
{𝓤} {X}
(_♯_ , a@(♯-pv , ♯-irrefl , ♯-sym , ♯-cot) , ♯-tight)
((x₀ , x₁) , x₀-apart-from-x₁)
α P P-is-prop = VI
where
_♯ᴾ_ : X → X → 𝓤 ̇
x ♯ᴾ y = P × (x ♯ y)
♯ᴾ-pv : is-prop-valued _♯ᴾ_
♯ᴾ-pv x y = ×-is-prop P-is-prop (♯-pv x y)
... | function | Exactly-One-Tight-Apartness-on-type-with-two-points-apart-gives-DNE | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"At-Most-One-Tight-Apartness",
"DNE",
"Idtofun",
"_♯_",
"ap",
"happly",
"has-two-points-apart",
"inl",
"inr",
"is-irreflexive",
"is-prop-valued",
"is-tight",
"pr₁",
"symmetric",
"x₀",
"¬¬",
"×-is-prop"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE'
: funext 𝓤 𝓤₀
→ {X : 𝓤 ̇ }
→ is-discrete X
→ has-two-distinct-points X
→ At-Most-One-Tight-Apartness X 𝓤
→ DNE 𝓤 | At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE'
fe δ
= Exactly-One-Tight-Apartness-on-type-with-two-points-apart-gives-DNE
(_≠_ , ≠-is-apartness-on-discrete-type fe δ , ≠-is-tight-on-discrete-type δ) | function | At-Most-One-Tight-Apartness-on-discrete-type-with-two-distinct-points-gives-DNE' | Apartness | source/Apartness/Properties.lagda | [
"UF.PropTrunc",
"Apartness.Definition",
"MLTT.Spartan",
"MLTT.Two-Properties",
"Naturals.Properties",
"NotionsOfDecidability.DecidableClassifier",
"Taboos.LPO",
"Taboos.WLPO",
"TypeTopology.Cantor",
"TypeTopology.TotallySeparated",
"UF.Base",
"UF.ClassicalLogic",
"UF.DiscreteAndSeparated",
... | [
"At-Most-One-Tight-Apartness",
"DNE",
"Exactly-One-Tight-Apartness-on-type-with-two-points-apart-gives-DNE",
"_≠_",
"fe",
"funext",
"has-two-distinct-points",
"is-discrete",
"≠-is-apartness-on-discrete-type",
"≠-is-tight-on-discrete-type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝔹 : Type₀ where
L R : 𝔹
l r : 𝔹 → 𝔹
eqL : L ＝ l L
eqM : l R ＝ r L
eqR : R ＝ r R | data | 𝔹 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
𝔻 : Type₀ where
middle : 𝔻
left : 𝔻 → 𝔻
right : 𝔻 → 𝔻 | data | 𝔻 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"right"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
𝔹' : Type₀ where
L' : 𝔹'
R' : 𝔹'
η : 𝔻 → 𝔹' | data | 𝔹' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
𝑟𝒆𝑓𝑙 : {X : Type ℓ} (x : X) → x ＝ x | 𝑟𝒆𝑓𝑙 x = refl | function | 𝑟𝒆𝑓𝑙 | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
id : {X : Type ℓ} → X → X | id = idfun _ | function | id | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Sigma : (X : Type ℓ) (A : X → Type ℓ') → Type (ℓ-max ℓ ℓ') | Sigma = Σ | function | Sigma | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Exists! : ∀ {ℓ ℓ'} (A : Type ℓ) (B : A → Type ℓ') → Type (ℓ-max ℓ ℓ') | Exists! = ∃! | function | Exists! | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"∃!"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
_∼_ : {X : Type ℓ} {A : X → Type ℓ'}
→ ((x : X) → A x)
→ ((x : X) → A x)
→ Type (ℓ-max ℓ ℓ') | function | _∼_ | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
isoFunInjective' : {A : Type ℓ} {B : Type ℓ'} (f : Iso A B) → (x y : A)
→ Iso.fun f x ＝ Iso.fun f y → x ＝ y | isoFunInjective' f x y h = sym (Iso.leftInv f x) ∙∙ cong (Iso.inv f) h ∙∙ Iso.leftInv f y | function | isoFunInjective' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"Type",
"sym"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
M : 𝔹 | M = l R | function | M | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
l' : 𝔹' → 𝔹' | l' L' = L'
l' R' = η middle
l' (η x) = η (left x) | function | l' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"left",
"middle",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
r' : 𝔹' → 𝔹' | r' L' = η middle
r' R' = R'
r' (η x) = η (right x) | function | r' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"middle",
"right",
"𝔹'"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
eqL' : L' ＝ l' L' | eqL' = refl | function | eqL' | BinarySystems | source/BinarySystems/CubicalBinarySystem.lagda | [
"Cubical.Foundations.Prelude",
"Cubical.Foundations.Function",
"Cubical.Foundations.Equiv",
"Cubical.Foundations.Isomorphism",
"Cubical.Foundations.HLevels",
"Cubical.Data.Sigma",
"Cubical.Relation.Nullary",
"Cubical.Relation.Nullary.DecidableEq",
"Cubical.Data.Empty",
"Cubical.Data.Unit",
"Cubi... | [
"l'",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e |
End of preview. Expand in Data Studio
Agda-TypeTopology
Structured declarations from TypeTopology - Martín Escardó's Agda development exploring logical manifestations of topological concepts via the univalent point of view. Source: github.com/martinescardo/TypeTopology
Source
- Repository: https://github.com/martinescardo/TypeTopology
- Commit:
66c36add15b0583ae0e96523f29bb6b56e68459e - Files: 959
- License: gpl-3.0
Schema
| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
Statistics
- Entries: 12,433
- With proof: 11,004 (88.5%)
- With docstring: 0 (0.0%)
- Libraries: 103
By type
| Type | Count |
|---|---|
| function | 12,230 |
| data | 114 |
| record | 86 |
| postulate | 3 |
Example
ap₃ : {A B C D : Type} (f : A → B → C → D) {a₁ a₂ : A} {b₁ b₂ : B} {c₁ c₂ : C}
→ a₁ ＝ a₂
→ b₁ ＝ b₂
→ c₁ ＝ c₂
→ f a₁ b₁ c₁ ＝ f a₂ b₂ c₂
ap₃ f refl refl refl = refl
- type: function | symbolic_name:
ap₃| source/AlgebraicStructuresForcingSethood/Majority.lagda
Use
Each declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
Citation
@misc{agda_typetopology_dataset,
title = {Agda-TypeTopology},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/martinescardo/TypeTopology, commit 66c36add15b0},
url = {https://huggingface.co/datasets/phanerozoic/Agda-TypeTopology}
}
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