type stringlengths 1 143k | tools listlengths 1 1 | tactic stringlengths 1 8.81k | removals listlengths 0 28 | name stringlengths 1 85 | messages listlengths 2 2 | kind stringclasses 2
values | goal stringlengths 11 205k |
|---|---|---|---|---|---|---|---|
β (val : Nat) (property : LT.lt 0 val),
Eq (β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) b) : Nat)
(HAdd.hAdd (HSub.hSub (HSub.hSub (β(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) : Nat) 1) (βb : Nat))
1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases a | [
"a"
] | mk | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : PNat\nβ’ Eq (β(HSub.hSub a b) : Nat) (HAdd.hAdd (HSub.hSub (HSub.hSub (βa : Nat) 1) (βb : Nat)) 1)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal ... | goal | a b : PNat
β’ Eq (β(HSub.hSub a b) : Nat) (HAdd.hAdd (HSub.hSub (HSub.hSub (βa : Nat) 1) (βb : Nat)) 1) |
β (val_1 : Nat) (property_1 : LT.lt 0 val_1),
Eq
(β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property)
(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val_1 property_1)) :
Nat)
(HAdd.hAdd
(HSub.hSub (HSub.hSub (β(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) : ... | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases b | [
"b"
] | mk | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nb : PNat\nval : Nat\nproperty : LT.lt 0 val\nβ’ Eq (β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) b) : Nat)\n (HAdd.hAdd (HSub.hSub (HSub.hSub (β(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) : ... | goal | b : PNat
val : Nat
property : LT.lt 0 val
β’ Eq (β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) b) : Nat)
(HAdd.hAdd (HSub.hSub (HSub.hSub (β(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property) : Nat) 1) (βb : Nat))
1) |
Eq (ite (LT.lt val_1 val) (HSub.hSub val val_1) 1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [PNat.mk_coe, _root_.PNat.sub_coe, β _root_.PNat.coe_lt_coe] | [] | mk | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nval : Nat\nproperty : LT.lt 0 val\nval_1 : Nat\nproperty_1 : LT.lt 0 val_1\nβ’ Eq\n (β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property)\n (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val_1 property_1)) ... | goal | val : Nat
property : LT.lt 0 val
val_1 : Nat
property_1 : LT.lt 0 val_1
β’ Eq
(β(HSub.hSub (Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val property)
(Subtype.mk (p := fun (n : Nat) β¦ LT.lt 0 n) val_1 property_1)) :
Nat)
(HAdd.hAdd
(HSub.hSub (HSub.hSub (β(Subtype.mk (p := fun (n : Nat) β¦ L... |
LT.lt val_1 val β Eq (HSub.hSub val val_1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs | [] | pos | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nval : Nat\nproperty : LT.lt 0 val\nval_1 : Nat\nproperty_1 : LT.lt 0 val_1\nβ’ Eq (ite (LT.lt val_1 val) (HSub.hSub val val_1) 1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would... | goal | val : Nat
property : LT.lt 0 val
val_1 : Nat
property_1 : LT.lt 0 val_1
β’ Eq (ite (LT.lt val_1 val) (HSub.hSub val val_1) 1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1) |
Not (LT.lt val_1 val) β Eq 1 (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs | [] | neg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nval : Nat\nproperty : LT.lt 0 val\nval_1 : Nat\nproperty_1 : LT.lt 0 val_1\npos : LT.lt val_1 val β Eq (HSub.hSub val val_1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1)\nβ’ Eq (ite (LT.lt val_1 val) (HSub.hSub val val_1) 1) (HAd... | goal | val : Nat
property : LT.lt 0 val
val_1 : Nat
property_1 : LT.lt 0 val_1
pos : LT.lt val_1 val β Eq (HSub.hSub val val_1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1)
β’ Eq (ite (LT.lt val_1 val) (HSub.hSub val val_1) 1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1) |
And (Eq a a') (Eq b b') β p | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rwa [Nat.pair_eq_pair, and_imp] | [] | rwa | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Prop\na b a' b' : Nat\nh : Eq a a' β Eq b b' β p\nβ’ Eq (Nat.pair a b) (Nat.pair a' b') β p\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is e... | goal | p : Prop
a b a' b' : Nat
h : Eq a a' β Eq b b' β p
β’ Eq (Nat.pair a b) (Nat.pair a' b') β p |
Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd (f.coeff n) (g.coeff n)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst βΉ_βΊ βΉ_βΊ | [
"a",
"b",
"h_add_left",
"h_add_right"
] | subst | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nn : Nat\na b : R\nf g : Polynomial R\nh_add_left : Eq (f.coeff n) a\nh_add_right : Eq (g.coeff n) b\nβ’ Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd a b)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe wou... | goal | R : Type u_1
inst : Semiring R
n : Nat
a b : R
f g : Polynomial R
h_add_left : Eq (f.coeff n) a
h_add_right : Eq (g.coeff n) b
β’ Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd a b) |
β (h : Eq d (HAdd.hAdd df dg)), Eq ((HMul.hMul f g).coeff d) (HMul.hMul a b) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs with h | [] | pos | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
β’ Eq ((HMul.hMul f g).coeff d) (ite (Eq d (HAdd.hAdd df dg)) (HMul.hMul a b) 0) |
β (h : Not (Eq d (HAdd.hAdd df dg))), Eq ((HMul.hMul f g).coeff d) 0 | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs with h | [] | neg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
pos : β (h : Eq d (HAdd.hAdd df dg)), Eq ((HMul.hMul f g).coeff d) (HMul.hMul a ... |
β (ddf : LE.le (HAdd.hAdd df dg) (HAdd.hAdd df dg)),
Eq ((HMul.hMul f g).coeff (HAdd.hAdd df dg)) (HMul.hMul (f.coeff df) (g.coeff dg)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst h_mul_left h_mul_right h | [
"d",
"a",
"b",
"h_mul_left",
"h_mul_right",
"ddf",
"h"
] | pos | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Eq d (HAdd.hAdd df dg)
β’ Eq ((HMul.hMul f g).coeff d) (HMul.hMul a b) |
LT.lt (HMul.hMul f g).natDegree d | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply coeff_eq_zero_of_natDegree_lt | [] | neg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
β’ Eq ((HMul.hMul f g).coeff d) 0 |
LE.le (HMul.hMul f g).natDegree (HAdd.hAdd df dg) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_) | [] | apply | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
β’ LT.lt (HMul.hMul f g).natDegree d |
Ne (HAdd.hAdd df dg) d | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne ddf ?_) | [] | applyβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nd df dg : Nat\na b : R\nf g : Polynomial R\nh_mul_left_1 : LE.le f.natDegree df\nh_mul_right_1 : LE.le g.natDegree dg\nh_mul_left : Eq (f.coeff df) a\nh_mul_right : Eq (g.coeff dg) b\nddf : LE.le (HAd... | goal | R : Type u_1
