-- Mathlib/.lake/packages/lean4/src/lean/lake/Lake/Build/Key.lean
theorem eq_of_quickCmp {k k' : BuildKey}  :
quickCmp k k' = Ordering.eq → k = k'

Proof state:
case moduleFacet.customTarget
m f package✝ target✝ : Name
a✝ :
  (match moduleFacet m f with
    | moduleFacet m f =>
      match customTarget package✝ target✝ with
      | moduleFacet m' f' =>
        match m.quickCmp m' with
        | Ordering.eq => f.quickCmp f'
        | ord => ord
      | x => Ordering.lt
    | packageFacet p f =>
      match customTarget package✝ target✝ with
      | moduleFacet module facet => Ordering.gt
      | packageFacet p' f' =>
        match p.quickCmp p' with
        | Ordering.eq => f.quickCmp f'
        | ord => ord
      | x => Ordering.lt
    | targetFacet p t f =>
      match customTarget package✝ target✝ with
      | customTarget package target => Ordering.lt
      | targetFacet p' t' f' =>
        match p.quickCmp p' with
        | Ordering.eq =>
          match t.quickCmp t' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | ord => ord
      | x => Ordering.gt
    | customTarget p t =>
      match customTarget package✝ target✝ with
      | customTarget p' t' =>
        match p.quickCmp p' with
        | Ordering.eq => t.quickCmp t'
        | ord => ord
      | x => Ordering.gt) =
    Ordering.eq
⊢ moduleFacet m f = customTarget package✝ target✝

Tactic:
contradiction

Result:
no goals

Proof state:
case packageFacet.moduleFacet
p f module✝ facet✝ : Name
⊢ (match packageFacet p f with
      | moduleFacet m f =>
        match moduleFacet module✝ facet✝ with
        | moduleFacet m' f' =>
          match m.quickCmp m' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | packageFacet p f =>
        match moduleFacet module✝ facet✝ with
        | moduleFacet module facet => Ordering.gt
        | packageFacet p' f' =>
          match p.quickCmp p' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | targetFacet p t f =>
        match moduleFacet module✝ facet✝ with
        | customTarget package target => Ordering.lt
        | targetFacet p' t' f' =>
          match p.quickCmp p' with
          | Ordering.eq =>
            match t.quickCmp t' with
            | Ordering.eq => f.quickCmp f'
            | ord => ord
          | ord => ord
        | x => Ordering.gt
      | customTarget p t =>
        match moduleFacet module✝ facet✝ with
        | customTarget p' t' =>
          match p.quickCmp p' with
          | Ordering.eq => t.quickCmp t'
          | ord => ord
        | x => Ordering.gt) =
      Ordering.eq →
    packageFacet p f = moduleFacet module✝ facet✝

case packageFacet.targetFacet
p f package✝ target✝ facet✝ : Name
⊢ (match packageFacet p f with
      | moduleFacet m f =>
        match targetFacet package✝ target✝ facet✝ with
        | moduleFacet m' f' =>
          match m.quickCmp m' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | packageFacet p f =>
        match targetFacet package✝ target✝ facet✝ with
        | moduleFacet module facet => Ordering.gt
        | packageFacet p' f' =>
          match p.quickCmp p' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | targetFacet p t f =>
        match targetFacet package✝ target✝ facet✝ with
        | customTarget package target => Ordering.lt
        | targetFacet p' t' f' =>
          match p.quickCmp p' with
          | Ordering.eq =>
            match t.quickCmp t' with
            | Ordering.eq => f.quickCmp f'
            | ord => ord
          | ord => ord
        | x => Ordering.gt
      | customTarget p t =>
        match targetFacet package✝ target✝ facet✝ with
        | customTarget p' t' =>
          match p.quickCmp p' with
          | Ordering.eq => t.quickCmp t'
          | ord => ord
        | x => Ordering.gt) =
      Ordering.eq →
    packageFacet p f = targetFacet package✝ target✝ facet✝

