statement stringlengths 7 403 | proof stringlengths 0 2.93k | type stringclasses 5
values | symbolic_name stringlengths 2 45 | library stringclasses 7
values | filename stringclasses 24
values | imports listlengths 1 10 | deps listlengths 0 5 | docstring stringclasses 10
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
arrowEliminationTop {A B : Prop} (a : A) (ab : A → B) : OurGoal | by
let h := ab a
sorry | theorem | arrowEliminationTop | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
arrowEliminationBottom {A B : Prop} (a : A) : B | by
revert a
sorry | theorem | arrowEliminationBottom | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
forallIntroductionBottom {α : Type} {A : α → Prop} : ∀ y, A y | by
intro x
sorry | theorem | forallIntroductionBottom | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"intro"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
forallEliminationTop {α : Type} {A : α → Prop}
(h : ∀ x, A x) (t : α) : OurGoal | by
let s := h t
sorry | theorem | forallEliminationTop | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsIntroductionTop {α : Type} {A : α → Prop} {t: α} (h : A t) : OurGoal | by
have h2 : ∃ x, A x := Exists.intro t h
sorry | theorem | existsIntroductionTop | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsIntroductionBottom {α : Type} {A : α → Prop} {t: α} : ∃ x, A x | by
apply Exists.intro (w := t)
sorry | theorem | existsIntroductionBottom | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"apply"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsEliminationTop {α : Type} {A : α → Prop} (h : ∃ x, A x) : OurGoal | by
cases' h with y h1
sorry | theorem | existsEliminationTop | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsEliminationBottom {α : Type} {A : α → Prop} (h : ∃ x, A x) (t : α) : A t | by
apply Exists.elim (p := A)
all_goals sorry | theorem | existsEliminationBottom | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"apply"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsEliminationTopWILD {α : Type} {B : Prop} {A : α → Prop} (h : ∃ x, A x) (hi : ∀ y, A y → B) : OurGoal | by
cases' h with s h2
let h3 := hi s h2
sorry | theorem | existsEliminationTopWILD | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
existsEliminationBottomWILD {α : Type} {A : α → Prop} {B: Prop} : B | by
apply Exists.elim (p := A)
sorry
intro y
sorry | theorem | existsEliminationBottomWILD | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"apply",
"intro"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
page66_evenOdd_TOP {even : ℕ → Prop} {odd : ℕ → Prop} (h1: ∀n, (¬ even n → odd n)) : ∀ n, (even n) ∨ (odd n) | by
intro x
have h := Classical.em (even x)
cases' h with hi hello
left
exact hi
right
exact h1 x hello | theorem | page66_evenOdd_TOP | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"intro"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
page66_evenOdd_BOTTOM {even : ℕ → Prop} {odd : ℕ → Prop} (h1: ∀n, (¬ even n → odd n)) : ∀ n, (even n) ∨ (odd n) | by
intro x
have h := Classical.em
apply Or.elim (a := even x) (b := ¬ even x)
apply h
intro hi
left
exact hi
intro hello
right
exact h1 x hello | theorem | page66_evenOdd_BOTTOM | _examples.demos | _examples/demos/NaturalDeduction.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Mathlib.Data.List.Chain",
"Mathlib.Tactic.Linarith",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Finset.Fold",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Lean",
"Paperproof"
] | [
"apply",
"intro"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
fake_mean_value_theorem {i1 i2 : ℝ} (f : ℝ → ℝ) (h : i1 < i2) : ∃ c ∈ Set.Icc i1 i2, deriv f c = (f i2 - f i1) / (i2 - i1) | by
sorry | theorem | fake_mean_value_theorem | _examples.demos | _examples/demos/NaturalLanguage.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof",
"Mathlib.Data.Real.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Tactic.Cases",
"Mathlib.Algebra.Order.Field.Basic"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
interval_child { a b c i1 i2 : ℝ } (aInI: a ∈ Set.Icc i1 i2) (bInI: b ∈ Set.Icc i1 i2) (cInAB: c ∈ Set.Icc a b) : c ∈ Set.Icc i1 i2 | by
let abInI := Set.Icc_subset_Icc aInI.1 bInI.2