inst : Semiring R
d df dg : Nat
a b : R
f g : Polynomial R
h_mul_left_1 : LE.le f.natDegree df
h_mul_right_1 : LE.le g.natDegree dg
h_mul_left : Eq (f.coeff df) a
h_mul_right : Eq (g.coeff dg) b
ddf : LE.le (HAdd.hAdd df dg) d
h : Not (Eq d (HAdd.hAdd df dg))
apply : LE.le (HMul.hMul f g).natDegree (HAdd.h... |
β (h : Eq o (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff o) (HPow.hPow a m) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs with h | [] | pos | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\nβ’ Eq ((HPow.hPow p m).coeff o) (ite (Eq o (HMul.hMul m n)) (HPow.hPow ... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
β’ Eq ((HPow.hPow p m).coeff o) (ite (Eq o (HMul.hMul m n)) (HPow.hPow a m) 0) |
β (h : Not (Eq o (HMul.hMul m n))), Eq ((HPow.hPow p m).coeff o) 0 | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split_ifs with h | [] | neg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\npos : β (h : Eq o (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff o) (HPow... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
pos : β (h : Eq o (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff o) (HPow.hPow a m)
β’ Eq ((HPow.hPow p m).coeff o) (ite (Eq o (HMul.hMul m n)) (HPow.hPow a m) 0) |
β (h_exp : LE.le (HMul.hMul m n) (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff (HMul.hMul m n)) (HPow.hPow (p.coeff n) m) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst h h_pow_bas | [
"o",
"a",
"h_exp",
"h_pow_bas",
"h"
] | pos | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\nh : Eq o (HMul.hMul m n)\nβ’ Eq ((HPow.hPow p m).coeff o) (HPow.hPow a ... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Eq o (HMul.hMul m n)
β’ Eq ((HPow.hPow p m).coeff o) (HPow.hPow a m) |
LT.lt (HPow.hPow p m).natDegree o | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply coeff_eq_zero_of_natDegree_lt | [] | neg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\nh : Not (Eq o (HMul.hMul m n))\nβ’ Eq ((HPow.hPow p m).coeff o) 0\n```\... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
β’ Eq ((HPow.hPow p m).coeff o) 0 |
LE.le (HPow.hPow p m).natDegree (HMul.hMul m n) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne βΉ_βΊ ?_) | [] | apply | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\nh : Not (Eq o (HMul.hMul m n))\nβ’ LT.lt (HPow.hPow p m).natDegree o\n`... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
β’ LT.lt (HPow.hPow p m).natDegree o |
Ne (HMul.hMul m n) o | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | apply lt_of_le_of_lt ?_ (lt_of_le_of_ne βΉ_βΊ ?_) | [] | applyβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm n o : Nat\na : R\np : Polynomial R\nh_pow : LE.le p.natDegree n\nh_exp : LE.le (HMul.hMul m n) o\nh_pow_bas : Eq (p.coeff n) a\nh : Not (Eq o (HMul.hMul m n))\napply : LE.le (HPow.hPow p m).natDegre... | goal | R : Type u_1
inst : Semiring R
m n o : Nat
a : R
p : Polynomial R
h_pow : LE.le p.natDegree n
h_exp : LE.le (HMul.hMul m n) o
h_pow_bas : Eq (p.coeff n) a
h : Not (Eq o (HMul.hMul m n))
apply : LE.le (HPow.hPow p m).natDegree (HMul.hMul m n)
β’ LT.lt (HPow.hPow p m).natDegree o |
β (coeff_ne_zero : Ne (p.coeff o) 0) (h_natDeg_le : LE.le p.natDegree o), Eq p.natDegree o | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst coeff_eq deg_eq_deg coeff_eq_deg | [
"deg",
"m",
"c",
"h_natDeg_le",
"coeff_eq",
"coeff_ne_zero",
"deg_eq_deg",
"coeff_eq_deg"
] | subst | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\ndeg m o : Nat\nc : R\np : Polynomial R\nh_natDeg_le : LE.le p.natDegree m\ncoeff_eq : Eq (p.coeff o) c\ncoeff_ne_zero : Ne c 0\ndeg_eq_deg : Eq m deg\ncoeff_eq_deg : Eq o deg\nβ’ Eq p.natDegree deg\n``... | goal | R : Type u_1
inst : Semiring R
deg m o : Nat
c : R
p : Polynomial R
h_natDeg_le : LE.le p.natDegree m
coeff_eq : Eq (p.coeff o) c
coeff_ne_zero : Ne c 0
deg_eq_deg : Eq m deg
coeff_eq_deg : Eq o deg
β’ Eq p.natDegree deg |
β (coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0), Eq p.degree m | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst coeff_eq coeff_eq_deg deg_eq_deg | [
"deg",
"o",
"c",
"coeff_eq",
"coeff_ne_zero",
"deg_eq_deg",
"coeff_eq_deg"
] | subst | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\ndeg m o : WithBot Nat\nc : R\np : Polynomial R\nh_deg_le : LE.le p.degree m\ncoeff_eq : Eq (p.coeff (WithBot.unbotD 0 deg)) c\ncoeff_ne_zero : Ne c 0\ndeg_eq_deg : Eq m deg\ncoeff_eq_deg : Eq o deg\nβ’... | goal | R : Type u_1
inst : Semiring R
deg m o : WithBot Nat
c : R
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_eq : Eq (p.coeff (WithBot.unbotD 0 deg)) c
coeff_ne_zero : Ne c 0
deg_eq_deg : Eq m deg
coeff_eq_deg : Eq o deg
β’ Eq p.degree deg |
β (h_deg_le : LE.le p.degree Bot.bot) (coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 Bot.bot)) 0), Eq p.degree Bot.bot | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rcases eq_or_ne m β₯ with rfl | hh | [
"m",
"h_deg_le",
"coeff_ne_zero"
] | inl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm : WithBot Nat\np : Polynomial R\nh_deg_le : LE.le p.degree m\ncoeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0\nβ’ Eq p.degree m\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to gene... | goal | R : Type u_1
inst : Semiring R
m : WithBot Nat
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0
β’ Eq p.degree m |
β (hh : Ne m Bot.bot), Eq p.degree m | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rcases eq_or_ne m β₯ with rfl | hh | [] | inr | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm : WithBot Nat\np : Polynomial R\nh_deg_le : LE.le p.degree m\ncoeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0\ninl :\n β (h_deg_le : LE.le p.degree Bot.bot) (coeff_ne_zero : Ne (p.coeff (WithBo... | goal | R : Type u_1
inst : Semiring R
m : WithBot Nat
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0
inl :
β (h_deg_le : LE.le p.degree Bot.bot) (coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 Bot.bot)) 0), Eq p.degree Bot.bot
β’ Eq p.degree m |
β (m : Nat) (h_deg_le : LE.le p.degree (βm : WithBot Nat))
(coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 (βm : WithBot Nat))) 0) (hh : Ne (βm : WithBot Nat) Bot.bot),
Eq p.degree (βm : WithBot Nat) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | obtain β¨m, rflβ© := WithBot.ne_bot_iff_exists.mp hh | [
"m",
"h_deg_le",
"coeff_ne_zero",
"hh"
] | inr | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Semiring R\nm : WithBot Nat\np : Polynomial R\nh_deg_le : LE.le p.degree m\ncoeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0\nhh : Ne m Bot.bot\nβ’ Eq p.degree m\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe... | goal | R : Type u_1
