| {"id":"definition : VectorSpace","label":"VectorSpace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A vector space is a space over a field K with an abelian group V. It has four main properties: smul_add: \\(\\forall (a : K) (x y : V), a \\bullet (x + y) = a \\bullet x + a \\bullet y \\) add_smul: \\(\\forall (a b : K) (x : V), (a + b) \\bullet x = a \\bullet x + b \\bullet x \\) mul_smul: \\(\\forall (a b : K) (x : V), (a * b) \\bullet x = a \\bullet (b \\bullet x) \\) one_smul: \\(\\forall (x : V), (1 : K) \\bullet x = x \\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.VectorSpace","latex_link":"sect0001.html#definition : VectorSpace","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.VectorSpace","lean_decl_name":"LinearAlgebraGame.VectorSpace","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LBPTester","source_path":"Game/Levels/VectorSpaceWorld/Level01.lean","lean4_code":"class VectorSpace (K V : Type) [Field K] [AddCommGroup V] extends Module K V where\n smul_add_explicit : ∀ (a : K) (x y : V), a • (x + y) = a • x + a • y := smul_add\n add_smul_explicit : ∀ (a b : K) (x : V), (a + b) • x = a • x + b • x := add_smul\n mul_smul_explicit : ∀ (a b : K) (x : V), (a * b) • x = a • (b • x) := mul_smul\n one_smul_explicit : ∀ (x : V), (1 : K) • x = x := one_smul\n","line":100,"end_line":104,"proof_status":"formalized"} |
| {"id":"theorem : smul_zero","label":"smul_zero","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, any scalar a multiplied by the zero vector gives the zero vector:\\( (a : K) : a \\bullet (0 : V) = (0 : V)\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.smul_zero_v","latex_link":"sect0001.html#theorem : smul_zero","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.smul_zero_v","lean_decl_name":"LinearAlgebraGame.smul_zero_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level02.lean","lean4_code":"/--\nIn any vector space V over K, any scalar a multiplied by the zero vector gives the zero vector.\n-/\nStatement smul_zero_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (a : K) : a • (0 : V) = (0 : V) := by\n Hint (hidden := true) \"Try `apply add_right_cancel (b := a • (0 : V))`\"\n apply add_right_cancel (b := a • (0 : V))\n Hint (hidden := true) \"Try `rw[(smul_add a (0 : V) (0 : V)).symm]`\"\n rw[(smul_add a (0 : V) (0 : V)).symm]\n Hint (hidden := true) \"Try `rw[zero_add]`\"\n rw[zero_add]\n Hint (hidden := true) \"Try `rw[zero_add]`\"\n rw[zero_add]","line":31,"end_line":42,"proof_status":"formalized"} |
| {"id":"theorem : zero_smul","label":"zero_smul","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, the scalar 0 multiplied by any vector gives the zero vector: (0 : K) • w = (0 : V)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.zero_smul_v","latex_link":"sect0001.html#theorem : zero_smul","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.zero_smul_v","lean_decl_name":"LinearAlgebraGame.zero_smul_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level01.lean","lean4_code":"/--\nIn any vector space V over K, the scalar 0 multiplied by any vector gives the zero vector.\n-/\nStatement zero_smul_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (w : V) : (0 : K) • w = (0 : V) := by\n Hint \"Remember, we are trying to prove this backwards. The last step in the normal proof was to\n cancel out 0 • w from both sides, so what should the first step of the backwards proof be?\"\n Hint (hidden := true) \"Try `apply add_right_cancel (b := (0 : K) • w)`\"\n apply add_right_cancel (b := (0 : K) • w)\n Hint \"Now, there is a theorem we have from the vector space definition that can change the left\n side of the equation. Think about the second to last step in the normal proof. You may also need to\n use `.symm` here.\"\n Hint (hidden := true) \"Try `rw[(add_smul (0 : K) (0 : K) w).symm]`\"\n rw[(add_smul (0 : K) (0 : K) w).symm]\n Hint \"Now, we just have to cancel out zeros.\"\n Hint (hidden := true) \"Try rw[zero_add]\"\n rw[zero_add]\n Hint (hidden := true) \"Try rw[zero_add]\"\n rw[zero_add]","line":184,"end_line":201,"proof_status":"formalized"} |
| {"id":"theorem : neg_one_smul","label":"neg_one_smul","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["definition : VectorSpace"],"nlp_text":"In any vector space V over K, multiplying a vector by -1 gives its additive inverse: \\((v : V) : (-1 : K) \\bullet v = -v\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.neg_one_smul_v","latex_link":"sect0001.html#theorem : neg_one_smul","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.neg_one_smul_v","lean_decl_name":"LinearAlgebraGame.neg_one_smul_v","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level03.lean","lean4_code":"/--\nIn any vector space V over K, multiplying a vector by -1 gives its additive inverse.\n-/\nStatement neg_one_smul_v (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) (v : V) : (-1 : K) • v = -v := by\n Hint \"A good first step is cancelling out the `-v` term on the right.\"\n Hint (hidden := true) \"Try `apply add_right_cancel (b := v)`\"\n apply add_right_cancel (b := v)\n Hint \"Remember the `nth_rw m [theorem]` tactic to only rewrite the mth instance.\"\n Hint (hidden := true) \"Try `nth_rw 2 [(one_smul (K := K) v).symm]`\"\n nth_rw 2 [(one_smul K v).symm]\n Hint (hidden := true) \"Try `rw[(add_smul (-1 : K) (1 : K) v).symm]`\"\n rw [(add_smul (-1 : K) (1 : K) v).symm]\n Hint (hidden := true) \"Try `rw[neg_add_self]`\"\n rw[neg_add_self]\n Hint (hidden := true) \"Try `rw[neg_add_self]`\"\n rw[neg_add_self]\n Hint \"This looks like something we've done before. Either the `rw` or `exact` tactics should solve the goal\"\n Hint (hidden := true) \"Try `rw [zero_smul_v]`\"\n rw [zero_smul_v]","line":52,"end_line":70,"proof_status":"formalized"} |