exact Set.mem_of_mem_of_subset cInAB abInI | theorem | interval_child | _examples.demos | _examples/demos/NaturalLanguage.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof",
"Mathlib.Data.Real.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Tactic.Cases",
"Mathlib.Algebra.Order.Field.Basic"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
posTop {n m : ℝ} (dPos: n > 0) (divPos: m / n > 0) : m > 0 | by
exact (div_pos_iff_of_pos_right dPos).mp divPos | theorem | posTop | _examples.demos | _examples/demos/NaturalLanguage.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof",
"Mathlib.Data.Real.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Tactic.Cases",
"Mathlib.Algebra.Order.Field.Basic"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
corollary_increasing_function (f : ℝ → ℝ) (i1 i2 : ℝ) :
(∀ x ∈ Set.Icc i1 i2, (deriv f x) > 0) →
(∀ x ∈ Set.Icc i1 i2, ∀ y ∈ Set.Icc i1 i2, x < y → f x < f y) | by
intros fact a aIn b bIn ab
let meanValueTheorem := fake_mean_value_theorem f ab
rcases meanValueTheorem with ⟨c, ⟨ cIn, derivEquals ⟩ ⟩
have derivPos := fact c (interval_child aIn bIn cIn)
have quotientPos : (f b - f a) / (b - a) > 0 := by
rw [derivEquals] at derivPos
exact derivPos
exact lt_... | theorem | corollary_increasing_function | _examples.demos | _examples/demos/NaturalLanguage.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof",
"Mathlib.Data.Real.Basic",
"Mathlib.Analysis.Calculus.Deriv.Basic",
"Mathlib.Tactic.Cases",
"Mathlib.Algebra.Order.Field.Basic"
] | [
"fake_mean_value_theorem",
"interval_child",
"posTop",
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleAnd (X: Prop) (Y: Prop) (h: X ∧ Y) : False | by
rcases h with ⟨h1, h2⟩
all_goals sorry | theorem | simpleAnd | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleOr (X: Prop) (Y: Prop) (h: X ∨ Y) : False | by
rcases h with h1 | h2
all_goals sorry | theorem | simpleOr | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleThen (X: Prop) (Y: Prop) (h: X → Y) : False | by
rw [imp_iff_not_or] at h
rcases h with h1 | h2
all_goals sorry | theorem | simpleThen | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleNotOr (X: Prop) (Y: Prop) (h: ¬(X ∨ Y)) : False | by
rw [not_or] at h
rcases h with ⟨h1, h2⟩
sorry | theorem | simpleNotOr | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleNotAnd (X: Prop) (Y: Prop) (h: ¬(X ∧ Y)) : False | by
rw [not_and_or] at h
rcases h with h1 | h2
all_goals sorry | theorem | simpleNotAnd | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleNotThen (X: Prop) (Y: Prop) (h: ¬(X → Y)) : False | by
rw [Classical.not_imp] at h
rcases h with ⟨h1, h2⟩
sorry | theorem | simpleNotThen | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleForall {α : Type} (φ : α → Prop) (a : α) (h: ∀ x, φ x) : False | by
let h1 := h a
sorry | theorem | simpleForall | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleExists {α : Type} (φ : α → Prop) (h: ∃ x, φ x) : False | by
rcases h with ⟨a, h1⟩
sorry | theorem | simpleExists | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleNotExists {α : Type} (φ : α → Prop) (a : α) (h: ¬ ∃ x, φ x) : False | by
rw [not_exists] at h
let h1 := h a
sorry | theorem | simpleNotExists | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
simpleNotForall {α : Type} (φ : α → Prop) (h: ¬ ∀ x, φ x) : False | by
rw [not_forall] at h
rcases h with ⟨a, h1⟩
sorry | theorem | simpleNotForall | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
deMorgan (p : Prop) (q : Prop) : ¬(p ∨ q) → (¬p ∧ ¬q) | by
-- Creating 1.
by_contra h1
-- Creating 2. and 3.
rw [Classical.not_imp] at h1
rcases h1 with ⟨h2, h3⟩
-- Creating 4. and 5.
rw [not_or] at h2
rcases h2 with ⟨h4, h5⟩
-- Creating 6. and 7.
rw [not_and, imp_iff_not_or] at h3
rcases h3 with h6 | h7
-- Close the branch!
exact h6 h4
-- C... | theorem | deMorgan | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
manyGoals (p : Prop) (q : Prop) : ((p ∨ q) ∧ (q → p)) → p | by
-- Creating 1.
by_contra h1
-- Creating 2. and 3.
rw [Classical.not_imp] at h1
rcases h1 with ⟨h2, h3⟩
-- Creating 4. and 5.
rcases h2 with ⟨h4, h5⟩
-- Creating 6. and 7.
rw [imp_iff_not_or] at h5
rcases h5 with h6 | h7
-- Creating 8. and 9.
rcases h4 with h8 | h9
-- Close the branch!