inst : Semiring R
m : WithBot Nat
p : Polynomial R
h_deg_le : LE.le p.degree m
coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 m)) 0
hh : Ne m Bot.bot
β’ Eq p.degree m |
Eq ((HSub.hSub f g).coeff n) (HSub.hSub (f.coeff n) (g.coeff n)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | subst hf hg | [
"a",
"b",
"hf",
"hg"
] | subst | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_1\ninst : Ring R\nn : Nat\na b : R\nf g : Polynomial R\nhf : Eq (f.coeff n) a\nhg : Eq (g.coeff n) b\nβ’ Eq ((HSub.hSub f g).coeff n) (HSub.hSub a b)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a... | goal | R : Type u_1
inst : Ring R
n : Nat
a b : R
f g : Polynomial R
hf : Eq (f.coeff n) a
hg : Eq (g.coeff n) b
β’ Eq ((HSub.hSub f g).coeff n) (HSub.hSub a b) |
LT.lt (Ξ± := Int) 0 (βn : Int) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | have := natCast_pos.2 <| Nat.le_of_ble_eq_true hn | [] | this | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nβ’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)\n (And (LT.lt (HMod.hMod a (βn : Int)).toNat n) (Int.ModEq (βn ... | hypothesis | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
β’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)
(And (LT.lt (HMod.hMod a (βn : Int)).toNat n) (Int.ModEq (βn : Int) a (β(HMod.hMod a (βn : Int)).toNat : Int))) |
LE.le 0 (HMod.hMod a (βn : Int)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | have nonneg := emod_nonneg a <| Int.ne_of_gt this | [] | nonneg | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nβ’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)\n (And (LT.lt (HMod.hMod ... | hypothesis | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
β’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)
(And (LT.lt (HMod.hMod a (βn : Int)).toNat n) (Int.ModEq (βn : Int) a (β(HMod.hMod a (βn : Int)).toNat : Int))) |
LT.lt (HMod.hMod a (βn : Int)).toNat n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine β¨Nat.zero_le _, ?_, ?_β© | [] | refine_1 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nβ’ And (LE.le 0 (HMod.hMod a (βn ... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
β’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)
(And (LT.lt (HMod.hMod a (βn : Int)).toNat n) (Int.ModEq (βn : Int) a (β(HM... |
Int.ModEq (βn : Int) a (β(HMod.hMod a (βn : Int)).toNat : Int) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine β¨Nat.zero_le _, ?_, ?_β© | [] | refine_2 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nrefine_1 : LT.lt (HMod.hMod a (β... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
refine_1 : LT.lt (HMod.hMod a (βn : Int)).toNat n
β’ And (LE.le 0 (HMod.hMod a (βn : Int)).toNat)
(And (LT.lt (HMod.hMod a ... |
LT.lt (HMod.hMod a (βn : Int)) (βn : Int) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Int.toNat_lt nonneg] | [] | refine_1 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nβ’ LT.lt (HMod.hMod a (βn : Int))... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
β’ LT.lt (HMod.hMod a (βn : Int)).toNat n |
Eq (HMod.hMod a (βn : Int)) (HMod.hMod (Ξ± := Int) (β(HMod.hMod a (βn : Int)).toNat : Int) (βn : Int)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nβ’ Int.ModEq (βn : Int) a (β(HMod... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
β’ Int.ModEq (βn : Int) a (β(HMod.hMod a (βn : Int)).toNat : Int) |
Eq (HMod.hMod a (βn : Int)) (HMod.hMod (HMod.hMod a (βn : Int)) (βn : Int)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2β | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nrefine_2 : Eq (HMod.hMod a (βn :... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
refine_2 : Eq (HMod.hMod a (βn : Int)) (HMod.hMod (Ξ± := Int) (β(HMod.hMod a (βn : Int)).toNat : Int) (βn : Int))
β’ Int.ModEq (... |
Eq (HMod.hMod a (βn : Int)) (HMod.hMod a (βn : Int)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Int.ModEq, Int.toNat_of_nonneg nonneg, emod_emod] | [] | refine_2β | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na : Int\nn : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p\nthis : LT.lt (Ξ± := Int) 0 (βn : Int)\nnonneg : LE.le 0 (HMod.hMod a (βn : Int))\nrefine_2 : Eq (HMod.hMod a (βn :... | goal | p : Sort u_1
a : Int
n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.IntMod.OnModCases n a 0 p
this : LT.lt (Ξ± := Int) 0 (βn : Int)
nonneg : LE.le 0 (HMod.hMod a (βn : Int))
refine_2 : Eq (HMod.hMod a (βn : Int)) (HMod.hMod (Ξ± := Int) (β(HMod.hMod a (βn : Int)).toNat : Int) (βn : Int))
refine_2β : E... |
LT.lt (HMod.hMod a n) n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine β¨Nat.zero_le _, ?_, ?_β© | [] | refine_1 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na n : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p\nβ’ And (LE.le 0 (HMod.hMod a n)) (And (LT.lt (HMod.hMod a n) n) (n.ModEq a (HMod.hMod a n)))\n```\n\nLibrary theorems:\n`... | goal | p : Sort u_1
a n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p
β’ And (LE.le 0 (HMod.hMod a n)) (And (LT.lt (HMod.hMod a n) n) (n.ModEq a (HMod.hMod a n))) |
n.ModEq a (HMod.hMod a n) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine β¨Nat.zero_le _, ?_, ?_β© | [] | refine_2 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na n : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p\nrefine_1 : LT.lt (HMod.hMod a n) n\nβ’ And (LE.le 0 (HMod.hMod a n)) (And (LT.lt (HMod.hMod a n) n) (n.ModEq a (HMod.hMod... | goal | p : Sort u_1
a n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p
refine_1 : LT.lt (HMod.hMod a n) n
β’ And (LE.le 0 (HMod.hMod a n)) (And (LT.lt (HMod.hMod a n) n) (n.ModEq a (HMod.hMod a n))) |
Eq (HMod.hMod a n) (HMod.hMod (HMod.hMod a n) n) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Nat.ModEq, Nat.mod_mod] | [] | refine_2 | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na n : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p\nβ’ n.ModEq a (HMod.hMod a n)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, ... | goal | p : Sort u_1
a n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p
β’ n.ModEq a (HMod.hMod a n) |
Eq (HMod.hMod a n) (HMod.hMod a n) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [Nat.ModEq, Nat.mod_mod] | [] | refine_2β | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np : Sort u_1\na n : Nat\nhn : Eq (Nat.ble 1 n) Bool.true\nH : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p\nrefine_2 : Eq (HMod.hMod a n) (HMod.hMod (HMod.hMod a n) n)\nβ’ n.ModEq a (HMod.hMod a n)\n```\n\nLibrary theorems:\n```l... | goal | p : Sort u_1
