url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_provably_subtheory | [9, 1] | [12, 54] | exact consequence_iff'.{u, w}.mp (sound! ⟨this⟩) M | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
this : U ⊢ p
⊢ inst✝.toStruc ⊧ p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
this : U ⊢ p
⊢ inst✝.toStruc ⊧ p
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_add_left | [16, 1] | [16, 112] | simp [Theory.add_def] | L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ T ⊆ T + U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ T ⊆ T + U
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_add_right | [18, 1] | [18, 113] | simp [Theory.add_def] | L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ U ⊆ T + U | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ U ⊆ T + U
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.add_zero | [25, 1] | [26, 74] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.addZero | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x + 0 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x + 0 = x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.add_assoc | [28, 1] | [29, 75] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.addAssoc | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x + y + z = x + (y + z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x + y + z = x + (y + z)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.add_comm | [31, 1] | [32, 74] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.addComm | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x + y = y + x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x + y = y + x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.add_eq_of_lt | [34, 1] | [35, 76] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.addEqOfLt | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x < y → ∃ z, x + z = y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x < y → ∃ z, x + z = y
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.zero_le | [37, 1] | [38, 103] | simpa[models_iff, Structure.le_iff_of_eq_of_lt] using ModelsTheory.models M Theory.peanoMinus.zeroLe | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), 0 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), 0 ≤ x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.zero_lt_one | [40, 1] | [41, 76] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.zeroLtOne | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ 0 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ 0 < 1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.one_le_of_zero_lt | [43, 1] | [44, 110] | simpa[models_iff, Structure.le_iff_of_eq_of_lt] using ModelsTheory.models M Theory.peanoMinus.oneLeOfZeroLt | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), 0 < x → 1 ≤ x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), 0 < x → 1 ≤ x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.add_lt_add | [46, 1] | [47, 75] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.addLtAdd | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → x + z < y + z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → x + z < y + z
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.mul_zero | [49, 1] | [50, 74] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.mulZero | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x * 0 = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x * 0 = 0
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.mul_one | [52, 1] | [53, 73] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.mulOne | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x * 1 = x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), x * 1 = x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.mul_assoc | [55, 1] | [56, 75] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.mulAssoc | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x * y * z = x * (y * z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x * y * z = x * (y * z)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.mul_comm | [58, 1] | [59, 74] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.mulComm | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x * y = y * x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x * y = y * x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.mul_lt_mul | [61, 1] | [62, 75] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.mulLtMul | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → 0 < z → x * z < y * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → 0 < z → x * z < y * z
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.distr | [64, 1] | [65, 72] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.distr | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x * (y + z) = x * y + x * z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x * (y + z) = x * y + x * z
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.lt_irrefl | [67, 1] | [68, 75] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.ltIrrefl | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), ¬x < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x : M), ¬x < x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.lt_trans | [70, 1] | [71, 74] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.ltTrans | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → y < z → x < z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y z : M), x < y → y < z → x < z
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.lt_tri | [73, 1] | [74, 72] | simpa[models_iff] using ModelsTheory.models M Theory.peanoMinus.ltTri | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x < y ∨ x = y ∨ y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ∀ (x y : M), x < y ∨ x = y ∨ y < x
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.zero_mul | [107, 1] | [107, 100] | simpa[mul_comm] using Model.mul_zero x | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
⊢ 0 * x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
⊢ 0 * x = 0
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.numeral_eq_natCast | [144, 1] | [147, 101] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ORingSymbol.numeral 1 = ↑1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
⊢ ORingSymbol.numeral 1 = ↑1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.numeral_eq_natCast | [144, 1] | [147, 101] | simp[ORingSymbol.numeral, numeral_eq_natCast (n + 1), add_assoc, one_add_one_eq_two] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
