statement stringlengths 1 16.5k | proof stringlengths 0 117k | type stringclasses 12
values | symbolic_name stringlengths 1 16.4k | library stringclasses 22
values | filename stringlengths 29 118 | imports listlengths 0 20 | deps listlengths 0 22 | docstring stringclasses 234
values | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
rule_yyxy (x y : FreeMagma α) :
rule (((y ⋆ y) ⋆ x) ⋆ y) = ((y ⋆ y) ⋆ x) ⋆ y | by
prove rule | theorem | rw3143.rule_yyxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyxyy (x y : FreeMagma α) :
rule ((((y ⋆ y) ⋆ x) ⋆ y) ⋆ y) = x | by
simp [rule] | theorem | rw3143.rule_yyxyy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [3143] [1629, 1832, 2644] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw3143.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule : FreeMagma α → FreeMagma α | | m@((((y₁ ⋆ y₂) ⋆ y₃) ⋆ x) ⋆ y₄) =>
if y₁ = y₂ ∧ y₁ = y₃ ∧ y₁ = y₄ then
x
else
m
| m => m | def | rw3150.rule | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyx (y : FreeMagma α) :
rule ((y ⋆ y) ⋆ y) = (y ⋆ y) ⋆ y | by
prove rule | theorem | rw3150.rule_yyx | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyyx (x y : FreeMagma α) :
rule (((y ⋆ y) ⋆ y) ⋆ x) = ((y ⋆ y) ⋆ y) ⋆ x | by
prove rule | theorem | rw3150.rule_yyyx | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyyxy (x y : FreeMagma α) :
rule ((((y ⋆ y) ⋆ y) ⋆ x) ⋆ y) = x | by
simp [rule] | theorem | rw3150.rule_yyyxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [3150] [411, 1426, 2035] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw3150.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule : FreeMagma α → FreeMagma α | | m@(y₁ ⋆ ((y₂ ⋆ (x ⋆ x₁)) ⋆ y₃)) =>
if y₁ = y₂ ∧ y₁ = y₃ ∧ x = x₁ then
x
else
m
| m => m | def | rw1110.rule | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yxx (x y : FreeMagma α) :
rule (y ⋆ (x ⋆ x)) = y ⋆ (x ⋆ x) | by
prove rule | theorem | rw1110.rule_yxx | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yxxy (x y : FreeMagma α) :
rule ((y ⋆ (x ⋆ x)) ⋆ y) = (y ⋆ (x ⋆ x)) ⋆ y | by
prove rule | theorem | rw1110.rule_yxxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyxxy (x y : FreeMagma α) :
rule (y ⋆ ((y ⋆ (x ⋆ x)) ⋆ y)) = x | by
simp [rule] | theorem | rw1110.rule_yyxxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [1110] [8, 411, 1629, 1832, 3253, 3319, 3862, 3915] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw1110.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule : FreeMagma α → FreeMagma α | | m@(y₁ ⋆ ((x ⋆ (y₂ ⋆ y₃)) ⋆ y₄)) =>
if y₁ = y₂ ∧ y₁ = y₃ ∧ y₁ = y₄ then
x
else
m
| m => m | def | rw1086.rule | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_xyyy (x y : FreeMagma α) :
rule ((x ⋆ (y ⋆ y)) ⋆ y) = (x ⋆ (y ⋆ y)) ⋆ y | by
prove rule | theorem | rw1086.rule_xyyy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yxyyy (x y : FreeMagma α) :
rule (y ⋆ ((x ⋆ (y ⋆ y)) ⋆ y)) = x | by
simp [rule] | theorem | rw1086.rule_yxyyy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [1086] [1832, 1898, 2644, 2710] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw1086.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule : FreeMagma α → FreeMagma α | | m@((y₁ ⋆ (((x ⋆ x₁) ⋆ y₂))) ⋆ y₃) =>
if y₁ = y₂ ∧ y₁ = y₃ ∧ x = x₁ then
x
else
m
| m => m | def | rw2497.rule | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_xx (x : FreeMagma α) : rule (x ⋆ x) = x ⋆ x | by
prove rule | theorem | rw2497.rule_xx | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_xxy (x y : FreeMagma α) :