a n : Nat
hn : Eq (Nat.ble 1 n) Bool.true
H : Mathlib.Tactic.ModCases.NatMod.OnModCases n a 0 p
refine_2 : Eq (HMod.hMod a n) (HMod.hMod (HMod.hMod a n) n)
β’ n.ModEq a (HMod.hMod a n) |
β (x : Ξ²) (hx : condΞ²Ξ± (Prod.mk (coeΞ²Ξ± x) y).fst) (hy : condδγ (Prod.mk (coeΞ²Ξ± x) y).snd),
Exists fun (y_1 : Prod β δ) ⦠Eq (Prod.map coeβα coeδγ y_1) (Prod.mk (coeβα x) y) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rcases CanLift.prf (Ξ² := Ξ²) x hx with β¨x, rflβ© | [
"x",
"hx",
"hy"
] | rcases | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞ± : Type u_1\nΞ² : Type u_2\nΞ³ : Type u_3\nΞ΄ : Type u_4\ncoeΞ²Ξ± : Ξ² β Ξ±\ncondΞ²Ξ± : Ξ± β Prop\ncoeδγ : Ξ΄ β Ξ³\ncondδγ : Ξ³ β Prop\ninst : CanLift Ξ± Ξ² coeΞ²Ξ± condΞ²Ξ±\ninst_1 : CanLift Ξ³ Ξ΄ coeδγ condδγ\nx : Ξ±\ny : Ξ³\nhx : condΞ²Ξ± (Prod.mk x y).f... | goal | Ξ± : Type u_1
Ξ² : Type u_2
Ξ³ : Type u_3
Ξ΄ : Type u_4
coeΞ²Ξ± : Ξ² β Ξ±
condΞ²Ξ± : Ξ± β Prop
coeδγ : Ξ΄ β Ξ³
condδγ : Ξ³ β Prop
inst : CanLift Ξ± Ξ² coeΞ²Ξ± condΞ²Ξ±
inst_1 : CanLift γ δ coeδγ condδγ
x : Ξ±
y : Ξ³
hx : condΞ²Ξ± (Prod.mk x y).fst
hy : condδγ (Prod.mk x y).snd
⒠Exists fun (y_1 : Prod β δ) ⦠Eq (Prod.map coeβα coeδγ y_1) (Pro... |
β (y : Ξ΄) (hx : condΞ²Ξ± (Prod.mk (coeΞ²Ξ± x) (coeδγ y)).fst) (hy : condδγ (Prod.mk (coeΞ²Ξ± x) (coeδγ y)).snd),
Exists fun (y_1 : Prod β δ) ⦠Eq (Prod.map coeβα coeδγ y_1) (Prod.mk (coeβα x) (coeδγ y)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rcases CanLift.prf (Ξ² := Ξ΄) y hy with β¨y, rflβ© | [
"y",
"hx",
"hy"
] | rcases | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞ± : Type u_1\nΞ² : Type u_2\nΞ³ : Type u_3\nΞ΄ : Type u_4\ncoeΞ²Ξ± : Ξ² β Ξ±\ncondΞ²Ξ± : Ξ± β Prop\ncoeδγ : Ξ΄ β Ξ³\ncondδγ : Ξ³ β Prop\ninst : CanLift Ξ± Ξ² coeΞ²Ξ± condΞ²Ξ±\ninst_1 : CanLift Ξ³ Ξ΄ coeδγ condδγ\ny : Ξ³\nx : Ξ²\nhx : condΞ²Ξ± (Prod.mk (coeΞ²Ξ±... | goal | Ξ± : Type u_1
Ξ² : Type u_2
Ξ³ : Type u_3
Ξ΄ : Type u_4
coeΞ²Ξ± : Ξ² β Ξ±
condΞ²Ξ± : Ξ± β Prop
coeδγ : Ξ΄ β Ξ³
condδγ : Ξ³ β Prop
inst : CanLift Ξ± Ξ² coeΞ²Ξ± condΞ²Ξ±
inst_1 : CanLift γ δ coeδγ condδγ
y : Ξ³
x : Ξ²
hx : condΞ²Ξ± (Prod.mk (coeΞ²Ξ± x) y).fst
hy : condδγ (Prod.mk (coeβα x) y).snd
β’ Exists fun (y_1 : Prod Ξ² Ξ΄) β¦ Eq (Prod.map coeΞ²Ξ±... |
β (this : DecidablePred p), Exists fun (g : (i : ΞΉ) β Ξ± i) β¦ Eq (fun (i : Subtype p) β¦ g (i.val (p := p))) f | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | haveI : DecidablePred p := fun i β¦ Classical.propDecidable (p i) | [] | haveI | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞΉ : Sort u_1\nΞ± : ΞΉ β Sort u_2\nne : β (i : ΞΉ), Nonempty.{u_2} (Ξ± i)\np : ΞΉ β Prop\nf : (i : Subtype p) β Ξ± (i.val (p := p))\nβ’ Exists fun (g : (i : ΞΉ) β Ξ± i) β¦ Eq (fun (i : Subtype p) β¦ g (i.val (p := p))) f\n```\n\nLibrary theorems... | goal | ΞΉ : Sort u_1
Ξ± : ΞΉ β Sort u_2
ne : β (i : ΞΉ), Nonempty.{u_2} (Ξ± i)
p : ΞΉ β Prop
f : (i : Subtype p) β Ξ± (i.val (p := p))
β’ Exists fun (g : (i : ΞΉ) β Ξ± i) β¦ Eq (fun (i : Subtype p) β¦ g (i.val (p := p))) f |
Not (LT.lt (Ξ± := ENat) Top.top Top.top) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases x | [
"x"
] | top | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nx : ENat\nβ’ Not (LT.lt Top.top x)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor instance, if you wis... | goal | x : ENat
β’ Not (LT.lt Top.top x) |
β (a : Nat), Not (LT.lt (Ξ± := ENat) Top.top (βa : ENat)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases x | [
"x"
] | coe | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nx : ENat\ntop : Not (LT.lt (Ξ± := ENat) Top.top Top.top)\nβ’ Not (LT.lt Top.top x)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove... | goal | x : ENat
top : Not (LT.lt (Ξ± := ENat) Top.top Top.top)
β’ Not (LT.lt Top.top x) |
β (h : Not r), r β p β And r q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\nβ’ And r p β And r q\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor insta... | goal | p q r : Prop
h : r β p β q
β’ And r p β And r q |
β (h : p β q), r β p β And r q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inr | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\ninl : β (h : Not r), r β p β And r q\nβ’ And r p β And r q\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to... | goal | p q r : Prop
h : r β p β q
inl : β (h : Not r), r β p β And r q
β’ And r p β And r q |
β (h : p β q), r β p β r | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inrβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\ninl : β (h : Not r), r β p β And r q\ninr inrβ : β (h : p β q), r β p β And r q\nβ’ And r p β And r q\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representin... | goal | p q r : Prop
h : r β p β q
inl : β (h : Not r), r β p β And r q
inr inrβ : β (h : p β q), r β p β And r q
β’ And r p β And r q |
β (h : p β q), r β p β q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inrβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\ninl : β (h : Not r), r β p β And r q\ninr inrβ : β (h : p β q), r β p β And r q\ninrβ : β (h : p β q), r β p β r\nβ’ And r p β And r q\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generat... | goal | p q r : Prop
h : r β p β q
inl : β (h : Not r), r β p β And r q
inr inrβ : β (h : p β q), r β p β And r q
inrβ : β (h : p β q), r β p β r
β’ And r p β And r q |
r β p β β (h : Not p), q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inrβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\ninl : β (h : Not r), r β p β And r q\ninr inrβ : β (h : p β q), r β p β And r q\ninrβ : β (h : p β q), r β p β r\ninrβ : β (h : p β q), r β p β q\nβ’ And r p β And r q\n```\n\nLibrary theorems:\n```lean4\n... | goal | p q r : Prop
h : r β p β q
inl : β (h : Not r), r β p β And r q
inr inrβ : β (h : p β q), r β p β And r q
inrβ : β (h : p β q), r β p β r
inrβ : β (h : p β q), r β p β q
β’ And r p β And r q |
r β p β β (h : q), q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inrβ
| [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r β p β q\ninl : β (h : Not r), r β p β And r q\ninr inrβ : β (h : p β q), r β p β And r q\ninrβ : β (h : p β q), r β p β r\ninrβ : β (h : p β q), r β p β q\ninrβ : r β p β β (h : Not p), q\nβ’ And r p β And r q\n```... | goal | p q r : Prop
h : r β p β q
inl : β (h : Not r), r β p β And r q
inr inrβ : β (h : p β q), r β p β And r q
inrβ : β (h : p β q), r β p β r
inrβ : β (h : p β q), r β p β q
inrβ : r β p β β (h : Not p), q
β’ And r p β And r q |
β (h' : β (a : Ξ±), Eq Ξ² Ξ²'), Eq (Ξ± := Sort (imax u v)) (Ξ± β Ξ²) (Ξ± β Ξ²') | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases h | [
"Ξ±'",
"h",
"h'"
