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 |
|---|---|---|---|---|---|---|---|
∀ (val : Nat) (property : LT.lt 0 val),
Eq (↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) b) : Nat)
(HAdd.hAdd (HSub.hSub (HSub.hSub (↑(Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) : Nat) 1) (↑b : 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\n⊢ Eq (↑(HSub.hSub a b) : Nat) (HAdd.hAdd (HSub.hSub (HSub.hSub (↑a : Nat) 1) (↑b : 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
⊢ Eq (↑(HSub.hSub a b) : Nat) (HAdd.hAdd (HSub.hSub (HSub.hSub (↑a : Nat) 1) (↑b : Nat)) 1) |
∀ (val_1 : Nat) (property_1 : LT.lt 0 val_1),
Eq
(↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property)
(Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val_1 property_1)) :
Nat)
(HAdd.hAdd
(HSub.hSub (HSub.hSub (↑(Subtype.mk (p := fun (n : Nat) ↦ 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\n⊢ Eq (↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) b) : Nat)\n (HAdd.hAdd (HSub.hSub (HSub.hSub (↑(Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) : ... | goal | b : PNat
val : Nat
property : LT.lt 0 val
⊢ Eq (↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) b) : Nat)
(HAdd.hAdd (HSub.hSub (HSub.hSub (↑(Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property) : Nat) 1) (↑b : 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, ← _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\n⊢ Eq\n (↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property)\n (Subtype.mk (p := fun (n : Nat) ↦ 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
⊢ Eq
(↑(HSub.hSub (Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val property)
(Subtype.mk (p := fun (n : Nat) ↦ LT.lt 0 n) val_1 property_1)) :
Nat)
(HAdd.hAdd
(HSub.hSub (HSub.hSub (↑(Subtype.mk (p := fun (n : Nat) ↦ L... |
LT.lt val_1 val → 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\n⊢ 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
⊢ 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) → 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 → Eq (HSub.hSub val val_1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 1)\n⊢ 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 → Eq (HSub.hSub val val_1) (HAdd.hAdd (HSub.hSub (HSub.hSub val 1) val_1) 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) |
And (Eq a a') (Eq b b') → 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' → Eq b b' → p\n⊢ Eq (Nat.pair a b) (Nat.pair a' b') → 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' → Eq b b' → p
⊢ Eq (Nat.pair a b) (Nat.pair a' b') → 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 ‹_› ‹_› | [
"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\n⊢ 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
⊢ Eq ((HAdd.hAdd f g).coeff n) (HAdd.hAdd a b) |
∀ (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
⊢ Eq ((HMul.hMul f g).coeff d) (ite (Eq d (HAdd.hAdd df dg)) (HMul.hMul a b) 0) |
∀ (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 : ∀ (h : Eq d (HAdd.hAdd df dg)), Eq ((HMul.hMul f g).coeff d) (HMul.hMul a ... |
∀ (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)
⊢ 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))
⊢ 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))
⊢ 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 ?_) | [] | apply₁ | [
{
"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... |
∀ (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\n⊢ 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
⊢ Eq ((HPow.hPow p m).coeff o) (ite (Eq o (HMul.hMul m n)) (HPow.hPow a m) 0) |
∀ (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 : ∀ (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 : ∀ (h : Eq o (HMul.hMul m n)), Eq ((HPow.hPow p m).coeff o) (HPow.hPow a m)
⊢ Eq ((HPow.hPow p m).coeff o) (ite (Eq o (HMul.hMul m n)) (HPow.hPow a m) 0) |
∀ (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)\n⊢ 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)