⊢ ORingSymbol.numeral (n + 2) = ↑(n + 2) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
⊢ ORingSymbol.numeral (n + 2) = ↑(n + 2)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.not_neg | [149, 1] | [149, 42] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
⊢ ¬x < 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
⊢ ¬x < 0
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_succ_of_pos | [151, 1] | [153, 26] | rcases le_iff_exists_add.mp (one_le_of_zero_lt x h) with ⟨y, rfl⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
h : 0 < x
⊢ ∃ y, x = y + 1 | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
y : M
h : 0 < 1 + y
⊢ ∃ y_1, 1 + y = y_1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
h : 0 < x
⊢ ∃ y, x = y + 1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_succ_of_pos | [151, 1] | [153, 26] | exact ⟨y, add_comm 1 y⟩ | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
y : M
h : 0 < 1 + y
⊢ ∃ y_1, 1 + y = y_1 + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
y : M
h : 0 < 1 + y
⊢ ∃ y_1, 1 + y = y_1 + 1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | intro h | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
⊢ x ≤ y → x < y + 1 | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h : x ≤ y
⊢ x < y + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
⊢ x ≤ y → x < y + 1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | exact lt_of_le_of_lt h (lt_add_one y) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h : x ≤ y
⊢ x < y + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h : x ≤ y
⊢ x < y + 1
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | rcases lt_iff_exists_add.mp h with ⟨z, hz, h⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h : x < y + 1
⊢ x ≤ y | case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z : M
hz : z > 0
h : y + 1 = x + z
⊢ x ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h : x < y + 1
⊢ x ≤ y
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | rcases eq_succ_of_pos hz with ⟨z', rfl⟩ | case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z : M
hz : z > 0
h : y + 1 = x + z
⊢ x ≤ y | case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
⊢ x ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z : M
hz : z > 0
h : y + 1 = x + z
⊢ x ≤ y
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | have : y = x + z' := by simpa[←add_assoc] using h | case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
⊢ x ≤ y | case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
this : y = x + z'
⊢ x ≤ y | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
⊢ x ≤ y
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | simp[this] | case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
this : y = x + z'
⊢ x ≤ y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
this : y = x + z'
⊢ x ≤ y
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.le_iff_lt_succ | [155, 1] | [161, 16] | simpa[←add_assoc] using h | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
⊢ y = x + z' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x y : M
h✝ : x < y + 1
z' : M
hz : z' + 1 > 0
h : y + 1 = x + (z' + 1)
⊢ y = x + z'
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | simp[not_neg] at hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
hx : x < ↑0
⊢ ∃ m, x = ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
x : M
hx : x < ↑0
⊢ ∃ m, x = ↑m
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | have : x ≤ n := by simpa[le_iff_lt_succ] using hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
⊢ ∃ m, x = ↑m | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
this : x ≤ ↑n
⊢ ∃ m, x = ↑m | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
⊢ ∃ m, x = ↑m
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | rcases this with (rfl | hx) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
this : x ≤ ↑n
⊢ ∃ m, x = ↑m | case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
hx : ↑n < ↑(n + 1)
⊢ ∃ m, ↑n = ↑m
case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx✝ : x < ↑(n + 1)
hx : x < ↑n
⊢ ∃ m, x = ↑m | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
this : x ≤ ↑n
⊢ ∃ m, x = ↑m
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | simpa[le_iff_lt_succ] using hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
⊢ x ≤ ↑n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx : x < ↑(n + 1)
⊢ x ≤ ↑n
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | exact ⟨n, rfl⟩ | case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
hx : ↑n < ↑(n + 1)
⊢ ∃ m, ↑n = ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
hx : ↑n < ↑(n + 1)
⊢ ∃ m, ↑n = ↑m
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.eq_nat_of_lt_nat | [163, 1] | [169, 32] | exact eq_nat_of_lt_nat hx | case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx✝ : x < ↑(n + 1)
hx : x < ↑n
⊢ ∃ m, x = ↑m | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
x : M
hx✝ : x < ↑(n + 1)
hx : x < ↑n
⊢ ∃ m, x = ↑m
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.val_numeral | [173, 1] | [179, 110] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin n
x✝ : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(x✝ x)) Empty.elim #a✝ = ↑(Semiterm.valm ℕ x✝ Empty.elim #a✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin n
x✝ : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(x✝ x)) Empty.elim #a✝ = ↑(Semiterm.valm ℕ x✝ Empty.elim #a✝)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.val_numeral | [173, 1] | [179, 110] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin 0 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Zero.zero a✝) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Zero.zero a✝)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin 0 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Zero.zero a✝) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Zero.zero a✝))
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.val_numeral | [173, 1] | [179, 110] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin 0 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.One.one a✝) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.One.one a✝)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