] | refl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞ± Ξ±' : Sort u\nΞ² Ξ²' : Sort v\nh : Eq Ξ± Ξ±'\nh' : β (a : Ξ±'), Eq Ξ² Ξ²'\nβ’ Eq (Ξ± := Sort (imax u v)) (Ξ± β Ξ²) (Ξ±' β Ξ²')\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermedia... | goal | Ξ± Ξ±' : Sort u
Ξ² Ξ²' : Sort v
h : Eq Ξ± Ξ±'
h' : β (a : Ξ±'), Eq Ξ² Ξ²'
β’ Eq (Ξ± := Sort (imax u v)) (Ξ± β Ξ²) (Ξ±' β Ξ²') |
Eq (Ξ± := Sort (imax u v)) ((x : Ξ±) β (fun (x : Ξ±) β¦ Ξ²') x) (Ξ± β Ξ²') | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [funext h'] | [] | refl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞ± : Sort u\nΞ² Ξ²' : Sort v\nh' : β (a : Ξ±), Eq Ξ² Ξ²'\nβ’ Eq (Ξ± := Sort (imax u v)) ((x : Ξ±) β (fun (x : Ξ±) β¦ Ξ²) x) (Ξ± β Ξ²')\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an inte... | goal | Ξ± : Sort u
Ξ² Ξ²' : Sort v
h' : β (a : Ξ±), Eq Ξ² Ξ²'
β’ Eq (Ξ± := Sort (imax u v)) ((x : Ξ±) β (fun (x : Ξ±) β¦ Ξ²) x) (Ξ± β Ξ²') |
β {p : HEq x x β Prop} (h : β (he : Eq x x), p sorry), p sorry | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases he | [
"y",
"p",
"h",
"he"
] | refl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΞ± : Sort u_1\nx y : Ξ±\np : HEq x y β Prop\nh : β (he : Eq x y), p β―\nhe : HEq x y\nβ’ p he\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier... | goal | Ξ± : Sort u_1
x y : Ξ±
p : HEq x y β Prop
h : β (he : Eq x y), p β―
he : HEq x y
β’ p he |
β {p : Eq x x β Prop} (h : β (he : Iff x x), p sorry), p sorry | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases he | [
"y",
"p",
"h",
"he"
] | refl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nx y : Prop\np : Eq x y β Prop\nh : β (he : Iff x y), p β―\nhe : Eq x y\nβ’ p he\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove fr... | goal | x y : Prop
p : Eq x y β Prop
h : β (he : Iff x y), p β―
he : Eq x y
β’ p he |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases p | [
"p"
] | var | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor instance... | goal | p q : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases p | [
"p"
] | and' | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q : Mathlib.Tactic.ITauto.IProp\nvar : Nat β Ordering\ntrue false : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easie... | goal | p q : Mathlib.Tactic.ITauto.IProp
var : Nat β Ordering
true false : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | var | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\na : Nat\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor i... | goal | q : Mathlib.Tactic.ITauto.IProp
a : Nat
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | varβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\na : Nat\nvar : Nat β Ordering\nvarβ varβ : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is... | goal | q : Mathlib.Tactic.ITauto.IProp
a : Nat
var : Nat β Ordering
varβ varβ : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | true | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor instance, ... | goal | q : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | trueβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\ntrue : Nat β Ordering\ncases trueβ : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easie... | goal | q : Mathlib.Tactic.ITauto.IProp
true : Nat β Ordering
cases trueβ : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | false | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor instance, ... | goal | q : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | falseβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\nfalse : Nat β Ordering\nfalseβ cases : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is eas... | goal | q : Mathlib.Tactic.ITauto.IProp
false : Nat β Ordering
falseβ cases : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | and' | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\na : Mathlib.Tactic.ITauto.AndKind\na_1 a_2 : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an interme... | goal | q : Mathlib.Tactic.ITauto.IProp
a : Mathlib.Tactic.ITauto.AndKind
a_1 a_2 : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | and'β | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\na : Mathlib.Tactic.ITauto.AndKind\na_1 a_2 : Mathlib.Tactic.ITauto.IProp\nand' : Nat β Ordering\nand'β and'β : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to gener... | goal | q : Mathlib.Tactic.ITauto.IProp
a : Mathlib.Tactic.ITauto.AndKind
a_1 a_2 : Mathlib.Tactic.ITauto.IProp
and' : Nat β Ordering
and'β and'β : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | or | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor inst... | goal | q a a_1 : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | orβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nor : Nat β Ordering\norβ orβ : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easie... | goal | q a a_1 : Mathlib.Tactic.ITauto.IProp
or : Nat β Ordering
orβ orβ : Ordering
β’ Ordering |
Nat β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | imp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises.\n\nFor inst... | goal | q a a_1 : Mathlib.Tactic.ITauto.IProp
β’ Ordering |
Mathlib.Tactic.ITauto.AndKind β Mathlib.Tactic.ITauto.IProp β Mathlib.Tactic.ITauto.IProp β Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | impβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nimp : Nat β Ordering\nimpβ impβ : Ordering\nβ’ Ordering\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is ea... | goal | q a a_1 : Mathlib.Tactic.ITauto.IProp
imp : Nat β Ordering
impβ impβ : Ordering
β’ Ordering |
Eq
(HMul.hMul (Mathlib.Tactic.FieldSimp.NF.eval List.nil)
(Mathlib.Tactic.FieldSimp.zpow' (Prod.mk Β«$rΒ» Β«$xΒ»).2 (Prod.mk Β«$rΒ» Β«$xΒ»).1))
Β«$eΒ» | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [NF.eval_cons] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nΒ«$vΒ» : Lean.Level\nΒ«$MΒ» : Type v\nΒ«$iMΒ» : CommGroupWithZero Β«$MΒ»\nΒ«$rΒ» : Int\nΒ«$xΒ» : Β«$MΒ»\nΒ«$sndΒ» : Nat\nΒ«$eΒ» : Β«$MΒ»\nΒ«$pfΒ» : Eq (Mathlib.Tactic.FieldSimp.zpow' Β«$xΒ» Β«$rΒ») Β«$eΒ»\nβ’ Eq (Mathlib.Tactic.FieldSimp.NF.cons (Prod.mk Β«$rΒ» Β«$... | goal | Β«$vΒ» : Lean.Level
Β«$MΒ» : Type v
Β«$iMΒ» : CommGroupWithZero Β«$MΒ»
Β«$rΒ» : Int
Β«$xΒ» : Β«$MΒ»
Β«$sndΒ» : Nat
Β«$eΒ» : Β«$MΒ»
Β«$pfΒ» : Eq (Mathlib.Tactic.FieldSimp.zpow' Β«$xΒ» Β«$rΒ») Β«$eΒ»
β’ Eq (Mathlib.Tactic.FieldSimp.NF.cons (Prod.mk Β«$rΒ» Β«$xΒ») List.nil).eval Β«$eΒ» |
Iff (Or (And (Not p) q) (And p (Not q))) (Or (And p (Not q)) (And (Not p) q)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [not_not, or_comm] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q : Prop\nβ’ Iff (Or (And (Not p) q) (And (Not (Not p)) (Not q))) (Or (And p (Not q)) (And (Not p) q))\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal ... | goal | p q : Prop
