url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
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| start
stringlengths 6
10
| end
stringlengths 6
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| tactic
stringlengths 1
11.2k
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stringlengths 3
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| state_after
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|---|---|---|---|---|---|---|---|---|
https://github.com/opencompl/egg-tactic-code.git | 8b4aa748047a43213fc2c0dfca6b7af4a475f785 | Evaluation/FunctionalProgramming.lean | Test.all_deforestation | [87, 1] | [93, 48] | induction xs with
| nil => simp [all, all', List.foldr, List.map]
| cons a as ih => simp [all, all', List.foldr, List.map, ih] | Ξ± : Sort u_1
p : Ξ± β Bool
xs : List Ξ±
β’ all p xs = all' p xs | no goals |
https://github.com/opencompl/egg-tactic-code.git | 8b4aa748047a43213fc2c0dfca6b7af4a475f785 | Evaluation/FunctionalProgramming.lean | Test.all_deforestation | [87, 1] | [93, 48] | simp [all, all', List.foldr, List.map] | case nil
Ξ± : Sort u_1
p : Ξ± β Bool
β’ all p [] = all' p [] | no goals |
https://github.com/opencompl/egg-tactic-code.git | 8b4aa748047a43213fc2c0dfca6b7af4a475f785 | Evaluation/FunctionalProgramming.lean | Test.all_deforestation | [87, 1] | [93, 48] | simp [all, all', List.foldr, List.map, ih] | case cons
Ξ± : Sort u_1
p : Ξ± β Bool
a : Ξ±
as : List Ξ±
ih : all p as = all' p as
β’ all p (a::as) = all' p (a::as) | case cons
Ξ± : Sort u_1
p : Ξ± β Bool
a : Ξ±
as : List Ξ±
ih : all p as = all' p as
β’ (p a && List.foldr and true (List.map p as)) = (p a && all' p as) |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.lt_mul_cancel_left | [171, 1] | [171, 102] | sorry_arith | a b x : β
H : a < b
XNEQ0 : 0 < x
β’ a * x < b * x | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_index_twice_is_id | [282, 1] | [288, 2] | cases ix | n m : β
ix : TensorIndex2D n m
β’ TensorIndex2D.transpose (TensorIndex2D.transpose ix) = ix | case mk
n m ix0β ix1β : β
IX0β : ix0β < n
IX1β : ix1β < m
β’ TensorIndex2D.transpose (TensorIndex2D.transpose { ix0 := ix0β, ix1 := ix1β, IX0 := IX0β, IX1 := IX1β }) =
{ ix0 := ix0β, ix1 := ix1β, IX0 := IX0β, IX1 := IX1β } |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_index_twice_is_id | [282, 1] | [288, 2] | case mk ix0 ix1 IX0 IX1 => {
simp[TensorIndex2D.transpose];
} | n m ix0 ix1 : β
IX0 : ix0 < n
IX1 : ix1 < m
β’ TensorIndex2D.transpose (TensorIndex2D.transpose { ix0 := ix0, ix1 := ix1, IX0 := IX0, IX1 := IX1 }) =
{ ix0 := ix0, ix1 := ix1, IX0 := IX0, IX1 := IX1 } | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_index_twice_is_id | [282, 1] | [288, 2] | simp[TensorIndex2D.transpose] | n m ix0 ix1 : β
IX0 : ix0 < n
IX1 : ix1 < m
β’ TensorIndex2D.transpose (TensorIndex2D.transpose { ix0 := ix0, ix1 := ix1, IX0 := IX0, IX1 := IX1 }) =
{ ix0 := ix0, ix1 := ix1, IX0 := IX0, IX1 := IX1 } | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_twice_is_id | [290, 1] | [296, 2] | cases t | t : Tensor2D
β’ Tensor2D.transpose (Tensor2D.transpose t) = t | case mk
size0β size1β : β
dataβ : TensorIndex2D size0β size1β β β€
β’ Tensor2D.transpose (Tensor2D.transpose { size0 := size0β, size1 := size1β, data := dataβ }) =
{ size0 := size0β, size1 := size1β, data := dataβ } |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_twice_is_id | [290, 1] | [296, 2] | case mk size0 size1 data => {
simp[Tensor2D.transpose];
simp[transpose_index_twice_is_id];
} | size0 size1 : β
data : TensorIndex2D size0 size1 β β€
β’ Tensor2D.transpose (Tensor2D.transpose { size0 := size0, size1 := size1, data := data }) =
{ size0 := size0, size1 := size1, data := data } | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_twice_is_id | [290, 1] | [296, 2] | simp[Tensor2D.transpose] | size0 size1 : β
data : TensorIndex2D size0 size1 β β€