a✝ : Fin 0 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.One.one a✝) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.One.one a✝))
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.val_numeral | [173, 1] | [179, 110] | simp[Semiterm.val_func, val_numeral (v 0), val_numeral (v 1)] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Add.add v) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Add.add v)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Add.add v) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Add.add v))
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.val_numeral | [173, 1] | [179, 110] | simp[Semiterm.val_func, val_numeral (v 0), val_numeral (v 1)] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Mul.mul v) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Mul.mul v)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n
e : Fin n → ℕ
⊢ Semiterm.valm M (fun x => ↑(e x)) Empty.elim (Semiterm.func Language.Mul.mul v) =
↑(Semiterm.valm ℕ e Empty.elim (Semiterm.func Language.Mul.mul v))
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
x✝ : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ x✝) ⊤ → (Semiformula.Evalbm M fun x => ↑(x✝ x)) ⊤ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
x✝ : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ x✝) ⊤ → (Semiformula.Evalbm M fun x => ↑(x✝ x)) ⊤
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
x✝ : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ x✝) ⊥ → (Semiformula.Evalbm M fun x => ↑(x✝ x)) ⊥ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
x✝ : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ x✝) ⊥ → (Semiformula.Evalbm M fun x => ↑(x✝ x)) ⊥
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[Semiformula.eval_rel, Matrix.comp_vecCons', val_numeral] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.rel op(=) v) → (Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.rel op(=) v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.rel op(=) v) → (Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.rel op(=) v)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[Semiformula.eval_nrel, Matrix.comp_vecCons', val_numeral] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.nrel op(=) v) →
(Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.nrel op(=) v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.nrel op(=) v) →
(Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.nrel op(=) v)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[Semiformula.eval_rel, Matrix.comp_vecCons', val_numeral] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.rel op(<) v) → (Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.rel op(<) v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.rel op(<) v) → (Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.rel op(<) v)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[Semiformula.eval_nrel, Matrix.comp_vecCons', val_numeral] | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.nrel op(<) v) →
(Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.nrel op(<) v) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
v : Fin 2 → Semiterm ℒₒᵣ Empty n✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (Semiformula.nrel op(<) v) →
(Semiformula.Evalbm M fun x => ↑(e x)) (Semiformula.nrel op(<) v)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (p✝ ⋏ q✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (p✝ ⋏ q✝) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ →
(Semiformula.Evalbm ℕ e) q✝ → (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (p✝ ⋏ q✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (p✝ ⋏ q✝)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | intro ep eq | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ →
(Semiformula.Evalbm ℕ e) q✝ → (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
ep : (Semiformula.Evalbm ℕ e) p✝
eq : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ →
(Semiformula.Evalbm ℕ e) q✝ → (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact ⟨pval_of_pval_nat_of_sigma_one hp ep, pval_of_pval_nat_of_sigma_one hq eq⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
ep : (Semiformula.Evalbm ℕ e) p✝
eq : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
ep : (Semiformula.Evalbm ℕ e) p✝
eq : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∧ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (p✝ ⋎ q✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (p✝ ⋎ q✝) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ ∨ (Semiformula.Evalbm ℕ e) q✝ →
(Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (p✝ ⋎ q✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (p✝ ⋎ q✝)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | rintro (h | h) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ ∨ (Semiformula.Evalbm ℕ e) q✝ →
(Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) p✝ ∨ (Semiformula.Evalbm ℕ e) q✝ →
(Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | left | case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | case inl.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact pval_of_pval_nat_of_sigma_one hp h | case inl.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) p✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | right | case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | case inr.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) p✝ ∨ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact pval_of_pval_nat_of_sigma_one hq h | case inr.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) q✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.h
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
a✝ : ℕ
p✝ q✝ : Semiformula ℒₒᵣ Empty a✝
hp : Hierarchy 𝚺 1 p✝
hq : Hierarchy 𝚺 1 q✝
e : Fin a✝ → ℕ
h : (Semiformula.Evalbm ℕ e) q✝
⊢ (Semiformula.Evalbm M fun x => ↑(e x)) q✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | rcases Rew.positive_iff.mp pt with ⟨t, rfl⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
t✝ : Semiterm ℒₒᵣ Empty (n✝ + 1)
pt : t✝.Positive
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (“(∀[#0 < !!t✝] !p✝)”) → (Semiformula.Evalbm M fun x => ↑(e x)) (“(∀[#0 < !!t✝] !p✝)”) | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
t✝ : Semiterm ℒₒᵣ Empty (n✝ + 1)
pt : t✝.Positive