β’ Iff (Or (And (Not p) q) (And (Not (Not p)) (Not q))) (Or (And p (Not q)) (And (Not p) q)) |
Iff (Or (And p (Not q)) (And (Not p) q)) (Or (And p (Not q)) (And (Not p) q)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [not_not, or_comm] | [] | rwβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q : Prop\nrw : Iff (Or (And (Not p) q) (And p (Not q))) (Or (And p (Not q)) (And (Not p) q))\nβ’ Iff (Or (And (Not p) q) (And (Not (Not p)) (Not q))) (Or (And p (Not q)) (And (Not p) q))\n```\n\nLibrary theorems:\n```lean4\n\n```\n\... | goal | p q : Prop
rw : Iff (Or (And (Not p) q) (And p (Not q))) (Or (And p (Not q)) (And (Not p) q))
β’ Iff (Or (And (Not p) q) (And (Not (Not p)) (Not q))) (Or (And p (Not q)) (And (Not p) q)) |
β (a b : Nat) (h : Ne (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).1 0),
LT.lt (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).2
((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | obtain β¨β¨a, bβ©, rflβ© := Nat.pairEquiv.surjective n | [
"n",
"h"
] | obtain | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nh : Ne (Nat.unpair n).1 0\nβ’ LT.lt (Nat.unpair n).2 n\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises... | goal | n : Nat
h : Ne (Nat.unpair n).1 0
β’ LT.lt (Nat.unpair n).2 n |
β (h : Ne a 0), LT.lt b (Nat.pair a b) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [Nat.pairEquiv_apply, Function.uncurry_apply_pair, Nat.unpair_pair] at * | [
"h"
] | simp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Nat\nh : Ne (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).1 0\nβ’ LT.lt (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).2\n ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.m... | goal | a b : Nat
h : Ne (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).1 0
β’ LT.lt (Nat.unpair ((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b))).2
((Nat.pairEquiv : (a : Prod Nat Nat) β Nat) (Prod.mk a b)) |
LE.le a (HMul.hMul a a) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | have := Nat.le_mul_self a | [] | this | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Nat\nh : Ne a 0\nβ’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b))\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an ... | hypothesis | a b : Nat
h : Ne a 0
β’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b)) |
LE.le b (HMul.hMul b b) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | have := Nat.le_mul_self b | [] | thisβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Nat\nh : Ne a 0\nthis : LE.le a (HMul.hMul a a)\nβ’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b))\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `h... | hypothesis | a b : Nat
h : Ne a 0
this : LE.le a (HMul.hMul a a)
β’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b)) |
LT.lt a b β LT.lt b (HAdd.hAdd (HMul.hMul b b) a) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split | [] | isTrue | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Nat\nh : Ne a 0\nthis_1 : LE.le a (HMul.hMul a a)\nthis : LE.le b (HMul.hMul b b)\nβ’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b))\n```\n\nLibrary theorems:\n```lean4\n\n```... | goal | a b : Nat
h : Ne a 0
this_1 : LE.le a (HMul.hMul a a)
this : LE.le b (HMul.hMul b b)
β’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b)) |
Not (LT.lt a b) β LT.lt b (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split | [] | isFalse | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Nat\nh : Ne a 0\nthis_1 : LE.le a (HMul.hMul a a)\nthis : LE.le b (HMul.hMul b b)\nisTrue : LT.lt a b β LT.lt b (HAdd.hAdd (HMul.hMul b b) a)\nβ’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul... | goal | a b : Nat
h : Ne a 0
this_1 : LE.le a (HMul.hMul a a)
this : LE.le b (HMul.hMul b b)
isTrue : LT.lt a b β LT.lt b (HAdd.hAdd (HMul.hMul b b) a)
β’ LT.lt b (ite (LT.lt a b) (HAdd.hAdd (HMul.hMul b b) a) (HAdd.hAdd (HAdd.hAdd (HMul.hMul a a) a) b)) |
Eq
(let p := Nat.unpair (Nat.pair 0 n);
dite (Eq p.1 0) (fun (h : Eq p.1 0) β¦ Mathlib.Deriving.Encodable.S.nat p.2) fun (h : Not (Eq p.1 0)) β¦
have this : LE.le p.1 (Nat.pair 0 n) := sorry;
have this := sorry;
have this := sorry;
(Mathlib.Deriving.Encodable.S.decode (HSub.hSub p.1 1)).cons (Mathlib.... | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | unfold S.encode S.decode | [] | nat | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode (Mathlib.Deriving.Encodable.S.nat n).encode)\n (Mathlib.Deriving.Encodable.S.nat n)\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, rep... | goal | n : Nat
β’ Eq (Mathlib.Deriving.Encodable.S.decode (Mathlib.Deriving.Encodable.S.nat n).encode)
(Mathlib.Deriving.Encodable.S.nat n) |
Eq
(let p := Nat.unpair (Nat.pair (HAdd.hAdd a.encode 1) b.encode);
dite (Eq p.1 0) (fun (h : Eq p.1 0) β¦ Mathlib.Deriving.Encodable.S.nat p.2) fun (h : Not (Eq p.1 0)) β¦
have this : LE.le p.1 (Nat.pair (HAdd.hAdd a.encode 1) b.encode) := sorry;
have this := sorry;
have this := sorry;
(Mathlib.Deriv... | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | unfold S.encode S.decode | [] | cons | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\na b : Mathlib.Deriving.Encodable.S\niha : Eq (Mathlib.Deriving.Encodable.S.decode a.encode) a\nihb : Eq (Mathlib.Deriving.Encodable.S.decode b.encode) b\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode (a.cons b).encode) (a.cons b)\n```\n\... | goal | a b : Mathlib.Deriving.Encodable.S
iha : Eq (Mathlib.Deriving.Encodable.S.decode a.encode) a
ihb : Eq (Mathlib.Deriving.Encodable.S.decode b.encode) b
β’ Eq (Mathlib.Deriving.Encodable.S.decode (a.cons b).encode) (a.cons b) |
β (n : Nat),
Eq (Mathlib.Deriving.Encodable.S.decode (Mathlib.Deriving.Encodable.S.nat n).encode)
(Mathlib.Deriving.Encodable.S.nat n) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | induction s with
| nat n =>
unfold S.encode S.decode
simp
| cons a b iha ihb =>
unfold S.encode S.decode
simp [iha, ihb] | [
"s"
] | nat | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\ns : Mathlib.Deriving.Encodable.S\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode s.encode) s\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier t... | goal | s : Mathlib.Deriving.Encodable.S
β’ Eq (Mathlib.Deriving.Encodable.S.decode s.encode) s |
β (a b : Mathlib.Deriving.Encodable.S) (iha : Eq (Mathlib.Deriving.Encodable.S.decode a.encode) a)
(ihb : Eq (Mathlib.Deriving.Encodable.S.decode b.encode) b),