rule ((x ⋆ x) ⋆ y) = (x ⋆ x) ⋆ y | by
prove rule | theorem | rw2497.rule_xxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yxxy (x y : FreeMagma α) :
rule (y ⋆ ((x ⋆ x) ⋆ y)) = (y ⋆ ((x ⋆ x) ⋆ y)) | by
prove rule | theorem | rw2497.rule_yxxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yxxyy (x y : FreeMagma α) :
rule ((y ⋆ ((x ⋆ x) ⋆ y)) ⋆ y) = x | by
simp [rule] | theorem | rw2497.rule_yxxyy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [2497] [23, 1629, 1832, 3050, 3456, 3522, 4065, 4118] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw2497.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule : FreeMagma α → FreeMagma α | | m@((y₁ ⋆ (((y₂ ⋆ y₃) ⋆ x))) ⋆ y₄) =>
if y₁ = y₂ ∧ y₁ = y₃ ∧ y₁ = y₄ then
x
else
m
| m => m | def | rw2541.rule | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyyx (x y : FreeMagma α) :
rule (y ⋆ ((y ⋆ y) ⋆ x)) = (y ⋆ ((y ⋆ y) ⋆ x)) | by
prove rule | theorem | rw2541.rule_yyyx | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
rule_yyyxy (x y : FreeMagma α) :
rule ((y ⋆ ((y ⋆ y) ⋆ x)) ⋆ y) = x | by
simp [rule] | theorem | rw2541.rule_yyyxy | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» : ∃ (G : Type) (_ : Magma G), Facts G [2541] [817, 917, 1629, 1729] | by
use ConfMagma (@rule Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp [Magma.op, bu, hx, hy]
all_goals refute | theorem | rw2541.«Facts» | Root | equational_theories/Confluence1.lean | [
"equational_theories.Confluence"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp2 {x y : FreeMagma α} (hx: NF rules x):
rules (y ⋆ rules (x ⋆ x ⋆ y)) = x | by
generalize h: rules (x ⋆ x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rw_eq_self_of_NF rules hx
· apply rules.eq1 | theorem | rw115.comp2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp3 {x y : FreeMagma α} (hx: NF rules x):
rules (y ⋆ rules (rules (x ⋆ x) ⋆ y)) = x | by
generalize h: rules (x ⋆ x) = e
simp only [rules.elim] at h
separate
exact comp2 hx | theorem | rw115.comp3 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [115] [2707, 4273, 4332] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp3 ((NF_iff_buFixed rules).mpr hx)
all_goals refute | theorem | rw115.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α}:
rules (y ⋆ rules (x ⋆ (y ⋆ y ⋆ y))) = x | by
generalize h: rules (x ⋆ (y ⋆ y ⋆ y)) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq1 | theorem | rw680.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y : FreeMagma α}:
rules (y ⋆ rules (x ⋆ rules (y ⋆ y ⋆ y))) = x | by
generalize h: rules (y ⋆ y ⋆ y) = e
simp only [rules.elim] at h
separate
apply comp1_2 | theorem | rw680.comp1_3 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y : FreeMagma α}:
rules (y ⋆ rules (x ⋆ rules (rules (y ⋆ y) ⋆ y))) = x | by
generalize h: rules (y ⋆ y) = e
simp only [rules.elim] at h
separate
apply comp1_3 | theorem | rw680.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [680] [1629, 1695, 1832, 2847, 2947] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw680.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α}:
rules (y ⋆ rules (x ⋆ x ⋆ x ⋆ y)) = x | by
generalize h: rules (x ⋆ x ⋆ x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq1 | theorem | rw1276.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y : FreeMagma α}:
rules (y ⋆ rules (rules (x ⋆ x ⋆ x) ⋆ y)) = x | by
generalize h: rules (x ⋆ x ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp1_2 | theorem | rw1276.comp1_3 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y : FreeMagma α}:
rules (y ⋆ rules (rules (rules (x ⋆ x) ⋆ x) ⋆ y)) = x | by
generalize h: rules (x ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp1_3 | theorem | rw1276.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1276] [4273, 4332] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1276.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α}:
rules ((x ⋆ x) ⋆ rules (y ⋆ (x ⋆ x))) = x | by
generalize h: rules (y ⋆ (x ⋆ x)) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq1
· apply rules.eq1 | theorem | rw1431.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y : FreeMagma α}:
rules (rules (x ⋆ x) ⋆ rules (y ⋆ rules (x ⋆ x))) = x | by
generalize h: rules (x ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp1_2 | theorem | rw1431.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1431] [1428, 4269, 4316] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1431.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α} (hx: NF rules x):
rules (y ⋆ y ⋆ rules (x ⋆ (y ⋆ y))) = x | by
generalize h: rules (x ⋆ (y ⋆ y)) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply hx.top
· apply rules.eq1 | theorem | rw1519.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y : FreeMagma α} (hx: NF rules x):
rules (rules (y ⋆ y) ⋆ rules (x ⋆ rules (y ⋆ y))) = x | by
generalize h: rules (y ⋆ y) = e
simp only [rules.elim] at h
separate
apply comp1_2 hx | theorem | rw1519.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1519] [2035, 2128] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4 ((NF_iff_buFixed rules).mpr hx)
all_goals refute | theorem | rw1519.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x : FreeMagma α}:
rules (default ⋆ default ⋆ rules (x ⋆ (default ⋆ default))) = x | by
generalize h: rules (x ⋆ (default ⋆ default)) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq1 | theorem | rw1523.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y z : FreeMagma α}:
rules (rules (y ⋆ y) ⋆ rules (x ⋆ rules (z ⋆ z))) = x | by
simp [rules.eq2]
apply comp1_2 | theorem | rw1523.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1523] [2035, 2038, 2124, 2128, 2132] | by
use ConfMagma (@rules Nat _ _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1523.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α}:
rules (x ⋆ x ⋆ rules (x ⋆ x ⋆ y)) = x | by
generalize h: rules (x ⋆ x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq2
· apply rules.eq1 | theorem | rw1630.comp1_2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y : FreeMagma α}:
rules (rules (x ⋆ x) ⋆ rules (rules (x ⋆ x) ⋆ y)) = x | by
generalize h: rules (x ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp1_2 | theorem | rw1630.comp1_4 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1630] [4268, 4282] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1630.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp2 {x y z : FreeMagma α} (hxy: NF rules (x ⋆ y)):
rules (z ⋆ rules (x ⋆ y ⋆ z)) = x ⋆ y | by
generalize h: rules (x ⋆ y ⋆ z) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rw_eq_self_of_NF rules hxy
· apply rules.eq1 | theorem | rw3588.comp2 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp3 {x y z : FreeMagma α} (hx: NF rules x) (hy: NF rules y):
rules (z ⋆ rules (rules (x ⋆ y) ⋆ z)) = rules (x ⋆ y) | by
generalize h: rules (x ⋆ y) = e
simp only [rules.elim] at h
separate
all_goals apply comp2
· apply Everywhere.left hy
· apply Everywhere.right hx
· rw [NF, Everywhere]
refine ⟨hx, hy, ?_⟩
simp only [rules.elim]
right