⊢ 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))\n⊢ 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))
⊢ 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 ‹_› ?_) | [] | 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))\n⊢ 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))
⊢ 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 ‹_› ?_) | [] | apply₁ | [
{
"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)
⊢ LT.lt (HPow.hPow p m).natDegree o |
∀ (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\n⊢ 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
⊢ Eq p.natDegree deg |
∀ (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\n⊢... | 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
⊢ Eq p.degree deg |
∀ (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 ⊥ 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\n⊢ 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
⊢ Eq p.degree m |
∀ (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 ⊥ 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 ∀ (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 :
∀ (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
⊢ Eq p.degree m |
∀ (m : Nat) (h_deg_le : LE.le p.degree (↑m : WithBot Nat))
(coeff_ne_zero : Ne (p.coeff (WithBot.unbotD 0 (↑m : WithBot Nat))) 0) (hh : Ne (↑m : WithBot Nat) Bot.bot),
Eq p.degree (↑m : WithBot Nat) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | obtain ⟨m, rfl⟩ := 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\n⊢ 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
⊢ 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\n⊢ 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
⊢ Eq ((HSub.hSub f g).coeff n) (HSub.hSub a b) |
LT.lt (α := Int) 0 (↑n : 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\n⊢ And (LE.le 0 (HMod.hMod a (↑n : Int)).toNat)\n (And (LT.lt (HMod.hMod a (↑n : Int)).toNat n) (Int.ModEq (↑n ... | 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
⊢ And (LE.le 0 (HMod.hMod a (↑n : Int)).toNat)
(And (LT.lt (HMod.hMod a (↑n : Int)).toNat n) (Int.ModEq (↑n : Int) a (↑(HMod.hMod a (↑n : Int)).toNat : Int))) |
LE.le 0 (HMod.hMod a (↑n : 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 (α := Int) 0 (↑n : Int)\n⊢ And (LE.le 0 (HMod.hMod a (↑n : 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 (α := Int) 0 (↑n : Int)
⊢ And (LE.le 0 (HMod.hMod a (↑n : Int)).toNat)
(And (LT.lt (HMod.hMod a (↑n : Int)).toNat n) (Int.ModEq (↑n : Int) a (↑(HMod.hMod a (↑n : Int)).toNat : Int))) |
LT.lt (HMod.hMod a (↑n : Int)).toNat n | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | 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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\n⊢ And (LE.le 0 (HMod.hMod a (↑n ... | 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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
⊢ And (LE.le 0 (HMod.hMod a (↑n : Int)).toNat)
(And (LT.lt (HMod.hMod a (↑n : Int)).toNat n) (Int.ModEq (↑n : Int) a (↑(HM... |
Int.ModEq (↑n : Int) a (↑(HMod.hMod a (↑n : Int)).toNat : Int) | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | refine ⟨Nat.zero_le _, ?_, ?_⟩ | [] | 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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\nrefine_1 : LT.lt (HMod.hMod a (↑... | 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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
refine_1 : LT.lt (HMod.hMod a (↑n : Int)).toNat n
⊢ And (LE.le 0 (HMod.hMod a (↑n : Int)).toNat)
(And (LT.lt (HMod.hMod a ... |
LT.lt (HMod.hMod a (↑n : Int)) (↑n : 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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\n⊢ LT.lt (HMod.hMod a (↑n : 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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
⊢ LT.lt (HMod.hMod a (↑n : Int)).toNat n |