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (“(∀[#0 < !!t✝] !p✝)”) → (Semiformula.Evalbm M fun x => ↑(e x)) (“(∀[#0 < !!t✝] !p✝)”)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[val_numeral] | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”) | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝) →
∀ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∀[#0 < !!(Rew.bShift t)] !p✝)”)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | intro h x hx | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝) →
∀ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : M
hx : x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝) →
∀ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | rcases eq_nat_of_lt_nat hx with ⟨x, rfl⟩ | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : M
hx : x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : ℕ
hx : ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : M
hx : x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one hp (h x (by simpa using hx)) | case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : ℕ
hx : ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : ℕ
hx : ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa using hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : ℕ
hx : ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ x < Semiterm.val (standardModel ℕ) e Empty.elim t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
h : ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t, (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
x : ℕ
hx : ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
⊢ x < Semiterm.val (standardModel ℕ) e Empty.elim t
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | rcases Rew.positive_iff.mp pt with ⟨t, rfl⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
t✝ : Semiterm ℒₒᵣ Empty (n✝ + 1)
pt : t✝.Positive
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (“(∃[#0 < !!t✝] !p✝)”) → (Semiformula.Evalbm M fun x => ↑(e x)) (“(∃[#0 < !!t✝] !p✝)”) | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”) | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
t✝ : Semiterm ℒₒᵣ Empty (n✝ + 1)
pt : t✝.Positive
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (“(∃[#0 < !!t✝] !p✝)”) → (Semiformula.Evalbm M fun x => ↑(e x)) (“(∃[#0 < !!t✝] !p✝)”)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp[val_numeral] | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”) | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t,
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ (Semiformula.Evalbm ℕ e) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”) →
(Semiformula.Evalbm M fun x => ↑(e x)) (“(∃[#0 < !!(Rew.bShift t)] !p✝)”)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | intro x hx h | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t,
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
⊢ ∀ x < Semiterm.val (standardModel ℕ) e Empty.elim t,
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact ⟨x, by simpa using hx, by simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one hp h⟩ | case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t),
(Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa using hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ↑x < ↑(Semiterm.valm ℕ (fun x => e x) Empty.elim t)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one hp h | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 1 p✝
e : Fin n✝ → ℕ
t : Semiterm ℒₒᵣ Empty n✝
pt : (Rew.bShift t).Positive
x : ℕ
hx : x < Semiterm.val (standardModel ℕ) e Empty.elim t
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (∃' p) → (Semiformula.Evalbm M fun x => ↑(e x)) (∃' p) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (∃' p) → (Semiformula.Evalbm M fun x => ↑(e x)) (∃' p)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | intro x h | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | have : Hierarchy 𝚺 1 p := hp.accum _ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
this : Hierarchy 𝚺 1 p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact ⟨x, by simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one this h⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
this : Hierarchy 𝚺 1 p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
this : Hierarchy 𝚺 1 p
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one this h | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
this : Hierarchy 𝚺 1 p
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚷 0 p
e : Fin n✝ → ℕ
x : ℕ
h : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p
this : Hierarchy 𝚺 1 p
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simp | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (∃' p✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (∃' p✝) | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
⊢ (Semiformula.Evalbm ℕ e) (∃' p✝) → (Semiformula.Evalbm M fun x => ↑(e x)) (∃' p✝)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | intro x hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
x : ℕ
hx : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
⊢ ∀ (x : ℕ),
(Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝ →
∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | exact ⟨x, by simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one hp hx⟩ | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
x : ℕ
hx : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
x : ℕ
hx : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ ∃ x, (Semiformula.Eval (standardModel M) (x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.Model.pval_of_pval_nat_of_sigma_one | [181, 1] | [209, 106] | simpa[Matrix.comp_vecCons'] using pval_of_pval_nat_of_sigma_one hp hx | M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
x : ℕ
hx : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐏𝐀⁻
n✝ : ℕ
p✝ : Semiformula ℒₒᵣ Empty (n✝ + 1)
hp : Hierarchy 𝚺 (0 + 1) p✝
e : Fin n✝ → ℕ
x : ℕ
hx : (Semiformula.Eval (standardModel ℕ) (x :> e) Empty.elim) p✝
⊢ (Semiformula.Eval (standardModel M) (↑x :> fun x => ↑(e x)) Empty.elim) p✝
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.sigma_one_completeness | [215, 1] | [219, 86] | haveI : M ⊧ₘ* 𝐏𝐀⁻ := ModelsTheory.of_provably_subtheory M 𝐏𝐀⁻ T inferInstance (by assumption) | M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
⊢ M ⊧ₘ σ | M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
this : M ⊧ₘ* 𝐏𝐀⁻
⊢ M ⊧ₘ σ | Please generate a tactic in lean4 to solve the state.
