Theorem: Reactor.Equivalent.symm (ReactorModel/Objects/Reactor/Theorems/Basic.lean)

State before:
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ : α
e : rtr₁ ≈ rtr₂
⊢ rtr₂ ≈ rtr₁
Tactic:
induction e
State after:
case refl
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr✝ : α
⊢ rtr✝ ≈ rtr✝

case struct
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
mem_get?_iff✝ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
get?_rcn_some_eq✝ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
get?_rtr_some_equiv_ih✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
⊢ rtr₂✝ ≈ rtr₁✝

State before:
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr✝ : α
⊢ rtr✝ ≈ rtr✝
Tactic:
case refl => exact .refl _
State after:
no goals

State before:
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr✝ : α
⊢ rtr✝ ≈ rtr✝
Tactic:
exact .refl _
State after:
no goals

State before:
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
⊢ rtr₂✝ ≈ rtr₁✝
Tactic:
constructor <;> intros
State after:
case mem_get?_iff
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
cpt✝ : Component
i✝ : ID
⊢ i✝ ∈ rtr₂✝{cpt✝} ↔ i✝ ∈ rtr₁✝{cpt✝}

case get?_rcn_some_eq
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
i✝ : ID
r₁✝ r₂✝ : Component.type α Component.rcn
a✝¹ : rtr₂✝{Component.rcn}{i✝} = some r₁✝
a✝ : rtr₁✝{Component.rcn}{i✝} = some r₂✝
⊢ r₁✝ = r₂✝

case get?_rtr_some_equiv
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
i✝ : ID
n₁✝ n₂✝ : Component.type α Component.rtr
a✝¹ : rtr₂✝{Component.rtr}{i✝} = some n₁✝
a✝ : rtr₁✝{Component.rtr}{i✝} = some n₂✝
⊢ Equivalent n₁✝ n₂✝

State before:
case mem_get?_iff
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
cpt✝ : Component
i✝ : ID
⊢ i✝ ∈ rtr₂✝{cpt✝} ↔ i✝ ∈ rtr₁✝{cpt✝}
Tactic:
exact h₁ ‹_› ‹_› |>.symm
State after:
no goals

State before:
case get?_rcn_some_eq
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
i✝ : ID
r₁✝ r₂✝ : Component.type α Component.rcn
a✝¹ : rtr₂✝{Component.rcn}{i✝} = some r₁✝
a✝ : rtr₁✝{Component.rcn}{i✝} = some r₂✝
⊢ r₁✝ = r₂✝
Tactic:
exact h₂ ‹_› ‹_› |>.symm
State after:
no goals

State before:
case get?_rtr_some_equiv
α : Type
inst✝ : Reactor α
rtr₁ rtr₂ rtr₁✝ rtr₂✝ : α
h₁ : ∀ (cpt : Component) (i : ID), i ∈ rtr₁✝{cpt} ↔ i ∈ rtr₂✝{cpt}
h₂ :
  ∀ {i : ID} {r₁ r₂ : Component.type α Component.rcn},
    rtr₁✝{Component.rcn}{i} = some r₁ → rtr₂✝{Component.rcn}{i} = some r₂ → r₁ = r₂
get?_rtr_some_equiv✝ :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → Equivalent n₁ n₂
hi :
  ∀ {i : ID} {n₁ n₂ : Component.type α Component.rtr},
    rtr₁✝{Component.rtr}{i} = some n₁ → rtr₂✝{Component.rtr}{i} = some n₂ → n₂ ≈ n₁
i✝ : ID
n₁✝ n₂✝ : Component.type α Component.rtr
a✝¹ : rtr₂✝{Component.rtr}{i✝} = some n₁✝
a✝ : rtr₁✝{Component.rtr}{i✝} = some n₂✝
⊢ Equivalent n₁✝ n₂✝
Tactic:
exact hi ‹_› ‹_›
State after:
no goals