Eq (HMod.hMod a (↑n : Int)) (HMod.hMod (α := Int) (↑(HMod.hMod a (↑n : Int)).toNat : Int) (↑n : 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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\n⊢ Int.ModEq (↑n : Int) a (↑(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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
⊢ Int.ModEq (↑n : Int) a (↑(HMod.hMod a (↑n : Int)).toNat : Int) |
Eq (HMod.hMod a (↑n : Int)) (HMod.hMod (HMod.hMod a (↑n : Int)) (↑n : 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_2₁ | [
{
"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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\nrefine_2 : Eq (HMod.hMod a (↑n :... | 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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
refine_2 : Eq (HMod.hMod a (↑n : Int)) (HMod.hMod (α := Int) (↑(HMod.hMod a (↑n : Int)).toNat : Int) (↑n : Int))
⊢ Int.ModEq (... |
Eq (HMod.hMod a (↑n : Int)) (HMod.hMod a (↑n : 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_2₂ | [
{
"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 (α := Int) 0 (↑n : Int)\nnonneg : LE.le 0 (HMod.hMod a (↑n : Int))\nrefine_2 : Eq (HMod.hMod a (↑n :... | 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 (α := Int) 0 (↑n : Int)
nonneg : LE.le 0 (HMod.hMod a (↑n : Int))
refine_2 : Eq (HMod.hMod a (↑n : Int)) (HMod.hMod (α := Int) (↑(HMod.hMod a (↑n : Int)).toNat : Int) (↑n : Int))
refine_2₁ : 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 ⟨Nat.zero_le _, ?_, ?_⟩ | [] | 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\n⊢ 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
⊢ 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 ⟨Nat.zero_le _, ?_, ?_⟩ | [] | 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\n⊢ 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
⊢ 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\n⊢ 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
⊢ 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_2₁ | [
{
"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)\n⊢ 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)
⊢ n.ModEq a (HMod.hMod a n) |
∀ (x : β) (hx : condβα (Prod.mk (coeβα x) y).fst) (hy : condδγ (Prod.mk (coeβα 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 (β := β) x hx with ⟨x, rfl⟩ | [
"x",
"hx",
"hy"
] | rcases | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ncoeβα : β → α\ncondβα : α → Prop\ncoeδγ : δ → γ\ncondδγ : γ → Prop\ninst : CanLift α β coeβα condβα\ninst_1 : CanLift γ δ coeδγ condδγ\nx : α\ny : γ\nhx : condβα (Prod.mk x y).f... | goal | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
coeβα : β → α
condβα : α → Prop
coeδγ : δ → γ
condδγ : γ → Prop
inst : CanLift α β coeβα condβα
inst_1 : CanLift γ δ coeδγ condδγ
x : α
y : γ
hx : condβα (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... |
∀ (y : δ) (hx : condβα (Prod.mk (coeβα x) (coeδγ y)).fst) (hy : condδγ (Prod.mk (coeβα 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 (β := δ) y hy with ⟨y, rfl⟩ | [
"y",
"hx",
"hy"
] | rcases | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ncoeβα : β → α\ncondβα : α → Prop\ncoeδγ : δ → γ\ncondδγ : γ → Prop\ninst : CanLift α β coeβα condβα\ninst_1 : CanLift γ δ coeδγ condδγ\ny : γ\nx : β\nhx : condβα (Prod.mk (coeβα... | goal | α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
coeβα : β → α
condβα : α → Prop
coeδγ : δ → γ
condδγ : γ → Prop
inst : CanLift α β coeβα condβα
inst_1 : CanLift γ δ coeδγ condδγ
y : γ
x : β
hx : condβα (Prod.mk (coeβα x) y).fst
hy : condδγ (Prod.mk (coeβα x) y).snd
⊢ Exists fun (y_1 : Prod β δ) ↦ Eq (Prod.map coeβα... |
∀ (this : DecidablePred p), Exists fun (g : (i : ι) → α i) ↦ Eq (fun (i : Subtype p) ↦ 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 ↦ Classical.propDecidable (p i) | [] | haveI | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nι : Sort u_1\nα : ι → Sort u_2\nne : ∀ (i : ι), Nonempty.{u_2} (α i)\np : ι → Prop\nf : (i : Subtype p) → α (i.val (p := p))\n⊢ Exists fun (g : (i : ι) → α i) ↦ Eq (fun (i : Subtype p) ↦ g (i.val (p := p))) f\n```\n\nLibrary theorems... | goal | ι : Sort u_1
α : ι → Sort u_2