β’ Tensor2D.transpose (Tensor2D.transpose { size0 := size0, size1 := size1, data := data }) =
{ size0 := size0, size1 := size1, data := data } | size0 size1 : β
data : TensorIndex2D size0 size1 β β€
β’ (fun ix => data (TensorIndex2D.transpose (TensorIndex2D.transpose ix))) = data |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | transpose_twice_is_id | [290, 1] | [296, 2] | simp[transpose_index_twice_is_id] | size0 size1 : β
data : TensorIndex2D size0 size1 β β€
β’ (fun ix => data (TensorIndex2D.transpose (TensorIndex2D.transpose ix))) = data | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shape_prod_nil | [377, 1] | [378, 47] | induction l <;> simp [shapeProd, List.foldr] | l : List β
β’ shapeProd (0 :: l) = 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd.cons_unfold | [381, 1] | [385, 2] | intros x xs | β’ β (x : β) (xs : List β), shapeProd (x :: xs) = x * shapeProd xs | x : β
xs : List β
β’ shapeProd (x :: xs) = x * shapeProd xs |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd.cons_unfold | [381, 1] | [385, 2] | simp [shapeProd, List.foldr] | x : β
xs : List β
β’ shapeProd (x :: xs) = x * shapeProd xs | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_non_zero | [428, 1] | [432, 2] | intros BOUND | bound : β
flat : TensorFlatIndex bound
β’ bound β 0 | bound : β
flat : TensorFlatIndex bound
BOUND : bound = 0
β’ False |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_non_zero | [428, 1] | [432, 2] | have H_INBOUND := flat.h_ix_inbound | bound : β
flat : TensorFlatIndex bound
BOUND : bound = 0
β’ False | bound : β
flat : TensorFlatIndex bound
BOUND : bound = 0
H_INBOUND : flat.ix < bound
β’ False |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_non_zero | [428, 1] | [432, 2] | simp [BOUND] at H_INBOUND | bound : β
flat : TensorFlatIndex bound
BOUND : bound = 0
H_INBOUND : flat.ix < bound
β’ False | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_zero_absurd | [434, 1] | [437, 2] | have H_INBOUND := flat.h_ix_inbound | flat : TensorFlatIndex 0
β’ False | flat : TensorFlatIndex 0
H_INBOUND : flat.ix < 0
β’ False |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_zero_absurd | [434, 1] | [437, 2] | simp [Nat.not_lt_zero] at H_INBOUND | flat : TensorFlatIndex 0
H_INBOUND : flat.ix < 0
β’ False | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.succ_gt_zero | [440, 1] | [442, 2] | simp [GT.gt] | n : β
β’ succ n > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | intros n | β’ β (n : β), n β 0 β n > 0 | n : β
β’ n β 0 β n > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | constructor | n : β
β’ n β 0 β n > 0 | case mp
n : β
β’ n β 0 β n > 0
case mpr
n : β
β’ n > 0 β n β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case mp => {
intros NEQ_0;
cases n;
case zero => {
contradiction;
}
case succ n' => { simp [Nat.succ_gt_zero]; }
} | n : β
β’ n β 0 β n > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case mpr => {
intros GT_ZERO;
cases n;
case zero => {
simp at GT_ZERO;
}
case succ n' => { simp; }
} | n : β
β’ n > 0 β n β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | intros NEQ_0 | n : β
β’ n β 0 β n > 0 | n : β
NEQ_0 : n β 0
β’ n > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | cases n | n : β
NEQ_0 : n β 0
β’ n > 0 | case zero
NEQ_0 : zero β 0
β’ zero > 0
case succ
nβ : β
NEQ_0 : succ nβ β 0
β’ succ nβ > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case zero => {
contradiction;
} | NEQ_0 : zero β 0
β’ zero > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case succ n' => { simp [Nat.succ_gt_zero]; } | n' : β
NEQ_0 : succ n' β 0
β’ succ n' > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | contradiction | NEQ_0 : zero β 0
β’ zero > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | simp [Nat.succ_gt_zero] | n' : β
NEQ_0 : succ n' β 0
β’ succ n' > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | intros GT_ZERO | n : β