case packageFacet.customTarget
p f package✝ target✝ : Name
⊢ (match packageFacet p f with
      | moduleFacet m f =>
        match customTarget package✝ target✝ with
        | moduleFacet m' f' =>
          match m.quickCmp m' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | packageFacet p f =>
        match customTarget package✝ target✝ with
        | moduleFacet module facet => Ordering.gt
        | packageFacet p' f' =>
          match p.quickCmp p' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | x => Ordering.lt
      | targetFacet p t f =>
        match customTarget package✝ target✝ with
        | customTarget package target => Ordering.lt
        | targetFacet p' t' f' =>
          match p.quickCmp p' with
          | Ordering.eq =>
            match t.quickCmp t' with
            | Ordering.eq => f.quickCmp f'
            | ord => ord
          | ord => ord
        | x => Ordering.gt
      | customTarget p t =>
        match customTarget package✝ target✝ with
        | customTarget p' t' =>
          match p.quickCmp p' with
          | Ordering.eq => t.quickCmp t'
          | ord => ord
        | x => Ordering.gt) =
      Ordering.eq →
    packageFacet p f = customTarget package✝ target✝

Tactic:
all_goals (intro; contradiction)

Result:
no goals

Proof state:
p t p' t' : Name
x✝ : Ordering
p_eq : p.quickCmp p' = Ordering.eq
t_eq : t.quickCmp t' = Ordering.eq
⊢ customTarget p t = customTarget p' t'

Tactic:
rw [eq_of_cmp p_eq, eq_of_cmp t_eq]

Result:
no goals

Proof state:
p t f p' t' f' : Name
x✝ : Ordering
p_eq : p.quickCmp p' = Ordering.eq
⊢ (match t.quickCmp t' with
      | Ordering.eq => f.quickCmp f'
      | ord => ord) =
      Ordering.eq →
    targetFacet p t f = targetFacet p' t' f'

Tactic:
split

Result:
case h_1
p t f p' t' f' : Name
x✝¹ : Ordering
p_eq : p.quickCmp p' = Ordering.eq
x✝ : Ordering
heq✝ : t.quickCmp t' = Ordering.eq
⊢ f.quickCmp f' = Ordering.eq → targetFacet p t f = targetFacet p' t' f'

case h_2
p t f p' t' f' : Name
x✝² : Ordering
p_eq : p.quickCmp p' = Ordering.eq
x✝¹ : Ordering
x✝ : t.quickCmp t' = Ordering.eq → False
⊢ t.quickCmp t' = Ordering.eq → targetFacet p t f = targetFacet p' t' f'

Proof state:
case targetFacet.customTarget
p t f package✝ target✝ : Name
a✝ :
  (match targetFacet p t f with
    | moduleFacet m f =>
      match customTarget package✝ target✝ with
      | moduleFacet m' f' =>
        match m.quickCmp m' with
        | Ordering.eq => f.quickCmp f'
        | ord => ord
      | x => Ordering.lt
    | packageFacet p f =>
      match customTarget package✝ target✝ with
      | moduleFacet module facet => Ordering.gt
      | packageFacet p' f' =>
        match p.quickCmp p' with
        | Ordering.eq => f.quickCmp f'
        | ord => ord
      | x => Ordering.lt
    | targetFacet p t f =>
      match customTarget package✝ target✝ with
      | customTarget package target => Ordering.lt
      | targetFacet p' t' f' =>
        match p.quickCmp p' with
        | Ordering.eq =>
          match t.quickCmp t' with
          | Ordering.eq => f.quickCmp f'
          | ord => ord
        | ord => ord
      | x => Ordering.gt
    | customTarget p t =>
      match customTarget package✝ target✝ with
      | customTarget p' t' =>
        match p.quickCmp p' with
        | Ordering.eq => t.quickCmp t'
        | ord => ord
      | x => Ordering.gt) =
    Ordering.eq
⊢ targetFacet p t f = customTarget package✝ target✝

Tactic:
contradiction

Result:
no goals