ne : ∀ (i : ι), Nonempty.{u_2} (α i)
p : ι → Prop
f : (i : Subtype p) → α (i.val (p := p))
⊢ Exists fun (g : (i : ι) → α i) ↦ Eq (fun (i : Subtype p) ↦ g (i.val (p := p))) f |
Not (LT.lt (α := 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\n⊢ 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
⊢ Not (LT.lt Top.top x) |
∀ (a : Nat), Not (LT.lt (α := ENat) Top.top (↑a : 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 (α := ENat) Top.top Top.top)\n⊢ 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 (α := ENat) Top.top Top.top)
⊢ Not (LT.lt Top.top x) |
∀ (h : Not r), r → p → 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 → p → q\n⊢ And r p → 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 → p → q
⊢ And r p → And r q |
∀ (h : p → q), r → p → 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 → p → q\ninl : ∀ (h : Not r), r → p → And r q\n⊢ And r p → 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 → p → q
inl : ∀ (h : Not r), r → p → And r q
⊢ And r p → And r q |
∀ (h : p → q), r → p → r | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inr₂ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r → p → q\ninl : ∀ (h : Not r), r → p → And r q\ninr inr₁ : ∀ (h : p → q), r → p → And r q\n⊢ And r p → 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 → p → q
inl : ∀ (h : Not r), r → p → And r q
inr inr₁ : ∀ (h : p → q), r → p → And r q
⊢ And r p → And r q |
∀ (h : p → q), r → p → q | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | tauto | [
"h"
] | inr₃ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r → p → q\ninl : ∀ (h : Not r), r → p → And r q\ninr inr₁ : ∀ (h : p → q), r → p → And r q\ninr₂ : ∀ (h : p → q), r → p → r\n⊢ And r p → And r q\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generat... | goal | p q r : Prop
h : r → p → q
inl : ∀ (h : Not r), r → p → And r q
inr inr₁ : ∀ (h : p → q), r → p → And r q
inr₂ : ∀ (h : p → q), r → p → r
⊢ And r p → And r q |
r → p → ∀ (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"
] | inr₄ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r → p → q\ninl : ∀ (h : Not r), r → p → And r q\ninr inr₁ : ∀ (h : p → q), r → p → And r q\ninr₂ : ∀ (h : p → q), r → p → r\ninr₃ : ∀ (h : p → q), r → p → q\n⊢ And r p → And r q\n```\n\nLibrary theorems:\n```lean4\n... | goal | p q r : Prop
h : r → p → q
inl : ∀ (h : Not r), r → p → And r q
inr inr₁ : ∀ (h : p → q), r → p → And r q
inr₂ : ∀ (h : p → q), r → p → r
inr₃ : ∀ (h : p → q), r → p → q
⊢ And r p → And r q |
r → p → ∀ (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"
] | inr₅ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\np q r : Prop\nh : r → p → q\ninl : ∀ (h : Not r), r → p → And r q\ninr inr₁ : ∀ (h : p → q), r → p → And r q\ninr₂ : ∀ (h : p → q), r → p → r\ninr₃ : ∀ (h : p → q), r → p → q\ninr₄ : r → p → ∀ (h : Not p), q\n⊢ And r p → And r q\n```... | goal | p q r : Prop
h : r → p → q
inl : ∀ (h : Not r), r → p → And r q
inr inr₁ : ∀ (h : p → q), r → p → And r q
inr₂ : ∀ (h : p → q), r → p → r
inr₃ : ∀ (h : p → q), r → p → q
inr₄ : r → p → ∀ (h : Not p), q
⊢ And r p → And r q |
∀ (h' : ∀ (a : α), Eq β β'), Eq (α := Sort (imax u v)) (α → β) (α → β') | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases h | [
"α'",
"h",
"h'"
] | refl | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nα α' : Sort u\nβ β' : Sort v\nh : Eq α α'\nh' : ∀ (a : α'), Eq β β'\n⊢ Eq (α := Sort (imax u v)) (α → β) (α' → β')\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an intermedia... | goal | α α' : Sort u
β β' : Sort v
h : Eq α α'
h' : ∀ (a : α'), Eq β β'
⊢ Eq (α := Sort (imax u v)) (α → β) (α' → β') |
Eq (α := Sort (imax u v)) ((x : α) → (fun (x : α) ↦ β') x) (α → β') | [
{
"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\nα : Sort u\nβ β' : Sort v\nh' : ∀ (a : α), Eq β β'\n⊢ Eq (α := Sort (imax u v)) ((x : α) → (fun (x : α) ↦ β) x) (α → β')\n```\n\nLibrary theorems:\n```lean4\n\n```\n\nWe would like to generate a `have` statement, representing an inte... | goal | α : Sort u
β β' : Sort v