constructorm* _ ∧ _
all_goals trivial | theorem | rw3588.comp3 | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [3588] [3862, 3878, 3917, 3955] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp3 ((NF_iff_buFixed rules).mpr hx) ((NF_iff_buFixed rules).mpr hy)
all_goals refute | theorem | rw3588.«Facts» | Root | equational_theories/Confluence2.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y : FreeMagma α} (hx: NF rules x):
rules (y ⋆ rules (x ⋆ (y ⋆ (default ⋆ default)))) = x | by
generalize h: rules (x ⋆ (y ⋆ (default ⋆ default))) = e
simp only [rules.elim] at h
separate
· apply rules.eq3
· apply hx.top
· apply rules.eq1 | theorem | rw481.comp1_2 | Root | equational_theories/Confluence3.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp2_2 {x : FreeMagma α} (hx: NF rules x):
rules (default ⋆ default ⋆ rules (x ⋆ (default ⋆ default))) = x | by
generalize h: rules (x ⋆ (default ⋆ default)) = e
simp only [rules.elim] at h
separate
· apply rules.eq3
· apply rules.eq2 | theorem | rw481.comp2_2 | Root | equational_theories/Confluence3.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y : FreeMagma α} (hx: NF rules x):
rules (y ⋆ rules (x ⋆ rules (y ⋆ (default ⋆ default)))) = x | by
generalize h: rules (y ⋆ (default ⋆ default)) = e
simp only [rules.elim] at h
separate
· apply comp2_2 hx
· apply comp1_2 hx | theorem | rw481.comp1_3 | Root | equational_theories/Confluence3.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y z : FreeMagma α} (hx: NF rules x):
rules (y ⋆ rules (x ⋆ rules (y ⋆ rules (z ⋆ z)))) = x | by
generalize h: rules (z ⋆ z) = e
simp only [rules.elim] at h
separate
· apply comp1_3 hx
· exfalso
simp_all only [Fork.injEq, not_exists, not_and, exists_eq', not_true_eq_false] | theorem | rw481.comp1_4 | Root | equational_theories/Confluence3.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [481] [1488, 1492, 1496, 2035, 2038, 2041, 2124, 2128, 2132, 2135, 3050, 3056, 3139, 3150, 3161] | by
use ConfMagma (@rules Nat _ _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4 ((NF_iff_buFixed rules).mpr hx)
all_goals refute | theorem | rw481.«Facts» | Root | equational_theories/Confluence3.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y z : FreeMagma α}:
rules (rules (x ⋆ y) ⋆ (x ⋆ (x ⋆ z))) = x | by
generalize h: rules (x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply rules.eq3
· apply rules.eq1
· apply rules.eq1
· apply rules.eq3
· apply rules.eq1 | theorem | rw1443.comp1_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y z : FreeMagma α}:
rules (rules (x ⋆ y) ⋆ rules (x ⋆ (x ⋆ z))) = x | by
generalize h: rules (x ⋆ (x ⋆ z)) = e
simp only [rules.elim] at h
separate
· apply comp1_2 | theorem | rw1443.comp1_3 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp4_2 {x y z : FreeMagma α}:
rules (rules (x ⋆ y ⋆ z) ⋆ (x ⋆ y ⋆ x)) = x ⋆ y | by
generalize h: rules (x ⋆ y ⋆ z) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq4
· apply rules.eq4
· apply rules.eq2
· apply rules.eq4 | theorem | rw1443.comp4_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp4_3 {x y z : FreeMagma α}:
rules (rules (x ⋆ y ⋆ z) ⋆ rules (x ⋆ y ⋆ x)) = x ⋆ y | by
generalize h: rules (x ⋆ y ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp4_2 | theorem | rw1443.comp4_3 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y z : FreeMagma α}:
rules (rules (x ⋆ y) ⋆ rules (x ⋆ rules (x ⋆ z))) = x | by
generalize h: rules (x ⋆ z) = e
simp only [rules.elim] at h
separate
· apply comp4_3
· apply comp1_3
· apply comp1_3
· apply comp4_3