β’ n > 0 β n β 0 | n : β
GT_ZERO : n > 0
β’ n β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | cases n | n : β
GT_ZERO : n > 0
β’ n β 0 | case zero
GT_ZERO : zero > 0
β’ zero β 0
case succ
nβ : β
GT_ZERO : succ nβ > 0
β’ succ nβ β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case zero => {
simp at GT_ZERO;
} | GT_ZERO : zero > 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | case succ n' => { simp; } | n' : β
GT_ZERO : succ n' > 0
β’ succ n' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | simp at GT_ZERO | GT_ZERO : zero > 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.nonzero_iff_gt_zero | [445, 1] | [464, 2] | simp | n' : β
GT_ZERO : succ n' > 0
β’ succ n' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | have BOUND_NONZERO: bound β 0 := TensorFlatIndex.bound_non_zero flat | bound : β
flat : TensorFlatIndex bound
β’ bound > 0 | bound : β
flat : TensorFlatIndex bound
BOUND_NONZERO : bound β 0
β’ bound > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | cases bound | bound : β
flat : TensorFlatIndex bound
BOUND_NONZERO : bound β 0
β’ bound > 0 | case zero
flat : TensorFlatIndex Nat.zero
BOUND_NONZERO : Nat.zero β 0
β’ Nat.zero > 0
case succ
nβ : β
flat : TensorFlatIndex (Nat.succ nβ)
BOUND_NONZERO : Nat.succ nβ β 0
β’ Nat.succ nβ > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | case zero => {
simp [Nat.zero, BOUND_NONZERO];
contradiction;
} | flat : TensorFlatIndex Nat.zero
BOUND_NONZERO : Nat.zero β 0
β’ Nat.zero > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | case succ bound' => {
apply Nat.succ_gt_zero;
} | bound' : β
flat : TensorFlatIndex (Nat.succ bound')
BOUND_NONZERO : Nat.succ bound' β 0
β’ Nat.succ bound' > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | simp [Nat.zero, BOUND_NONZERO] | flat : TensorFlatIndex Nat.zero
BOUND_NONZERO : Nat.zero β 0
β’ Nat.zero > 0 | flat : TensorFlatIndex Nat.zero
BOUND_NONZERO : Nat.zero β 0
β’ False |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | contradiction | flat : TensorFlatIndex Nat.zero
BOUND_NONZERO : Nat.zero β 0
β’ False | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.bound_gt_zero | [467, 1] | [477, 2] | apply Nat.succ_gt_zero | bound' : β
flat : TensorFlatIndex (Nat.succ bound')
BOUND_NONZERO : Nat.succ bound' β 0
β’ Nat.succ bound' > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | intros a b NEQ | β’ β (a b : β), a * b β 0 β a β 0 | a b : β
NEQ : a * b β 0
β’ a β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | induction a | a b : β
NEQ : a * b β 0
β’ a β 0 | case zero
b : β
NEQ : zero * b β 0
β’ zero β 0
case succ
b nβ : β
n_ihβ : nβ * b β 0 β nβ β 0
NEQ : succ nβ * b β 0
β’ succ nβ β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | case zero => {
simp at NEQ;
} | b : β
NEQ : zero * b β 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | case succ a' IH => {
apply Nat.succ_ne_zero;
} | b a' : β
IH : a' * b β 0 β a' β 0
NEQ : succ a' * b β 0
β’ succ a' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | simp at NEQ | b : β
NEQ : zero * b β 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_left_nonzero | [480, 1] | [489, 2] | apply Nat.succ_ne_zero | b a' : β
IH : a' * b β 0 β a' β 0
NEQ : succ a' * b β 0
β’ succ a' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | intros a b NEQ | β’ β (a b : β), a * b β 0 β b β 0 | a b : β
NEQ : a * b β 0
β’ b β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | induction a | a b : β
NEQ : a * b β 0
β’ b β 0 | case zero
b : β
NEQ : zero * b β 0
β’ b β 0
case succ
b nβ : β
n_ihβ : nβ * b β 0 β b β 0
NEQ : succ nβ * b β 0
β’ b β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | case zero => {
simp at NEQ;
} | b : β
NEQ : zero * b β 0
β’ b β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | case succ a' IH => {
induction b;
case zero => {
simp at NEQ;
}
case succ b' IH => {
apply Nat.succ_ne_zero;
}
} | b a' : β