h' : ∀ (a : α), Eq β β'
⊢ Eq (α := Sort (imax u v)) ((x : α) → (fun (x : α) ↦ β) x) (α → β') |
∀ {p : HEq x x → Prop} (h : ∀ (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\nα : Sort u_1\nx y : α\np : HEq x y → Prop\nh : ∀ (he : Eq x y), p ⋯\nhe : HEq x y\n⊢ 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 | α : Sort u_1
x y : α
p : HEq x y → Prop
h : ∀ (he : Eq x y), p ⋯
he : HEq x y
⊢ p he |
∀ {p : Eq x x → Prop} (h : ∀ (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 → Prop\nh : ∀ (he : Iff x y), p ⋯\nhe : Eq x y\n⊢ 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 → Prop
h : ∀ (he : Iff x y), p ⋯
he : Eq x y
⊢ p he |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → 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 → Ordering\ntrue false : Ordering\n⊢ 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 → Ordering
true false : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | var₃ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\na : Nat\nvar : Nat → Ordering\nvar₁ var₂ : Ordering\n⊢ 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 → Ordering
var₁ var₂ : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | true₂ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\ntrue : Nat → Ordering\ncases true₁ : Ordering\n⊢ 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 → Ordering
cases true₁ : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | false₂ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq : Mathlib.Tactic.ITauto.IProp\nfalse : Nat → Ordering\nfalse₁ cases : Ordering\n⊢ 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 → Ordering
false₁ cases : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → 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'₃ | [
{
"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 → Ordering\nand'₁ and'₂ : Ordering\n⊢ 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 → Ordering
and'₁ and'₂ : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | or₃ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nor : Nat → Ordering\nor₁ or₂ : Ordering\n⊢ 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 → Ordering
or₁ or₂ : Ordering
⊢ Ordering |
Nat → 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\n⊢ 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
⊢ Ordering |
Mathlib.Tactic.ITauto.AndKind → Mathlib.Tactic.ITauto.IProp → Mathlib.Tactic.ITauto.IProp → Ordering | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | cases q | [
"q"
] | imp₃ | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\nq a a_1 : Mathlib.Tactic.ITauto.IProp\nimp : Nat → Ordering\nimp₁ imp₂ : Ordering\n⊢ 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 → Ordering
imp₁ imp₂ : Ordering
⊢ Ordering |
Eq
(HMul.hMul (Mathlib.Tactic.FieldSimp.NF.eval List.nil)
(Mathlib.Tactic.FieldSimp.zpow' (Prod.mk «$r» «$x»).2 (Prod.mk «$r» «$x»).1))
«$e» | [
{
"type": "function",
"function": {
"parameters": {
"type": "object",
"required": [
"type",
"name",
"clear"
],
"properties": {
"type": {
"type": "string",
"description": "An intermediate statement toward pr... | rw [NF.eval_cons] | [] | rw | [
{
"role": "user",
"content": "Consider the following Lean goal:\n```lean4\n«$v» : Lean.Level\n«$M» : Type v\n«$iM» : CommGroupWithZero «$M»\n«$r» : Int\n«$x» : «$M»\n«$snd» : Nat\n«$e» : «$M»\n«$pf» : Eq (Mathlib.Tactic.FieldSimp.zpow' «$x» «$r») «$e»\n⊢ Eq (Mathlib.Tactic.FieldSimp.NF.cons (Prod.mk «$r» «$... | goal | «$v» : Lean.Level
«$M» : Type v
«$iM» : CommGroupWithZero «$M»
«$r» : Int
«$x» : «$M»
«$snd» : Nat
«$e» : «$M»
«$pf» : Eq (Mathlib.Tactic.FieldSimp.zpow' «$x» «$r») «$e»
⊢ Eq (Mathlib.Tactic.FieldSimp.NF.cons (Prod.mk «$r» «$x») List.nil).eval «$e» |
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\n⊢ 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
⊢ Iff (Or (And (Not p) q) (And (Not (Not p)) (Not q))) (Or (And p (Not q)) (And (Not p) q)) |
End of preview. Expand in Data Studio
README.md exists but content is empty.
- Downloads last month
- 145