· apply comp1_3 | theorem | rw1443.comp1_4 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1443] [23, 1629, 1630, 2441, 3050, 3055, 3522, 4065, 4067, 4118, 4268, 4282] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1443.«Facts» | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y z : FreeMagma α}:
rules (x ⋆ x ⋆ rules (x ⋆ y ⋆ z)) = x | by
generalize h: rules (x ⋆ y ⋆ z) = e
simp only [rules.elim] at h
separate
· apply rules.eq2
· apply rules.eq2
· apply rules.eq2
· apply rules.eq2
· apply rules.eq1 | theorem | rw1633.comp1_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp3_2 {x y : FreeMagma α}:
rules (x ⋆ x ⋆ (x ⋆ x) ⋆ rules (x ⋆ y)) = x ⋆ x | by
generalize h: rules (x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply rules.eq4
· apply rules.eq4
· apply rules.eq4
· apply rules.eq4
· apply rules.eq3 | theorem | rw1633.comp3_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y z : FreeMagma α}:
rules (x ⋆ x ⋆ rules (rules (x ⋆ y) ⋆ z)) = x | by
generalize h: rules (x ⋆ y) = e
simp only [rules.elim] at h
separate
· apply comp3_2
· apply comp3_2
· apply comp3_2
· apply comp3_2
· apply comp1_2 | theorem | rw1633.comp1_3 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y z : FreeMagma α}:
rules (rules (x ⋆ x) ⋆ rules (rules (x ⋆ y) ⋆ z)) = x | by
generalize h: rules (x ⋆ x) = e
simp only [rules.elim] at h
separate
apply comp1_3 | theorem | rw1633.comp1_4 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [1633] [4268, 4282] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp (disch := bufixed) only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4
all_goals refute | theorem | rw1633.«Facts» | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_2 {x y z : FreeMagma α}:
rules (y ⋆ y ⋆ x ⋆ rules (x ⋆ z)) = x | by
generalize h: rules (x ⋆ z) = e
simp only [rules.elim] at h
separate
· apply rules.eq3
· apply rules.eq1
· apply rules.eq3
· apply rules.eq1
· apply rules.eq1 | theorem | rw2126.comp1_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp2_2 {x y z : FreeMagma α} (hxxy: NF rules (x ⋆ x ⋆ y)):
rules (x ⋆ x ⋆ rules (x ⋆ x ⋆ y ⋆ z)) = x ⋆ x ⋆ y | by
generalize h: rules (x ⋆ x ⋆ y ⋆ z) = e
simp only [rules.elim] at h
separate
· apply hxxy.top
· apply rules.eq2
· apply hxxy.top
· apply rules.eq2
· apply rules.eq2 | theorem | rw2126.comp2_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp4_2 {x y z w : FreeMagma α} (hxxyz: NF rules (x ⋆ x ⋆ y ⋆ z)):
rules (((x ⋆ x) ⋆ y) ⋆ rules ((((x ⋆ x) ⋆ y) ⋆ z) ⋆ w)) = (((x ⋆ x) ⋆ y) ⋆ z) | by
generalize h: rules ((((x ⋆ x) ⋆ y) ⋆ z) ⋆ w) = e
simp only [rules.elim] at h
separate
· apply hxxyz.top
· apply rules.eq4
· apply hxxyz.top
· apply rules.eq2
· apply rules.eq4 | theorem | rw2126.comp4_2 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_3 {x y z : FreeMagma α} (hx: NF rules x):
rules (rules (y ⋆ y ⋆ x) ⋆ rules (x ⋆ z)) = x | by
generalize h: rules (y ⋆ y ⋆ x) = e
simp only [rules.elim] at h
separate
· apply comp2_2 hx
· apply comp4_2 hx
· apply comp1_2
· apply comp4_2 hx
· apply comp1_2 | theorem | rw2126.comp1_3 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
comp1_4 {x y z : FreeMagma α} (hx: NF rules x):
rules (rules (rules (y ⋆ y) ⋆ x) ⋆ rules (x ⋆ z)) = x | by
generalize h: rules (y ⋆ y) = e
simp only [rules.elim] at h
separate
all_goals apply comp1_3 hx | theorem | rw2126.comp1_4 | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
«Facts» :