IH : a' * b β 0 β b β 0
NEQ : succ a' * b β 0
β’ b β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | simp at NEQ | b : β
NEQ : zero * b β 0
β’ b β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | induction b | b a' : β
IH : a' * b β 0 β b β 0
NEQ : succ a' * b β 0
β’ b β 0 | case zero
a' : β
IH : a' * zero β 0 β zero β 0
NEQ : succ a' * zero β 0
β’ zero β 0
case succ
a' nβ : β
n_ihβ : (a' * nβ β 0 β nβ β 0) β succ a' * nβ β 0 β nβ β 0
IH : a' * succ nβ β 0 β succ nβ β 0
NEQ : succ a' * succ nβ β 0
β’ succ nβ β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | case zero => {
simp at NEQ;
} | a' : β
IH : a' * zero β 0 β zero β 0
NEQ : succ a' * zero β 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | case succ b' IH => {
apply Nat.succ_ne_zero;
} | a' b' : β
IHβ : (a' * b' β 0 β b' β 0) β succ a' * b' β 0 β b' β 0
IH : a' * succ b' β 0 β succ b' β 0
NEQ : succ a' * succ b' β 0
β’ succ b' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | simp at NEQ | a' : β
IH : a' * zero β 0 β zero β 0
NEQ : succ a' * zero β 0
β’ zero β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_nonzero_implies_right_nonzero | [492, 1] | [507, 2] | apply Nat.succ_ne_zero | a' b' : β
IHβ : (a' * b' β 0 β b' β 0) β succ a' * b' β 0 β b' β 0
IH : a' * succ b' β 0 β succ b' β 0
NEQ : succ a' * succ b' β 0
β’ succ b' β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | intros xs x MEM | β’ β (xs : List β) (x : β), List.Mem x xs β shapeProd xs > 0 β x > 0 | xs : List β
x : β
MEM : List.Mem x xs
β’ shapeProd xs > 0 β x > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | induction MEM | xs : List β
x : β
MEM : List.Mem x xs
β’ shapeProd xs > 0 β x > 0 | case head
xs : List β
x : β
asβ : List β
β’ shapeProd (x :: asβ) > 0 β x > 0
case tail
xs : List β
x bβ : β
asβ : List β
aβ : List.Mem x asβ
a_ihβ : shapeProd asβ > 0 β x > 0
β’ shapeProd (bβ :: asβ) > 0 β x > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | case head as => {
simp [shapeProd, List.foldr];
intros H H2;
apply H;
} | xs : List β
x : β
as : List β
β’ shapeProd (x :: as) > 0 β x > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | case tail b bs MEM IH => {
intros H;
apply IH;
simp at H;
rewrite [<- Nat.nonzero_iff_gt_zero] at *;
simp[H];
} | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
β’ shapeProd (b :: bs) > 0 β x > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | simp [shapeProd, List.foldr] | xs : List β
x : β
as : List β
β’ shapeProd (x :: as) > 0 β x > 0 | xs : List β
x : β
as : List β
β’ 0 < x β 0 < List.foldr (fun x x_1 => x * x_1) 1 as β 0 < x |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | intros H H2 | xs : List β
x : β
as : List β
β’ 0 < x β 0 < List.foldr (fun x x_1 => x * x_1) 1 as β 0 < x | xs : List β
x : β
as : List β
H : 0 < x
H2 : 0 < List.foldr (fun x x_1 => x * x_1) 1 as
β’ 0 < x |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | apply H | xs : List β
x : β
as : List β
H : 0 < x
H2 : 0 < List.foldr (fun x x_1 => x * x_1) 1 as
β’ 0 < x | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | intros H | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
β’ shapeProd (b :: bs) > 0 β x > 0 | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : shapeProd (b :: bs) > 0
β’ x > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | apply IH | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : shapeProd (b :: bs) > 0
β’ x > 0 | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : shapeProd (b :: bs) > 0
β’ shapeProd bs > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | simp at H | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : shapeProd (b :: bs) > 0
β’ shapeProd bs > 0 | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : 0 < b β§ 0 < shapeProd bs
β’ shapeProd bs > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | rewrite [<- Nat.nonzero_iff_gt_zero] at * | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs > 0 β x > 0