STATE:
M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
⊢ M ⊧ₘ σ
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.sigma_one_completeness | [215, 1] | [219, 86] | simpa [Matrix.empty_eq] using Model.pval_of_pval_nat_of_sigma_one (M := M) hσ H | M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
this : M ⊧ₘ* 𝐏𝐀⁻
⊢ M ⊧ₘ σ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
this : M ⊧ₘ* 𝐏𝐀⁻
⊢ M ⊧ₘ σ
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Arith/PeanoMinus.lean | LO.FirstOrder.Arith.sigma_one_completeness | [215, 1] | [219, 86] | assumption | M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
⊢ M ⊧ₘ* T | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M✝ : Type u_1
inst✝⁷ : Zero M✝
inst✝⁶ : One M✝
inst✝⁵ : Add M✝
inst✝⁴ : Mul M✝
inst✝³ : LT M✝
inst✝² : M✝ ⊧ₘ* 𝐏𝐀⁻
T : Theory ℒₒᵣ
inst✝¹ : 𝐄𝐐 ≼ T
inst✝ : 𝐏𝐀⁻ ≼ T
σ : Sentence ℒₒᵣ
hσ : Hierarchy 𝚺 1 σ
H : ℕ ⊧ₘ σ
M : Type
x✝⁵ : Zero M
x✝⁴ : One M
x✝³ : Add M
x✝² : Mul M
x✝¹ : LT M
x✝ : M ⊧ₘ* T
⊢ M ⊧ₘ* T
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.subset_def | [19, 1] | [19, 93] | rfl | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ ⊆ t₂ ↔ t₁.1 ⊆ t₂.1 ∧ t₁.2 ⊆ t₂.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ ⊆ t₂ ↔ t₁.1 ⊆ t₂.1 ∧ t₁.2 ⊆ t₂.2
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | constructor | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ ↔ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ → t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂ | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ ↔ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | . intro h; cases h; simp; | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ → t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂ | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ → t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | . rintro ⟨h₁, h₂⟩; cases t₁; cases t₂; simp_all; | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | intro h | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ → t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h : t₁ = t₂
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁ = t₂ → t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | cases h | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h : t₁ = t₂
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 | case mp.refl
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ : Tableau α
⊢ t₁.1 = t₁.1 ∧ t₁.2 = t₁.2 | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h : t₁ = t₂
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | simp | case mp.refl
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ : Tableau α
⊢ t₁.1 = t₁.1 ∧ t₁.2 = t₁.2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.refl
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ : Tableau α
⊢ t₁.1 = t₁.1 ∧ t₁.2 = t₁.2
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | rintro ⟨h₁, h₂⟩ | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂ | case mpr.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h₁ : t₁.1 = t₂.1
h₂ : t₁.2 = t₂.2
⊢ t₁ = t₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
⊢ t₁.1 = t₂.1 ∧ t₁.2 = t₂.2 → t₁ = t₂
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | cases t₁ | case mpr.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h₁ : t₁.1 = t₂.1
h₂ : t₁.2 = t₂.2
⊢ t₁ = t₂ | case mpr.intro.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₂ : Tableau α
fst✝ snd✝ : Theory α
h₁ : (fst✝, snd✝).1 = t₂.1
h₂ : (fst✝, snd✝).2 = t₂.2
⊢ (fst✝, snd✝) = t₂ | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₁ t₂ : Tableau α
h₁ : t₁.1 = t₂.1
h₂ : t₁.2 = t₂.2
⊢ t₁ = t₂