∃ (G : Type) (_ : Magma G), Facts G [2126] [151, 152, 167, 168, 1426, 1427, 1428, 1429, 1430, 1478, 1479, 1480, 1481, 1482, 1483, 1484, 1485, 1486, 2050, 2051, 2052, 2087, 2088, 2089, 2161, 2162, 2163, 3456, 3457, 3511, 3512, 3513, 3862, 3877, 3918, 3955, 3993] | by
use ConfMagma (@rules Nat _), inferInstance
repeat' apply And.intro
· rintro ⟨x, hx⟩ ⟨y, hy⟩ ⟨z, hz⟩
simp only [Magma.op, bu, hx, hy, hz, buFixed_rw_op]
symm
apply Subtype.ext
apply comp1_4 ((NF_iff_buFixed rules).mpr hx)
all_goals refute | theorem | rw2126.«Facts» | Root | equational_theories/Confluence4.lean | [
"equational_theories.ConfluenceSystem"
] | [
"Facts",
"Magma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
not_eq_def | Not.eq_def | def | Confluence.not_eq_def | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
"separate" : tactic => `(tactic| (
try simp only [Not.eq_def]
try intros
try injections
try casesm* _ ∨ _, _ ∧ _, Exists _
try any_goals
try subst_eqs
try trivial
)) | macro | separate | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | ||
"prove_elim" : tactic => `(tactic| (
repeat' split
all_goals simp_all only [confluence_simps, exists_and_right, exists_eq_right_right', exists_eq_right', false_iff, not_exists, true_and]
separate
try simp_all only [not_true_eq_false, imp_false]
try
constructor
· intro h
subst_eqs
repeat co... | macro | prove_elim | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | ||
"prove_elim_not" : tactic => `(tactic| (
repeat' split
all_goals simp_all only [confluence_simps, false_iff, not_exists, not_and, true_iff, forall_eq', forall_apply_eq_imp_iff, true_iff, not_false_eq_true]
separate
try simp_all only [not_true_eq_false, imp_false]
try any_goals separate
try any_goals
try... | macro | prove_elim_not | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | ||
makePattern (inc: Nat) : Syntax → StateM (HashMap Name Nat) (TSyntax `term) | | .node info kind args => do
pure <| TSyntax.mk <| Syntax.node info kind (← args.mapM (makePattern inc ·))
| s@(.ident _ _ name _) =>
modifyGet (λ m ↦ match m[name]? with
| some n => ((mkIdentFrom s (.mkSimple s!"{name}{n}")), m.insert name (n + inc))
| _ => (TSyntax.mk s, m))
| s@_ => pure <| TSyntax.mk s | def | Confluence.makePattern | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
countVars : Syntax → StateM (HashMap Name Nat) Unit | | .node _ _ args => args.forM countVars
| .ident _ _ name _ => modify (λ m ↦ match m[name]? with
| some n => m.insert name (n + 1)
| _ => m)
| _ => pure () | def | Confluence.countVars | Root | equational_theories/ConfluenceSystem.lean | [
"equational_theories.Confluence"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
order : FreeMagma X → ℕ | | Lf _ => 0
| a ⋆ b => order a + order b + 1 | def | FreeMagma.order | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
order_leaf (a : X) : (Lf a).order = 0 | rfl | lemma | FreeMagma.order_leaf | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
order_fork (a b : FreeMagma X) : (a ⋆ b).order = a.order + b.order + 1 | rfl | lemma | FreeMagma.order_fork | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
pairwiseFork (a b : Finset (FreeMagma X)) : Finset (FreeMagma X) | (a ×ˢ b).map ⟨fun ⟨x, y⟩ ↦ x ⋆ y, fun ⟨a, b⟩ ⟨c, d⟩ ↦ by simp⟩ | abbrev | FreeMagma.pairwiseFork | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
elementsOfNumNodesEq : ℕ → Finset (FreeMagma X) | | 0 => Finset.univ.map ⟨Lf, fun a b h ↦ by simpa using h⟩
| n + 1 =>
(Finset.antidiagonal n).attach.biUnion fun ijh =>
pairwiseFork (elementsOfNumNodesEq ijh.1.1) (elementsOfNumNodesEq ijh.1.2)
-- Porting note: Add this to satisfy the linter.