H : 0 < b β§ 0 < shapeProd bs
β’ shapeProd bs > 0 | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs β 0 β x > 0
H : 0 < b β§ 0 < shapeProd bs
β’ shapeProd bs β 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | shapeProd_nonzero_implies_member_nonzero | [510, 1] | [526, 2] | simp[H] | xs : List β
x b : β
bs : List β
MEM : List.Mem x bs
IH : shapeProd bs β 0 β x > 0
H : 0 < b β§ 0 < shapeProd bs
β’ shapeProd bs β 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.shapeProd_member_nonzero | [530, 1] | [538, 2] | have PROD_NONZERO: shapeProd shape > 0 := flat.bound_gt_zero | shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
β’ n > 0 | shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ n > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.shapeProd_member_nonzero | [530, 1] | [538, 2] | apply shapeProd_nonzero_implies_member_nonzero | shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ n > 0 | case MEM
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ List.Mem n ?xs
case PROD
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ shapeProd ?xs > 0
case xs
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ List β |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.shapeProd_member_nonzero | [530, 1] | [538, 2] | exact MEMBER | case MEM
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ List.Mem n ?xs
case PROD
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ shapeProd ?xs > 0
case xs
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ List β | case PROD
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ shapeProd shape > 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | TensorFlatIndex.shapeProd_member_nonzero | [530, 1] | [538, 2] | exact PROD_NONZERO | case PROD
shape : List β
flat : TensorFlatIndex (shapeProd shape)
n : β
MEMBER : List.Mem n shape
PROD_NONZERO : shapeProd shape > 0
β’ shapeProd shape > 0 | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | have MULTIPLE: n = 0 + (n / modulus) * modulus := by {
rewrite [<- MODZERO];
rewrite [Nat.mul_comm];
simp [Nat.mod_add_div];
} | n modulus : β
MODZERO : n % modulus = 0
β’ n / modulus * modulus = n | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = 0 + n / modulus * modulus
β’ n / modulus * modulus = n |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | simp at MULTIPLE | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = 0 + n / modulus * modulus
β’ n / modulus * modulus = n | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = n / modulus * modulus
β’ n / modulus * modulus = n |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | rewrite [<- MULTIPLE] | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = n / modulus * modulus
β’ n / modulus * modulus = n | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = n / modulus * modulus
β’ n = n |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | rfl | n modulus : β
MODZERO : n % modulus = 0
MULTIPLE : n = n / modulus * modulus
β’ n = n | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | rewrite [<- MODZERO] | n modulus : β
MODZERO : n % modulus = 0
β’ n = 0 + n / modulus * modulus | n modulus : β
MODZERO : n % modulus = 0
β’ n = n % modulus + n / modulus * modulus |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | rewrite [Nat.mul_comm] | n modulus : β
MODZERO : n % modulus = 0
β’ n = n % modulus + n / modulus * modulus | n modulus : β
MODZERO : n % modulus = 0
β’ n = n % modulus + modulus * (n / modulus) |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mod_zero_implies_div_mul_equal | [542, 1] | [552, 2] | simp [Nat.mod_add_div] | n modulus : β
MODZERO : n % modulus = 0
β’ n = n % modulus + modulus * (n / modulus) | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_cancel_right | [555, 1] | [558, 2] | rewrite [Nat.mod_zero_implies_div_mul_equal n m MODZERO] | n m : β
MODZERO : n % m = 0