... | theorem | manyGoals | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
firstOrderLogic {α : Type} (P Q : α → Prop)
: (∃ x, P x ∧ Q x) → (∃ x, P x) ∧ (∃ x, Q x) | by
by_contra h1
-- Creating 2. and 3.
rw [Classical.not_imp] at h1
rcases h1 with ⟨h2, h3⟩
-- Creating 4.
rcases h2 with ⟨a, h4⟩
-- Creating 5. and 6.
rcases h4 with ⟨h5, h6⟩
-- Creating 7. and 8.
rw [not_and_or] at h3
rcases h3 with h7 | h8
-- Creating 9.
rw [not_exists] at h7
let h9 :... | theorem | firstOrderLogic | _examples.demos | _examples/demos/SemanticTableaux.lean | [
"Mathlib.Data.Nat.Prime.Basic",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
finFunctionFinEquiv_single11 {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) : (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) | by
rw [finFunctionFinEquiv_apply, Fintype.sum_eq_single i, Pi.single_eq_same]
rintro x hx
rw [Pi.single_eq_of_ne hx, Fin.coe_ofNat_eq_mod, Nat.zero_mod, zero_mul] | theorem | finFunctionFinEquiv_single11 | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
prod (hi: Prod Nat Nat) : SomeGoal | by
rcases hi with ⟨a, b⟩
sorry | theorem | OurInductives.prod | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
sum (hi: Sum Nat Nat) : SomeGoal | by
rcases hi with x | y
sorry; sorry | theorem | OurInductives.sum | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
Random where
| hi: ℕ → String → Random
| hello: (2 + 2 = 4) → Random
| wow: Random | inductive | OurInductives.Random | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | ||
casesRandom (h: Random) : SomeGoal | by
rcases h with ⟨a, b⟩ | ⟨c⟩
sorry; sorry; sorry | theorem | OurInductives.casesRandom | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
combs (p q r : Prop) (hp : p)
: (p ∨ q ∨ r) ∧ (q ∨ p ∨ r) ∧ (q ∨ r ∨ p) | by
repeat (first | apply And.intro | apply Or.inl; assumption | apply Or.inr | assumption) | theorem | OurInductives.combs | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [
"apply",
"combs"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
append_assoc {α : Type} (a b c : List α) : (a ++ b) ++ c = a ++ (b ++ c) | by
induction a with
| nil => simp
| cons x xs ih =>
simp [List.cons_append, ih] | theorem | OurInductives.append_assoc | _examples.demos | _examples/demos/Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [
"append_assoc",
"induction"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
commutativityOfIntersections
(s t : Set Nat) : s ∩ t = t ∩ s | by
ext x
apply Iff.intro
rw [Set.mem_inter_iff, and_comm] | theorem | commutativityOfIntersections | _examples.development | _examples/development/Rw.lean | [
"Mathlib.Tactic",
"Paperproof"
] | [
"apply",
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
sss
(s t : Set Nat) : s ∩ t = t ∩ s | by
ext x
apply Iff.intro
erw [Set.mem_inter_iff, and_comm] | theorem | sss | _examples.development | _examples/development/Rw.lean | [
"Mathlib.Tactic",
"Paperproof"
] | [
"apply"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
www
(s t : Set Nat) : s ∩ t = t ∩ s | by
ext x
apply Iff.intro
rw_mod_cast [Set.mem_inter_iff, and_comm] | theorem | www | _examples.development | _examples/development/Rw.lean | [
"Mathlib.Tactic",
"Paperproof"
] | [
"apply"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
ddd
(s t : Set Nat) : s ∩ t = t ∩ s | by
ext x
apply Iff.intro
rwa [Set.mem_inter_iff, and_comm] | theorem | ddd | _examples.development | _examples/development/Rw.lean | [
"Mathlib.Tactic",
"Paperproof"
] | [
"apply"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
TEST_encard_union_le (s t : Set α) : (s ∪ t).encard ≤ s.encard + t.encard | by