decreasing_by
· simp_wf; have := Finset.antidiagonal.fst_... | def | FreeMagma.elementsOfNumNodesEq | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
elementsOfNumNodesEq_zero : elementsOfNumNodesEq X 0 =
Finset.univ.map ⟨Lf, fun a b h ↦ by simpa using h⟩ | by rw [elementsOfNumNodesEq] | theorem | FreeMagma.elementsOfNumNodesEq_zero | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
elementsOfNumNodesEq_succ {n : ℕ} : elementsOfNumNodesEq X (n + 1) =
(Finset.antidiagonal n).biUnion fun ijh => pairwiseFork (elementsOfNumNodesEq X ijh.1)
(elementsOfNumNodesEq X ijh.2) | by
rw [elementsOfNumNodesEq]
ext
simp | theorem | FreeMagma.elementsOfNumNodesEq_succ | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
mem_elementsOfNumNodesEq {x : FreeMagma X} {n : ℕ} :
x ∈ elementsOfNumNodesEq X n ↔ order x = n | by
induction x generalizing n <;> cases n
· simp
· simp [elementsOfNumNodesEq_succ]
· simp
· simp [elementsOfNumNodesEq_succ, *] | theorem | FreeMagma.mem_elementsOfNumNodesEq | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [
"FreeMagma"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
elementsOfNumNodesEq_card_eq_catalan_mul_pow (n : ℕ) :
(elementsOfNumNodesEq X n).card = catalan n * (Fintype.card X) ^ (n + 1) | by
induction n using Nat.case_strong_induction_on
· simp
case hi n ih =>
rw [elementsOfNumNodesEq_succ, Finset.card_biUnion, catalan_succ', Finset.sum_mul]
· apply Finset.sum_congr rfl
rintro ⟨i, j⟩ h
rw [Finset.card_map, Finset.card_product, ih _ (Finset.antidiagonal.fst_le h),
ih _ (Finset.ant... | theorem | FreeMagma.elementsOfNumNodesEq_card_eq_catalan_mul_pow | Root | equational_theories/Counting.lean | [
"equational_theories.FreeMagma",
"Mathlib.Combinatorics.Enumerative.Catalan"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
preprocessPropToDecide (expectedType : Expr) : TermElabM Expr | do
let mut expectedType ← instantiateMVars expectedType
if expectedType.hasFVar then
expectedType ← zetaReduce expectedType
if expectedType.hasFVar || expectedType.hasMVar then
throwError "expected type must not contain free or meta variables{indentExpr expectedType}"
return expectedType | def | preprocessPropToDecide | Root | equational_theories/DecideBang.lean | [
"Lean"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
splitConjs (e : Lean.Expr) : Array (Lean.Expr) | Id.run do
let mut e := e
let mut r := #[]
while e.isAppOf `And do
r := r.push e.appFn!.appArg!
e := e.appArg!
r := r.push e
return r | def | splitConjs | Root | equational_theories/DecideBang.lean | [
"Lean"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
"decide!" : tactic => do
closeMainGoalUsing `decide fun expectedType _ => do
let expectedType ← preprocessPropToDecide expectedType
let expectedTypes | splitConjs expectedType
let proofs ← expectedTypes.mapM fun expectedType => do
let d ← mkDecide expectedType
let d ← instantiateMVars d
-- Get instance from `d`
let s := d.appArg!