β’ n / m * m = n | n m : β
MODZERO : n % m = 0
β’ n = n |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.mul_cancel_right | [555, 1] | [558, 2] | rfl | n m : β
MODZERO : n % m = 0
β’ n = n | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.div_lt_if_mod | [561, 1] | [567, 2] | rewrite [Nat.div_lt_iff_lt_mul, Nat.mul_cancel_right] | ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ ix / modulus < bound / modulus | ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ ix < bound
case MODZERO
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ bound % modulus = 0
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.div_lt_if_mod | [561, 1] | [567, 2] | apply IX | ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ ix < bound
case MODZERO
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ bound % modulus = 0
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus | case MODZERO
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ bound % modulus = 0
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.div_lt_if_mod | [561, 1] | [567, 2] | apply DIV | case MODZERO
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ bound % modulus = 0
ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus | ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.div_lt_if_mod | [561, 1] | [567, 2] | apply MODULUS | ix bound modulus : β
IX : ix < bound
MODULUS : modulus > 0
DIV : bound % modulus = 0
β’ 0 < modulus | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | cases n | x n : β
X_LT_N : x < n
β’ x β€ pred n | case zero
x : β
X_LT_N : x < zero
β’ x β€ pred zero
case succ
x nβ : β
X_LT_N : x < succ nβ
β’ x β€ pred (succ nβ) |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | case zero => { simp [Nat.not_lt_zero] at X_LT_N; } | x : β
X_LT_N : x < zero
β’ x β€ pred zero | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | case succ n' => {
rewrite [Nat.pred_succ];
apply Nat.le_of_lt_succ;
exact X_LT_N;
} | x n' : β
X_LT_N : x < succ n'
β’ x β€ pred (succ n') | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | simp [Nat.not_lt_zero] at X_LT_N | x : β
X_LT_N : x < zero
β’ x β€ pred zero | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | rewrite [Nat.pred_succ] | x n' : β
X_LT_N : x < succ n'
β’ x β€ pred (succ n') | x n' : β
X_LT_N : x < succ n'
β’ x β€ n' |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | apply Nat.le_of_lt_succ | x n' : β
X_LT_N : x < succ n'
β’ x β€ n' | case a
x n' : β
X_LT_N : x < succ n'
β’ x < succ n' |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_pred_if_lt | [580, 1] | [588, 2] | exact X_LT_N | case a
x n' : β
X_LT_N : x < succ n'
β’ x < succ n' | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_one_minus_if_lt | [590, 1] | [594, 2] | rewrite [<- Nat.sub_one] | x n : β
X_LT_N : x < n
β’ x β€ pred n | x n : β
X_LT_N : x < n
β’ x β€ n - 1 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_one_minus_if_lt | [590, 1] | [594, 2] | apply Nat.le_pred_if_lt | x n : β
X_LT_N : x < n
β’ x β€ n - 1 | case X_LT_N
x n : β
X_LT_N : x < n
β’ x < Nat.sub n 0 |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_one_minus_if_lt | [590, 1] | [594, 2] | simp | case X_LT_N
x n : β
X_LT_N : x < n
β’ x < Nat.sub n 0 | case X_LT_N
x n : β
X_LT_N : x < n
β’ x < n |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_one_minus_if_lt | [590, 1] | [594, 2] | exact X_LT_N | case X_LT_N
x n : β
X_LT_N : x < n
β’ x < n | no goals |
https://github.com/opencompl/lean-mlir.git | e43d21592801e5e40477b14b7a554e356060c40c | MLIR/Util/KDTensor.lean | Nat.le_mul_pred | [596, 1] | [610, 2] | cases H:n | x y n : β
LE : x β€ pred n
β’ x * y β€ n * y - y | case zero
x y n : β
LE : x β€ pred n
H : n = zero
β’ x * y β€ zero * y - y
case succ
x y n : β
LE : x β€ pred n
nβ : β
H : n = succ nβ
β’ x * y β€ succ nβ * y - y |
Subsets and Splits
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