rw [← encard_union_add_encard_inter]; exact le_self_add | theorem | TEST_encard_union_le | _examples.development | _examples/development/TEST.lean | [
"Paperproof",
"Mathlib.Data.Set.Card",
"Mathlib.Tactic.Explode"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
TEST_encard_insert_of_not_mem {a : α} (has : a ∉ s) : (insert a s).encard = s.encard + 1 | by
rw [← union_singleton, encard_union_eq (by simpa), encard_singleton] | theorem | TEST_encard_insert_of_not_mem | _examples.development | _examples/development/TEST.lean | [
"Paperproof",
"Mathlib.Data.Set.Card",
"Mathlib.Tactic.Explode"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
encard_le_encard_of_injOn (hf : MapsTo f s t) (f_inj : InjOn f s) :
s.encard ≤ t.encard | by
rw [← f_inj.encard_image]; apply encard_le_card; rintro _ ⟨x, hx, rfl⟩; exact hf hx | theorem | encard_le_encard_of_injOn | _examples.development | _examples/development/TEST.lean | [
"Paperproof",
"Mathlib.Data.Set.Card",
"Mathlib.Tactic.Explode"
] | [
"apply",
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
commutativityOfIntersections
(s t : Set ℕ) : s ∩ t = t ∩ s | by
sorry | theorem | commutativityOfIntersections | _examples.workshop-bonn-2025 | _examples/workshop-bonn-2025/1-Demo.lean | [
"Mathlib.Data.Set.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
finFunctionFinEquiv_single11 {m n : ℕ} [NeZero m] (i : Fin n) (j : Fin m) : (finFunctionFinEquiv (Pi.single i j) : ℕ) = j * m ^ (i : ℕ) | by
rw [finFunctionFinEquiv_apply, Fintype.sum_eq_single i]
· rw [Pi.single_eq_same]
· rintro x hx
rw [Pi.single_eq_of_ne hx, Fin.coe_ofNat_eq_mod, Nat.zero_mod, zero_mul] | theorem | finFunctionFinEquiv_single11 | _examples.workshop-bonn-2025 | _examples/workshop-bonn-2025/2-Formalization.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
append_assoc {α : Type} (a b c : List α) : (a ++ b) ++ c = a ++ (b ++ c) | by
induction a with
| nil => simp only [List.nil_append]
| cons x xs ih =>
simp [List.cons_append, ih] | theorem | append_assoc | _examples.workshop-bonn-2025 | _examples/workshop-bonn-2025/2-Formalization.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [
"induction"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
top : a ∧ b → a | by
sorry | theorem | top | _examples.workshop-tbilisi-2024 | _examples/workshop-tbilisi-2024/2-Tutorial.lean | [
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.Use",
"Mathlib.Data.Set.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
bottom : a → b → a ∧ b | by
sorry | theorem | bottom | _examples.workshop-tbilisi-2024 | _examples/workshop-tbilisi-2024/2-Tutorial.lean | [
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.Use",
"Mathlib.Data.Set.Basic",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
commutativityOfIntersections
(s t : Set Nat) : s ∩ t = t ∩ s | by
ext x
apply Iff.intro
intro h1
rw [Set.mem_inter_iff, and_comm] at h1
exact h1 | theorem | commutativityOfIntersections | _examples.workshop-tbilisi-2024 | _examples/workshop-tbilisi-2024/2-Tutorial.lean | [
"Mathlib.Tactic.Cases",
"Mathlib.Tactic.Use",
"Mathlib.Data.Set.Basic",
"Paperproof"
] | [
"apply",
"intro",
"rw"
] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 | |
casesRandom (h: Random) : SomeGoal | by
rcases h with ⟨a, b⟩ | ⟨c⟩ | _
sorry; sorry; sorry | theorem | OurInductives.casesRandom | _examples.workshop-tbilisi-2024 | _examples/workshop-tbilisi-2024/3-Tutorial.lean | [
"Mathlib.Data.Set.Basic",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Tactic.GCongr",
"Paperproof"
] | [] | https://github.com/Paper-Proof/paperproof | c944afd12974e745b26f803acf67af02e7820271 |
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