| {"id":"theorem : subspace_contains_zero","label":"subspace_contains_zero","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["theorem : zero_smul","definition : Subspace"],"nlp_text":"This is a proof that any subspace contains the zero vector:\\((0 : V) \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_contains_zero","latex_link":"sect0001.html#theorem : subspace_contains_zero","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_contains_zero","lean_decl_name":"LinearAlgebraGame.subspace_contains_zero","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level04.lean","lean4_code":"/--\nThis is a proof that any subspace contains the zero vector.\n-/\nStatement subspace_contains_zero (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) {W : Set V} (hW : isSubspace (K := K) (V := V) W) : (0 : V) ∈ W := by\n Hint \"Try to expand out your hypotheses using `obtain`.\"\n Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n obtain ⟨h1, _h2, h3⟩ := hW\n Hint \"Again, you can use `obtain` to simplify a hypothesis.\"\n Hint (hidden := true) \"Try `obtain ⟨w, hw⟩ := {h1}`\"\n obtain ⟨w, hw⟩ := h1\n Hint \"We know that `0 • {w} ∈ W`. If this was our goal, the level would be easy to solve. Also,\n remember that if you have to use a theorem you have proven in a previous level, you have to write\n `theorem_name fk acg vs theorem_args` to show Lean that K V is a vector space.\"\n Hint (hidden := true) \"Try `rw [(zero_smul_v fk acg vs {w}).symm]`\"\n rw [(zero_smul_v fk acg vs w).symm]\n Hint \"Now, apply the fact that subspaces are closed under scalar multiplication.\"\n Hint (hidden := true) \"Try `apply {h3}`\"\n apply h3\n Hint (hidden := true) \"Try `exact {hw}`\"\n exact hw","line":74,"end_line":93,"proof_status":"formalized"} |
| {"id":"theorem : subspace_neg","label":"subspace_neg","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":["theorem : neg_one_smul","definition : Subspace"],"nlp_text":"This is a proof that if a subspace contains a vector ‘x‘, it also contains ‘-x‘:\\( \\forall (x : V), x \\in W \\rightarrow (-x) \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_neg","latex_link":"sect0001.html#theorem : subspace_neg","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.subspace_neg","lean_decl_name":"LinearAlgebraGame.subspace_neg","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/AdamJasonKern/LBPTester/8eb9f5c46e81d41eb1a4490ee2c761793c3f839d/Game/Levels/VectorSpaceWorld/Level05.lean","lean4_code":"Statement subspace_neg (fk : Field K) (acg : AddCommGroup V) (vs : VectorSpace K V) {W : Set V} (hW : isSubspace (K := K) (V := V) W) : ∀ (x : V), x ∈ W → (-x) ∈ W := by\n Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n Branch\n intros _x _hx\n Hint (hidden := true) \"Try `obtain ⟨h1, h2, h3⟩ := hW`\"\n obtain ⟨_h1, _h2, h3⟩ := hW\n Hint (hidden := true) \"Try `intros x hx`\"\n intros x hx\n Hint (hidden := true) \"Try `rw [(neg_one_smul_v fk acg vs {x}).symm]`\"\n rw [(neg_one_smul_v fk acg vs x).symm]\n Hint (hidden := true) \"Try `apply {h3}`\"\n apply h3\n Hint (hidden := true) \"Try `exact {hx}`\"\n exact hx","line":24,"end_line":37,"proof_status":"formalized"} |
| {"id":"definition : Subspace","label":"Subspace","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://AdamJasonKern.github.io/LBPTester/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A subspace is a subset of a vector space that acts similarly to a vector space itself. It has three main properties: Nonempty: W.Nonempty Closure_Under_Addition: \\(\\forall (x y : V), x ∈ W \\rightarrow y \\in W \\rightarrow x + y \\in W\\) Closure_Under_Scalar_Multiplication: \\(\\forall (a : K) (x : V), x \\in W \\rightarrow a \\bullet x \\in W\\)","doc_url":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.isSubspace","latex_link":"sect0001.html#definition : Subspace","lean_link":"https://AdamJasonKern.github.io/LBPTester/docs/find/#doc/LinearAlgebraGame.isSubspace","lean_decl_name":"LinearAlgebraGame.isSubspace","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"source_repo":"LBPTester","source_path":"Game/Levels/VectorSpaceWorld/Level04.lean","lean4_code":"def isSubspace {K V : Type} [Field K] [AddCommGroup V] [VectorSpace K V] (W : Set V) : Prop :=\n W.Nonempty ∧ (∀ (x y : V), x ∈ W → y ∈ W → x + y ∈ W) ∧ (∀ (a : K) (x : V), x ∈ W → a • x ∈ W)\n","line":31,"end_line":32,"proof_status":"formalized"} |
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