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | cases t₂ | case mpr.intro.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₂ : Tableau α
fst✝ snd✝ : Theory α
h₁ : (fst✝, snd✝).1 = t₂.1
h₂ : (fst✝, snd✝).2 = t₂.2
⊢ (fst✝, snd✝) = t₂ | case mpr.intro.mk.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
fst✝¹ snd✝¹ fst✝ snd✝ : Theory α
h₁ : (fst✝¹, snd✝¹).1 = (fst✝, snd✝).1
h₂ : (fst✝¹, snd✝¹).2 = (fst✝, snd✝).2
⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝) | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
t₂ : Tableau α
fst✝ snd✝ : Theory α
h₁ : (fst✝, snd✝).1 = t₂.1
h₂ : (fst✝, snd✝).2 = t₂.2
⊢ (fst✝, snd✝) = t₂
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.equality_def | [21, 1] | [24, 51] | simp_all | case mpr.intro.mk.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
fst✝¹ snd✝¹ fst✝ snd✝ : Theory α
h₁ : (fst✝¹, snd✝¹).1 = (fst✝, snd✝).1
h₂ : (fst✝¹, snd✝¹).2 = (fst✝, snd✝).2
⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.mk.mk
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
fst✝¹ snd✝¹ fst✝ snd✝ : Theory α
h₁ : (fst✝¹, snd✝¹).1 = (fst✝, snd✝).1
h₂ : (fst✝¹, snd✝¹).2 = (fst✝, snd✝).2
⊢ (fst✝¹, snd✝¹) = (fst✝, snd✝)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | constructor | α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) ↔
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | Please generate a tactic in lean4 to solve the state.
STATE:
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) ↔
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | . intro h Γ Δ hΓ hΔ;
by_contra hC;
have : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj' := h (by simp; intro q hq; right; exact hΓ q hq;) hΔ;
have : 𝓓 ⊢! (p :: Γ).conj' ⟶ Δ.disj' := implyLeft_cons_conj'!.mpr hC;
contradiction; | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | . intro h Γ Δ hΓ hΔ;
simp_all only [Set.mem_insert_iff];
have : 𝓓 ⊬! p ⋏ (Γ.remove p).conj' ⟶ Δ.disj' := h (by
intro q hq;
have := by simpa using hΓ q $ List.mem_of_mem_remove hq;
cases this with
| inl h => simpa [h] using List.mem_remove_iff.mp hq;
| inr h => assumption;
) hΔ;
by_contra hC;
have : 𝓓 ⊢! p ⋏ (Γ.remove p).conj' ⟶ Δ.disj' := imp_trans! andComm! $ implyLeftRemoveConj' (p := p) hC;
contradiction; | case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj') →
(𝓓)-Consistent (insert p T, U)
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | intro h Γ Δ hΓ hΔ | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj' | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
⊢ (𝓓)-Consistent (insert p T, U) →
∀ {Γ Δ : List (Formula α)}, (∀ p ∈ Γ, p ∈ T) → (∀ p ∈ Δ, p ∈ U) → 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | by_contra hC | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj' | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
⊢ 𝓓 ⊬! p ⋏ Γ.conj' ⟶ Δ.disj'
TACTIC:
|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/Completeness.lean | LO.Propositional.Superintuitionistic.Tableau.iff_ParametricConsistent_insert₁ | [30, 1] | [48, 19] | have : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj' := h (by simp; intro q hq; right; exact hΓ q hq;) hΔ | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ False | case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
this : 𝓓 ⊬! (p :: Γ).conj' ⟶ Δ.disj'
⊢ False | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
α : Type u_1
inst✝² : DecidableEq α
inst✝¹ : Inhabited α
𝓓 : DeductionParameter α
inst✝ : 𝓓.IncludeEFQ
p : Formula α
T U : Theory α
h : (𝓓)-Consistent (insert p T, U)
Γ Δ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
hΔ : ∀ p ∈ Δ, p ∈ U
hC : 𝓓 ⊢! p ⋏ Γ.conj' ⟶ Δ.disj'
⊢ False
TACTIC:
|
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