Eq (Mathlib.Deriving.Encodable.S.decode (a.cons b).encode) (a.cons b) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | induction s with
| nat n =>
unfold S.encode S.decode
simp
| cons a b iha ihb =>
unfold S.encode S.decode
simp [iha, ihb] | [
"s"
] | cons | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\ns : Mathlib.Deriving.Encodable.S\nnat :\n β (n : Nat),\n Eq (Mathlib.Deriving.Encodable.S.decode (Mathlib.Deriving.Encodable.S.nat n).encode)\n (Mathlib.Deriving.Encodable.S.nat n)\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode ... | goal | s : Mathlib.Deriving.Encodable.S
nat :
β (n : Nat),
Eq (Mathlib.Deriving.Encodable.S.decode (Mathlib.Deriving.Encodable.S.nat n).encode)
(Mathlib.Deriving.Encodable.S.nat n)
β’ Eq (Mathlib.Deriving.Encodable.S.decode s.encode) s |
Eq
(Mathlib.Deriving.Encodable.S.encode
(let p := Nat.unpair n;
dite (Eq p.1 0) (fun (h : Eq p.1 0) β¦ Mathlib.Deriving.Encodable.S.nat p.2) fun (h : Not (Eq p.1 0)) β¦
have this := sorry;
have this := sorry;
have this := sorry;
(Mathlib.Deriving.Encodable.S.decode (HSub.hSub p.1 1)).con... | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | unfold S.decode | [] | ind | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode n).encode n\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statem... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
β’ Eq (Mathlib.Deriving.Encodable.S.decode n).encode n |
Eq (Nat.unpair n).1 0 β Eq (Mathlib.Deriving.Encodable.S.nat (Nat.unpair n).2).encode n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split | [] | ind | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nβ’ Eq\n (Mathlib.Deriving.Encodable.S.encode\n (dite (Eq (Nat.unpair n).1 0)\n (fun (h : Eq (Nat.unpair n).1 0) β¦ Mathlib.Deriv... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
β’ Eq
(Mathlib.Deriving.Encodable.S.encode
(dite (Eq (Nat.unpair n).1 0)
(fun (h : Eq (Nat.unpair n).1 0) β¦ Mathlib.Deriving.Encodable.S.nat (Nat.unpair n).2)
fun (h : Not (Eq (Nat.unpair n).1 0)) β¦
... |
Not (Eq (Nat.unpair n).1 0) β
Eq
((Mathlib.Deriving.Encodable.S.decode (HSub.hSub (Nat.unpair n).1 1)).cons
(Mathlib.Deriving.Encodable.S.decode (Nat.unpair n).2)).encode
n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | split | [] | indβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nind : Eq (Nat.unpair n).1 0 β Eq (Mathlib.Deriving.Encodable.S.nat (Nat.unpair n).2).encode n\nβ’ Eq\n (Mathlib.Deriving.Encodable.S.encode... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
ind : Eq (Nat.unpair n).1 0 β Eq (Mathlib.Deriving.Encodable.S.nat (Nat.unpair n).2).encode n
β’ Eq
(Mathlib.Deriving.Encodable.S.encode
(dite (Eq (Nat.unpair n).1 0)
(fun (h : Eq (Nat.unpair n).1 0) β¦ Mathlib.Der... |
Eq (Nat.pair (Nat.unpair n).1 (Nat.unpair n).2) n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [β h, Nat.pair_unpair] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Eq (Nat.unpair n).1 0\nβ’ Eq (Nat.pair 0 (Nat.unpair n).2) n\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `ha... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Eq (Nat.unpair n).1 0
β’ Eq (Nat.pair 0 (Nat.unpair n).2) n |
Eq n n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [β h, Nat.pair_unpair] | [] | rwβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Eq (Nat.unpair n).1 0\nrw : Eq (Nat.pair (Nat.unpair n).1 (Nat.unpair n).2) n\nβ’ Eq (Nat.pair 0 (Nat.unpair n).2) n\n```\n\nLibrary theor... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Eq (Nat.unpair n).1 0
rw : Eq (Nat.pair (Nat.unpair n).1 (Nat.unpair n).2) n
β’ Eq (Nat.pair 0 (Nat.unpair n).2) n |
LE.le 1 (Nat.unpair n).1 | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\nβ’ Eq\n (Nat.pair (HAdd.hAdd (Mathlib.Deriving.Encodable.S.decode (HSub.hSub (Nat.unpair n).1 1)).encode 1... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
β’ Eq
(Nat.pair (HAdd.hAdd (Mathlib.Deriving.Encodable.S.decode (HSub.hSub (Nat.unpair n).1 1)).encode 1)
(Mathlib.Deriving.Encodable.S.decode (Nat.unpair n).2).encode)
n |
LT.lt (Nat.unpair n).2 n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | a | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\nrw : LE.le 1 (Nat.unpair n).1\nβ’ Eq\n (Nat.pair (HAdd.hAdd (Mathlib.Deriving.Encodable.S.decode (HSub.hSu... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
rw : LE.le 1 (Nat.unpair n).1
β’ Eq
(Nat.pair (HAdd.hAdd (Mathlib.Deriving.Encodable.S.decode (HSub.hSub (Nat.unpair n).1 1)).encode 1)
(Mathlib.Deriving.Encodable.S.decode (Nat.unpair n).2... |
LT.lt (HSub.hSub (Nat.unpair n).1 1) n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair] | [] | aβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\nrw : LE.le 1 (Nat.unpair n).1\na : LT.lt (Nat.unpair n).2 n\nβ’ Eq\n (Nat.pair (HAdd.hAdd (Mathlib.Derivin... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
rw : LE.le 1 (Nat.unpair n).1
a : LT.lt (Nat.unpair n).2 n
β’ Eq
(Nat.pair (HAdd.hAdd (Mathlib.Deriving.Encodable.S.decode (HSub.hSub (Nat.unpair n).1 1)).encode 1)
(Mathlib.Deriving.Encoda... |
Ne (Nat.unpair n).1 0 | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rwa [Nat.one_le_iff_ne_zero] | [] | rwa | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\nβ’ LE.le 1 (Nat.unpair n).1\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` ... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
β’ LE.le 1 (Nat.unpair n).1 |
β (ih : β (m : Nat), LT.lt m 0 β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m) (h : Not (Eq (Nat.unpair 0).1 0)),
LT.lt (HSub.hSub (Nat.unpair 0).1 1) 0 | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | obtain _ | n' := n | [
"n",
"ih",
"h"
] | a | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\nβ’ LT.lt (HSub.hSub (Nat.unpair n).1 1) n\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to gene... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
β’ LT.lt (HSub.hSub (Nat.unpair n).1 1) n |
β (n' : Nat) (ih : β (m : Nat), LT.lt m (HAdd.hAdd n' 1) β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m)
(h : Not (Eq (Nat.unpair (HAdd.hAdd n' 1)).1 0)), LT.lt (HSub.hSub (Nat.unpair (HAdd.hAdd n' 1)).1 1) (HAdd.hAdd n' 1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | obtain _ | n' := n | [
"n",
"ih",
"h"
] | aβ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair n).1 0)\na :\n β (ih : β (m : Nat), LT.lt m 0 β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m)\n (h : Not (... | goal | n : Nat
ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair n).1 0)
a :
β (ih : β (m : Nat), LT.lt m 0 β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m)
(h : Not (Eq (Nat.unpair 0).1 0)), LT.lt (HSub.hSub (Nat.unpair 0).1 1) 0