let rflPrf ← mkEqRefl (toExpr true)
return mkApp3 (Lean.mkConst ``of_decide_eq_true) expectedType s ... | elab | decide! | Root | equational_theories/DecideBang.lean | [
"Lean"
] | [
"preprocessPropToDecide",
"splitConjs"
] | This is a variant of the `decide` tactic. It does not actually check the proposition at elaboration
time, and just assumes it is true. This is useful in generated lean code, where we know it will
succeed, and we really just want the check done once, in the kernel. | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 |
inferDecideFin (p : Expr) : MetaM Expr | do
let p ← whnfR p
match p with
| .forallE i d b bi =>
match_expr (← whnfR d) with
| Fin n =>
let inst ← withLocalDeclD i d fun x => do
let body := b.instantiate1 x
let inst ← inferDecideFin body
mkLambdaFVars #[x] inst
return mkApp3 (mkConst ``Nat.decidableForallFin) n... | def | inferDecideFin | Root | equational_theories/DecideBang.lean | [
"Lean"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
"decideFin!" : tactic => do
closeMainGoalUsing `decideFin fun expectedType _ => do
let expectedType ← preprocessPropToDecide expectedType
let expectedTypes | splitConjs expectedType
let proofs ← expectedTypes.mapM fun expectedType => do
let s ← inferDecideFin expectedType
let rflPrf ← mkEqRefl (toExpr true)
return mkApp3 (Lean.mkConst ``of_decide_eq_true) expectedType s rflPrf
let proof ← proofs.pop.foldrM (mkAppM ``And.intro #[·, ·]) proofs.back!
... | elab | decideFin! | Root | equational_theories/DecideBang.lean | [
"Lean"
] | [
"inferDecideFin",
"preprocessPropToDecide",
"splitConjs"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
EntryVariant where
| implication : Implication → EntryVariant
| facts : Facts → EntryVariant
/-- An equation that always holds. -/
| unconditional : String → EntryVariant
deriving Lean.ToJson, Lean.FromJson, Inhabited | inductive | EntryVariant | Root | equational_theories/EquationalResult.lean | [
"Lean.Elab.Exception",
"Lean.Elab.Declaration",
"Lean.Util.CollectAxioms",
"equational_theories.ParseImplications"
] | [
"Facts",
"Implication"
] | An entry in the equational results or conjecture environment extension. | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
Entry where
/-- Name of the declaration. -/
name : Name
/-- Name of the file where this declaration was found. -/
filename : String
/-- Line number of the declaration, if available -/
line : Option Nat
/-- Which kind of result is it? -/
variant : EntryVariant
/-- Is it proven? -/
proven : Bool
deriv... | structure | Entry | Root | equational_theories/EquationalResult.lean | [
"Lean.Elab.Exception",
"Lean.Elab.Declaration",
"Lean.Util.CollectAxioms",
"equational_theories.ParseImplications"
] | [
"EntryVariant"
] | An entry in the equational results environment extension | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
Entry.toConjecture : Entry → Entry | | .mk n f l v _ => ⟨n, f, l, v, false⟩ | def | Entry.toConjecture | Root | equational_theories/EquationalResult.lean | [
"Lean.Elab.Exception",
"Lean.Elab.Declaration",
"Lean.Util.CollectAxioms",
"equational_theories.ParseImplications"
] | [
"Entry"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
Entry.keepCore (e : Entry) : Option Entry | match e.variant with
| .implication ⟨lhs, rhs, _⟩ =>
if isCoreEquationName lhs && isCoreEquationName rhs
then some e
else none
| .facts ⟨satisfied, refuted, finite⟩ =>
let sat1 := satisfied.filterMap filterCoreEquationName
let ref1 := refuted.filterMap filterCoreEquationName
if ... | def | Entry.keepCore | Root | equational_theories/EquationalResult.lean | [
"Lean.Elab.Exception",
"Lean.Elab.Declaration",
"Lean.Util.CollectAxioms",
"equational_theories.ParseImplications"
] | [
"Entry",
"filterCoreEquationName",
"isCoreEquationName"
] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 | |
equationalResultHelpString : String | "tags theorems that provide implications or negated implications to the directed graph" | def | Result.equationalResultHelpString | Root | equational_theories/EquationalResult.lean | [
"Lean.Elab.Exception",
"Lean.Elab.Declaration",
"Lean.Util.CollectAxioms",
"equational_theories.ParseImplications"
] | [] | https://github.com/teorth/equational_theories | 3f3999d958c5e289c7f5a063479af9dac122a7a8 |
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