β’ LT.lt (HSub.hSub (Nat.unpa... |
LT.lt (Nat.unpair (HAdd.hAdd n' 1)).1 (HAdd.hAdd n' 1) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | have := Nat.unpair_lt (by lia : 1 β€ n' + 1) | [] | this | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn' : Nat\nih : β (m : Nat), LT.lt m (HAdd.hAdd n' 1) β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m\nh : Not (Eq (Nat.unpair (HAdd.hAdd n' 1)).1 0)\nβ’ LT.lt (HSub.hSub (Nat.unpair (HAdd.hAdd n' 1)).1 1) (HAdd.hAdd n' 1)\n```\n... | hypothesis | n' : Nat
ih : β (m : Nat), LT.lt m (HAdd.hAdd n' 1) β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m
h : Not (Eq (Nat.unpair (HAdd.hAdd n' 1)).1 0)
β’ LT.lt (HSub.hSub (Nat.unpair (HAdd.hAdd n' 1)).1 1) (HAdd.hAdd n' 1) |
β (n : Nat) (ih : β (m : Nat), LT.lt m n β Eq (Mathlib.Deriving.Encodable.S.decode m).encode m),
Eq (Mathlib.Deriving.Encodable.S.decode n).encode n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | induction n using Nat.strongRecOn with | _ n ih =>
unfold S.decode
dsimp only
split
next h =>
unfold S.encode
rw [β h, Nat.pair_unpair]
next h =>
unfold S.encode
rw [ih, ih, Nat.sub_add_cancel, Nat.pair_unpair]
Β· rwa [Nat.one_le_iff_ne_zero]
Β· exact nat_unpair_lt_... | [
"n"
] | ind | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nn : Nat\nβ’ Eq (Mathlib.Deriving.Encodable.S.decode n).encode n\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermediate subgoal that is easier to prove from the premises... | goal | n : Nat
β’ Eq (Mathlib.Deriving.Encodable.S.decode n).encode n |
Eq (HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd lβ.eval (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval)))
(HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd lβ.eval lβ.eval))) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [β h, eval_cons, add_smul, add_assoc] | [] | simp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\nrβ rβ : R\nx : M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval\nβ’ Eq\n (HAdd.hAdd (Mathlib.Tactic.M... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
rβ rβ : R
x : M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval
β’ Eq
(HAdd.hAdd (Mathlib.Tactic.Module.NF.cons (Prod.mk rβ x) lβ).eval
(Mathlib.Tactic.Module.NF.cons (Prod.mk rβ x) lβ).eva... |
Eq (HAdd.hAdd lβ.eval (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval)) (HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd lβ.eval lβ.eval)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | congr! 1 | [] | congr! | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\nrβ rβ : R\nx : M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval\nβ’ Eq (HAdd.hAdd (HSMul.hSMul rβ x) (HA... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
rβ rβ : R
x : M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd lβ.eval (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval)))
(HAdd.hAdd (HSMul.hSMul rβ x) (HAdd... |
Eq (HAdd.hAdd (HAdd.hAdd lβ.eval (HSMul.hSMul rβ x)) lβ.eval) (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) lβ.eval) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [β add_assoc] | [] | simp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\nrβ rβ : R\nx : M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval\nβ’ Eq (HAdd.hAdd lβ.eval (HAdd.hAdd (HS... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
rβ rβ : R
x : M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd lβ.eval (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval))
(HAdd.hAdd (HSMul.hSMul rβ x) (HAdd.hAdd lβ.eval lβ.eval)) |
Eq (HAdd.hAdd lβ.eval (HSMul.hSMul rβ x)) (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | congr! 1 | [] | congr! | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\nrβ rβ : R\nx : M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval\nβ’ Eq (HAdd.hAdd (HAdd.hAdd lβ.eval (HS... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
rβ rβ : R
x : M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd (HAdd.hAdd lβ.eval (HSMul.hSMul rβ x)) lβ.eval)
(HAdd.hAdd (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) lβ.eval) |
Eq (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [add_comm] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\nrβ rβ : R\nx : M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval\nβ’ Eq (HAdd.hAdd lβ.eval (HSMul.hSMul r... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
rβ rβ : R
x : M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd lβ.eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd lβ.eval (HSMul.hSMul rβ x)) (HAdd.hAdd (HSMul.hSMul rβ x) lβ.eval) |
Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval))
(HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) lβ.eval)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [eval_cons, β h] | [] | simp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\naβ aβ : Prod R M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval\nβ’ E... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
aβ aβ : Prod R M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval (Mathlib.Tactic.Module.NF.cons aβ lβ).e... |
Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval))
(HAdd.hAdd (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) lβ.eval) (HSMul.hSMul aβ.1 aβ.2)) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | nth_rw 4 [add_comm] | [] | nth_rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\naβ aβ : Prod R M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval\nβ’ E... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
aβ aβ : Prod R M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.... |
Eq (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) (HAdd.hAdd lβ.eval (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval)))
(HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) (HAdd.hAdd lβ.eval (HAdd.hAdd lβ.eval (HSMul.hSMul aβ.1 aβ.2)))) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | simp only [add_assoc] | [] | simp | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nR : Type u_2\nM : Type u_3\ninst : Semiring R\ninst_1 : AddCommMonoid M\ninst_2 : Module R M\naβ aβ : Prod R M\nlβ lβ l : Mathlib.Tactic.Module.NF R M\nh : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval\nβ’ E... | goal | R : Type u_2
M : Type u_3
inst : Semiring R
inst_1 : AddCommMonoid M
inst_2 : Module R M
aβ aβ : Prod R M
lβ lβ l : Mathlib.Tactic.Module.NF R M
h : Eq (HAdd.hAdd (Mathlib.Tactic.Module.NF.cons aβ lβ).eval lβ.eval) l.eval
β’ Eq (HAdd.hAdd (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.eval) (HAdd.hAdd (HSMul.hSMul aβ.1 aβ.2) lβ.... |
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